nag_kernel_density_estim (g10bac) : 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 and

b = n_{s} h

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,

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:

1.	discretize the data to give $n_{s}$ equally spaced points $t_{l}$ with weights $ξ_{l}$ (see Jones and Lotwick (1984));
2.	compute the FFT of the weights $ξ_{l}$ to give $Y_{l}$ ;
3.	compute $ζ_{l} = e^{- \frac{1}{2} h^{2} s_{l}^{2}} Y_{l}$ where $s_{l} = 2 π l / (b - a)$ ;
4.	find the inverse FFT of $ζ_{l}$ to give $\hat{f} (x)$ .

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: the number of observations in the sample,

n

Constraint:

n > 0

2: x[n] – const doubleInput

On entry: the

n

observations,

x_{i}

, for

i = 1, 2, \dots, n

3: window – doubleInput

On entry: the window width,

h

Constraint:

window > 0.0

4: low – doubleInput

On entry: the lower limit of the interval on which the estimate is calculated,

a

. For most applications low should be at least three window widths below the lowest data point.

Constraint:

low < high

5: high – doubleInput

On entry: the upper limit of the interval on which the estimate is calculated,

b

. For most applications high 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,

n_{s}

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.

7: smooth[ns] – doubleOutput

On exit: the

n_{s}

values of the density estimate,

\hat{f} (t_{l})

, for

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

8: t[ns] – doubleOutput

On exit: the points at which the estimate is calculated,

t_{l}

, for

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

9: fail – NagError *Input/Output

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

6 Error Indicators and Warnings

NE_2_REAL_ARG_LE

On entry,

high = ⟨value⟩

while

low = ⟨value⟩

. These arguments must satisfy

high > low

NE_ALLOC_FAIL

Dynamic memory allocation failed.

NE_C06_FACTORS

At least one of the prime factors of ns is greater than 19 or ns has more than 20 prime factors.

NE_G10BA_INTERVAL

On entry, the interval given by low to high does not extend beyond three window widths at either extreme of the dataset. This may distort the density estimate in some cases.

NE_INT_ARG_LE

On entry,

n = ⟨value⟩

.
Constraint:

n > 0

NE_INT_ARG_LT

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_REAL_ARG_LE

On entry, window must not be less than or equal to 0.0:

window = ⟨value⟩

NAG Library Function Document

nag_kernel_density_estim (g10bac)

+− 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_estim (g10bac)

+− 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_estim (g10bac)