fitctree

Fit binary decision tree for multiclass classification

Syntax

Mdl = fitctree(Tbl,ResponseVarName)

Mdl = fitctree(Tbl,formula)

Mdl = fitctree(Tbl,Y)

Mdl = fitctree(X,Y)

Mdl = fitctree(___,Name,Value)

[Mdl,AggregateOptimizationResults] = fitctree(___)

Description

Mdl = fitctree(Tbl,ResponseVarName) returns a fitted binary classification decision tree based on the input variables (also known as predictors, features, or attributes) contained in the table Tbl and output (response or labels) contained in Tbl.ResponseVarName. The returned binary tree splits branching nodes based on the values of a column of Tbl.

Mdl = fitctree(Tbl,formula) returns a fitted binary classification decision tree based on the input variables contained in the table Tbl. formula is an explanatory model of the response and a subset of predictor variables in Tbl used to fit Mdl.

Mdl = fitctree(Tbl,Y) returns a fitted binary classification decision tree based on the input variables contained in the table Tbl and output in vector Y.

Mdl = fitctree(X,Y) returns a fitted binary classification decision tree based on the input variables contained in matrix X and output Y. The returned binary tree splits branching nodes based on the values of a column of X.

Mdl = fitctree(___,Name,Value) fits a tree with additional options specified by one or more name-value pair arguments, using any of the previous syntaxes. For example, you can specify the algorithm used to find the best split on a categorical predictor, grow a cross-validated tree, or hold out a fraction of the input data for validation.

[Mdl,AggregateOptimizationResults] = fitctree(___) also returns AggregateOptimizationResults, which contains hyperparameter optimization results when you specify the OptimizeHyperparameters and HyperparameterOptimizationOptions name-value arguments. You must also specify the ConstraintType and ConstraintBounds options of HyperparameterOptimizationOptions. You can use this syntax to optimize on compact model size instead of cross-validation loss, and to perform a set of multiple optimization problems that have the same options but different constraint bounds.

Note

For a list of supported syntaxes when the input variables are tall arrays, see Tall Arrays.

Examples

Grow a Classification Tree

Grow a classification tree using the ionosphere data set.

load ionosphere
tc = fitctree(X,Y)

tc = 
  ClassificationTree
             ResponseName: 'Y'
    CategoricalPredictors: []
               ClassNames: {'b'  'g'}
           ScoreTransform: 'none'
          NumObservations: 351

Control Tree Depth

You can control the depth of the trees using the MaxNumSplits, MinLeafSize, or MinParentSize name-value pair parameters. fitctree grows deep decision trees by default. You can grow shallower trees to reduce model complexity or computation time.

Load the ionosphere data set.

load ionosphere

The default values of the tree depth controllers for growing classification trees are:

• n - 1 for MaxNumSplits. n is the training sample size.

• 1 for MinLeafSize.

• 10 for MinParentSize.

These default values tend to grow deep trees for large training sample sizes.

Train a classification tree using the default values for tree depth control. Cross-validate the model by using 10-fold cross-validation.

rng(1); % For reproducibility
MdlDefault = fitctree(X,Y,'CrossVal','on');

Draw a histogram of the number of imposed splits on the trees. Also, view one of the trees.

numBranches = @(x)sum(x.IsBranch);
mdlDefaultNumSplits = cellfun(numBranches, MdlDefault.Trained);

figure;
histogram(mdlDefaultNumSplits)

view(MdlDefault.Trained{1},'Mode','graph')

The average number of splits is around 15.

Suppose that you want a classification tree that is not as complex (deep) as the ones trained using the default number of splits. Train another classification tree, but set the maximum number of splits at 7, which is about half the mean number of splits from the default classification tree. Cross-validate the model by using 10-fold cross-validation.

Mdl7 = fitctree(X,Y,'MaxNumSplits',7,'CrossVal','on');
view(Mdl7.Trained{1},'Mode','graph')

Compare the cross-validation classification errors of the models.

classErrorDefault = kfoldLoss(MdlDefault)

classErrorDefault = 
0.1168

classError7 = kfoldLoss(Mdl7)

classError7 = 
0.1311

Mdl7 is much less complex and performs only slightly worse than MdlDefault.

Optimize Classification Tree

This example shows how to optimize hyperparameters automatically using fitctree. The example uses Fisher's iris data.

Load Fisher's iris data.

load fisheriris

Optimize the cross-validation loss of the classifier, using the data in meas to predict the response in species.

X = meas;
Y = species;
Mdl = fitctree(X,Y,'OptimizeHyperparameters','auto')

|======================================================================================|
| Iter | Eval   | Objective   | Objective   | BestSoFar   | BestSoFar   |  MinLeafSize |
|      | result |             | runtime     | (observed)  | (estim.)    |              |
|======================================================================================|
|    1 | Best   |    0.066667 |     0.71361 |    0.066667 |    0.066667 |           31 |
|    2 | Accept |    0.066667 |     0.31624 |    0.066667 |    0.066667 |           12 |
|    3 | Best   |        0.06 |    0.087234 |        0.06 |    0.060001 |            2 |
|    4 | Accept |     0.66667 |     0.15086 |        0.06 |     0.18681 |           73 |
|    5 | Accept |    0.066667 |    0.090292 |        0.06 |    0.060018 |           28 |
|    6 | Accept |        0.06 |    0.076536 |        0.06 |     0.06001 |            4 |
|    7 | Accept |    0.066667 |    0.040914 |        0.06 |    0.055628 |            7 |
|    8 | Accept |        0.06 |    0.076439 |        0.06 |    0.054969 |            1 |
|    9 | Accept |    0.066667 |    0.068903 |        0.06 |    0.060608 |           20 |
|   10 | Accept |        0.06 |    0.077669 |        0.06 |    0.061024 |            3 |
|   11 | Accept |    0.066667 |    0.062875 |        0.06 |        0.06 |           25 |
|   12 | Accept |    0.066667 |    0.077881 |        0.06 |    0.059937 |            5 |
|   13 | Accept |    0.066667 |      0.1081 |        0.06 |    0.059918 |            9 |
|   14 | Accept |        0.06 |    0.065452 |        0.06 |    0.059961 |            3 |
|   15 | Accept |        0.06 |     0.14076 |        0.06 |    0.059963 |            1 |
|   16 | Accept |        0.06 |    0.072872 |        0.06 |    0.059965 |            2 |
|   17 | Accept |    0.066667 |     0.20074 |        0.06 |    0.059959 |           15 |
|   18 | Accept |        0.06 |     0.15304 |        0.06 |     0.05996 |            4 |
|   19 | Accept |        0.06 |     0.15285 |        0.06 |    0.059974 |            3 |
|   20 | Accept |        0.06 |     0.11444 |        0.06 |    0.059976 |            1 |
|======================================================================================|
| Iter | Eval   | Objective   | Objective   | BestSoFar   | BestSoFar   |  MinLeafSize |
|      | result |             | runtime     | (observed)  | (estim.)    |              |
|======================================================================================|
|   21 | Accept |        0.06 |    0.063955 |        0.06 |    0.059977 |            2 |
|   22 | Accept |        0.06 |    0.038461 |        0.06 |    0.059977 |            4 |
|   23 | Accept |        0.06 |    0.050634 |        0.06 |    0.059984 |            3 |
|   24 | Accept |        0.06 |    0.047099 |        0.06 |    0.059984 |            2 |
|   25 | Accept |        0.06 |    0.034826 |        0.06 |    0.059985 |            1 |
|   26 | Accept |     0.33333 |     0.10805 |        0.06 |    0.059995 |           49 |
|   27 | Accept |        0.06 |    0.047514 |        0.06 |    0.059995 |            4 |
|   28 | Accept |    0.066667 |    0.034601 |        0.06 |    0.059993 |            6 |
|   29 | Accept |    0.066667 |    0.079854 |        0.06 |        0.06 |           38 |
|   30 | Accept |    0.066667 |    0.039293 |        0.06 |        0.06 |           35 |

__________________________________________________________
Optimization completed.
MaxObjectiveEvaluations of 30 reached.
Total function evaluations: 30
Total elapsed time: 18.4909 seconds
Total objective function evaluation time: 3.392

Best observed feasible point:
    MinLeafSize
    ___________

         2     

Observed objective function value = 0.06
Estimated objective function value = 0.06
Function evaluation time = 0.087234

Best estimated feasible point (according to models):
    MinLeafSize
    ___________

         3     

Estimated objective function value = 0.06
Estimated function evaluation time = 0.084126

Mdl = 
  ClassificationTree
                         ResponseName: 'Y'
                CategoricalPredictors: []
                           ClassNames: {'setosa'  'versicolor'  'virginica'}
                       ScoreTransform: 'none'
                      NumObservations: 150
    HyperparameterOptimizationResults: [1x1 BayesianOptimization]

Unbiased Predictor Importance Estimates

Load the census1994 data set. Consider a model that predicts a person's salary category given their age, working class, education level, martial status, race, sex, capital gain and loss, and number of working hours per week.

load census1994
X = adultdata(:,{'age','workClass','education_num','marital_status','race',...
    'sex','capital_gain','capital_loss','hours_per_week','salary'});

Display the number of categories represented in the categorical variables using summary.

summary(X)

X: 32561x10 table

Variables:

    age: double
    workClass: categorical (8 categories)
    education_num: double
    marital_status: categorical (7 categories)
    race: categorical (5 categories)
    sex: categorical (2 categories)
    capital_gain: double
    capital_loss: double
    hours_per_week: double
    salary: categorical (2 categories)

Statistics for applicable variables:

                      NumMissing     Min      Median       Max            Mean              Std      

    age                     0         17        37           90           38.5816           13.6404  
    workClass            1836                                                                        
    education_num           0          1        10           16           10.0807            2.5727  
    marital_status          0                                                                        
    race                    0                                                                        
    sex                     0                                                                        
    capital_gain            0          0         0        99999        1.0776e+03        7.3853e+03  
    capital_loss            0          0         0         4356           87.3038          402.9602  
    hours_per_week          0          1        40           99           40.4375           12.3474  
    salary                  0                                                                        

Because there are few categories represented in the categorical variables compared to levels in the continuous variables, the standard CART, predictor-splitting algorithm prefers splitting a continuous predictor over the categorical variables.

Train a classification tree using the entire data set. To grow unbiased trees, specify usage of the curvature test for splitting predictors. Because there are missing observations in the data, specify usage of surrogate splits.

Mdl = fitctree(X,'salary','PredictorSelection','curvature',...
    'Surrogate','on');

Estimate predictor importance values by summing changes in the risk due to splits on every predictor and dividing the sum by the number of branch nodes. Compare the estimates using a bar graph.

imp = predictorImportance(Mdl);

figure;
bar(imp);
title('Predictor Importance Estimates');
ylabel('Estimates');
xlabel('Predictors');
h = gca;
h.XTickLabel = Mdl.PredictorNames;
h.XTickLabelRotation = 45;
h.TickLabelInterpreter = 'none';

In this case, capital_gain is the most important predictor, followed by education_num.

Optimize Classification Tree on Tall Array

This example shows how to optimize hyperparameters of a classification tree automatically using a tall array. The sample data set airlinesmall.csv is a large data set that contains a tabular file of airline flight data. This example creates a tall table containing the data and uses it to run the optimization procedure.

When you perform calculations on tall arrays, MATLAB® uses either a parallel pool (default if you have Parallel Computing Toolbox™) or the local MATLAB session. If you want to run the example using the local MATLAB session when you have Parallel Computing Toolbox, you can change the global execution environment by using the mapreducer function.

Create a datastore that references the folder location with the data. Select a subset of the variables to work with, and treat 'NA' values as missing data so that datastore replaces them with NaN values. Create a tall table that contains the data in the datastore.

ds = datastore('airlinesmall.csv');
ds.SelectedVariableNames = {'Month','DayofMonth','DayOfWeek',...
                            'DepTime','ArrDelay','Distance','DepDelay'};
ds.TreatAsMissing = 'NA';
tt  = tall(ds) % Tall table

Starting parallel pool (parpool) using the 'local' profile ...
Connected to the parallel pool (number of workers: 6).

tt =

  M×7 tall table

    Month    DayofMonth    DayOfWeek    DepTime    ArrDelay    Distance    DepDelay
    _____    __________    _________    _______    ________    ________    ________

     10          21            3          642          8         308          12   
     10          26            1         1021          8         296           1   
     10          23            5         2055         21         480          20   
     10          23            5         1332         13         296          12   
     10          22            4          629          4         373          -1   
     10          28            3         1446         59         308          63   
     10           8            4          928          3         447          -2   
     10          10            6          859         11         954          -1   
      :          :             :           :          :           :           :
      :          :             :           :          :           :           :

Determine the flights that are late by 10 minutes or more by defining a logical variable that is true for a late flight. This variable contains the class labels. A preview of this variable includes the first few rows.

Y = tt.DepDelay > 10 % Class labels

Y =

  M×1 tall logical array

   1
   0
   1
   1
   0
   1
   0
   0
   :
   :

Create a tall array for the predictor data.

X = tt{:,1:end-1} % Predictor data

X =

  M×6 tall double matrix

          10          21           3         642           8         308
          10          26           1        1021           8         296
          10          23           5        2055          21         480
          10          23           5        1332          13         296
          10          22           4         629           4         373
          10          28           3        1446          59         308
          10           8           4         928           3         447
          10          10           6         859          11         954
          :           :            :          :           :           :
          :           :            :          :           :           :

Remove rows in X and Y that contain missing data.

R = rmmissing([X Y]); % Data with missing entries removed
X = R(:,1:end-1); 
Y = R(:,end); 

Standardize the predictor variables.

Z = zscore(X);

Optimize hyperparameters automatically using the 'OptimizeHyperparameters' name-value pair argument. Find the optimal 'MinLeafSize' value that minimizes holdout cross-validation loss. (Specifying 'auto' uses 'MinLeafSize'.) For reproducibility, use the 'expected-improvement-plus' acquisition function and set the seeds of the random number generators using rng and tallrng. The results can vary depending on the number of workers and the execution environment for the tall arrays. For details, see Control Where Your Code Runs.

rng('default') 
tallrng('default')
[Mdl,FitInfo,HyperparameterOptimizationResults] = fitctree(Z,Y,...
    'OptimizeHyperparameters','auto',...
    'HyperparameterOptimizationOptions',struct('Holdout',0.3,...
    'AcquisitionFunctionName','expected-improvement-plus'))

Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 3: Completed in 5.6 sec
- Pass 2 of 3: Completed in 2.1 sec
- Pass 3 of 3: Completed in 3.4 sec
Evaluation completed in 13 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 1: Completed in 0.73 sec
Evaluation completed in 0.9 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 1: Completed in 1.2 sec
Evaluation completed in 1.5 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.64 sec
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Evaluation completed in 5.9 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.53 sec
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- Pass 3 of 4: Completed in 0.58 sec
- Pass 4 of 4: Completed in 0.8 sec
Evaluation completed in 3.5 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.54 sec
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- Pass 3 of 4: Completed in 0.54 sec
- Pass 4 of 4: Completed in 0.79 sec
Evaluation completed in 3.5 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.51 sec
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- Pass 3 of 4: Completed in 0.59 sec
- Pass 4 of 4: Completed in 0.87 sec
Evaluation completed in 3.3 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.56 sec
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- Pass 3 of 4: Completed in 0.54 sec
- Pass 4 of 4: Completed in 0.89 sec
Evaluation completed in 3.4 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.57 sec
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Evaluation completed in 3.5 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.54 sec
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- Pass 3 of 4: Completed in 0.68 sec
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Evaluation completed in 4.3 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.65 sec
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Evaluation completed in 4.4 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.63 sec
- Pass 2 of 4: Completed in 0.85 sec
- Pass 3 of 4: Completed in 0.58 sec
- Pass 4 of 4: Completed in 2.2 sec
Evaluation completed in 4.9 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 1.2 sec
- Pass 2 of 4: Completed in 0.88 sec
- Pass 3 of 4: Completed in 0.58 sec
- Pass 4 of 4: Completed in 3 sec
Evaluation completed in 6.9 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.72 sec
- Pass 2 of 4: Completed in 0.96 sec
- Pass 3 of 4: Completed in 0.59 sec
- Pass 4 of 4: Completed in 4.2 sec
Evaluation completed in 7.2 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.77 sec
- Pass 2 of 4: Completed in 0.95 sec
- Pass 3 of 4: Completed in 0.65 sec
- Pass 4 of 4: Completed in 4.8 sec
Evaluation completed in 7.9 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.79 sec
- Pass 2 of 4: Completed in 1 sec
- Pass 3 of 4: Completed in 0.61 sec
- Pass 4 of 4: Completed in 5.1 sec
Evaluation completed in 8.4 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.89 sec
- Pass 2 of 4: Completed in 1.1 sec
- Pass 3 of 4: Completed in 0.59 sec
- Pass 4 of 4: Completed in 5.8 sec
Evaluation completed in 9.3 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 1 sec
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- Pass 3 of 4: Completed in 0.63 sec
- Pass 4 of 4: Completed in 5.2 sec
Evaluation completed in 8.9 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 1.6 sec
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- Pass 3 of 4: Completed in 0.74 sec
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Evaluation completed in 9.2 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 1 sec
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Evaluation completed in 7.7 sec
Evaluating tall expression using the Parallel Pool 'local':
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Evaluation completed in 7.3 sec
Evaluating tall expression using the Parallel Pool 'local':
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Evaluation completed in 6.1 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 1.1 sec
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- Pass 3 of 4: Completed in 0.66 sec
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Evaluation completed in 5.9 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 1.1 sec
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- Pass 3 of 4: Completed in 0.69 sec
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Evaluation completed in 5.5 sec
Evaluating tall expression using the Parallel Pool 'local':
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Evaluation completed in 5.5 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 1.3 sec
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Evaluation completed in 5.4 sec
Evaluating tall expression using the Parallel Pool 'local':
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Evaluation completed in 5 sec
Evaluating tall expression using the Parallel Pool 'local':
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Evaluation completed in 5.1 sec
Evaluating tall expression using the Parallel Pool 'local':
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Evaluation completed in 5.1 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 1: Completed in 2.4 sec
Evaluation completed in 2.6 sec
|======================================================================================|
| Iter | Eval   | Objective   | Objective   | BestSoFar   | BestSoFar   |  MinLeafSize |
|      | result |             | runtime     | (observed)  | (estim.)    |              |
|======================================================================================|
|    1 | Best   |     0.11572 |      197.12 |     0.11572 |     0.11572 |           10 |
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 1: Completed in 0.4 sec
Evaluation completed in 0.56 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 1: Completed in 0.93 sec
Evaluation completed in 1.1 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.5 sec
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Evaluation completed in 3.7 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 1: Completed in 1.4 sec
Evaluation completed in 1.6 sec
|    2 | Accept |     0.19635 |      10.496 |     0.11572 |     0.12008 |        48298 |
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 1: Completed in 0.33 sec
Evaluation completed in 0.47 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 1: Completed in 0.83 sec
Evaluation completed in 0.99 sec
Evaluating tall expression using the Parallel Pool 'local':
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Evaluation completed in 3 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.48 sec
- Pass 2 of 4: Completed in 0.69 sec
- Pass 3 of 4: Completed in 0.51 sec
- Pass 4 of 4: Completed in 0.73 sec
Evaluation completed in 3 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.51 sec
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- Pass 3 of 4: Completed in 0.68 sec
- Pass 4 of 4: Completed in 0.77 sec
Evaluation completed in 3.2 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.48 sec
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- Pass 3 of 4: Completed in 0.55 sec
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Evaluation completed in 3.1 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.5 sec
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Evaluation completed in 3.1 sec
Evaluating tall expression using the Parallel Pool 'local':
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Evaluation completed in 3 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.54 sec
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Evaluation completed in 4 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.53 sec
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Evaluation completed in 3.1 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.56 sec
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- Pass 3 of 4: Completed in 0.56 sec
- Pass 4 of 4: Completed in 0.78 sec
Evaluation completed in 3.2 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 1: Completed in 0.75 sec
Evaluation completed in 0.87 sec
|    3 | Best   |      0.1048 |      44.614 |      0.1048 |     0.11431 |         3166 |
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 1: Completed in 0.3 sec
Evaluation completed in 0.45 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 1: Completed in 0.83 sec
Evaluation completed in 0.97 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.99 sec
- Pass 2 of 4: Completed in 0.68 sec
- Pass 3 of 4: Completed in 0.52 sec
- Pass 4 of 4: Completed in 0.73 sec
Evaluation completed in 3.5 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.47 sec
- Pass 2 of 4: Completed in 0.76 sec
- Pass 3 of 4: Completed in 0.54 sec
- Pass 4 of 4: Completed in 0.82 sec
Evaluation completed in 3.2 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.52 sec
- Pass 2 of 4: Completed in 0.74 sec
- Pass 3 of 4: Completed in 0.54 sec
- Pass 4 of 4: Completed in 0.81 sec
Evaluation completed in 3.2 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.55 sec
- Pass 2 of 4: Completed in 0.7 sec
- Pass 3 of 4: Completed in 0.53 sec
- Pass 4 of 4: Completed in 0.81 sec
Evaluation completed in 3.1 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.52 sec
- Pass 2 of 4: Completed in 0.77 sec
- Pass 3 of 4: Completed in 0.58 sec
- Pass 4 of 4: Completed in 0.89 sec
Evaluation completed in 3.3 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.51 sec
- Pass 2 of 4: Completed in 0.74 sec
- Pass 3 of 4: Completed in 0.6 sec
- Pass 4 of 4: Completed in 1.5 sec
Evaluation completed in 3.9 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 1 sec
- Pass 2 of 4: Completed in 0.75 sec
- Pass 3 of 4: Completed in 0.55 sec
- Pass 4 of 4: Completed in 1.1 sec
Evaluation completed in 4.1 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.52 sec
- Pass 2 of 4: Completed in 0.78 sec
- Pass 3 of 4: Completed in 0.5 sec
- Pass 4 of 4: Completed in 1.3 sec
Evaluation completed in 3.6 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.63 sec
- Pass 2 of 4: Completed in 1.3 sec
- Pass 3 of 4: Completed in 0.61 sec
- Pass 4 of 4: Completed in 1.4 sec
Evaluation completed in 4.5 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.59 sec
- Pass 2 of 4: Completed in 0.8 sec
- Pass 3 of 4: Completed in 0.57 sec
- Pass 4 of 4: Completed in 1.5 sec
Evaluation completed in 4.1 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.66 sec
- Pass 2 of 4: Completed in 0.81 sec
- Pass 3 of 4: Completed in 0.52 sec
- Pass 4 of 4: Completed in 1.5 sec
Evaluation completed in 4.2 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.62 sec
- Pass 2 of 4: Completed in 0.75 sec
- Pass 3 of 4: Completed in 0.61 sec
- Pass 4 of 4: Completed in 1.4 sec
Evaluation completed in 3.9 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.61 sec
- Pass 2 of 4: Completed in 0.78 sec
- Pass 3 of 4: Completed in 1.1 sec
- Pass 4 of 4: Completed in 1.6 sec
Evaluation completed in 4.7 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.6 sec
- Pass 2 of 4: Completed in 0.81 sec
- Pass 3 of 4: Completed in 0.55 sec
- Pass 4 of 4: Completed in 1.5 sec
Evaluation completed in 4 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.61 sec
- Pass 2 of 4: Completed in 0.78 sec
- Pass 3 of 4: Completed in 0.55 sec
- Pass 4 of 4: Completed in 0.9 sec
Evaluation completed in 3.4 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.59 sec
- Pass 2 of 4: Completed in 0.81 sec
- Pass 3 of 4: Completed in 0.53 sec
- Pass 4 of 4: Completed in 0.81 sec
Evaluation completed in 3.3 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.59 sec
- Pass 2 of 4: Completed in 0.78 sec
- Pass 3 of 4: Completed in 1 sec
- Pass 4 of 4: Completed in 1.3 sec
Evaluation completed in 4.3 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 1: Completed in 0.8 sec
Evaluation completed in 0.94 sec
|    4 | Best   |     0.10094 |      91.723 |     0.10094 |     0.10574 |          180 |
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 1: Completed in 0.3 sec
Evaluation completed in 0.42 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 1: Completed in 0.93 sec
Evaluation completed in 1.1 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.55 sec
- Pass 2 of 4: Completed in 0.66 sec
- Pass 3 of 4: Completed in 0.55 sec
- Pass 4 of 4: Completed in 0.83 sec
Evaluation completed in 3.1 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.49 sec
- Pass 2 of 4: Completed in 0.71 sec
- Pass 3 of 4: Completed in 0.54 sec
- Pass 4 of 4: Completed in 0.76 sec
Evaluation completed in 3.1 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.5 sec
- Pass 2 of 4: Completed in 0.7 sec
- Pass 3 of 4: Completed in 0.56 sec
- Pass 4 of 4: Completed in 0.78 sec
Evaluation completed in 3.1 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.56 sec
- Pass 2 of 4: Completed in 0.72 sec
- Pass 3 of 4: Completed in 0.51 sec
- Pass 4 of 4: Completed in 0.81 sec
Evaluation completed in 3.2 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.52 sec
- Pass 2 of 4: Completed in 1.3 sec
- Pass 3 of 4: Completed in 1.1 sec
- Pass 4 of 4: Completed in 0.88 sec
Evaluation completed in 4.3 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.52 sec
- Pass 2 of 4: Completed in 0.7 sec
- Pass 3 of 4: Completed in 0.5 sec
- Pass 4 of 4: Completed in 0.98 sec
Evaluation completed in 3.3 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.56 sec
- Pass 2 of 4: Completed in 0.74 sec
- Pass 3 of 4: Completed in 0.5 sec
- Pass 4 of 4: Completed in 1.1 sec
Evaluation completed in 3.5 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.61 sec
- Pass 2 of 4: Completed in 0.81 sec
- Pass 3 of 4: Completed in 0.56 sec
- Pass 4 of 4: Completed in 1.2 sec
Evaluation completed in 3.8 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.59 sec
- Pass 2 of 4: Completed in 0.81 sec
- Pass 3 of 4: Completed in 0.7 sec
- Pass 4 of 4: Completed in 1.4 sec
Evaluation completed in 4.1 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.56 sec
- Pass 2 of 4: Completed in 0.73 sec
- Pass 3 of 4: Completed in 0.59 sec
- Pass 4 of 4: Completed in 1.4 sec
Evaluation completed in 3.8 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.58 sec
- Pass 2 of 4: Completed in 0.8 sec
- Pass 3 of 4: Completed in 0.52 sec
- Pass 4 of 4: Completed in 1.2 sec
Evaluation completed in 3.7 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.64 sec
- Pass 2 of 4: Completed in 0.79 sec
- Pass 3 of 4: Completed in 0.54 sec
- Pass 4 of 4: Completed in 1.1 sec
Evaluation completed in 3.6 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.57 sec
- Pass 2 of 4: Completed in 0.78 sec
- Pass 3 of 4: Completed in 0.56 sec
- Pass 4 of 4: Completed in 0.97 sec
Evaluation completed in 3.5 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.55 sec
- Pass 2 of 4: Completed in 0.75 sec
- Pass 3 of 4: Completed in 0.55 sec
- Pass 4 of 4: Completed in 0.89 sec
Evaluation completed in 3.3 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.6 sec
- Pass 2 of 4: Completed in 1.3 sec
- Pass 3 of 4: Completed in 0.61 sec
- Pass 4 of 4: Completed in 0.85 sec
Evaluation completed in 3.9 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.6 sec
- Pass 2 of 4: Completed in 0.82 sec
- Pass 3 of 4: Completed in 0.56 sec
- Pass 4 of 4: Completed in 0.79 sec
Evaluation completed in 3.4 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 1: Completed in 1.3 sec
Evaluation completed in 1.4 sec
|    5 | Best   |     0.10087 |       82.84 |     0.10087 |     0.10085 |          219 |
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 1: Completed in 0.32 sec
Evaluation completed in 0.45 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 1: Completed in 0.87 sec
Evaluation completed in 1 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.5 sec
- Pass 2 of 4: Completed in 0.7 sec
- Pass 3 of 4: Completed in 0.56 sec
- Pass 4 of 4: Completed in 0.76 sec
Evaluation completed in 3.1 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.53 sec
- Pass 2 of 4: Completed in 0.74 sec
- Pass 3 of 4: Completed in 0.54 sec
- Pass 4 of 4: Completed in 0.79 sec
Evaluation completed in 3.2 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.49 sec
- Pass 2 of 4: Completed in 0.66 sec
- Pass 3 of 4: Completed in 0.5 sec
- Pass 4 of 4: Completed in 0.78 sec
Evaluation completed in 3 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.49 sec
- Pass 2 of 4: Completed in 0.68 sec
- Pass 3 of 4: Completed in 0.51 sec
- Pass 4 of 4: Completed in 0.81 sec
Evaluation completed in 3.1 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.55 sec
- Pass 2 of 4: Completed in 0.68 sec
- Pass 3 of 4: Completed in 0.54 sec
- Pass 4 of 4: Completed in 0.86 sec
Evaluation completed in 3.2 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.51 sec
- Pass 2 of 4: Completed in 0.71 sec
- Pass 3 of 4: Completed in 0.53 sec
- Pass 4 of 4: Completed in 1 sec
Evaluation completed in 3.4 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.59 sec
- Pass 2 of 4: Completed in 1.2 sec
- Pass 3 of 4: Completed in 0.55 sec
- Pass 4 of 4: Completed in 0.85 sec
Evaluation completed in 3.8 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.6 sec
- Pass 2 of 4: Completed in 0.74 sec
- Pass 3 of 4: Completed in 0.6 sec
- Pass 4 of 4: Completed in 0.84 sec
Evaluation completed in 3.3 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.51 sec
- Pass 2 of 4: Completed in 0.77 sec
- Pass 3 of 4: Completed in 0.58 sec
- Pass 4 of 4: Completed in 0.87 sec
Evaluation completed in 3.3 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.53 sec
- Pass 2 of 4: Completed in 0.78 sec
- Pass 3 of 4: Completed in 1.1 sec
- Pass 4 of 4: Completed in 0.92 sec
Evaluation completed in 3.9 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.54 sec
- Pass 2 of 4: Completed in 0.81 sec
- Pass 3 of 4: Completed in 0.59 sec
- Pass 4 of 4: Completed in 0.77 sec
Evaluation completed in 3.2 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.53 sec
- Pass 2 of 4: Completed in 1.2 sec
- Pass 3 of 4: Completed in 0.68 sec
- Pass 4 of 4: Completed in 0.86 sec
Evaluation completed in 3.9 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 1: Completed in 0.77 sec
Evaluation completed in 0.93 sec
|    6 | Accept |     0.10155 |      61.043 |     0.10087 |     0.10089 |         1089 |
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 1: Completed in 0.33 sec
Evaluation completed in 0.46 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 1: Completed in 0.89 sec
Evaluation completed in 1 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.56 sec
- Pass 2 of 4: Completed in 0.72 sec
- Pass 3 of 4: Completed in 0.56 sec
- Pass 4 of 4: Completed in 0.8 sec
Evaluation completed in 3.2 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.53 sec
- Pass 2 of 4: Completed in 0.69 sec
- Pass 3 of 4: Completed in 0.54 sec
- Pass 4 of 4: Completed in 0.85 sec
Evaluation completed in 3.1 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.51 sec
- Pass 2 of 4: Completed in 1.2 sec
- Pass 3 of 4: Completed in 0.59 sec
- Pass 4 of 4: Completed in 0.83 sec
Evaluation completed in 3.8 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 1 sec
- Pass 2 of 4: Completed in 0.76 sec
- Pass 3 of 4: Completed in 0.56 sec
- Pass 4 of 4: Completed in 0.87 sec
Evaluation completed in 4 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 1 sec
- Pass 2 of 4: Completed in 0.77 sec
- Pass 3 of 4: Completed in 0.53 sec
- Pass 4 of 4: Completed in 0.9 sec
Evaluation completed in 3.8 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 1 sec
- Pass 2 of 4: Completed in 0.77 sec
- Pass 3 of 4: Completed in 0.51 sec
- Pass 4 of 4: Completed in 0.98 sec
Evaluation completed in 3.9 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 1 sec
- Pass 2 of 4: Completed in 0.78 sec
- Pass 3 of 4: Completed in 0.62 sec
- Pass 4 of 4: Completed in 1.1 sec
Evaluation completed in 4.2 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.52 sec
- Pass 2 of 4: Completed in 0.71 sec
- Pass 3 of 4: Completed in 0.53 sec
- Pass 4 of 4: Completed in 1.5 sec
Evaluation completed in 3.9 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.59 sec
- Pass 2 of 4: Completed in 0.72 sec
- Pass 3 of 4: Completed in 0.59 sec
- Pass 4 of 4: Completed in 2 sec
Evaluation completed in 4.5 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.59 sec
- Pass 2 of 4: Completed in 0.81 sec
- Pass 3 of 4: Completed in 0.57 sec
- Pass 4 of 4: Completed in 2.7 sec
Evaluation completed in 5.4 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.64 sec
- Pass 2 of 4: Completed in 0.87 sec
- Pass 3 of 4: Completed in 1.2 sec
- Pass 4 of 4: Completed in 3.7 sec
Evaluation completed in 7.1 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.73 sec
- Pass 2 of 4: Completed in 0.92 sec
- Pass 3 of 4: Completed in 0.6 sec
- Pass 4 of 4: Completed in 4.4 sec
Evaluation completed in 7.3 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.86 sec
- Pass 2 of 4: Completed in 1.5 sec
- Pass 3 of 4: Completed in 0.64 sec
- Pass 4 of 4: Completed in 4.8 sec
Evaluation completed in 8.6 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.9 sec
- Pass 2 of 4: Completed in 1.1 sec
- Pass 3 of 4: Completed in 0.65 sec
- Pass 4 of 4: Completed in 5.2 sec
Evaluation completed in 8.5 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 1 sec
- Pass 2 of 4: Completed in 1.3 sec
- Pass 3 of 4: Completed in 0.73 sec
- Pass 4 of 4: Completed in 5.6 sec
Evaluation completed in 9.4 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 1.5 sec
- Pass 2 of 4: Completed in 1.6 sec
- Pass 3 of 4: Completed in 0.75 sec
- Pass 4 of 4: Completed in 5.8 sec
Evaluation completed in 10 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 1.3 sec
- Pass 2 of 4: Completed in 1.4 sec
- Pass 3 of 4: Completed in 1.2 sec
- Pass 4 of 4: Completed in 5.1 sec
Evaluation completed in 9.9 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 1.4 sec
- Pass 2 of 4: Completed in 1.5 sec
- Pass 3 of 4: Completed in 0.7 sec
- Pass 4 of 4: Completed in 4.1 sec
Evaluation completed in 8.5 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 1.4 sec
- Pass 2 of 4: Completed in 1.6 sec
- Pass 3 of 4: Completed in 0.71 sec
- Pass 4 of 4: Completed in 3.6 sec
Evaluation completed in 7.9 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 1.5 sec
- Pass 2 of 4: Completed in 1.8 sec
- Pass 3 of 4: Completed in 0.74 sec
- Pass 4 of 4: Completed in 3.2 sec
Evaluation completed in 7.9 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 1.4 sec
- Pass 2 of 4: Completed in 1.7 sec
- Pass 3 of 4: Completed in 0.73 sec
- Pass 4 of 4: Completed in 2.8 sec
Evaluation completed in 7.3 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 1.5 sec
- Pass 2 of 4: Completed in 1.7 sec
- Pass 3 of 4: Completed in 0.82 sec
- Pass 4 of 4: Completed in 2.4 sec
Evaluation completed in 7 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 2 sec
- Pass 2 of 4: Completed in 1.9 sec
- Pass 3 of 4: Completed in 0.79 sec
- Pass 4 of 4: Completed in 2.3 sec
Evaluation completed in 7.6 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 1.6 sec
- Pass 2 of 4: Completed in 1.8 sec
- Pass 3 of 4: Completed in 0.73 sec
- Pass 4 of 4: Completed in 2.2 sec
Evaluation completed in 6.9 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 1.6 sec
- Pass 2 of 4: Completed in 1.7 sec
- Pass 3 of 4: Completed in 0.79 sec
- Pass 4 of 4: Completed in 2.3 sec
Evaluation completed in 7 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 1.7 sec
- Pass 2 of 4: Completed in 1.9 sec
- Pass 3 of 4: Completed in 0.8 sec
- Pass 4 of 4: Completed in 1.8 sec
Evaluation completed in 6.8 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 1.7 sec
- Pass 2 of 4: Completed in 1.8 sec
- Pass 3 of 4: Completed in 0.77 sec
- Pass 4 of 4: Completed in 1.8 sec
Evaluation completed in 6.6 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 1.4 sec
- Pass 2 of 4: Completed in 1.6 sec
- Pass 3 of 4: Completed in 0.73 sec
- Pass 4 of 4: Completed in 1.8 sec
Evaluation completed in 6.1 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 1.5 sec
- Pass 2 of 4: Completed in 1.7 sec
- Pass 3 of 4: Completed in 1.3 sec
- Pass 4 of 4: Completed in 1.7 sec
Evaluation completed in 6.7 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 1.5 sec
- Pass 2 of 4: Completed in 1.7 sec
- Pass 3 of 4: Completed in 0.73 sec
- Pass 4 of 4: Completed in 1.8 sec
Evaluation completed in 6.3 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 1: Completed in 1.3 sec
Evaluation completed in 1.5 sec
|    7 | Accept |     0.13495 |      241.76 |     0.10087 |     0.10089 |            1 |
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 1: Completed in 0.31 sec
Evaluation completed in 0.44 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 1: Completed in 0.87 sec
Evaluation completed in 1 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.47 sec
- Pass 2 of 4: Completed in 0.67 sec
- Pass 3 of 4: Completed in 0.54 sec
- Pass 4 of 4: Completed in 0.74 sec
Evaluation completed in 3 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.51 sec
- Pass 2 of 4: Completed in 0.79 sec
- Pass 3 of 4: Completed in 0.56 sec
- Pass 4 of 4: Completed in 0.76 sec
Evaluation completed in 3.2 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.49 sec
- Pass 2 of 4: Completed in 0.69 sec
- Pass 3 of 4: Completed in 1.1 sec
- Pass 4 of 4: Completed in 0.78 sec
Evaluation completed in 3.6 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 1 sec
- Pass 2 of 4: Completed in 0.78 sec
- Pass 3 of 4: Completed in 0.53 sec
- Pass 4 of 4: Completed in 0.85 sec
Evaluation completed in 3.8 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.54 sec
- Pass 2 of 4: Completed in 0.77 sec
- Pass 3 of 4: Completed in 0.52 sec
- Pass 4 of 4: Completed in 0.89 sec
Evaluation completed in 3.3 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.51 sec
- Pass 2 of 4: Completed in 0.73 sec
- Pass 3 of 4: Completed in 0.6 sec
- Pass 4 of 4: Completed in 1.1 sec
Evaluation completed in 3.5 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.53 sec
- Pass 2 of 4: Completed in 0.77 sec
- Pass 3 of 4: Completed in 0.54 sec
- Pass 4 of 4: Completed in 1.3 sec
Evaluation completed in 3.7 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.62 sec
- Pass 2 of 4: Completed in 0.8 sec
- Pass 3 of 4: Completed in 0.61 sec
- Pass 4 of 4: Completed in 1.6 sec
Evaluation completed in 4.4 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 1.2 sec
- Pass 2 of 4: Completed in 1.5 sec
- Pass 3 of 4: Completed in 1.1 sec
- Pass 4 of 4: Completed in 1.9 sec
Evaluation completed in 6.3 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.69 sec
- Pass 2 of 4: Completed in 0.88 sec
- Pass 3 of 4: Completed in 0.75 sec
- Pass 4 of 4: Completed in 2.1 sec
Evaluation completed in 5.1 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.61 sec
- Pass 2 of 4: Completed in 0.79 sec
- Pass 3 of 4: Completed in 0.54 sec
- Pass 4 of 4: Completed in 2.2 sec
Evaluation completed in 4.8 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.61 sec
- Pass 2 of 4: Completed in 0.84 sec
- Pass 3 of 4: Completed in 0.58 sec
- Pass 4 of 4: Completed in 2.2 sec
Evaluation completed in 4.9 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.68 sec
- Pass 2 of 4: Completed in 0.85 sec
- Pass 3 of 4: Completed in 0.59 sec
- Pass 4 of 4: Completed in 2.2 sec
Evaluation completed in 4.9 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.68 sec
- Pass 2 of 4: Completed in 0.91 sec
- Pass 3 of 4: Completed in 0.58 sec
- Pass 4 of 4: Completed in 2.4 sec
Evaluation completed in 5.3 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.92 sec
- Pass 2 of 4: Completed in 0.86 sec
- Pass 3 of 4: Completed in 0.57 sec
- Pass 4 of 4: Completed in 1.6 sec
Evaluation completed in 4.7 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.69 sec
- Pass 2 of 4: Completed in 0.91 sec
- Pass 3 of 4: Completed in 0.63 sec
- Pass 4 of 4: Completed in 1.3 sec
Evaluation completed in 4.1 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.67 sec
- Pass 2 of 4: Completed in 0.86 sec
- Pass 3 of 4: Completed in 0.56 sec
- Pass 4 of 4: Completed in 0.99 sec
Evaluation completed in 3.6 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 1.2 sec
- Pass 2 of 4: Completed in 0.9 sec
- Pass 3 of 4: Completed in 0.57 sec
- Pass 4 of 4: Completed in 0.95 sec
Evaluation completed in 4.2 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.73 sec
- Pass 2 of 4: Completed in 0.91 sec
- Pass 3 of 4: Completed in 0.57 sec
- Pass 4 of 4: Completed in 0.91 sec
Evaluation completed in 3.7 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.76 sec
- Pass 2 of 4: Completed in 0.93 sec
- Pass 3 of 4: Completed in 0.57 sec
- Pass 4 of 4: Completed in 0.9 sec
Evaluation completed in 3.7 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 1: Completed in 0.91 sec
Evaluation completed in 1.1 sec
|    8 | Accept |     0.10246 |      115.31 |     0.10087 |     0.10089 |           58 |
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 1: Completed in 0.34 sec
Evaluation completed in 0.49 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 1: Completed in 0.87 sec
Evaluation completed in 1 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.55 sec
- Pass 2 of 4: Completed in 0.72 sec
- Pass 3 of 4: Completed in 0.57 sec
- Pass 4 of 4: Completed in 0.8 sec
Evaluation completed in 3.2 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.48 sec
- Pass 2 of 4: Completed in 0.7 sec
- Pass 3 of 4: Completed in 0.52 sec
- Pass 4 of 4: Completed in 0.76 sec
Evaluation completed in 3 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.51 sec
- Pass 2 of 4: Completed in 0.69 sec
- Pass 3 of 4: Completed in 0.54 sec
- Pass 4 of 4: Completed in 0.79 sec
Evaluation completed in 3.1 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.53 sec
- Pass 2 of 4: Completed in 0.72 sec
- Pass 3 of 4: Completed in 0.56 sec
- Pass 4 of 4: Completed in 0.81 sec
Evaluation completed in 3.2 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 1 sec
- Pass 2 of 4: Completed in 0.75 sec
- Pass 3 of 4: Completed in 0.56 sec
- Pass 4 of 4: Completed in 1.4 sec
Evaluation completed in 4.3 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.55 sec
- Pass 2 of 4: Completed in 0.77 sec
- Pass 3 of 4: Completed in 0.6 sec
- Pass 4 of 4: Completed in 0.9 sec
Evaluation completed in 3.3 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.5 sec
- Pass 2 of 4: Completed in 0.73 sec
- Pass 3 of 4: Completed in 1.1 sec
- Pass 4 of 4: Completed in 1.1 sec
Evaluation completed in 4 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.56 sec
- Pass 2 of 4: Completed in 0.75 sec
- Pass 3 of 4: Completed in 0.53 sec
- Pass 4 of 4: Completed in 1.2 sec
Evaluation completed in 3.5 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.61 sec
- Pass 2 of 4: Completed in 0.76 sec
- Pass 3 of 4: Completed in 1.1 sec
- Pass 4 of 4: Completed in 1.1 sec
Evaluation completed in 4.1 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.6 sec
- Pass 2 of 4: Completed in 0.74 sec
- Pass 3 of 4: Completed in 0.53 sec
- Pass 4 of 4: Completed in 0.95 sec
Evaluation completed in 3.4 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.55 sec
- Pass 2 of 4: Completed in 0.76 sec
- Pass 3 of 4: Completed in 0.57 sec
- Pass 4 of 4: Completed in 0.94 sec
Evaluation completed in 3.4 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.58 sec
- Pass 2 of 4: Completed in 0.77 sec
- Pass 3 of 4: Completed in 0.54 sec
- Pass 4 of 4: Completed in 0.83 sec
Evaluation completed in 3.3 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.59 sec
- Pass 2 of 4: Completed in 0.78 sec
- Pass 3 of 4: Completed in 0.57 sec
- Pass 4 of 4: Completed in 0.83 sec
Evaluation completed in 3.4 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.6 sec
- Pass 2 of 4: Completed in 0.76 sec
- Pass 3 of 4: Completed in 0.55 sec
- Pass 4 of 4: Completed in 0.77 sec
Evaluation completed in 3.3 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.72 sec
- Pass 2 of 4: Completed in 0.81 sec
- Pass 3 of 4: Completed in 0.6 sec
- Pass 4 of 4: Completed in 0.76 sec
Evaluation completed in 3.6 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 1: Completed in 1.3 sec
Evaluation completed in 1.4 sec
|    9 | Accept |     0.10173 |      77.229 |     0.10087 |     0.10086 |          418 |
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 1: Completed in 0.32 sec
Evaluation completed in 0.46 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 1: Completed in 0.84 sec
Evaluation completed in 1 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.5 sec
- Pass 2 of 4: Completed in 0.72 sec
- Pass 3 of 4: Completed in 0.57 sec
- Pass 4 of 4: Completed in 0.75 sec
Evaluation completed in 3.1 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.49 sec
- Pass 2 of 4: Completed in 0.68 sec
- Pass 3 of 4: Completed in 0.55 sec
- Pass 4 of 4: Completed in 0.76 sec
Evaluation completed in 3 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.52 sec
- Pass 2 of 4: Completed in 0.71 sec
- Pass 3 of 4: Completed in 0.54 sec
- Pass 4 of 4: Completed in 0.91 sec
Evaluation completed in 3.3 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.53 sec
- Pass 2 of 4: Completed in 0.69 sec
- Pass 3 of 4: Completed in 0.52 sec
- Pass 4 of 4: Completed in 0.82 sec
Evaluation completed in 3.1 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.51 sec
- Pass 2 of 4: Completed in 0.7 sec
- Pass 3 of 4: Completed in 0.61 sec
- Pass 4 of 4: Completed in 0.82 sec
Evaluation completed in 3.2 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 1.1 sec
- Pass 2 of 4: Completed in 0.78 sec
- Pass 3 of 4: Completed in 0.54 sec
- Pass 4 of 4: Completed in 0.95 sec
Evaluation completed in 3.9 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.55 sec
- Pass 2 of 4: Completed in 0.71 sec
- Pass 3 of 4: Completed in 0.53 sec
- Pass 4 of 4: Completed in 1.1 sec
Evaluation completed in 3.5 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.54 sec
- Pass 2 of 4: Completed in 0.7 sec
- Pass 3 of 4: Completed in 0.58 sec
- Pass 4 of 4: Completed in 1.3 sec
Evaluation completed in 3.7 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.55 sec
- Pass 2 of 4: Completed in 0.73 sec
- Pass 3 of 4: Completed in 0.59 sec
- Pass 4 of 4: Completed in 1.7 sec
Evaluation completed in 4.2 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.58 sec
- Pass 2 of 4: Completed in 0.79 sec
- Pass 3 of 4: Completed in 0.56 sec
- Pass 4 of 4: Completed in 1.7 sec
Evaluation completed in 4.2 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.71 sec
- Pass 2 of 4: Completed in 1.7 sec
- Pass 3 of 4: Completed in 0.59 sec
- Pass 4 of 4: Completed in 1.7 sec
Evaluation completed in 5.2 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.65 sec
- Pass 2 of 4: Completed in 0.83 sec
- Pass 3 of 4: Completed in 0.61 sec
- Pass 4 of 4: Completed in 1.4 sec
Evaluation completed in 4.1 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.67 sec
- Pass 2 of 4: Completed in 0.87 sec
- Pass 3 of 4: Completed in 0.58 sec
- Pass 4 of 4: Completed in 1.4 sec
Evaluation completed in 4.1 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.61 sec
- Pass 2 of 4: Completed in 0.82 sec
- Pass 3 of 4: Completed in 0.55 sec
- Pass 4 of 4: Completed in 1.1 sec
Evaluation completed in 3.7 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.65 sec
- Pass 2 of 4: Completed in 0.84 sec
- Pass 3 of 4: Completed in 0.62 sec
- Pass 4 of 4: Completed in 0.89 sec
Evaluation completed in 3.6 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.64 sec
- Pass 2 of 4: Completed in 0.81 sec
- Pass 3 of 4: Completed in 0.56 sec
- Pass 4 of 4: Completed in 0.88 sec
Evaluation completed in 3.6 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.62 sec
- Pass 2 of 4: Completed in 0.9 sec
- Pass 3 of 4: Completed in 0.55 sec
- Pass 4 of 4: Completed in 0.86 sec
Evaluation completed in 3.5 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.65 sec
- Pass 2 of 4: Completed in 0.81 sec
- Pass 3 of 4: Completed in 0.56 sec
- Pass 4 of 4: Completed in 0.8 sec
Evaluation completed in 3.3 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 1: Completed in 0.77 sec
Evaluation completed in 0.89 sec
|   10 | Accept |     0.10114 |      94.532 |     0.10087 |     0.10091 |          123 |
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 1: Completed in 0.86 sec
Evaluation completed in 1 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 1: Completed in 0.83 sec
Evaluation completed in 0.99 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.48 sec
- Pass 2 of 4: Completed in 0.7 sec
- Pass 3 of 4: Completed in 0.54 sec
- Pass 4 of 4: Completed in 0.8 sec
Evaluation completed in 3 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.5 sec
- Pass 2 of 4: Completed in 0.72 sec
- Pass 3 of 4: Completed in 0.53 sec
- Pass 4 of 4: Completed in 0.79 sec
Evaluation completed in 3.1 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.56 sec
- Pass 2 of 4: Completed in 0.73 sec
- Pass 3 of 4: Completed in 0.54 sec
- Pass 4 of 4: Completed in 0.85 sec
Evaluation completed in 3.3 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.5 sec
- Pass 2 of 4: Completed in 0.69 sec
- Pass 3 of 4: Completed in 0.55 sec
- Pass 4 of 4: Completed in 0.81 sec
Evaluation completed in 3.2 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.55 sec
- Pass 2 of 4: Completed in 0.82 sec
- Pass 3 of 4: Completed in 0.64 sec
- Pass 4 of 4: Completed in 0.94 sec
Evaluation completed in 3.5 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.49 sec
- Pass 2 of 4: Completed in 0.77 sec
- Pass 3 of 4: Completed in 0.53 sec
- Pass 4 of 4: Completed in 0.97 sec
Evaluation completed in 3.3 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.56 sec
- Pass 2 of 4: Completed in 0.78 sec
- Pass 3 of 4: Completed in 0.58 sec
- Pass 4 of 4: Completed in 1.3 sec
Evaluation completed in 3.8 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.55 sec
- Pass 2 of 4: Completed in 0.81 sec
- Pass 3 of 4: Completed in 0.56 sec
- Pass 4 of 4: Completed in 1.3 sec
Evaluation completed in 3.8 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.55 sec
- Pass 2 of 4: Completed in 0.76 sec
- Pass 3 of 4: Completed in 0.56 sec
- Pass 4 of 4: Completed in 1.5 sec
Evaluation completed in 3.9 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.59 sec
- Pass 2 of 4: Completed in 0.76 sec
- Pass 3 of 4: Completed in 0.55 sec
- Pass 4 of 4: Completed in 1.5 sec
Evaluation completed in 4 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.59 sec
- Pass 2 of 4: Completed in 0.8 sec
- Pass 3 of 4: Completed in 0.59 sec
- Pass 4 of 4: Completed in 1.4 sec
Evaluation completed in 3.9 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.59 sec
- Pass 2 of 4: Completed in 0.77 sec
- Pass 3 of 4: Completed in 0.55 sec
- Pass 4 of 4: Completed in 1.2 sec
Evaluation completed in 3.7 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 1.1 sec
- Pass 2 of 4: Completed in 0.8 sec
- Pass 3 of 4: Completed in 0.58 sec
- Pass 4 of 4: Completed in 1.1 sec
Evaluation completed in 4.2 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.65 sec
- Pass 2 of 4: Completed in 0.84 sec
- Pass 3 of 4: Completed in 1.1 sec
- Pass 4 of 4: Completed in 1 sec
Evaluation completed in 4.3 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.63 sec
- Pass 2 of 4: Completed in 0.84 sec
- Pass 3 of 4: Completed in 0.59 sec
- Pass 4 of 4: Completed in 0.9 sec
Evaluation completed in 3.5 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 1.2 sec
- Pass 2 of 4: Completed in 0.83 sec
- Pass 3 of 4: Completed in 0.56 sec
- Pass 4 of 4: Completed in 0.81 sec
Evaluation completed in 4 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.65 sec
- Pass 2 of 4: Completed in 0.79 sec
- Pass 3 of 4: Completed in 0.59 sec
- Pass 4 of 4: Completed in 0.8 sec
Evaluation completed in 3.4 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 1: Completed in 0.77 sec
Evaluation completed in 0.89 sec
|   11 | Best   |      0.1008 |      90.637 |      0.1008 |     0.10088 |          178 |
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 1: Completed in 0.38 sec
Evaluation completed in 0.52 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 1: Completed in 0.88 sec
Evaluation completed in 1 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.49 sec
- Pass 2 of 4: Completed in 0.69 sec
- Pass 3 of 4: Completed in 0.51 sec
- Pass 4 of 4: Completed in 0.78 sec
Evaluation completed in 3 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.59 sec
- Pass 2 of 4: Completed in 0.72 sec
- Pass 3 of 4: Completed in 0.53 sec
- Pass 4 of 4: Completed in 0.79 sec
Evaluation completed in 3.1 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.58 sec
- Pass 2 of 4: Completed in 0.74 sec
- Pass 3 of 4: Completed in 0.55 sec
- Pass 4 of 4: Completed in 0.93 sec
Evaluation completed in 3.4 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.57 sec
- Pass 2 of 4: Completed in 0.79 sec
- Pass 3 of 4: Completed in 0.58 sec
- Pass 4 of 4: Completed in 0.83 sec
Evaluation completed in 3.3 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.53 sec
- Pass 2 of 4: Completed in 1.2 sec
- Pass 3 of 4: Completed in 0.59 sec
- Pass 4 of 4: Completed in 0.91 sec
Evaluation completed in 3.8 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.58 sec
- Pass 2 of 4: Completed in 0.85 sec
- Pass 3 of 4: Completed in 0.58 sec
- Pass 4 of 4: Completed in 1 sec
Evaluation completed in 3.6 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.56 sec
- Pass 2 of 4: Completed in 0.77 sec
- Pass 3 of 4: Completed in 0.55 sec
- Pass 4 of 4: Completed in 1.2 sec
Evaluation completed in 3.6 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 1.1 sec
- Pass 2 of 4: Completed in 0.81 sec
- Pass 3 of 4: Completed in 0.52 sec
- Pass 4 of 4: Completed in 1.4 sec
Evaluation completed in 4.4 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.54 sec
- Pass 2 of 4: Completed in 0.74 sec
- Pass 3 of 4: Completed in 0.54 sec
- Pass 4 of 4: Completed in 1.5 sec
Evaluation completed in 3.9 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.61 sec
- Pass 2 of 4: Completed in 0.79 sec
- Pass 3 of 4: Completed in 0.58 sec
- Pass 4 of 4: Completed in 1.5 sec
Evaluation completed in 4 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.61 sec
- Pass 2 of 4: Completed in 0.82 sec
- Pass 3 of 4: Completed in 0.61 sec
- Pass 4 of 4: Completed in 1.4 sec
Evaluation completed in 4 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.66 sec
- Pass 2 of 4: Completed in 0.77 sec
- Pass 3 of 4: Completed in 0.54 sec
- Pass 4 of 4: Completed in 1.2 sec
Evaluation completed in 3.8 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.61 sec
- Pass 2 of 4: Completed in 0.79 sec
- Pass 3 of 4: Completed in 0.56 sec
- Pass 4 of 4: Completed in 1.2 sec
Evaluation completed in 3.7 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.62 sec
- Pass 2 of 4: Completed in 0.85 sec
- Pass 3 of 4: Completed in 0.56 sec
- Pass 4 of 4: Completed in 1 sec
Evaluation completed in 3.6 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.61 sec
- Pass 2 of 4: Completed in 0.86 sec
- Pass 3 of 4: Completed in 1.1 sec
- Pass 4 of 4: Completed in 0.96 sec
Evaluation completed in 4.1 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.65 sec
- Pass 2 of 4: Completed in 0.8 sec
- Pass 3 of 4: Completed in 0.59 sec
- Pass 4 of 4: Completed in 0.86 sec
Evaluation completed in 3.5 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.69 sec
- Pass 2 of 4: Completed in 0.84 sec
- Pass 3 of 4: Completed in 0.53 sec
- Pass 4 of 4: Completed in 0.83 sec
Evaluation completed in 3.5 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 1: Completed in 0.76 sec
Evaluation completed in 0.89 sec
|   12 | Accept |      0.1008 |      90.267 |      0.1008 |     0.10086 |          179 |
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 1: Completed in 0.32 sec
Evaluation completed in 0.45 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 1: Completed in 0.9 sec
Evaluation completed in 1.1 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.58 sec
- Pass 2 of 4: Completed in 0.71 sec
- Pass 3 of 4: Completed in 0.53 sec
- Pass 4 of 4: Completed in 0.77 sec
Evaluation completed in 3.2 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.49 sec
- Pass 2 of 4: Completed in 0.69 sec
- Pass 3 of 4: Completed in 0.58 sec
- Pass 4 of 4: Completed in 0.77 sec
Evaluation completed in 3.1 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.52 sec
- Pass 2 of 4: Completed in 0.71 sec
- Pass 3 of 4: Completed in 0.51 sec
- Pass 4 of 4: Completed in 0.78 sec
Evaluation completed in 3.1 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.54 sec
- Pass 2 of 4: Completed in 0.7 sec
- Pass 3 of 4: Completed in 0.54 sec
- Pass 4 of 4: Completed in 0.72 sec
Evaluation completed in 3.1 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.54 sec
- Pass 2 of 4: Completed in 0.74 sec
- Pass 3 of 4: Completed in 0.51 sec
- Pass 4 of 4: Completed in 1.3 sec
Evaluation completed in 3.6 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.54 sec
- Pass 2 of 4: Completed in 0.78 sec
- Pass 3 of 4: Completed in 0.59 sec
- Pass 4 of 4: Completed in 0.74 sec
Evaluation completed in 3.2 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 1: Completed in 0.83 sec
Evaluation completed in 0.97 sec
|   13 | Accept |     0.11126 |      32.134 |      0.1008 |     0.10084 |        10251 |
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 1: Completed in 0.32 sec
Evaluation completed in 0.45 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 1: Completed in 0.85 sec
Evaluation completed in 0.99 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.51 sec
- Pass 2 of 4: Completed in 0.75 sec
- Pass 3 of 4: Completed in 0.55 sec
- Pass 4 of 4: Completed in 0.74 sec
Evaluation completed in 3.1 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.5 sec
- Pass 2 of 4: Completed in 0.7 sec
- Pass 3 of 4: Completed in 0.57 sec
- Pass 4 of 4: Completed in 0.78 sec
Evaluation completed in 3.1 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.5 sec
- Pass 2 of 4: Completed in 0.68 sec
- Pass 3 of 4: Completed in 0.53 sec
- Pass 4 of 4: Completed in 0.79 sec
Evaluation completed in 3 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.5 sec
- Pass 2 of 4: Completed in 1.3 sec
- Pass 3 of 4: Completed in 0.54 sec
- Pass 4 of 4: Completed in 0.91 sec
Evaluation completed in 3.7 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.53 sec
- Pass 2 of 4: Completed in 1.2 sec
- Pass 3 of 4: Completed in 0.59 sec
- Pass 4 of 4: Completed in 0.86 sec
Evaluation completed in 3.7 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.51 sec
- Pass 2 of 4: Completed in 0.71 sec
- Pass 3 of 4: Completed in 0.59 sec
- Pass 4 of 4: Completed in 1 sec
Evaluation completed in 3.4 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.56 sec
- Pass 2 of 4: Completed in 0.71 sec
- Pass 3 of 4: Completed in 0.64 sec
- Pass 4 of 4: Completed in 0.99 sec
Evaluation completed in 3.4 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.54 sec
- Pass 2 of 4: Completed in 1.2 sec
- Pass 3 of 4: Completed in 0.58 sec
- Pass 4 of 4: Completed in 0.94 sec
Evaluation completed in 3.8 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.51 sec
- Pass 2 of 4: Completed in 0.77 sec
- Pass 3 of 4: Completed in 0.59 sec
- Pass 4 of 4: Completed in 0.9 sec
Evaluation completed in 3.3 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.56 sec
- Pass 2 of 4: Completed in 0.78 sec
- Pass 3 of 4: Completed in 0.56 sec
- Pass 4 of 4: Completed in 0.9 sec
Evaluation completed in 3.3 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.55 sec
- Pass 2 of 4: Completed in 0.72 sec
- Pass 3 of 4: Completed in 0.52 sec
- Pass 4 of 4: Completed in 0.89 sec
Evaluation completed in 3.2 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.58 sec
- Pass 2 of 4: Completed in 0.76 sec
- Pass 3 of 4: Completed in 0.54 sec
- Pass 4 of 4: Completed in 0.8 sec
Evaluation completed in 3.3 sec
Evaluating tall expression using the Parallel Pool 'local':
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|   14 | Accept |     0.10154 |      66.262 |      0.1008 |     0.10085 |          736 |
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Mdl = 
  CompactClassificationTree
             ResponseName: 'Y'
    CategoricalPredictors: []
               ClassNames: [0 1]
           ScoreTransform: 'none'


  Properties, Methods

FitInfo = struct with no fields.

HyperparameterOptimizationResults = 
  BayesianOptimization with properties:

                      ObjectiveFcn: @createObjFcn/tallObjFcn
              VariableDescriptions: [4×1 optimizableVariable]
                           Options: [1×1 struct]
                      MinObjective: 0.1004
                   XAtMinObjective: [1×1 table]
             MinEstimatedObjective: 0.1008
          XAtMinEstimatedObjective: [1×1 table]
           NumObjectiveEvaluations: 30
                  TotalElapsedTime: 3.0367e+03
                         NextPoint: [1×1 table]
                            XTrace: [30×1 table]
                    ObjectiveTrace: [30×1 double]
                  ConstraintsTrace: []
                     UserDataTrace: {30×1 cell}
      ObjectiveEvaluationTimeTrace: [30×1 double]
                IterationTimeTrace: [30×1 double]
                        ErrorTrace: [30×1 double]
                  FeasibilityTrace: [30×1 logical]
       FeasibilityProbabilityTrace: [30×1 double]
               IndexOfMinimumTrace: [30×1 double]
             ObjectiveMinimumTrace: [30×1 double]
    EstimatedObjectiveMinimumTrace: [30×1 double]

Input Arguments

Tbl — Sample data

table

Sample data used to train the model, specified as a table. Each row of Tbl corresponds to one observation, and each column corresponds to one predictor variable. Optionally, Tbl can contain one additional column for the response variable. Multicolumn variables and cell arrays other than cell arrays of character vectors are not allowed.

• If Tbl contains the response variable, and you want to use all remaining variables in Tbl as predictors, then specify the response variable by using ResponseVarName.

• If Tbl contains the response variable, and you want to use only a subset of the remaining variables in Tbl as predictors, then specify a formula by using formula.

• If Tbl does not contain the response variable, then specify a response variable by using Y. The length of the response variable and the number of rows in Tbl must be equal.

ResponseVarName — Response variable name

name of variable in Tbl

Response variable name, specified as the name of a variable in Tbl.

You must specify ResponseVarName as a character vector or string scalar. For example, if the response variable Y is stored as Tbl.Y, then specify it as "Y". Otherwise, the software treats all columns of Tbl, including Y, as predictors when training the model.

The response variable must be a categorical, character, or string array; a logical or numeric vector; or a cell array of character vectors. If Y is a character array, then each element of the response variable must correspond to one row of the array.

A good practice is to specify the order of the classes by using the ClassNames name-value argument.

Data Types: char | string

formula — Explanatory model of response variable and subset of predictor variables

character vector | string scalar

Explanatory model of the response variable and a subset of the predictor variables, specified as a character vector or string scalar in the form "Y~x1+x2+x3". In this form, Y represents the response variable, and x1, x2, and x3 represent the predictor variables.

To specify a subset of variables in Tbl as predictors for training the model, use a formula. If you specify a formula, then the software does not use any variables in Tbl that do not appear in formula.

The variable names in the formula must be both variable names in Tbl (Tbl.Properties.VariableNames) and valid MATLAB^® identifiers. You can verify the variable names in Tbl by using the isvarname function. If the variable names are not valid, then you can convert them by using the matlab.lang.makeValidName function.

Data Types: char | string

Y — Class labels

numeric vector | categorical vector | logical vector | character array | string array | cell array of character vectors

Class labels, specified as a numeric vector, categorical vector, logical vector, character array, string array, or cell array of character vectors. Each row of Y represents the classification of the corresponding row of X.

When fitting the tree, fitctree considers NaN, '' (empty character vector), "" (empty string), <missing>, and <undefined> values in Y to be missing values. fitctree does not use observations with missing values for Y in the fit.

For numeric Y, consider fitting a regression tree using fitrtree instead.

Data Types: single | double | categorical | logical | char | string | cell

X — Predictor data

numeric matrix

Predictor data, specified as a numeric matrix. Each row of X corresponds to one observation, and each column corresponds to one predictor variable.

fitctree considers NaN values in X as missing values. fitctree does not use observations with all missing values for X in the fit. fitctree uses observations with some missing values for X to find splits on variables for which these observations have valid values.

Data Types: single | double

Name-Value Arguments

Specify optional pairs of arguments as Name1=Value1,...,NameN=ValueN, where Name is the argument name and Value is the corresponding value. Name-value arguments must appear after other arguments, but the order of the pairs does not matter.

Before R2021a, use commas to separate each name and value, and enclose Name in quotes.

Example: 'CrossVal','on','MinLeafSize',40 specifies a cross-validated classification tree with a minimum of 40 observations per leaf.

Note

You cannot use any cross-validation name-value argument together with the OptimizeHyperparameters name-value argument. You can modify the cross-validation for OptimizeHyperparameters only by using the HyperparameterOptimizationOptions name-value argument.

Model Parameters

AlgorithmForCategorical — Algorithm for best categorical predictor split

'Exact' | 'PullLeft' | 'PCA' | 'OVAbyClass'

Algorithm to find the best split on a categorical predictor with C categories for data and K ≥ 3 classes, specified as the comma-separated pair consisting of 'AlgorithmForCategorical' and one of the following values.

Value	Description
'Exact'	Consider all 2C–1 – 1 combinations.
'PullLeft'	Start with all C categories on the right branch. Consider moving each category to the left branch as it achieves the minimum impurity for the K classes among the remaining categories. From this sequence, choose the split that has the lowest impurity.
'PCA'	Compute a score for each category using the inner product between the first principal component of a weighted covariance matrix (of the centered class probability matrix) and the vector of class probabilities for that category. Sort the scores in ascending order, and consider all C – 1 splits.
'OVAbyClass'	Start with all C categories on the right branch. For each class, order the categories based on their probability for that class. For the first class, consider moving each category to the left branch in order, recording the impurity criterion at each move. Repeat for the remaining classes. From this sequence, choose the split that has the minimum impurity.

fitctree automatically selects the optimal subset of algorithms for each split using the known number of classes and levels of a categorical predictor. For K = 2 classes, fitctree always performs the exact search. To specify a particular algorithm, use the 'AlgorithmForCategorical' name-value pair argument.

For more details, see Splitting Categorical Predictors in Classification Trees.

Example: 'AlgorithmForCategorical','PCA'

CategoricalPredictors — Categorical predictors list

vector of positive integers | logical vector | character matrix | string array | cell array of character vectors | 'all'

Categorical predictors list, specified as one of the values in this table.

Value	Description
Vector of positive integers	Each entry in the vector is an index value indicating that the corresponding predictor is categorical. The index values are between 1 and p, where p is the number of predictors used to train the model. If fitctree uses a subset of input variables as predictors, then the function indexes the predictors using only the subset. The CategoricalPredictors values do not count any response variable, observation weights variable, or other variable that the function does not use.
Logical vector	A true entry means that the corresponding predictor is categorical. The length of the vector is p.
Character matrix	Each row of the matrix is the name of a predictor variable. The names must match the entries in PredictorNames. Pad the names with extra blanks so each row of the character matrix has the same length.
String array or cell array of character vectors	Each element in the array is the name of a predictor variable. The names must match the entries in PredictorNames.
"all"	All predictors are categorical.

By default, if the predictor data is a table (Tbl), fitctree assumes that a variable is categorical if it is a logical vector, unordered categorical vector, character array, string array, or cell array of character vectors. If the predictor data is a matrix (X), fitctree assumes that all predictors are continuous. To identify any other predictors as categorical predictors, specify them by using the CategoricalPredictors name-value argument.

Example: 'CategoricalPredictors','all'

Data Types: single | double | logical | char | string | cell

ClassNames — Names of classes to use for training

categorical array | character array | string array | logical vector | numeric vector | cell array of character vectors

Names of classes to use for training, specified as a categorical, character, or string array; a logical or numeric vector; or a cell array of character vectors. ClassNames must have the same data type as the response variable in Tbl or Y.

If ClassNames is a character array, then each element must correspond to one row of the array.

Use ClassNames to:

• Specify the order of the classes during training.

• Specify the order of any input or output argument dimension that corresponds to the class order. For example, use ClassNames to specify the order of the dimensions of Cost or the column order of classification scores returned by predict.

• Select a subset of classes for training. For example, suppose that the set of all distinct class names in Y is ["a","b","c"]. To train the model using observations from classes "a" and "c" only, specify "ClassNames",["a","c"].

The default value for ClassNames is the set of all distinct class names in the response variable in Tbl or Y.

Example: "ClassNames",["b","g"]

Data Types: categorical | char | string | logical | single | double | cell

Cost — Cost of misclassification

square matrix | structure

Cost of misclassification of a point, specified as the comma-separated pair consisting of 'Cost' and one of the following:

• Square matrix, where Cost(i,j) is the cost of classifying a point into class j if its true class is i (i.e., the rows correspond to the true class and the columns correspond to the predicted class). To specify the class order for the corresponding rows and columns of Cost, also specify the ClassNames name-value pair argument.

• Structure S having two fields: S.ClassNames containing the group names as a variable of the same data type as Y, and S.ClassificationCosts containing the cost matrix.

The default is Cost(i,j)=1 if i~=j, and Cost(i,j)=0 if i=j.

Data Types: single | double | struct

MaxDepth — Maximum tree depth

positive integer

Maximum tree depth, specified as the comma-separated pair consisting of 'MaxDepth' and a positive integer. Specify a value for this argument to return a tree that has fewer levels and requires fewer passes through the tall array to compute. Generally, the algorithm of fitctree takes one pass through the data and an additional pass for each tree level. The function does not set a maximum tree depth, by default.

Note

This option applies only when you use fitctree on tall arrays. See Tall Arrays for more information.

MaxNumCategories — Maximum category levels

10 (default) | nonnegative scalar value

Maximum category levels, specified as the comma-separated pair consisting of 'MaxNumCategories' and a nonnegative scalar value. fitctree splits a categorical predictor using the exact search algorithm if the predictor has at most MaxNumCategories levels in the split node. Otherwise, fitctree finds the best categorical split using one of the inexact algorithms.

Passing a small value can lead to loss of accuracy and passing a large value can increase computation time and memory overload.

Example: 'MaxNumCategories',8

MaxNumSplits — Maximal number of decision splits

size(X,1) - 1 (default) | nonnegative scalar

Maximal number of decision splits (or branch nodes), specified as the comma-separated pair consisting of 'MaxNumSplits' and a nonnegative scalar. fitctree splits MaxNumSplits or fewer branch nodes. For more details on splitting behavior, see Algorithms.

Example: 'MaxNumSplits',5

Data Types: single | double

MergeLeaves — Leaf merge flag

'on' (default) | 'off'

Leaf merge flag, specified as the comma-separated pair consisting of 'MergeLeaves' and 'on' or 'off'.

If MergeLeaves is 'on', then fitctree:

• Merges leaves that originate from the same parent node, and that yields a sum of risk values greater than or equal to the risk associated with the parent node

• Estimates the optimal sequence of pruned subtrees, but does not prune the classification tree

Otherwise, fitctree does not merge leaves.

Example: 'MergeLeaves','off'

MinLeafSize — Minimum number of leaf node observations

1 (default) | positive integer value

Minimum number of leaf node observations, specified as the comma-separated pair consisting of 'MinLeafSize' and a positive integer value. Each leaf has at least MinLeafSize observations per tree leaf. If you supply both MinParentSize and MinLeafSize, fitctree uses the setting that gives larger leaves: MinParentSize = max(MinParentSize,2*MinLeafSize).

Example: 'MinLeafSize',3

Data Types: single | double

MinParentSize — Minimum number of branch node observations

10 (default) | positive integer value

Minimum number of branch node observations, specified as the comma-separated pair consisting of 'MinParentSize' and a positive integer value. Each branch node in the tree has at least MinParentSize observations. If you supply both MinParentSize and MinLeafSize, fitctree uses the setting that gives larger leaves: MinParentSize = max(MinParentSize,2*MinLeafSize).

Example: 'MinParentSize',8

Data Types: single | double

NumBins — Number of bins for numeric predictors

[](empty) (default) | positive integer scalar

Number of bins for numeric predictors, specified as a positive integer scalar.

• If the NumBins value is empty (default), then fitctree does not bin any predictors.

• If you specify the NumBins value as a positive integer scalar (numBins), then fitctree bins every numeric predictor into at most numBins equiprobable bins, and then grows trees on the bin indices instead of the original data.

  • The number of bins can be less than numBins if a predictor has fewer than numBins unique values.

  • fitctree does not bin categorical predictors.

When you use a large training data set, this binning option speeds up training but might cause a potential decrease in accuracy. You can try "NumBins",50 first, and then change the value depending on the accuracy and training speed.

A trained model stores the bin edges in the BinEdges property.

Example: "NumBins",50

Data Types: single | double

NumVariablesToSample — Number of predictors to select at random for each split

'all' (default) | positive integer value

Number of predictors to select at random for each split, specified as the comma-separated pair consisting of 'NumVariablesToSample' and a positive integer value. Alternatively, you can specify 'all' to use all available predictors.

If the training data includes many predictors and you want to analyze predictor importance, then specify 'NumVariablesToSample' as 'all'. Otherwise, the software might not select some predictors, underestimating their importance.

To reproduce the random selections, you must set the seed of the random number generator by using rng and specify 'Reproducible',true.

Example: 'NumVariablesToSample',3

Data Types: char | string | single | double

PredictorNames — Predictor variable names

string array of unique names | cell array of unique character vectors

Predictor variable names, specified as a string array of unique names or cell array of unique character vectors. The functionality of PredictorNames depends on the way you supply the training data.

• If you supply X and Y, then you can use PredictorNames to assign names to the predictor variables in X.

  • The order of the names in PredictorNames must correspond to the column order of X. That is, PredictorNames{1} is the name of X(:,1), PredictorNames{2} is the name of X(:,2), and so on. Also, size(X,2) and numel(PredictorNames) must be equal.

  • By default, PredictorNames is {'x1','x2',...}.

• If you supply Tbl, then you can use PredictorNames to choose which predictor variables to use in training. That is, fitctree uses only the predictor variables in PredictorNames and the response variable during training.

  • PredictorNames must be a subset of Tbl.Properties.VariableNames and cannot include the name of the response variable.

  • By default, PredictorNames contains the names of all predictor variables.

  • A good practice is to specify the predictors for training using either PredictorNames or formula, but not both.

Example: "PredictorNames",["SepalLength","SepalWidth","PetalLength","PetalWidth"]

Data Types: string | cell

PredictorSelection — Algorithm used to select the best split predictor

'allsplits' (default) | 'curvature' | 'interaction-curvature'

Algorithm used to select the best split predictor at each node, specified as the comma-separated pair consisting of 'PredictorSelection' and a value in this table.

Value	Description
'allsplits'	Standard CART — Selects the split predictor that maximizes the split-criterion gain over all possible splits of all predictors [1].
'curvature'	Curvature test — Selects the split predictor that minimizes the p-value of chi-square tests of independence between each predictor and the response [4]. Training speed is similar to standard CART.
'interaction-curvature'	Interaction test — Chooses the split predictor that minimizes the p-value of chi-square tests of independence between each predictor and the response, and that minimizes the p-value of a chi-square test of independence between each pair of predictors and response [3]. Training speed can be slower than standard CART.

For 'curvature' and 'interaction-curvature', if all tests yield p-values greater than 0.05, then fitctree stops splitting nodes.

Tip

• Standard CART tends to select split predictors containing many distinct values, e.g., continuous variables, over those containing few distinct values, e.g., categorical variables [4]. Consider specifying the curvature or interaction test if any of the following are true:

  • If there are predictors that have relatively fewer distinct values than other predictors, for example, if the predictor data set is heterogeneous.

  • If an analysis of predictor importance is your goal. For more on predictor importance estimation, see predictorImportance and Introduction to Feature Selection.

• Trees grown using standard CART are not sensitive to predictor variable interactions. Also, such trees are less likely to identify important variables in the presence of many irrelevant predictors than the application of the interaction test. Therefore, to account for predictor interactions and identify importance variables in the presence of many irrelevant variables, specify the interaction test [3].

• Prediction speed is unaffected by the value of 'PredictorSelection'.

For details on how fitctree selects split predictors, see Node Splitting Rules and Choose Split Predictor Selection Technique.

Example: 'PredictorSelection','curvature'

Prior — Prior probabilities

'empirical' (default) | 'uniform' | vector of scalar values | structure

Prior probabilities for each class, specified as one of the following:

• Character vector or string scalar.

  • 'empirical' determines class probabilities from class frequencies in the response variable in Y or Tbl. If you pass observation weights, fitctree uses the weights to compute the class probabilities.

  • 'uniform' sets all class probabilities to be equal.

• Vector (one scalar value for each class). To specify the class order for the corresponding elements of 'Prior', set the 'ClassNames' name-value argument.

• Structure S with two fields.

  • S.ClassNames contains the class names as a variable of the same type as the response variable in Y or Tbl.

  • S.ClassProbs contains a vector of corresponding probabilities.

fitctree normalizes the weights in each class ('Weights') to add up to the value of the prior probability of the respective class.

Example: 'Prior','uniform'

Data Types: char | string | single | double | struct

Prune — Flag to estimate optimal sequence of pruned subtrees

'on' (default) | 'off'

Flag to estimate the optimal sequence of pruned subtrees, specified as the comma-separated pair consisting of 'Prune' and 'on' or 'off'.

If Prune is 'on', then fitctree grows the classification tree without pruning it, but estimates the optimal sequence of pruned subtrees. If Prune is 'off' and MergeLeaves is also 'off', then fitctree grows the classification tree without estimating the optimal sequence of pruned subtrees.

To prune a trained ClassificationTree model, pass it to prune.

Example: 'Prune','off'

PruneCriterion — Pruning criterion

'error' (default) | 'impurity'

Pruning criterion, specified as the comma-separated pair consisting of 'PruneCriterion' and 'error' or 'impurity'.

If you specify 'impurity', then fitctree uses the impurity measure specified by the 'SplitCriterion' name-value pair argument.

For details, see Impurity and Node Error.

Example: 'PruneCriterion','impurity'

Reproducible — Flag to enforce reproducibility

false (logical 0) (default) | true (logical 1)

Flag to enforce reproducibility over repeated runs of training a model, specified as the comma-separated pair consisting of 'Reproducible' and either false or true.

If 'NumVariablesToSample' is not 'all', then the software selects predictors at random for each split. To reproduce the random selections, you must specify 'Reproducible',true and set the seed of the random number generator by using rng. Note that setting 'Reproducible' to true can slow down training.

Example: 'Reproducible',true

Data Types: logical

ResponseName — Response variable name

'Y' (default) | character vector | string scalar

Response variable name, specified as the comma-separated pair consisting of 'ResponseName' and a character vector or string scalar representing the name of the response variable.

This name-value pair is not valid when using the ResponseVarName or formula input arguments.

Example: 'ResponseName','IrisType'

Data Types: char | string

ScoreTransform — Score transformation

"none" (default) | "doublelogit" | "invlogit" | "ismax" | "logit" | function handle | ...

Score transformation, specified as a character vector, string scalar, or function handle.

This table summarizes the available character vectors and string scalars.

Value	Description
"doublelogit"	1/(1 + e–2x)
"invlogit"	log(x / (1 – x))
"ismax"	Sets the score for the class with the largest score to 1, and sets the scores for all other classes to 0
"logit"	1/(1 + e–x)
"none" or "identity"	x (no transformation)
"sign"	–1 for x < 0 0 for x = 0 1 for x > 0
"symmetric"	2x – 1
"symmetricismax"	Sets the score for the class with the largest score to 1, and sets the scores for all other classes to –1
"symmetriclogit"	2/(1 + e–x) – 1

For a MATLAB function or a function you define, use its function handle for the score transform. The function handle must accept a matrix (the original scores) and return a matrix of the same size (the transformed scores).

Example: "ScoreTransform","logit"

Data Types: char | string | function_handle

SplitCriterion — Split criterion

'gdi' (default) | 'twoing' | 'deviance'

Split criterion, specified as the comma-separated pair consisting of 'SplitCriterion' and 'gdi' (Gini's diversity index), 'twoing' for the twoing rule, or 'deviance' for maximum deviance reduction (also known as cross entropy).

For details, see Impurity and Node Error.

Example: 'SplitCriterion','deviance'

Surrogate — Surrogate decision splits flag

'off' (default) | 'on' | 'all' | positive integer value

Surrogate decision splits flag, specified as the comma-separated pair consisting of 'Surrogate' and 'on', 'off', 'all', or a positive integer value.

• When set to 'on', fitctree finds at most 10 surrogate splits at each branch node.

• When set to 'all', fitctree finds all surrogate splits at each branch node. The 'all' setting can use considerable time and memory.

• When set to a positive integer value, fitctree finds at most the specified number of surrogate splits at each branch node.

Use surrogate splits to improve the accuracy of predictions for data with missing values. The setting also lets you compute measures of predictive association between predictors. For more details, see Node Splitting Rules.

Example: 'Surrogate','on'

Data Types: single | double | char | string

Weights — Observation weights

ones(size(X,1),1) (default) | vector of scalar values | name of variable in Tbl

Observation weights, specified as the comma-separated pair consisting of 'Weights' and a vector of scalar values or the name of a variable in Tbl. The software weights the observations in each row of X or Tbl with the corresponding value in Weights. The size of Weights must equal the number of rows in X or Tbl.

If you specify the input data as a table Tbl, then Weights can be the name of a variable in Tbl that contains a numeric vector. In this case, you must specify Weights as a character vector or string scalar. For example, if the weights vector W is stored as Tbl.W, then specify it as 'W'. Otherwise, the software treats all columns of Tbl, including W, as predictors when training the model.

fitctree normalizes the weights in each class to add up to the value of the prior probability of the respective class.

Data Types: single | double | char | string

Cross-Validation Options

CrossVal — Flag to grow cross-validated decision tree

'off' (default) | 'on'

Flag to grow a cross-validated decision tree, specified as the comma-separated pair consisting of 'CrossVal' and 'on' or 'off'.

If 'on', fitctree grows a cross-validated decision tree with 10 folds. You can override this cross-validation setting using one of the 'KFold', 'Holdout', 'Leaveout', or 'CVPartition' name-value pair arguments. You can only use one of these four arguments at a time when creating a cross-validated tree.

Alternatively, cross-validate Mdl later using the crossval method.

Example: 'CrossVal','on'

CVPartition — Partition for cross-validated tree

cvpartition object

Partition to use in a cross-validated tree, specified as the comma-separated pair consisting of 'CVPartition' and an object created using cvpartition.

If you use 'CVPartition', you cannot use any of the 'KFold', 'Holdout', or 'Leaveout' name-value pair arguments.

Holdout — Fraction of data for holdout validation

0 (default) | scalar value in the range (0,1)

Fraction of data used for holdout validation, specified as the comma-separated pair consisting of 'Holdout' and a scalar value in the range (0,1). Holdout validation tests the specified fraction of the data, and uses the rest of the data for training.

If you use 'Holdout', you cannot use any of the 'CVPartition', 'KFold', or 'Leaveout' name-value pair arguments.

Example: 'Holdout',0.1

Data Types: single | double

KFold — Number of folds

10 (default) | positive integer value greater than 1

Number of folds to use in a cross-validated classifier, specified as the comma-separated pair consisting of 'KFold' and a positive integer value greater than 1. If you specify, e.g., 'KFold',k, then the software:

1. Randomly partitions the data into k sets

2. For each set, reserves the set as validation data, and trains the model using the other k – 1 sets

3. Stores the k compact, trained models in the cells of a k-by-1 cell vector in the Trained property of the cross-validated model.

To create a cross-validated model, you can use one of these four options only: CVPartition, Holdout, KFold, or Leaveout.

Example: 'KFold',8

Data Types: single | double

Leaveout — Leave-one-out cross-validation flag

'off' (default) | 'on'

Leave-one-out cross-validation flag, specified as the comma-separated pair consisting of 'Leaveout' and 'on' or 'off'. Specify 'on' to use leave-one-out cross-validation.

If you use 'Leaveout', you cannot use any of the 'CVPartition', 'Holdout', or 'KFold' name-value pair arguments.

Example: 'Leaveout','on'

Hyperparameter Optimization Options

OptimizeHyperparameters — Parameters to optimize

'none' (default) | 'auto' | 'all' | string array or cell array of eligible parameter names | vector of optimizableVariable objects

Parameters to optimize, specified as the comma-separated pair consisting of 'OptimizeHyperparameters' and one of the following:

• 'none' — Do not optimize.

• 'auto' — Use {'MinLeafSize'}

• 'all' — Optimize all eligible parameters.

• String array or cell array of eligible parameter names

• Vector of optimizableVariable objects, typically the output of hyperparameters

The optimization attempts to minimize the cross-validation loss (error) for fitctree by varying the parameters. To control the cross-validation type and other aspects of the optimization, use the HyperparameterOptimizationOptions name-value argument. When you use HyperparameterOptimizationOptions, you can use the (compact) model size instead of the cross-validation loss as the optimization objective by setting the ConstraintType and ConstraintBounds options.

Note

The values of OptimizeHyperparameters override any values you specify using other name-value arguments. For example, setting OptimizeHyperparameters to "auto" causes fitctree to optimize hyperparameters corresponding to the "auto" option and to ignore any specified values for the hyperparameters.

The eligible parameters for fitctree are:

• MaxNumSplits — fitctree searches among integers, by default log-scaled in the range [1,max(2,NumObservations-1)].

• MinLeafSize — fitctree searches among integers, by default log-scaled in the range [1,max(2,floor(NumObservations/2))].

• SplitCriterion — For two classes, fitctree searches among 'gdi' and 'deviance'. For three or more classes, fitctree also searches among 'twoing'.

• NumVariablesToSample — fitctree does not optimize over this hyperparameter. If you pass NumVariablesToSample as a parameter name, fitctree simply uses the full number of predictors. However, fitcensemble does optimize over this hyperparameter.

Set nondefault parameters by passing a vector of optimizableVariable objects that have nondefault values. For example,

load fisheriris
params = hyperparameters('fitctree',meas,species);
params(1).Range = [1,30];

Pass params as the value of OptimizeHyperparameters.

By default, the iterative display appears at the command line, and plots appear according to the number of hyperparameters in the optimization. For the optimization and plots, the objective function is the misclassification rate. To control the iterative display, set the Verbose option of the HyperparameterOptimizationOptions name-value argument. To control the plots, set the ShowPlots field of the HyperparameterOptimizationOptions name-value argument.

For an example, see Optimize Classification Tree.

Example: 'auto'

HyperparameterOptimizationOptions — Options for optimization

HyperparameterOptimizationOptions object | structure

Options for optimization, specified as a HyperparameterOptimizationOptions object or a structure. This argument modifies the effect of the OptimizeHyperparameters name-value argument. If you specify HyperparameterOptimizationOptions, you must also specify OptimizeHyperparameters. All the options are optional. However, you must set ConstraintBounds and ConstraintType to return AggregateOptimizationResults. The options that you can set in a structure are the same as those in the HyperparameterOptimizationOptions object.

Option	Values	Default
Optimizer	"bayesopt" — Use Bayesian optimization. Internally, this setting calls bayesopt. "gridsearch" — Use grid search with NumGridDivisions values per dimension. "gridsearch" searches in a random order, using uniform sampling without replacement from the grid. After optimization, you can get a table in grid order by using the command sortrows(Mdl.HyperparameterOptimizationResults). "randomsearch" — Search at random among MaxObjectiveEvaluations points.	"bayesopt"
ConstraintBounds	Constraint bounds for N optimization problems, specified as an N-by-2 numeric matrix or []. The columns of ConstraintBounds contain the lower and upper bound values of the optimization problems. If you specify ConstraintBounds as a numeric vector, the software assigns the values to the second column of ConstraintBounds, and zeros to the first column. If you specify ConstraintBounds, you must also specify ConstraintType.	[]
ConstraintTarget	Constraint target for the optimization problems, specified as "matlab" or "coder". If ConstraintBounds and ConstraintType are [] and you set ConstraintTarget, then the software sets ConstraintTarget to []. The values of ConstraintTarget and ConstraintType determine the objective and constraint functions. For more information, see HyperparameterOptimizationOptions.	If you specify ConstraintBounds and ConstraintType, then the default value is "matlab". Otherwise, the default value is [].
ConstraintType	Constraint type for the optimization problems, specified as "size" or "loss". If you specify ConstraintType, you must also specify ConstraintBounds. The values of ConstraintTarget and ConstraintType determine the objective and constraint functions. For more information, see HyperparameterOptimizationOptions.	[]
AcquisitionFunctionName	Type of acquisition function: "expected-improvement-per-second-plus" "expected-improvement" "expected-improvement-plus" "expected-improvement-per-second" "lower-confidence-bound" "probability-of-improvement" Acquisition functions whose names include per-second do not yield reproducible results, because the optimization depends on the run time of the objective function. Acquisition functions whose names include plus modify their behavior when they overexploit an area. For more details, see Acquisition Function Types.	"expected-improvement-per-second-plus"
MaxObjectiveEvaluations	Maximum number of objective function evaluations. If you specify multiple optimization problems using ConstraintBounds, the value of MaxObjectiveEvaluations applies to each optimization problem individually.	30 for "bayesopt" and "randomsearch", and the entire grid for "gridsearch"
MaxTime	Time limit for the optimization, specified as a nonnegative real scalar. The time limit is in seconds, as measured by tic and toc. The software performs at least one optimization iteration, regardless of the value of MaxTime. The run time can exceed MaxTime because MaxTime does not interrupt function evaluations. If you specify multiple optimization problems using ConstraintBounds, the time limit applies to each optimization problem individually.	Inf
NumGridDivisions	For Optimizer="gridsearch", the number of values in each dimension. The value can be a vector of positive integers giving the number of values for each dimension, or a scalar that applies to all dimensions. This option is ignored for categorical variables.	10
ShowPlots	Logical value indicating whether to show plots of the optimization progress. If this option is true, the software plots the best observed objective function value against the iteration number. If you use Bayesian optimization (Optimizer="bayesopt"), then the software also plots the best estimated objective function value. The best observed objective function values and best estimated objective function values correspond to the values in the BestSoFar (observed) and BestSoFar (estim.) columns of the iterative display, respectively. You can find these values in the properties ObjectiveMinimumTrace and EstimatedObjectiveMinimumTrace of Mdl.HyperparameterOptimizationResults. If the problem includes one or two optimization parameters for Bayesian optimization, then ShowPlots also plots a model of the objective function against the parameters.	true
SaveIntermediateResults	Logical value indicating whether to save the optimization results. If this option is true, the software overwrites a workspace variable named "BayesoptResults" at each iteration. The variable is a BayesianOptimization object. If you specify multiple optimization problems using ConstraintBounds, the workspace variable is an AggregateBayesianOptimization object named "AggregateBayesoptResults".	false
Verbose	Display level at the command line: 0 — No iterative display 1 — Iterative display 2 — Iterative display with additional information For details, see the bayesopt Verbose name-value argument and the example Optimize Classifier Fit Using Bayesian Optimization.	1
UseParallel	Logical value indicating whether to run the Bayesian optimization in parallel, which requires Parallel Computing Toolbox™. Due to the nonreproducibility of parallel timing, parallel Bayesian optimization does not necessarily yield reproducible results. For details, see Parallel Bayesian Optimization.	false
Repartition	Logical value indicating whether to repartition the cross-validation at every iteration. If this option is false, the optimizer uses a single partition for the optimization. A value of true usually gives the most robust results because this setting takes partitioning noise into account. However, for optimal results, true requires at least twice as many function evaluations.	false
Specify only one of the following three options.
CVPartition	cvpartition object created by cvpartition	Kfold=5 if you do not specify a cross-validation option
Holdout	Scalar in the range (0,1) representing the holdout fraction
Kfold	Integer greater than 1

Example: HyperparameterOptimizationOptions=struct(UseParallel=true)

Output Arguments

Mdl — Classification tree

classification tree object

Classification tree, returned as a classification tree object.

Using the 'CrossVal', 'KFold', 'Holdout', 'Leaveout', or 'CVPartition' options results in a tree of class ClassificationPartitionedModel. You cannot use a partitioned tree for prediction, so this kind of tree does not have a predict method. Instead, use kfoldPredict to predict responses for observations not used for training.

Otherwise, Mdl is of class ClassificationTree, and you can use the predict method to make predictions.

If you specify OptimizeHyperparameters and set the ConstraintType and ConstraintBounds options of HyperparameterOptimizationOptions, then Mdl is an N-by-1 cell array of model objects, where N is equal to the number of rows in ConstraintBounds. If none of the optimization problems yields a feasible model, then each cell array value is [].

AggregateOptimizationResults — Aggregate optimization results

AggregateBayesianOptimization object

Aggregate optimization results for multiple optimization problems, returned as an AggregateBayesianOptimization object. To return AggregateOptimizationResults, you must specify OptimizeHyperparameters and HyperparameterOptimizationOptions. You must also specify the ConstraintType and ConstraintBounds options of HyperparameterOptimizationOptions. For an example that shows how to produce this output, see Hyperparameter Optimization with Multiple Constraint Bounds.

More About

Curvature Test

The curvature test is a statistical test assessing the null hypothesis that two variables are unassociated.

The curvature test between predictor variable x and y is conducted using this process.

1. If x is continuous, then partition it into its quartiles. Create a nominal variable that bins observations according to which section of the partition they occupy. If there are missing values, then create an extra bin for them.

2. For each level in the partitioned predictor j = 1...J and class in the response k = 1,...,K, compute the weighted proportion of observations in class k

   $${\widehat{\pi }}_{jk}={\displaystyle \sum _{i=1}^{n}I\{y_i=k\}}{w}_i.$$

   w_i is the weight of observation i, $${\displaystyle \sum w_i}=1$$, I is the indicator function, and n is the sample size. If all observations have the same weight, then $${\widehat{\pi }}_{jk}=\frac{n_{jk}}{n}$$, where n_jk is the number of observations in level j of the predictor that are in class k.

3. Compute the test statistic

   $$t=n{\displaystyle \sum _{k=1}^K{\displaystyle \sum _{j=1}^J\frac{{\left({\widehat{\pi }}_{jk}-{\widehat{\pi }}_{j+}{\widehat{\pi }}_{+k}\right)}^2}{{\widehat{\pi }}_{j+}{\widehat{\pi }}_{+k}}}}$$

   $${\widehat{\pi }}_{j+}={\displaystyle \sum _k{\widehat{\pi }}_{jk}}$$, that is, the marginal probability of observing the predictor at level j. $${\widehat{\pi }}_{+k}={\displaystyle \sum _j{\widehat{\pi }}_{jk}}$$, that is the marginal probability of observing class k. If n is large enough, then t is distributed as a χ² with (K – 1)(J – 1) degrees of freedom.

4. If the p-value for the test is less than 0.05, then reject the null hypothesis that there is no association between x and y.

When determining the best split predictor at each node, the standard CART algorithm prefers to select continuous predictors that have many levels. Sometimes, such a selection can be spurious and can also mask more important predictors that have fewer levels, such as categorical predictors.

The curvature test can be applied instead of standard CART to determine the best split predictor at each node. In that case, the best split predictor variable is the one that minimizes the significant p-values (those less than 0.05) of curvature tests between each predictor and the response variable. Such a selection is robust to the number of levels in individual predictors.

Note

If levels of a predictor are pure for a particular class, then fitctree merges those levels. Therefore, in step 3 of the algorithm, J can be less than the actual number of levels in the predictor. For example, if x has 4 levels, and all observations in bins 1 and 2 belong to class 1, then those levels are pure for class 1. Consequently, fitctree merges the observations in bins 1 and 2, and J reduces to 3.

For more details on how the curvature test applies to growing classification trees, see Node Splitting Rules and [4].

Impurity and Node Error

A decision tree splits nodes based on either impurity or node error.

Impurity means one of several things, depending on your choice of the SplitCriterion name-value argument:

• Gini's Diversity Index (gdi) — The Gini index of a node is

  $$1-{\displaystyle \sum _i{p}^2(i)},$$

  where the sum is over the classes i at the node, and p(i) is the observed fraction of classes with class i that reach the node. A node with just one class (a pure node) has Gini index 0; otherwise, the Gini index is positive. So the Gini index is a measure of node impurity.

• Deviance ("deviance") — With p(i) defined the same as for the Gini index, the deviance of a node is

  $$-{\displaystyle \sum _{i}p(i){\log }_{2}p(i)}.$$

  A pure node has deviance 0; otherwise, the deviance is positive.

• Twoing rule ("twoing") — Twoing is not a purity measure of a node, but is a different measure for deciding how to split a node. Let L(i) denote the fraction of members of class i in the left child node after a split, and R(i) denote the fraction of members of class i in the right child node after a split. Choose the split criterion to maximize

  $$P(L)P(R){\left({\displaystyle \sum _i\left|L(i)-R(i)\right|}\right)}^2,$$

  where P(L) and P(R) are the fractions of observations that split to the left and right, respectively. If the expression is large, the split made each child node purer. Similarly, if the expression is small, the split made each child node similar to each other and, therefore, similar to the parent node. The split did not increase node purity.

• Node error — The node error is the fraction of misclassified classes at a node. If j is the class with the largest number of training samples at a node, the node error is

  1 – p(j).

Interaction Test

The interaction test is a statistical test that assesses the null hypothesis that there is no interaction between a pair of predictor variables and the response variable.

The interaction test assessing the association between predictor variables x₁ and x₂ with respect to y is conducted using this process.

1. If x₁ or x₂ is continuous, then partition that variable into its quartiles. Create a nominal variable that bins observations according to which section of the partition they occupy. If there are missing values, then create an extra bin for them.

2. Create the nominal variable z with J = J₁J₂ levels that assigns an index to observation i according to which levels of x₁ and x₂ it belongs. Remove any levels of z that do not correspond to any observations.

3. Conduct a curvature test between z and y.

When growing decision trees, if there are important interactions between pairs of predictors, but there are also many other less important predictors in the data, then standard CART tends to miss the important interactions. However, conducting curvature and interaction tests for predictor selection instead can improve detection of important interactions, which can yield more accurate decision trees.

For more details on how the interaction test applies to growing decision trees, see Curvature Test, Node Splitting Rules and [3].

Predictive Measure of Association

The predictive measure of association is a value that indicates the similarity between decision rules that split observations. Among all possible decision splits that are compared to the optimal split (found by growing the tree), the best surrogate decision split yields the maximum predictive measure of association. The second-best surrogate split has the second-largest predictive measure of association.

Suppose x_j and x_k are predictor variables j and k, respectively, and j ≠ k. At node t, the predictive measure of association between the optimal split x_j < u and a surrogate split x_k < v is

$${\lambda }_{jk}=\frac{\text{min}\left(P_L,P_R\right)-\left(1-P_{L_j{L}_k}-P_{R_j{R}_k}\right)}{\text{min}\left(P_L,P_R\right)}.$$

• P_L is the proportion of observations in node t, such that x_j < u. The subscript L stands for the left child of node t.

• P_R is the proportion of observations in node t, such that x_j ≥ u. The subscript R stands for the right child of node t.

• $$P_{L_j{L}_k}$$ is the proportion of observations at node t, such that x_j < u and x_k < v.

• $$P_{R_j{R}_k}$$ is the proportion of observations at node t, such that x_j ≥ u and x_k ≥ v.

• Observations with missing values for x_j or x_k do not contribute to the proportion calculations.

λ_jk is a value in (–∞,1]. If λ_jk > 0, then x_k < v is a worthwhile surrogate split for x_j < u.

Surrogate Decision Splits

A surrogate decision split is an alternative to the optimal decision split at a given node in a decision tree. The optimal split is found by growing the tree; the surrogate split uses a similar or correlated predictor variable and split criterion.

When the value of the optimal split predictor for an observation is missing, the observation is sent to the left or right child node using the best surrogate predictor. When the value of the best surrogate split predictor for the observation is also missing, the observation is sent to the left or right child node using the second-best surrogate predictor, and so on. Candidate splits are sorted in descending order by their predictive measure of association.

Tips

• By default, Prune is 'on'. However, this specification does not prune the classification tree. To prune a trained classification tree, pass the classification tree to prune.

• After training a model, you can generate C/C++ code that predicts labels for new data. Generating C/C++ code requires MATLAB Coder™. For details, see Introduction to Code Generation.

Algorithms

Node Splitting Rules

fitctree uses these processes to determine how to split node t.

• For standard CART (that is, if PredictorSelection is 'allpairs') and for all predictors x_i, i = 1,...,p:

  1. fitctree computes the weighted impurity of node t, i_t. For supported impurity measures, see SplitCriterion.

  2. fitctree estimates the probability that an observation is in node t using

     $$P\left(T\right)={\displaystyle \sum _{j\in T}{w}_j}.$$

     w_j is the weight of observation j, and T is the set of all observation indices in node t. If you do not specify Prior or Weights, then w_j = 1/n, where n is the sample size.

  3. fitctree sorts x_i in ascending order. Each element of the sorted predictor is a splitting candidate or cut point. fitctree stores any indices corresponding to missing values in the set T_U, which is the unsplit set.

  4. fitctree determines the best way to split node t using x_i by maximizing the impurity gain (ΔI) over all splitting candidates. That is, for all splitting candidates in x_i:

     1. fitctree splits the observations in node t into left and right child nodes (t_L and t_R, respectively).

     2. fitctree computes ΔI. Suppose that for a particular splitting candidate, t_L and t_R contain observation indices in the sets T_L and T_R, respectively.

        • If x_i does not contain any missing values, then the impurity gain for the current splitting candidate is

          $$\Delta I=P\left(T\right)i_t-P\left(T_L\right)i_{t_L}-P\left(T_R\right)i_{t_R}.$$

        • If x_i contains missing values then, assuming that the observations are missing at random, the impurity gain is

          $$\Delta I_U=P\left(T-T_U\right)i_t-P\left(T_L\right)i_{t_L}-P\left(T_R\right)i_{t_R}.$$

          T – T_U is the set of all observation indices in node t that are not missing.

        • If you use surrogate decision splits, then:

          1. fitctree computes the predictive measures of association between the decision split x_j < u and all possible decision splits x_k < v, j ≠ k.

          2. fitctree sorts the possible alternative decision splits in descending order by their predictive measure of association with the optimal split. The surrogate split is the decision split yielding the largest measure.

          3. fitctree decides the child node assignments for observations with a missing value for x_i using the surrogate split. If the surrogate predictor also contains a missing value, then fitctree uses the decision split with the second largest measure, and so on, until there are no other surrogates. It is possible for fitctree to split two different observations at node t using two different surrogate splits. For example, suppose the predictors x₁ and x₂ are the best and second best surrogates, respectively, for the predictor x_i, i ∉ {1,2}, at node t. If observation m of predictor x_i is missing (i.e., x_mi is missing), but x_m1 is not missing, then x₁ is the surrogate predictor for observation x_mi. If observations x_{(m + 1),i} and x(m + 1),1 are missing, but x_{(m + 1),2} is not missing, then x₂ is the surrogate predictor for observation m + 1.

          4. fitctree uses the appropriate impurity gain formula. That is, if fitctree fails to assign all missing observations in node t to children nodes using surrogate splits, then the impurity gain is ΔI_U. Otherwise, fitctree uses ΔI for the impurity gain.

     3. fitctree chooses the candidate that yields the largest impurity gain.

  fitctree splits the predictor variable at the cut point that maximizes the impurity gain.

• For the curvature test (that is, if PredictorSelection is 'curvature'):

  1. fitctree conducts curvature tests between each predictor and the response for observations in node t.

     • If all p-values are at least 0.05, then fitctree does not split node t.

     • If there is a minimal p-value, then fitctree chooses the corresponding predictor to split node t.

     • If more than one p-value is zero due to underflow, then fitctree applies standard CART to the corresponding predictors to choose the split predictor.

  2. If fitctree chooses a split predictor, then it uses standard CART to choose the cut point (see step 4 in the standard CART process).

• For the interaction test (that is, if PredictorSelection is 'interaction-curvature' ):

  1. For observations in node t, fitctree conducts curvature tests between each predictor and the response and interaction tests between each pair of predictors and the response.

     • If all p-values are at least 0.05, then fitctree does not split node t.

     • If there is a minimal p-value and it is the result of a curvature test, then fitctree chooses the corresponding predictor to split node t.

     • If there is a minimal p-value and it is the result of an interaction test, then fitctree chooses the split predictor using standard CART on the corresponding pair of predictors.

     • If more than one p-value is zero due to underflow, then fitctree applies standard CART to the corresponding predictors to choose the split predictor.

  2. If fitctree chooses a split predictor, then it uses standard CART to choose the cut point (see step 4 in the standard CART process).

Tree Depth Control

• If MergeLeaves is 'on' and PruneCriterion is 'error' (which are the default values for these name-value pair arguments), then the software applies pruning only to the leaves and by using classification error. This specification amounts to merging leaves that share the most popular class per leaf.

• To accommodate MaxNumSplits, fitctree splits all nodes in the current layer, and then counts the number of branch nodes. A layer is the set of nodes that are equidistant from the root node. If the number of branch nodes exceeds MaxNumSplits, fitctree follows this procedure:

  1. Determine how many branch nodes in the current layer must be unsplit so that there are at most MaxNumSplits branch nodes.

  2. Sort the branch nodes by their impurity gains.

  3. Unsplit the number of least successful branches.

  4. Return the decision tree grown so far.

  This procedure produces maximally balanced trees.

• The software splits branch nodes layer by layer until at least one of these events occurs:

  • There are MaxNumSplits branch nodes.

  • A proposed split causes the number of observations in at least one branch node to be fewer than MinParentSize.

  • A proposed split causes the number of observations in at least one leaf node to be fewer than MinLeafSize.

  • The algorithm cannot find a good split within a layer (i.e., the pruning criterion (see PruneCriterion), does not improve for all proposed splits in a layer). A special case is when all nodes are pure (i.e., all observations in the node have the same class).

  • For values 'curvature' or 'interaction-curvature' of PredictorSelection, all tests yield p-values greater than 0.05.

  MaxNumSplits and MinLeafSize do not affect splitting at their default values. Therefore, if you set 'MaxNumSplits', splitting might stop due to the value of MinParentSize, before MaxNumSplits splits occur.

Parallelization

For dual-core systems and above, fitctree parallelizes training decision trees using Intel^® Threading Building Blocks (TBB). For details on Intel TBB, see https://www.intel.com/content/www/us/en/developer/tools/oneapi/onetbb.html.

Cost, Prior, and Weights

If you specify the Cost, Prior, and Weights name-value arguments, the output model object stores the specified values in the Cost, Prior, and W properties, respectively. The Cost property stores the user-specified cost matrix as is. The Prior and W properties store the prior probabilities and observation weights, respectively, after normalization. For details, see Misclassification Cost Matrix, Prior Probabilities, and Observation Weights.

References

[1] Breiman, L., J. Friedman, R. Olshen, and C. Stone. Classification and Regression Trees. Boca Raton, FL: CRC Press, 1984.

[2] Coppersmith, D., S. J. Hong, and J. R. M. Hosking. “Partitioning Nominal Attributes in Decision Trees.” Data Mining and Knowledge Discovery, Vol. 3, 1999, pp. 197–217.

[3] Loh, W.Y. “Regression Trees with Unbiased Variable Selection and Interaction Detection.” Statistica Sinica, Vol. 12, 2002, pp. 361–386.

[4] Loh, W.Y. and Y.S. Shih. “Split Selection Methods for Classification Trees.” Statistica Sinica, Vol. 7, 1997, pp. 815–840.

Extended Capabilities

Tall Arrays

Calculate with arrays that have more rows than fit in memory.

The fitctree function supports tall arrays with the following usage notes and limitations:

• Supported syntaxes are:

  • Mdl = fitctree(Tbl,Y)

  • Mdl = fitctree(X,Y)

  • Mdl = fitctree(___,Name=Value)

  • [Mdl,FitInfo,HyperparameterOptimizationResults] = fitctree(___,Name=Value) — fitctree returns the additional output arguments FitInfo and HyperparameterOptimizationResults when you specify the 'OptimizeHyperparameters' name-value argument.

• The FitInfo output argument is an empty structure array currently reserved for possible future use.

• The HyperparameterOptimizationResults output argument is usually a BayesianOptimization object or a table of hyperparameters with associated values that describe the cross-validation optimization of hyperparameters. However, if you specify HyperparameterOptimizationOptions and set ConstraintType and ConstraintBounds, then HyperparameterOptimizationResults is an AggregateBayesianOptimization object.

  If you specify HyperparameterOptimizationOptions and do not set ConstraintType, ConstraintBounds, or Optimizer="bayesopt", then HyperparameterOptimizationResults is a table of the hyperparameters used, observed objective function values, and rank of observations from lowest (best) to highest (worst).

  HyperparameterOptimizationResults is nonempty when the OptimizeHyperparameters name-value argument is nonempty at the time you create the model. The values in HyperparameterOptimizationResults depend on the value you specify for the HyperparameterOptimizationOptions name-value argument when you create the model.

• Supported name-value pair arguments, and any differences, are:

  • 'AlgorithmForCategorical'

  • 'CategoricalPredictors'

  • 'ClassNames'

  • 'Cost'

  • 'HyperparameterOptimizationOptions' — For cross-validation, tall optimization supports only 'Holdout' validation. By default, the software selects and reserves 20% of the data as holdout validation data, and trains the model using the rest of the data. You can specify a different value for the holdout fraction by using this argument. For example, specify 'HyperparameterOptimizationOptions',struct('Holdout',0.3) to reserve 30% of the data as validation data.

  • 'MaxNumCategories'

  • 'MaxNumSplits'— for tall optimization, fitctree searches among integers, by default log-scaled in the range [1,max(2,min(10000,NumObservations-1))].

  • 'MergeLeaves'

  • 'MinLeafSize'

  • 'MinParentSize'

  • 'NumVariablesToSample'

  • 'OptimizeHyperparameters'

  • 'PredictorNames'

  • 'Prior'

  • 'ResponseName'

  • 'ScoreTransform'

  • 'SplitCriterion'

  • 'Weights'

• This additional name-value pair argument is specific to tall arrays:

  • 'MaxDepth' — A positive integer specifying the maximum depth of the output tree. Specify a value for this argument to return a tree that has fewer levels and requires fewer passes through the tall array to compute. Generally, the algorithm of fitctree takes one pass through the data and an additional pass for each tree level. The function does not set a maximum tree depth, by default.

For more information, see Tall Arrays.

Automatic Parallel Support

Accelerate code by automatically running computation in parallel using Parallel Computing Toolbox™.

To perform parallel hyperparameter optimization, use the UseParallel=true option in the HyperparameterOptimizationOptions name-value argument in the call to the fitctree function.

For more information on parallel hyperparameter optimization, see Parallel Bayesian Optimization.

For general information about parallel computing, see Run MATLAB Functions with Automatic Parallel Support (Parallel Computing Toolbox).

GPU Arrays

Accelerate code by running on a graphics processing unit (GPU) using Parallel Computing Toolbox™.

Usage notes and limitations:

• fitctree does not support surrogate splits. You can specify the name-value argument Surrogate only as "off".

• For data with categorical predictors, the following apply:

  • For multiclass classification, fitctree supports only the OVAbyClass algorithm for finding the best split.

  • You can specify the name-value argument NumVariablesToSample only as "all".

• You can specify the name-value argument PredictorSelection only as "allsplits".

• fitctree fits the model on a GPU if one of the following applies:

  • The input argument X is a gpuArray object.

  • The input argument Tbl contains gpuArray predictor variables.

• Note that fitctree might not execute faster on a GPU than a CPU for deeper decision trees.

For more information, see Run MATLAB Functions on a GPU (Parallel Computing Toolbox).

Version History

Introduced in R2014a