ksdensity

Kernel smoothing function estimate for univariate and bivariate data

Syntax

[f,xi] = ksdensity(x)

[f,xi] = ksdensity(x,pts)

[f,xi] = ksdensity(___,Name,Value)

[f,xi,bw] = ksdensity(___)

ksdensity(___)

ksdensity(ax,___)

Description

[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.

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

[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.

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

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

Estimate Density

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.

Estimate Density with Boundary Correction

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.

Estimate Cumulative Distribution Function at Specified Values

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.

Plot Estimated Cumulative Distribution Function for Given Number of Points

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')

Estimate Survivor and Cumulative Hazard for Censored Failure Data

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');

Estimate Inverse Cumulative Distribution Function for Specified Probability Values

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');

Return Bandwidth of Smoothing Window

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.

Plot Kernel Density Estimate of Bivariate Data

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

x — Sample data

column vector | two-column matrix

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

pts — Points at which to evaluate f

vector | two-column matrix

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

ax — Axes handle

handle

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 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: '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.

Bandwidth — Bandwidth of kernel smoothing window

"normal-approx" (default) | "plug-in" | scalar value | two-element vector

Bandwidth of the kernel smoothing window, which is a function of the number of points in x, specified as "normal-approx" or "plug-in". If the sample data is bivariate, Bandwidth can also be a two-element vector [L,U].

• When Bandwidth is "normal-approx", ksdensity uses the normal approximation method, or Silverman's rule of thumb, to calculate the bandwidth. The calculated value is optimal for estimating normal densities [2], but you might want to specify a larger value for more smoothing or a smaller value for less.

• When Bandwidth is "plug-in", ksdensity uses the improved plug-in method described in [1] to calculate the bandwidth. The plug-in method is sometimes called the Sheather-Jones method.

• When Bandwidth is a positive scalar, its value controls the smoothness of the probability function estimate. As the value increases, the probability function estimate gets smoother.

If you specify BoundaryCorrection as "log"(default) and Support as either "positive" or a vector [L U], ksdensity converts bounded data to 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

BoundaryCorrection — Boundary correction method

'log' (default) | 'reflection'

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

Value	Description
'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'

Censoring — Logical vector

vector of 0s (default) | vector of 0s and 1s

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 — Function to estimate

'pdf' (default) | 'cdf' | 'icdf' | 'survivor' | 'cumhazard'

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

Value	Description
'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'

Kernel — Type of kernel smoother

'normal' (default) | 'box' | 'triangle' | 'epanechnikov' | function handle | character vector | string scalar

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'

NumPoints — Number of equally spaced points

100 (default) | scalar value

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 — Support for density

"unbounded" (default) | "positive" | "nonnegative" | "negative" | two-element vector | two-by-two matrix

Support for the density, specified as one of the following values.

Value	Description
"unbounded"	Allow the density to extend over the whole real line (default).
"positive"	Restrict the density to positive values.
"nonnegative"	Restrict the density to positive values and 0.
"negative"	Restrict the density to negative values.
Two-element vector [L U]	Give the finite lower and upper bounds for the support of the density. This option is valid only 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 valid only for bivariate sample data.

For bivariate data, Support can be a combination of positive, unbounded, and 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

PlotFcn — Function used to create kernel density plot

'surf' (default) | 'contour' | 'plot3' | 'surfc'

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

Value	Description
'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 — Weights for sample data

vector

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

f — Estimated function values

vector

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

xi — Evaluation points

100 equally spaced points | 900 equally spaced points | vector | two-column matrix

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.

bw — Bandwidth of smoothing window

scalar value

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

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

$${\widehat{f}}_h\left(x\right)=\frac{1}{nh}{\displaystyle \sum _{i=1}^{n}K\left(\frac{x-x_i}{h}\right)},$$

where x₁, x₂, …, x_n are random samples from an unknown distribution, n is the sample size, $$K(·)$$ 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

$${\widehat{F}}_h\left(x\right)={\displaystyle {\int }_{-\infty }^x{\widehat{f}}_h(t)dt}=\frac{1}{n}{\displaystyle \sum _{i=1}^{n}G\left(\frac{x-x_i}{h}\right)},$$

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

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

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

  where $$x_i^{-}=2L-x_i$$, $$x_i^{+}=2U-x_i$$, and x_i is the ith sample data.

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

       $${\widehat{F}}_h(x)=\frac{1}{n}{\displaystyle \sum _{i=1}^n\left[G\left(\frac{x-x_i^{-}}{h}\right)+G\left(\frac{x-x_i}{h}\right)+G\left(\frac{x-x_i^{+}}{h}\right)\right]}-\frac{1}{n}{\displaystyle \sum _{i=1}^n\left[G\left(\frac{L-x_i^{-}}{h}\right)+G\left(\frac{L-x_i}{h}\right)+G\left(\frac{L-x_i^{+}}{h}\right)\right]}$$ 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 $${\widehat{f}}_h(x)$$ and $${\widehat{F}}_h(x)$$.

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.

Alternative Functionality

You can also estimate the pdf or cdf for univariate data by using the MATLAB^® kde function. Unlike ksdensity, kde does not support boundary correction methods or data censoring.

References

[1] Botev, Z. I., J. F. Grotowski, and D. P. Kroese. "Kernel Density Estimation via Diffusion." The Annals of Statistics, vol. 38, no. 5 (October 1, 2010). https://projecteuclid.org/journals/annals-of-statistics/volume-38/issue-5/Kernel-density-estimation-via-diffusion/10.1214/10-AOS799.full

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

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

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

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

Extended Capabilities

Tall Arrays

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

This function supports tall arrays for out-of-memory data with some limitations.

• Some options that require extra passes or sorting of the input data are not supported:

  • 'BoundaryCorrection'

  • 'Censoring'

  • 'Support' (support is always unbounded).

• Uses standard deviation (instead of median absolute deviation) to compute the bandwidth.

For more information, see Tall Arrays for Out-of-Memory Data.

C/C++ Code Generation

Generate C and C++ code using MATLAB® Coder™.

Usage notes and limitations:

• Plotting is not supported.

• Names in name-value pair arguments must be compile-time constants.

• Values in the following name-value pair arguments must also be compile-time constants: 'BoundaryCorrection', 'Function', and 'Kernel'. For example, to use the 'Function','cdf' name-value pair argument in the generated code, include {coder.Constant('Function'),coder.Constant('cdf')} in the -args value of codegen.

• The value of the 'Kernel' name-value pair argument cannot be a custom function handle. To specify a custom kernel function, use a character vector or string scalar.

• For the value of the 'Support' name-value pair argument, the compile-time data type must match the runtime data type.

For more information on code generation, see Introduction to Code Generation and General Code Generation Workflow.

GPU Arrays

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

This function fully supports GPU arrays. For more information, see Run MATLAB Functions on a GPU (Parallel Computing Toolbox).

Version History

Introduced before R2006a