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Predict responses of linear regression model


ypred = predict(mdl,Xnew)
[ypred,yci] = predict(mdl,Xnew)
[ypred,yci] = predict(mdl,Xnew,Name,Value)



ypred = predict(mdl,Xnew) returns the predicted response values of the linear regression model mdl to the points in Xnew.

[ypred,yci] = predict(mdl,Xnew) also returns confidence intervals for the responses at Xnew.


[ypred,yci] = predict(mdl,Xnew,Name,Value) specifies additional options using one or more name-value pair arguments. For example, you can specify the confidence level of the confidence interval and the prediction type.


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Create a quadratic model of car mileage as a function of weight from the carsmall data set.

load carsmall
X = Weight;
y = MPG;
mdl = fitlm(X,y,'quadratic');

Create predicted responses to the data.

ypred = predict(mdl,X);

Plot the original responses and the predicted responses to see how they differ.


Fit a linear regression model, and then save the model by using saveCompactModel. Define an entry-point function that loads the model by using loadCompactModel and calls the predict function of the fitted model. Then use codegen to generate C/C++ code. Note that generating C/C++ code requires MATLAB® Coder™.

This example briefly explains the code generation workflow for the prediction of linear regression models at the command line. For more details, see Code Generation for Prediction of Machine Learning Model at Command Line. You can also generate code using the MATLAB Coder app. For details, see Code Generation for Prediction of Machine Learning Model Using MATLAB Coder App.

Train Model

Load the carsmall data set, and then fit the quadratic regression model.

load carsmall
X = Weight;
y = MPG;
mdl = fitlm(X,y,'quadratic');

Save Model

Save the fitted quadratic model to the file QLMMdl.mat by using saveCompactModel.


Define Entry-Point Function

Define an entry-point function named mypredictQLM that does the following:

  • Accept measurements corresponding to X and optional, valid name-value pair arguments.

  • Load the fitted quadratic model in QLMMdl.mat.

  • Return predictions and confidence interval bounds.

type mypredictQLM.m % Display contents of mypredictQLM.m
function [yhat,ci] = mypredictQLM(x,varargin) %#codegen
%MYPREDICTQLM Predict response using linear model
%   MYPREDICTQLM predicts responses for the n observations in the n-by-1
%   vector x using the linear model stored in the MAT-file QLMMdl.mat, and
%   then returns the predictions in the n-by-1 vector yhat. MYPREDICTQLM
%   also returns confidence interval bounds for the predictions in the
%   n-by-2 vector ci.
CompactMdl = loadCompactModel('QLMMdl');
[yhat,ci] = predict(CompactMdl,x,varargin{:});

Add the %#codegen compiler directive (or pragma) to the entry-point function after the function signature to indicate that you intend to generate code for the MATLAB algorithm. Adding this directive instructs the MATLAB Code Analyzer to help you diagnose and fix violations that would result in errors during code generation.

Note: If you click the button located in the upper-right section of this example and open the example in MATLAB®, then MATLAB® opens the example folder. This folder includes the entry-point function file.

Generate Code

Generate code for the entry-point function using codegen. Because C and C++ are statically typed languages, you must determine the properties of all variables in the entry-point function at compile time. To specify the data type and exact input array size, pass a MATLAB® expression that represents the set of values with a certain data type and array size. Use coder.Constant for the names of name-value pair arguments.

If the number of observations is unknown at compile time, you can also specify the input as variable-size by using coder.typeof. For details, see Specify Variable-Size Arguments for Code Generation and Specify Properties of Entry-Point Function Inputs (MATLAB Coder).

codegen mypredictQLM -args {X,coder.Constant('Alpha'),0.1,coder.Constant('Simultaneous'),true}

codegen generates the MEX function mypredictQLM_mex with a platform-dependent extension.

Verify Generated Code

Compare predictions and confidence intervals using predict and mypredictQLM_mex. Specify name-value pair arguments in the same order as in the -args argument in the call to codegen.

Xnew = sort(X);
[yhat1,ci1] = predict(mdl,Xnew,'Alpha',0.1,'Simultaneous',true);
[yhat2,ci2] = mypredictQLM_mex(Xnew,'Alpha',0.1,'Simultaneous',true);

The returned values from mypredictQLM_mex might include round-off differences compared to the values from predict. In this case, compare the values allowing a small tolerance.

find(abs(yhat1-yhat2) > 1e-6)
ans =

  0x1 empty double column vector
find(abs(ci1-ci2) > 1e-6)
ans =

  0x1 empty double column vector

The comparison confirms that the returned values are equal within the tolerance 1e–6.

Plot the returned values for comparison.

h1 = plot(X,y,'k.');
hold on
h2 = plot(Xnew,yhat1,'ro',Xnew,yhat2,'gx');
h3 = plot(Xnew,ci1,'r-','LineWidth',4);
h4 = plot(Xnew,ci2,'g--','LineWidth',2);
legend([h1; h2; h3(1); h4(1)], ...
    {'Data','predict estimates','MEX estimates','predict CIs','MEX CIs'});

Input Arguments

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Linear regression model object, specified as a LinearModel object created by using fitlm or stepwiselm, or a CompactLinearModel object created by using compact.

New predictor input values, specified as a table, dataset array, or matrix. Each row of Xnew corresponds to one observation, and each column corresponds to one variable.

  • If Xnew is a table or dataset array, it must contain predictors that have the same predictor names as in the PredictorNames property of mdl.

  • If Xnew is a matrix, it must have the same number of variables (columns) in the same order as the predictor input used to create mdl. Note that Xnew must also contain any predictor variables that are not used as predictors in the fitted model. Also, all variables used in creating mdl must be numeric or logical. To treat numerical predictors as categorical, identify the predictors using the 'CategoricalVars' name-value pair argument when you create mdl.

Data Types: single | double | table

Name-Value Pair Arguments

Specify optional comma-separated pairs of Name,Value arguments. Name is the argument name and Value is the corresponding value. Name must appear inside quotes. You can specify several name and value pair arguments in any order as Name1,Value1,...,NameN,ValueN.

Example: [ypred,yci] = predict(Mdl,Xnew,'Alpha',0.01,'Simultaneous',true) returns the confidence interval yci with a 99% confidence level, computed simultaneously for all predictor values.

Significance level for the confidence interval, specified as the comma-separated pair consisting of 'Alpha' and a numeric value in the range [0,1]. The confidence level of yci is equal to 100(1 – Alpha)%. Alpha is the probability that the confidence interval does not contain the true value.

Example: 'Alpha',0.01

Data Types: single | double

Prediction type, specified as the comma-separated pair consisting of 'Prediction' and either 'curve' or 'observation'.

A regression model for the predictor variables X and the response variable y has the form

y = f(X) + ε,

where f is a fitted regression function and ε is a random noise term.

  • If 'Prediction' is 'curve', then predict predicts confidence bounds for f(Xnew), the fitted responses at Xnew.

  • If 'Prediction' is 'observation', then predict predicts confidence bounds for y, the response observations at Xnew.

The bounds for y are wider than the bounds for f(X) because of the additional variability of the noise term.

Example: 'Prediction','observation'

Flag to compute simultaneous confidence bounds, specified as the comma-separated pair consisting of 'Simultaneous' and either true or false.

  • truepredict computes confidence bounds for the curve of response values corresponding to all predictor values in Xnew, using Scheffe's method. The range between the upper and lower bounds contains the curve consisting of true response values with 100(1 – α)% confidence.

  • falsepredict computes confidence bounds for the response value at each observation in Xnew. The confidence interval for a response value at a specific predictor value contains the true response value with 100(1 – α)% confidence.

Simultaneous bounds are wider than separate bounds, because requiring the entire curve of response values to be within the bounds is stricter than requiring the response value at a single predictor value to be within the bounds.

Example: 'Simultaneous',true

Output Arguments

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Predicted response values evaluated at Xnew, returned as a numeric vector.

Confidence intervals for the responses, returned as a two-column matrix with each row providing one interval. The meaning of the confidence interval depends on the settings of the name-value pair arguments 'Alpha', 'Prediction', and 'Simultaneous'.

Alternative Functionality

  • feval returns the same predictions as predict. The feval function can take multiple input arguments, with one input for each predictor variable, and does not give confidence intervals on its predictions.

  • random returns predictions with added noise.

  • Use plotSlice to create a figure containing a series of plots, each representing a slice through the predicted regression surface. Each plot shows the fitted response values as a function of a single predictor variable, with the other predictor variables held constant.

Extended Capabilities

Introduced in R2012a