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Kernel smoothing function estimate for univariate and bivariate data
[f,xi]
= ksdensity(x)
[f,xi]
= ksdensity(x,pts)
[f,xi]
= ksdensity(___,Name,Value)
[f,xi,bw]
= ksdensity(___)
ksdensity(___)
ksdensity(ax,___)
[
returns a probability
density estimate, f
,xi
]
= ksdensity(x
)f
, for the sample data in the
vector or twocolumn matrix x
. The estimate is
based on a normal kernel function, and is evaluated at equallyspaced
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.
[
uses additional options specified by one or more namevalue pair arguments in
addition to any of the input arguments in the previous syntaxes. For example,
you can define the function type f
,xi
]
= ksdensity(___,Name,Value
)ksdensity
evaluates, such
as probability density, cumulative probability, survivor function, and so on. Or
you can specify the bandwidth of the smoothing window.
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 halfnormal 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'
namevalue 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','kernelbw: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','kernelbw:default','kernelbw: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 twocolumn 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 30by2 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);
x
— Sample dataSample data for which ksdensity
returns f
values,
specified as a column vector or twocolumn matrix. Use a column vector
for univariate data, and a twocolumn matrix for bivariate data.
Example: [f,xi] = ksdensity(x)
Data Types: single
 double
pts
— Points to evaluate f
Points to evaluate f
at, specified as a
vector or twocolumn matrix. For univariate data, pts
can
be a row or column vector. The returned output f
has
the same dimensions as pts
.
Example: pts = (0:1:25); ksdensity(x,pts);
Data Types: single
 double
ax
— Axes handleAxes 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)
Specify optional
commaseparated 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
.
'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 the kernel smoothing windowThe bandwidth of the kernelsmoothing window, which is a function of
the number of points in x
, specified as the
commaseparated pair consisting of 'Bandwidth'
and a
scalar value. If the sample data is bivariate,
Bandwidth
can also be a twoelement 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
'BoundaryCorrection'
— Boundary correction methodBoundary correction method, specified as the commaseparated pair
consisting of 'BoundaryCorrection'
and
'log'
or 'reflection'
.
Value  Description 

'log' 
The value of 
'reflection' 

ksdensity
applies boundary correction only when
you specify 'Support'
as a value other than
'unbounded'
.
Example: 'BoundaryCorrection','reflection'
'Censoring'
— Logical vectorLogical vector indicating which entries are censored, specified as the
commaseparated 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 namevalue 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 commaseparated pair consisting
of 'Function'
and one of the following.
Value  Description 

'pdf'  Probability density function. 
'cdf'  Cumulative distribution function. 
'icdf' 
Inverse cumulative distribution function.
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 scalarType of kernel smoother, specified as the commaseparated pair
consisting of 'Kernel'
and one of the
following.
'normal'
(default)
'box'
'triangle'
'epanechnikov'
You can also specify a custom kernel function as a
function handle (for example, @normpdf
)
or as either a character vector or string scalar (for
example, 'normpdf'
). This calls the
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'
, this 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 pointsNumber of equally spaced points in xi
, specified
as the commaseparated pair consisting of 'NumPoints'
and a scalar value. This namevalue 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 the density'unbounded'
(default)  'positive'
 twoelement vector, [L U]
 twobytwo matrix, [L1 L2 ; U1 U2]
Support for the density, specified as the commaseparated pair
consisting of 'support'
and one of the
following.
Value  Description 

'unbounded'  Default. Allow the density to extend over the whole real line. 
'positive'  Restrict the density to positive values. 
Twoelement 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. 
Twobytwo 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
'PlotFcn'
— Function used to create kernel density plot'surf'
(default)  'contour'
 'plot3'
 'surfc'
Function used to create kernel density plot, specified as the
commaseparated pair consisting of 'PlotFcn'
and one
of the following.
Value  Description 

'surf'  3D shaded surface plot, created using surf 
'contour'  Contour plot, created using contour 
'plot3'  3D line plot, created using plot3 
'surfc'  Contour plot under a 3D shaded surface plot, created
using surfc 
This namevalue pair is only valid for bivariate sample data.
Example: 'PlotFcn','contour'
'Weights'
— Weights for sample dataWeights for sample data, specified as the commaseparated 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
xi
— Evaluation pointsEvaluation points at which ksdensity
calculates f
,
returned as a vector. For univariate data, the default is 100 equallyspaced
points that cover the range of data in x
. For
bivariate data, the default is 900 equallyspaced points created using meshgrid
from 30 equallyspaced points
in each dimension.
bw
— Bandwidth of smoothing windowBandwidth 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.
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
$${\widehat{f}}_{h}\left(x\right)=\frac{1}{nh}{\displaystyle \sum _{i=1}^{n}K\left(\frac{x{x}_{i}}{h}\right)}\text{\hspace{0.17em}},$$
where x_{1}, x_{2}, …, x_{n} are random samples from an unknown distribution, n is the sample size, $$K(\xb7)$$ 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
$${\widehat{F}}_{h}\left(x\right)={\displaystyle {\int}_{\infty}^{x}{\widehat{f}}_{h}(t)dt}=\frac{1}{n}{\displaystyle \sum _{i=1}^{n}G\left(\frac{x{x}_{i}}{h}\right)}\text{\hspace{0.17em}},$$
where $$G(x)={\displaystyle {\int}_{\infty}^{x}K(t)dt}$$.
For more details, see Kernel Distribution.
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 i
th 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.
[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. 605620.
[3] Jones, M. C. “Simple boundary correction for kernel density estimation.” Statistics and Computing. Vol. 3, Issue 3, 1993, pp. 135146.
[4] Silverman, B. W. Density Estimation for Statistics and Data Analysis. Chapman & Hall/CRC, 1986.
This function supports tall arrays for outofmemory 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 (MATLAB).
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