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# ksdensity

Kernel smoothing function estimate for univariate and bivariate data

## Description

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[f,xi] = ksdensity(x) returns a probability density estimate, f, for the sample data in the vector or two-column matrix x. The estimate is based on a normal kernel function, and is evaluated at equally-spaced points, xi, that cover the range of the data in x. ksdensity estimates the density at 100 points for univariate data, or 900 points for bivariate data.

ksdensity works best with continuously distributed samples.

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[f,xi] = ksdensity(x,pts) specifies points (pts) to evaluate f. Here, xi and pts contain identical values.

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[f,xi] = ksdensity(___,Name,Value) uses additional options specified by one or more name-value pair arguments in addition to any of the input arguments in the previous syntaxes. For example, you can define the function type ksdensity evaluates, such as probability density, cumulative probability, survivor function, and so on. Or you can specify the bandwidth of the smoothing window.

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[f,xi,bw] = ksdensity(___) also returns the bandwidth of the kernel smoothing window, bw. The default bandwidth is the optimal for normal densities.

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ksdensity(___) plots the kernel smoothing function estimate.

ksdensity(ax,___) plots the results using axes with the handle, ax, instead of the current axes returned by gca.

## Examples

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Generate a sample data set from a mixture of two normal distributions.

rng('default')  % For reproducibility
x = [randn(30,1); 5+randn(30,1)];

Plot the estimated density.

[f,xi] = ksdensity(x);
figure
plot(xi,f);

The density estimate shows the bimodality of the sample.

Generate a nonnegative sample data set from the half-normal distribution.

rng('default') % For reproducibility
pd = makedist('HalfNormal','mu',0,'sigma',1);
x = random(pd,100,1);

Estimate pdfs with two different boundary correction methods, log transformation and reflection, by using the 'BoundaryCorrection' name-value pair argument.

pts = linspace(0,5,1000); % points to evaluate the estimator
[f1,xi1] = ksdensity(x,pts,'Support','positive');
[f2,xi2] = ksdensity(x,pts,'Support','positive','BoundaryCorrection','reflection');

Plot the two estimated pdfs.

plot(xi1,f1,xi2,f2)
lgd = legend('log','reflection');
title(lgd, 'Boundary Correction Method')
xl = xlim;
xlim([xl(1)-0.25 xl(2)])

ksdensity uses a boundary correction method when you specify either positive or bounded support. The default boundary correction method is log transformation. When ksdensity transforms the support back, it introduces the 1/x term in the kernel density estimator. Therefore, the estimate has a peak near x = 0. On the other hand, the reflection method does not cause undesirable peaks near the boundary.

Load the sample data.

load hospital

Compute and plot the estimated cdf evaluated at a specified set of values.

pts = (min(hospital.Weight):2:max(hospital.Weight));
figure()
ecdf(hospital.Weight)
hold on
[f,xi,bw] = ksdensity(hospital.Weight,pts,'Support','positive',...
'Function','cdf');
plot(xi,f,'-g','LineWidth',2)
legend('empirical cdf','kernel-bw:default','Location','northwest')
xlabel('Patient weights')
ylabel('Estimated cdf')

ksdensity seems to smooth the cumulative distribution function estimate too much. An estimate with a smaller bandwidth might produce a closer estimate to the empirical cumulative distribution function.

Return the bandwidth of the smoothing window.

bw
bw = 0.1070

Plot the cumulative distribution function estimate using a smaller bandwidth.

[f,xi] = ksdensity(hospital.Weight,pts,'Support','positive',...
'Function','cdf','Bandwidth',0.05);
plot(xi,f,'--r','LineWidth',2)
legend('empirical cdf','kernel-bw:default','kernel-bw:0.05',...
'Location','northwest')
hold off

The ksdensity estimate with a smaller bandwidth matches the empirical cumulative distribution function better.

Load the sample data.

load hospital

Plot the estimated cdf evaluated at 50 equally spaced points.

figure()
ksdensity(hospital.Weight,'Support','positive','Function','cdf',...
'NumPoints',50)
xlabel('Patient weights')
ylabel('Estimated cdf')

Generate sample data from an exponential distribution with mean 3.

rng('default')  % For reproducibility
x = random('exp',3,100,1);

Create a logical vector that indicates censoring. Here, observations with lifetimes longer than 10 are censored.

T = 10;
cens = (x>T);

Compute and plot the estimated density function.

figure
ksdensity(x,'Support','positive','Censoring',cens);

Compute and plot the survivor function.

figure
ksdensity(x,'Support','positive','Censoring',cens,...
'Function','survivor');

Compute and plot the cumulative hazard function.

figure
ksdensity(x,'Support','positive','Censoring',cens,...
'Function','cumhazard');

Generate a mixture of two normal distributions, and plot the estimated inverse cumulative distribution function at a specified set of probability values.

rng('default')  % For reproducibility
x = [randn(30,1); 5+randn(30,1)];
pi = linspace(.01,.99,99);
figure
ksdensity(x,pi,'Function','icdf');

Generate a mixture of two normal distributions.

rng('default')  % For reproducibility
x = [randn(30,1); 5+randn(30,1)];

Return the bandwidth of the smoothing window for the probability density estimate.

[f,xi,bw] = ksdensity(x);
bw
bw = 1.5141

The default bandwidth is optimal for normal densities.

Plot the estimated density.

figure
plot(xi,f);
xlabel('xi')
ylabel('f')
hold on

Plot the density using an increased bandwidth value.

[f,xi] = ksdensity(x,'Bandwidth',1.8);
plot(xi,f,'--r','LineWidth',1.5)

A higher bandwidth further smooths the density estimate, which might mask some characteristics of the distribution.

Now, plot the density using a decreased bandwidth value.

[f,xi] = ksdensity(x,'Bandwidth',0.8);
plot(xi,f,'-.k','LineWidth',1.5)
legend('bw = default','bw = 1.8','bw = 0.8')
hold off

A smaller bandwidth smooths the density estimate less, which exaggerates some characteristics of the sample.

Create a two-column vector of points at which to evaluate the density.

gridx1 = -0.25:.05:1.25;
gridx2 = 0:.1:15;
[x1,x2] = meshgrid(gridx1, gridx2);
x1 = x1(:);
x2 = x2(:);
xi = [x1 x2];

Generate a 30-by-2 matrix containing random numbers from a mixture of bivariate normal distributions.

rng('default')  % For reproducibility
x = [0+.5*rand(20,1) 5+2.5*rand(20,1);
.75+.25*rand(10,1) 8.75+1.25*rand(10,1)];

Plot the estimated density of the sample data.

figure
ksdensity(x,xi);

## Input Arguments

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Sample data for which ksdensity returns f values, specified as a column vector or two-column matrix. Use a column vector for univariate data, and a two-column matrix for bivariate data.

Example: [f,xi] = ksdensity(x)

Data Types: single | double

Points at which to evaluate f, specified as a vector or two-column matrix. For univariate data, pts can be a row or column vector. The length of the returned output f is equal to the number of points in pts.

Example: pts = (0:1:25); ksdensity(x,pts);

Data Types: single | double

Axes handle for the figure ksdensity plots to, specified as a handle.

For example, if h is a handle for a figure, then ksdensity can plot to that figure as follows.

Example: ksdensity(h,x)

### Name-Value 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: 'Censoring',cens,'Kernel','triangle','NumPoints',20,'Function','cdf' specifies that ksdensity estimates the cdf by evaluating at 20 equally spaced points that covers the range of data, using the triangle kernel smoothing function and accounting for the censored data information in vector cens.

The bandwidth of the kernel-smoothing window, which is a function of the number of points in x, specified as the comma-separated pair consisting of 'Bandwidth' and a scalar value. If the sample data is bivariate, Bandwidth can also be a two-element vector. The default is optimal for estimating normal densities [1], but you might want to choose a larger or smaller value to smooth more or less.

If you specify 'BoundaryCorrection' as 'log'(default) and 'Support' as either 'positive' or a vector [L U], ksdensity converts bounded data to be unbounded by using log transformation. The value of 'Bandwidth' is on the scale of the transformed values.

Example: 'Bandwidth',0.8

Data Types: single | double

Boundary correction method, specified as the comma-separated pair consisting of 'BoundaryCorrection' and 'log' or 'reflection'.

ValueDescription
'log'

ksdensity converts bounded data x to be unbounded by one of the following transformations. Then, it transforms back to the original bounded scale after density estimation.

• For univariate data, if you specify 'Support','positive', then ksdensity applies log(x).

• For univariate data, if you specify 'Support',[L U], where L and U are numeric scalars and L < U, then ksdensity applies log((x-L)/(U–x)).

• For bivariate data, ksdensity transforms each column of x in the same way with the univariate data.

The value of 'Bandwidth' and the bw output are on the scale of the transformed values.

'reflection'

ksdensity augments bounded data by adding reflected data near the boundaries, then it returns estimates corresponding to the original support. For details, see Reflection Method.

ksdensity applies boundary correction only when you specify 'Support' as a value other than 'unbounded'.

Example: 'BoundaryCorrection','reflection'

Logical vector indicating which entries are censored, specified as the comma-separated pair consisting of 'Censoring' and a vector of binary values. A value of 0 indicates there is no censoring, 1 indicates that observation is censored. Default is there is no censoring. This name-value pair is only valid for univariate data.

Example: 'Censoring',censdata

Data Types: logical

Function to estimate, specified as the comma-separated pair consisting of 'Function' and one of the following.

ValueDescription
'pdf'Probability density function.
'cdf'Cumulative distribution function.
'icdf'

Inverse cumulative distribution function. ksdensity computes the estimated inverse cdf of the values in x, and evaluates it at the probability values specified in pi.

This value is valid only for univariate data.

'survivor'Survivor function.
'cumhazard'

Cumulative hazard function.

This value is valid only for univariate data.

Example: 'Function','icdf'

Type of kernel smoother, specified as the comma-separated pair consisting of 'Kernel' and one of the following.

• 'normal' (default)

• 'box'

• 'triangle'

• 'epanechnikov'

• A kernel function that is a custom or built-in function. Specify the function as a function handle (for example, @myfunction or @normpdf) or as a character vector or string scalar (for example, 'myfunction' or 'normpdf'). The software calls the specified function with one argument that is an array of distances between data values and locations where the density is evaluated. The function must return an array of the same size containing corresponding values of the kernel function.

When 'Function' is 'pdf', the kernel function returns density values. Otherwise, it returns cumulative probability values.

Specifying a custom kernel when 'Function' is 'icdf' returns an error.

For bivariate data, ksdensity applies the same kernel to each dimension.

Example: 'Kernel','box'

Number of equally spaced points in xi, specified as the comma-separated pair consisting of 'NumPoints' and a scalar value. This name-value pair is only valid for univariate data.

For example, for a kernel smooth estimate of a specified function at 80 equally spaced points within the range of sample data, input:

Example: 'NumPoints',80

Data Types: single | double

Support for the density, specified as the comma-separated pair consisting of 'support' and one of the following.

ValueDescription
'unbounded'Default. Allow the density to extend over the whole real line.
'positive'Restrict the density to positive values.
Two-element vector, [L U]Give the finite lower and upper bounds for the support of the density. This option is only valid for univariate sample data.
Two-by-two matrix, [L1 L2; U1 U2]Give the finite lower and upper bounds for the support of the density. The first row contains the lower limits and the second row contains the upper limits. This option is only valid for bivariate sample data.

For bivariate data, 'Support' can be a combination of positive, unbounded, or bounded variables specified as [0 -Inf; Inf Inf] or [0 L; Inf U].

Example: 'Support','positive'

Example: 'Support',[0 10]

Data Types: single | double | char | string

Function used to create kernel density plot, specified as the comma-separated pair consisting of 'PlotFcn' and one of the following.

ValueDescription
'surf'3-D shaded surface plot, created using surf
'contour'Contour plot, created using contour
'plot3'3-D line plot, created using plot3
'surfc'Contour plot under a 3-D shaded surface plot, created using surfc

This name-value pair is only valid for bivariate sample data.

Example: 'PlotFcn','contour'

Weights for sample data, specified as the comma-separated pair consisting of 'Weights' and a vector of length size(x,1), where x is the sample data.

Example: 'Weights',xw

Data Types: single | double

## Output Arguments

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Estimated function values, returned as a vector whose length is equal to the number of points in xi or pts.

Evaluation points at which ksdensity calculates f, returned as a vector or a two-column matrix. For univariate data, the default is 100 equally-spaced points that cover the range of data in x. For bivariate data, the default is 900 equally-spaced points created using meshgrid from 30 equally-spaced points in each dimension.

Bandwidth of smoothing window, returned as a scalar value.

If you specify 'BoundaryCorrection' as 'log'(default) and 'Support' as either 'positive' or a vector [L U], ksdensity converts bounded data to be unbounded by using log transformation. The value of bw is on the scale of the transformed values.

## More About

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### Kernel Distribution

A kernel distribution is a nonparametric representation of the probability density function (pdf) of a random variable. You can use a kernel distribution when a parametric distribution cannot properly describe the data, or when you want to avoid making assumptions about the distribution of the data. A kernel distribution is defined by a smoothing function and a bandwidth value, which control the smoothness of the resulting density curve.

The kernel density estimator is the estimated pdf of a random variable. For any real values of x, the kernel density estimator's formula is given by

${\stackrel{^}{f}}_{h}\left(x\right)=\frac{1}{nh}\sum _{i=1}^{n}K\left(\frac{x-{x}_{i}}{h}\right)\text{\hspace{0.17em}},$

where x1, x2, …, xn are random samples from an unknown distribution, n is the sample size, $K\left(·\right)$ is the kernel smoothing function, and h is the bandwidth.

The kernel estimator for the cumulative distribution function (cdf), for any real values of x, is given by

${\stackrel{^}{F}}_{h}\left(x\right)={\int }_{-\infty }^{x}{\stackrel{^}{f}}_{h}\left(t\right)dt=\frac{1}{n}\sum _{i=1}^{n}G\left(\frac{x-{x}_{i}}{h}\right)\text{\hspace{0.17em}},$

where $G\left(x\right)={\int }_{-\infty }^{x}K\left(t\right)dt$.

For more details, see Kernel Distribution.

### Reflection Method

The reflection method is a boundary correction method that accurately finds kernel density estimators when a random variable has bounded support. If you specify 'BoundaryCorrection','reflection', ksdensity uses the reflection method. This method augments bounded data by adding reflected data near the boundaries, and estimates the pdf. Then, ksdensity returns the estimated pdf corresponding to the original support with proper normalization, so that the estimated pdf's integral over the original support is equal to one.

If you additionally specify 'Support',[L U], then ksdensity finds the kernel estimator as follows.

• If 'Function' is 'pdf', then the kernel density estimator is

${\stackrel{^}{f}}_{h}\left(x\right)=\frac{1}{nh}\sum _{i=1}^{n}\left[K\left(\frac{x-{x}_{i}^{-}}{h}\right)+K\left(\frac{x-{x}_{i}}{h}\right)+K\left(\frac{x-{x}_{i}^{+}}{h}\right)\right]$ for L ≤ x ≤ U,

where ${x}_{i}^{-}=2L-{x}_{i}$, ${x}_{i}^{+}=2U-{x}_{i}$, and xi is the ith sample data.

• If 'Function' is 'cdf', then the kernel estimator for cdf is

${\stackrel{^}{F}}_{h}\left(x\right)=\frac{1}{n}\sum _{i=1}^{n}\left[G\left(\frac{x-{x}_{i}^{-}}{h}\right)+G\left(\frac{x-{x}_{i}}{h}\right)+G\left(\frac{x-{x}_{i}^{+}}{h}\right)\right]-\frac{1}{n}\sum _{i=1}^{n}\left[G\left(\frac{L-{x}_{i}^{-}}{h}\right)+G\left(\frac{L-{x}_{i}}{h}\right)+G\left(\frac{L-{x}_{i}^{+}}{h}\right)\right]$ for L ≤ x ≤ U.

• To obtain a kernel estimator for an inverse cdf, a survivor function, or a cumulative hazard function (when 'Function' is 'icdf', 'survivor', or 'cumhazrd'), ksdensity uses both ${\stackrel{^}{f}}_{h}\left(x\right)$ and ${\stackrel{^}{F}}_{h}\left(x\right)$.

If you additionally specify 'Support' as 'positive' or [0 inf], then ksdensity finds the kernel estimator by replacing [L U] with [0 inf] in the above equations.

## References

[1] Bowman, A. W., and A. Azzalini. Applied Smoothing Techniques for Data Analysis. New York: Oxford University Press Inc., 1997.

[2] Hill, P. D. “Kernel estimation of a distribution function.” Communications in Statistics - Theory and Methods. Vol 14, Issue. 3, 1985, pp. 605-620.

[3] Jones, M. C. “Simple boundary correction for kernel density estimation.” Statistics and Computing. Vol. 3, Issue 3, 1993, pp. 135-146.

[4] Silverman, B. W. Density Estimation for Statistics and Data Analysis. Chapman & Hall/CRC, 1986.

## Extended Capabilities

Introduced before R2006a

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