nag_kernel_density_gauss (g10bbc) : NAG Library, Mark 24

3 Description

Given a sample of

n

observations,

x_{1}, x_{2}, \dots, x_{n}

, from a distribution with unknown density function,

f (x)

, an estimate of the density function,

\hat{f} (x)

, may be required. The simplest form of density estimator is the histogram. This may be defined by:

\hat{f} (x) = \frac{1}{n h} n_{j}, a + (j - 1) h < x < a + j h, j = 1, 2, \dots, n_{s},

where

n_{j}

is the number of observations falling in the interval

a + (j - 1) h

a + j h

a

is the lower bound to the histogram,

b = n_{s} h

is the upper bound and

n_{s}

is the total number of intervals. The value

h

is known as the window width. To produce a smoother density estimate a kernel method can be used. A kernel function,

K (t)

, satisfies the conditions:

\int_{- \infty}^{\infty} K (t) d t = 1 and K (t) \geq 0 .

The kernel density estimator is then defined as

\hat{f} (x) = \frac{1}{n h} \sum_{i = 1}^{n} K (\frac{x - x_{i}}{h}) .

The choice of

K

is usually not important but to ease the computational burden use can be made of the Gaussian kernel defined as

K (t) = \frac{1}{\sqrt{2 π}} e^{- t^{2} / 2} .

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

b

the following steps are required.

(i)	Discretize the data to give $n_{s}$ equally spaced points $t_{l}$ with weights $ξ_{l}$ (see Jones and Lotwick (1984)).
(ii)	Compute the FFT of the weights $ξ_{l}$ to give $Y_{l}$ .
(iii)	Compute $ζ_{l} = e^{- \frac{1}{2} h^{2} s_{l}^{2}} Y_{l}$ where $s_{l} = 2 π l / (b - a)$ .
(iv)	Find the inverse FFT of $ζ_{l}$ to give $\hat{f} (x)$ .

To compute the kernel density estimate for further values of

h

only steps (iii) and (iv) need be repeated.

4 References

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

5 Arguments

1: n – IntegerInput

On entry:

n

, the number of observations in the sample.

fcall = Nag_FALSE

, n must be unchanged since the last call to nag_kernel_density_gauss (g10bbc).

Constraint:

n > 0

2: x[n] – const doubleInput

On entry:

x_{i}

, for

i = 1, 2, \dots, n

fcall = Nag_FALSE

, x must be unchanged since the last call to nag_kernel_density_gauss (g10bbc).

3: wtype – Nag_WindowTypeInput

On entry: how the window width,

h

, is to be calculated:

$wtype = Nag_WindowSupplied$

h

is supplied in window.

$wtype = Nag_RuleOfThumb$

h

is to be calculated from the data, with

h = m \times (\frac{0.9 \times \min (q_{75} - q_{25}, σ)}{n^{0.2}})

where

q_{75} - q_{25}

is the inter-quartile range and

σ

the standard deviation of the sample,

x

, and

m

is a multipler supplied in window. The

25 %

and

75 %

quartiles,

q_{25}

and

q_{75}

, are calculated using nag_double_quantiles (g01amc). This is the "rule-of-thumb" suggested by Silverman (1990).

Suggested value:

wtype = Nag_RuleOfThumb

and

window = 1.0

Constraint:

wtype = Nag_WindowSupplied

Nag_RuleOfThumb

4: window – double *Input/Output

On entry: if

wtype = Nag_WindowSupplied

, then

h

, the window width. Otherwise,

m

, the multiplier used in the calculation of

h

Suggested value:

window = 1.0

and

wtype = Nag_RuleOfThumb

On exit:

h

, the window width actually used.

Constraint:

window > 0.0

5: slo – double *Input/Output

On entry: if

slo < shi

then

a

, the lower limit of the interval on which the estimate is calculated. Otherwise,

a

and

b

, the lower and upper limits of the interval, are calculated as follows:

\begin{matrix} a & = & \min_{i} \{x_{i}\} - slo \times h \\ b & = & \max_{i} \{x_{i}\} + slo \times h \end{matrix}

where

h

is the window width.

For most applications

a

should be at least three window widths below the lowest data point.

fcall = Nag_FALSE

, slo must be unchanged since the last call to nag_kernel_density_gauss (g10bbc).

Suggested value:

slo = 3.0

and

shi = 0.0

which would cause

a

and

b

to be set

3

window widths below and above the lowest and highest data points respectively.

On exit:

a

, the lower limit actually used.

6: shi – double *Input/Output

On entry: if

slo < shi

then

b

, the upper limit of the interval on which the estimate is calculated. Otherwise a value for

b

is calculated from the data as stated in the description of slo and the value supplied in shi is not used.

For most applications

b

should be at least three window widths above the highest data point.

fcall = Nag_FALSE

, shi must be unchanged since the last call to nag_kernel_density_gauss (g10bbc).

On exit:

b

, the upper limit actually used.

7: ns – IntegerInput

On entry:

n_{s}

, the number of points at which the estimate is calculated.

fcall = Nag_FALSE

, ns must be unchanged since the last call to nag_kernel_density_gauss (g10bbc).

Suggested value:

ns = 512

Constraints:

$ns \geq 2$ ;
The largest prime factor of ns must not exceed $19$ , and the total number of prime factors of ns, counting repetitions, must not exceed $20$ .

8: smooth[ns] – doubleOutput

On exit:

\hat{f} (t_{l})

, for

l = 1, 2, \dots, n_{s}

, the

n_{s}

values of the density estimate.

9: t[ns] – doubleOutput

On exit:

t_{l}

, for

l = 1, 2, \dots, n_{s}

, the points at which the estimate is calculated.

10: fcall – Nag_BooleanInput

On entry: If

fcall = Nag_TRUE

then the values of

Y_{l}

are to be calculated by this call to nag_kernel_density_gauss (g10bbc), otherwise it is assumed that the values of

Y_{l}

were calculated by a previous call to this routine and the relevant information is stored in rcomm.

11: rcomm[

ns + 20

] – doubleCommunication Array

On entry: communication array, used to store information between calls to nag_kernel_density_gauss (g10bbc).

fcall = Nag_FALSE

, rcomm must be unchanged since the last call to nag_kernel_density_gauss (g10bbc).

On exit: the last ns elements of rcomm contain the fast Fourier transform of the weights of the discretized data, that is

rcomm [l + 19] = Y_{l}

, for

l = 1, 2, \dots, n_{s}

12: fail – NagError *Input/Output

The NAG error argument (see Section 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_ALLOC_FAIL

Dynamic memory allocation failed.

NE_BAD_PARAM

On entry, argument

⟨value⟩

had an illegal value.

NE_ILLEGAL_COMM

rcomm has been corrupted between calls.

NE_INT

On entry,

n = ⟨value⟩

.
Constraint:

n > 0

On entry,

ns = ⟨value⟩

.
Constraint:

ns \geq 2

NE_INTERNAL_ERROR

An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.

NE_PREV_CALL

On entry,

n = ⟨value⟩

.
On entry at previous call,

n = ⟨value⟩

.
Constraint: if

fcall = Nag_FALSE

, n must be unchanged since previous call.

On entry,

ns = ⟨value⟩

.
On entry at previous call,

ns = ⟨value⟩

.
Constraint: if

fcall = Nag_FALSE

, ns must be unchanged since previous call.

On entry,

shi = ⟨value⟩

.
On exit from previous call,

shi = ⟨value⟩

.
Constraint: if

fcall = Nag_FALSE

, shi must be unchanged since previous call.

On entry,

slo = ⟨value⟩

.
On exit from previous call,

slo = ⟨value⟩

.
Constraint: if

fcall = Nag_FALSE

, slo must be unchanged since previous call.

NE_PRIME_FACTOR

On entry,

ns = ⟨value⟩

.
Constraint: Largest prime factor of ns must not exceed

19

On entry,

ns = ⟨value⟩

.
Constraint: Total number of prime factors of ns must not exceed

20

NE_REAL

On entry,

window = ⟨value⟩

.
Constraint:

window > 0.0

NW_POTENTIAL_PROBLEM

On entry,

slo = ⟨value⟩

and

shi = ⟨value⟩

.
On entry,

\min (x) = ⟨value⟩

and

\max (x) = ⟨value⟩

.
Expected values of at least

⟨value⟩

and

⟨value⟩

for slo and shi.
All output values have been returned.

10 Example

Data is read from a file and the density estimated. The first

20

values are then printed.

10.1 Program Text

10.2 Program Data

10.3 Program Results

This plot shows the estimated density function for the example data for several window widths.

NAG Library Function Document

nag_kernel_density_gauss (g10bbc)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG Library Function Documentnag_kernel_density_gauss (g10bbc)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG Library Function Document

nag_kernel_density_gauss (g10bbc)