Given a sample of observations, , from a distribution with unknown density function, , an estimate of the density function, , may be required. The simplest form of density estimator is the histogram. This may be defined by:
where is the number of observations falling in the interval to , is the lower bound to the histogram and is the upper bound. The value is known as the window width. To produce a smoother density estimate a kernel method can be used. A kernel function, , satisfies the conditions:
The kernel density estimator is then defined as
The choice of is usually not important but to ease the computational burden use can be made of the Gaussian kernel defined as
The smoothness of the estimator depends on the window width . The larger the value of the smoother the density estimate. The value of can be chosen by examining plots of the smoothed density for different values of or by using cross-validation methods (see Silverman (1990)).
Silverman (1982) and Silverman (1990) show how the Gaussian kernel density estimator can be computed using a fast Fourier transform (fft). In order to compute the kernel density estimate over the range to the following steps are required.
To compute the kernel density estimate for further values of only steps (iii) and (iv) need be repeated.
Jones M C and Lotwick H W (1984) Remark AS R50. A remark on algorithm AS 176. Kernel density estimation using the Fast Fourier Transform Appl. Statist.33 120–122
Silverman B W (1982) Algorithm AS 176. Kernel density estimation using the fast Fourier transform Appl. Statist.31 93–99
Silverman B W (1990) Density Estimation Chapman and Hall
1: – IntegerInput
On entry: , the number of observations in the sample.
2: – Real (Kind=nag_wp) arrayInput
On entry: the observations,
, for .
3: – Real (Kind=nag_wp)Input
On entry: , the window width.
4: – Real (Kind=nag_wp)Input
On entry: , the lower limit of the interval on which the estimate is calculated. For most applications slo should be at least three window widths below the lowest data point.
5: – Real (Kind=nag_wp)Input
On entry: , the upper limit of the interval on which the estimate is calculated. For most applications shi should be at least three window widths above the highest data point.
6: – IntegerInput
On entry: the number of points at which the estimate is calculated, .
7: – Real (Kind=nag_wp) arrayOutput
On exit: the values of the density estimate,
, for .
8: – Real (Kind=nag_wp) arrayOutput
On exit: the points at which the estimate is calculated,
, for .
9: – LogicalInput
On entry: must be set to .FALSE. if the values of are to be calculated by g10baf and to .TRUE. if they have been computed by a previous call to g10baf and are provided in fft. If then the arguments n, slo, shi, ns and fft must remain unchanged from the previous call to g10baf with .
10: – Real (Kind=nag_wp) arrayInput/Output
On entry: if , fft must contain the fast Fourier transform of the weights of the discretized data,
, for . Otherwise fft need not be set.
On exit: the fast Fourier transform of the weights of the discretized data,
, for .
11: – IntegerInput/Output
On entry: ifail must be set to , . If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value is recommended. If the output of error messages is undesirable, then the value is recommended. Otherwise, if you are not familiar with this argument, the recommended value is . When the value is used it is essential to test the value of ifail on exit.
On exit: unless the routine detects an error or a warning has been flagged (see Section 6).
Error Indicators and Warnings
If on entry or , explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
g10baf has been called with but the routine has not been called previously with ,
g10baf has been called with but some of the arguments n, slo, shi, ns have been changed since the previous call to g10baf with .
On entry, the interval given by slo to shi does not extend beyond three window widths at either extreme of the dataset. This may distort the density estimate in some cases.
An unexpected error has been triggered by this routine. Please
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.
Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.
Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.
g10baf is not thread safe and should not be called from a multithreaded user program. Please see Section 3.12.1 in How to Use the NAG Library and its Documentation for more information on thread safety.
g10baf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
g10baf makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.
The time for computing the weights of the discretized data is of order , while the time for computing the fft is of order , as is the time for computing the inverse of the fft.
Data is read from a file and the density estimated. The first values are then printed. The full estimated density function is shown in the accompanying plot.