G10BAF performs kernel density estimation using a Gaussian kernel.
Given a sample of
n observations,
x1,x2,…,xn, from a distribution with unknown density function,
fx, an estimate of the density function,
f^x, may be required. The simplest form of density estimator is the histogram. This may be defined by:
where
nj is the number of observations falling in the interval
a+j-1h to
a+jh,
a is the lower bound to the histogram and
b=nsh is the upper bound. The value
h is known as the window width. To produce a smoother density estimate a kernel method can be used. A kernel function,
Kt, satisfies the conditions:
The kernel density estimator is then defined as
The choice of
K 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
h. The larger the value of
h the smoother the density estimate. The value of
h can be chosen by examining plots of the smoothed density for different values of
h 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
a to
b the following steps are required.
| (i) |
Discretize the data to give ns equally spaced points tl with weights ξl (see Jones and Lotwick (1984)). |
| (ii) |
Compute the FFT of the weights ξl to give Yl. |
| (iii) |
Compute ζl=e-12h2sl2Yl where sl=2πl/b-a. |
| (iv) |
Find the inverse FFT of ζl to give f^x. |
To compute the kernel density estimate for further values of
h 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
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
- 1: N – INTEGERInput
On entry:
n, the number of observations in the sample.
Constraint:
N>0.
- 2: X(N) – double precision arrayInput
On entry: the n observations, xi, for i=1,2,…,n.
- 3: WINDOW – double precisionInput
On entry: h, the window width.
Constraint:
WINDOW>0.0.
- 4: SLO – double precisionInput
On entry:
a, 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.
Constraint:
SLO<SHI.
- 5: SHI – double precisionInput
On entry:
b, 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: NS – INTEGERInput
On entry:
the number of points at which the estimate is calculated, ns.
Constraints:
- NS≥2;
- The largest prime factor of NS must not exceed 19, and the total number of prime factors of NS, counting repetitions, must not exceed 20.
- 7: SMOOTH(NS) – double precision arrayOutput
On exit: the ns values of the density estimate, f^tl, for l=1,2,…,ns.
- 8: T(NS) – double precision arrayOutput
On exit: the points at which the estimate is calculated, tl, for l=1,2,…,ns.
- 9: USEFFT – LOGICALInput
On entry: must be set to .FALSE. if the values of
Yl 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
USEFFT=.TRUE. then the arguments
N,
SLO,
SHI,
NS and
FFT must remain unchanged from the previous call to G10BAF with
USEFFT=.FALSE..
- 10: FFT(NS) – double precision arrayInput/Output
On entry: if
USEFFT=.TRUE., then
FFT must contain the fast Fourier transform of the weights of the discretized data,
ξl, for
l=1,2,…,ns. Otherwise
FFT need not be set.
On exit: the fast Fourier transform of the weights of the discretized data, ξl, for l=1,2,…,ns.
- 11: IFAIL – INTEGERInput/Output
-
On entry:
IFAIL must be set to
0,
-1 or 1. If you are unfamiliar with this parameter you should refer to
Section 3.3 in the Essential Introduction for details.
On exit:
IFAIL=0 unless the routine detects an error (see
Section 6).
For environments where it might be inappropriate to halt program execution when an error is detected, the value
-1 or 1 is recommended. If the output of error messages is undesirable, then the value
1 is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is
0.
When the value -1 or 1 is used it is essential to test the value of IFAIL on exit.
If on entry
IFAIL=0 or
-1, explanatory error messages are output on the current error message unit (as defined by
X04AAF).
See
Jones and Lotwick (1984) for a discussion of the accuracy of this method.
The time for computing the weights of the discretized data is of order
n, while the time for computing the
FFT is of order
nslogns, as is the time for computing the inverse of the
FFT.
A sample of
1000 standard Normal
0,1 variates are generated using
G05SKF and the density estimated on
100 points with a window width of
0.1. The resulting estimate of the density function is plotted using
G01AGF.