pearspdf

Pearson probability density function

Since R2023b

Syntax

``f = pearspdf(X,mu,sigma,skew,kurt)``
``[f,type] = pearspdf(X,mu,sigma,skew,kurt)``
``[f,type,coefs] = pearspdf(X,mu,sigma,skew,kurt)``

Description

example

````f = pearspdf(X,mu,sigma,skew,kurt)` returns the probability density function (pdf) of the Pearson system evaluated at the values in `X`, using the mean `mu`, standard deviation `sigma`, skewness `skew`, and kurtosis `kurt`. ```

example

````[f,type] = pearspdf(X,mu,sigma,skew,kurt)` also returns the type of the specified distribution within the Pearson system.```

example

````[f,type,coefs] = pearspdf(X,mu,sigma,skew,kurt)` also returns the coefficients in the denominator of the differential equation $\frac{p\text{'}\left(x\right)}{p\left(x\right)}=-\frac{a+\left(x-\mu \right)}{{b}_{0}+{b}_{1}\left(x-\mu \right)+{b}_{2}{\left(x-\mu \right)}^{2}},$which defines the Pearson pdf $p\left(x\right)$.```

Examples

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Define the variables `mu`, `sigma`, `skew`, and `kurtosis`, which contain values for the mean, standard deviation, skewness, and kurtosis of a Pearson distribution, respectively.

```mu = 0; sigma = 2; skew = 0; kurtosis = 3;```

A Pearson distribution with a skewness of 0 and kurtosis of 3 is equivalent to the normal distribution.

Create a vector `X` of points from `—7` to `7` using the `linspace` function. Evaluate the pdf for the Pearson distribution given by `mu`, `sigma`, `skew`, and `kurtosis` at the points in `X`. Plot the result together with the pdf for the standard normal distribution.

```X = linspace(-7,7,1000); Fp = pearspdf(X,mu,sigma,skew,kurtosis); Fn = normpdf(X,mu,sigma); figure hold on plot(X,Fp) plot(X,Fn) legend(["Pearson PDF" "Normal PDF"])```

The plot shows that the blue curve for the Pearson distribution pdf is completely hidden by the red curve for the normal distribution pdf. This result indicates that the Pearson pdf is identical to the normal distribution pdf.

Define the variables `mu`, `sigma`, `skew`, and `kurtosis`, which contain values for the mean, standard deviation, skewness, and kurtosis of a Pearson distribution, respectively.

```mu = 2; sigma = 1; skew = 2; kurtosis = 10;```

Return the type of the Pearson distribution given by `mu`, `sigma`, `skew`, and `kurtosis`, and return the coefficients of the corresponding quadratic polynomial.

`[~,type,coefs] = pearspdf([],mu,sigma,skew,kurtosis)`
```type = 6 ```
```coefs = 1×3 0.8235 0.7647 0.0588 ```

The output shows that the distribution is of type 6, and displays the coefficients for the quadratic polynomial. Type 6 Pearson distributions are bounded on one side. The bound is calculated from the roots of the quadratic polynomial in the denominator of the differential equation that defines the pdf. For more information, see Probability Density Function and Support.

Find the roots of the quadratic polynomial by using the `fliplr` function to reverse the order of the coefficients in `coefs`. Pass the result to the `roots` function.

```coefs = fliplr(coefs); a = roots(coefs)```
```a = 2×1 -11.8151 -1.1849 ```

Both of the roots are negative. This result indicates that the distribution has a lower bound, which you can calculate by using `mu`, `sigma`, and the largest root in `a`.

Calculate the lower bound for the distribution.

`lower = sigma*max(a) + mu;`

Create a vector of points from `lower` to `10` by using the `linspace` function.

`X = linspace(lower,10,1000);`

Evaluate the pdf for the distribution at the points in `X`, and then plot the result.

```F = pearspdf(X,mu,sigma,skew,kurtosis); plot(X,F) hold on xlim([lower,10])```

The distribution pdf has a shape typical of an F-distribution.

Input Arguments

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Values at which to evaluate the Pearson pdf, specified as a scalar or a numeric array.

To evaluate the pdf at multiple values, specify `X` using an array. To evaluate the pdfs of multiple distributions, specify either `mu` or `sigma` (or both) using arrays. If one or more of the input arguments `X`, `mu`, and `sigma` are arrays, then the array sizes must be the same. In this case, `pearspdf` expands each scalar input into a constant array of the same size as the array inputs. Each element in `f` is the pdf value of the distribution specified by the corresponding elements in `mu` and `sigma`, evaluated at the corresponding element in `X`.

Example: `[0 0.4 0.8 0.12]`

Data Types: `single` | `double`

Mean of the Pearson distribution, specified as a scalar or a numeric array.

To evaluate the pdf at multiple values, specify `X` using an array. To evaluate the pdfs of multiple distributions, specify either `mu` or `sigma` (or both) using arrays. If one or more of the input arguments `X`, `mu`, and `sigma` are arrays, then the array sizes must be the same. In this case, `pearspdf` expands each scalar input into a constant array of the same size as the array inputs. Each element in `f` is the pdf value of the distribution specified by the corresponding elements in `mu` and `sigma`, evaluated at the corresponding element in `X`.

Example: `[0 1 2; 0 1 2]`

Data Types: `single` | `double`

Standard deviation of the Pearson distribution, specified as a positive scalar or an array of positive values.

To evaluate the pdf at multiple values, specify `X` using an array. To evaluate the pdfs of multiple distributions, specify either `mu` or `sigma` (or both) using arrays. If one or more of the input arguments `X`, `mu`, and `sigma` are arrays, then the array sizes must be the same. In this case, `pearspdf` expands each scalar input into a constant array of the same size as the array inputs. Each element in `f` is the pdf value of the distribution specified by the corresponding elements in `mu` and `sigma`, evaluated at the corresponding element in `X`.

Example: `[1 1 1; 2 2 2]`

Data Types: `single` | `double`

Skewness for the Pearson distribution, specified as a scalar. The value of `skew` must be less than ```sqrt(kurt - 1)```. For more information, see Skewness.

Example: `3`

Data Types: `single` | `double`

Kurtosis for the Pearson distribution, specified as a scalar. The value of `kurt` must be greater than ```skew^2 + 1```. For more information, see Kurtosis.

Example: `11`

Data Types: `single` | `double`

Output Arguments

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Pearson pdf evaluated at the values in `X`, returned as a scalar or a numeric array. `f` is the same size as `X`, `mu`, and `sigma` after any necessary scalar expansion. Each element in `f` is the pdf value of the distribution specified by `skew`, `kurt`, and the corresponding elements in `mu` and `sigma`, evaluated at the corresponding value in `X`.

Type of Pearson distribution used to calculate the pdf, returned as an integer from the interval `[0 7]` or `NaN`. If the distribution parameters are invalid, `pearspdf` returns `NaN`.

This table describes the distribution corresponding to each Pearson distribution type.

Pearson Distribution TypeDescription
`0`Normal
`1`4-parameter beta
`2`Symmetric 4-parameter beta
`3`3-parameter gamma
`4`Distribution specific to the Pearson system with pdf proportional to ${\left(1+{\left(\frac{x-\mu }{\sigma }\right)}^{2}\right)}^{-a}\mathrm{exp}\left(-b\mathrm{arctan}\left(\frac{x-\mu }{\sigma }\right)\right)$, where a and b are quantities related to the differential equation that defines the Pearson distribution
`5`Inverse 3-parameter gamma
`6`F location scale
`7`Student's t location scale

Quadratic polynomial coefficients, returned as a numeric 1-by-3 vector. The ith element of `coefs` is the coefficient ${b}_{i}$ in the differential equation

`$\frac{p\text{'}\left(x\right)}{p\left(x\right)}=-\frac{a+\left(x-\mu \right)}{{b}_{0}+{b}_{1}\left(x-\mu \right)+{b}_{2}{\left(x-\mu \right)}^{2}},$`

which defines the Pearson distribution pdf $p\left(x\right)$.

You can calculate the support for the Pearson distribution pdf using `coefs`. For more information, see Probability Density Function and Support.

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Skewness

Skewness is a measure of the asymmetry of the data around the sample mean. If skewness is negative, the data spreads out more to the left of the mean than to the right. If skewness is positive, the data spreads out more to the right. The skewness of the normal distribution (or any perfectly symmetric distribution) is zero.

The skewness of a distribution is defined as

`$s=\frac{E{\left(x-\mu \right)}^{3}}{{\sigma }^{3}},$`

where µ is the mean of x, σ is the standard deviation of x, and E(t) represents the expected value of the quantity t.

Kurtosis

Kurtosis is a measure of how prone a distribution is to outliers. The kurtosis of the normal distribution is 3. Distributions that are more prone to outliers than the normal distribution have a kurtosis value greater than 3; distributions that are less prone have a kurtosis value less than 3. Some definitions of kurtosis subtract 3 from the computed value, so that the normal distribution has a kurtosis of 0. `pearspdf` does not use this convention.

The kurtosis of a distribution is defined as

`$k=\frac{E{\left(x-\mu \right)}^{4}}{{\sigma }^{4}},$`

where μ is the mean of x, σ is the standard deviation of x, and E(t) represents the expected value of the quantity t.

Alternative Functionality

`pearspdf` is a function specific to the Pearson distribution. Statistics and Machine Learning Toolbox™ also offers the generic function `pdf`, which supports various probability distributions. To use `pdf`, specify the probability distribution name and its parameters.

References

[1] Johnson, Norman Lloyd, et al. "Continuous Univariate Distributions." 2nd ed, vol. 1, Wiley, 1994.

[2] Willink, R. "A Closed-Form Expression for the Pearson Type IV Distribution Function." Australian & New Zealand Journal of Statistics, vol. 50, no. 2, June 2008, pp. 199–205. https://onlinelibrary.wiley.com/doi/10.1111/j.1467-842X.2008.00508.x.

Version History

Introduced in R2023b