NAG Library Function Document

nag_kernel_density_gauss (g10bbc)

1  Purpose

2  Specification

3  Description

4  References

5  Arguments

1:    $\mathbf{n}$ – IntegerInput

On entry: $n$, the number of observations in the sample.

If $\mathbf{fcall}=\mathrm{Nag\_FALSE}$, n must be unchanged since the last call to nag_kernel_density_gauss (g10bbc).

Constraint: $\mathbf{n}>0$.

2:    $\mathbf{x}\left[\mathbf{n}\right]$ – const doubleInput

On entry: $x_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,n$.

If $\mathbf{fcall}=\mathrm{Nag\_FALSE}$, x must be unchanged since the last call to nag_kernel_density_gauss (g10bbc).

3:    $\mathbf{wtype}$ – Nag_WindowTypeInput

On entry: how the window width, $h$, is to be calculated:

$\mathbf{wtype}=\mathrm{Nag\_WindowSupplied}$

$h$ is supplied in window.

$\mathbf{wtype}=\mathrm{Nag\_RuleOfThumb}$

$h$ is to be calculated from the data, with

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

where $q_{75}-q_{25}$ is the inter-quartile range and $\sigma$ 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: $\mathbf{wtype}=\mathrm{Nag\_RuleOfThumb}$ and $\mathbf{window}=1.0$

Constraint: $\mathbf{wtype}=\mathrm{Nag\_WindowSupplied}$ or $\mathrm{Nag\_RuleOfThumb}$.

4:    $\mathbf{window}$ – double *Input/Output

On entry: if $\mathbf{wtype}=\mathrm{Nag\_WindowSupplied}$, then $h$, the window width. Otherwise, $m$, the multiplier used in the calculation of $h$.

Suggested value: $\mathbf{window}=1.0$ and $\mathbf{wtype}=\mathrm{Nag\_RuleOfThumb}$ 

On exit: $h$, the window width actually used.

Constraint: $\mathbf{window}>0.0$.

5:    $\mathbf{slo}$ – double *Input/Output

On entry: if $\mathbf{slo}<\mathbf{shi}$ 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:

$$\begin{array}{ccc}a& =& \underset{i}{\min }\left\{x_i\right\}-\mathbf{slo}\times h\\ b& =& \underset{i}{\max }\left\{x_i\right\}+\mathbf{slo}\times h\end{array}$$

where $h$ is the window width.

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

If $\mathbf{fcall}=\mathrm{Nag\_FALSE}$, slo must be unchanged since the last call to nag_kernel_density_gauss (g10bbc).

Suggested value: $\mathbf{slo}=3.0$ and $\mathbf{shi}=0.0$ 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:    $\mathbf{shi}$ – double *Input/Output

On entry: if $\mathbf{slo}<\mathbf{shi}$ 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.

If $\mathbf{fcall}=\mathrm{Nag\_FALSE}$, shi must be unchanged since the last call to nag_kernel_density_gauss (g10bbc).

On exit: $b$, the upper limit actually used.

7:    $\mathbf{ns}$ – IntegerInput

On entry: $n_s$, the number of points at which the estimate is calculated.

If $\mathbf{fcall}=\mathrm{Nag\_FALSE}$, ns must be unchanged since the last call to nag_kernel_density_gauss (g10bbc).

Suggested value: $\mathbf{ns}=512$

Constraints:

• $\mathbf{ns}\ge 2$;
• 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:    $\mathbf{smooth}\left[\mathbf{ns}\right]$ – doubleOutput

On exit: $\hat{f}\left(t_{\mathit{l}}\right)$, for $\mathit{l}=1,2,\dots ,n_s$, the $n_s$ values of the density estimate.

9:    $\mathbf{t}\left[\mathbf{ns}\right]$ – doubleOutput

On exit: $t_{\mathit{l}}$, for $\mathit{l}=1,2,\dots ,n_s$, the points at which the estimate is calculated.

10:  $\mathbf{fcall}$ – Nag_BooleanInput

On entry: If $\mathbf{fcall}=\mathrm{Nag\_TRUE}$ 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:  $\mathbf{rcomm}\left[\mathbf{ns}+20\right]$ – doubleCommunication Array

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

If $\mathbf{fcall}=\mathrm{Nag\_FALSE}$, 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 $\mathbf{rcomm}\left[\mathit{l}+19\right]=Y_{\mathit{l}}$, for $\mathit{l}=1,2,\dots ,n_s$.

12:  $\mathbf{fail}$ – 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.
See Section 3.2.1.2 in the Essential Introduction for further information.

NE_BAD_PARAM

On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.

NE_ILLEGAL_COMM

rcomm has been corrupted between calls.

NE_INT

On entry, $\mathbf{n}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{n}>0$.

On entry, $\mathbf{ns}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{ns}\ge 2$.

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.

An unexpected error has been triggered by this function. Please contact NAG.
See Section 3.6.6 in the Essential Introduction for further information.

NE_NO_LICENCE

Your licence key may have expired or may not have been installed correctly.
See Section 3.6.5 in the Essential Introduction for further information.

NE_PREV_CALL

On entry, $\mathbf{n}=〈\mathit{\text{value}}〉$.
On entry at previous call, $\mathbf{n}=〈\mathit{\text{value}}〉$.
Constraint: if $\mathbf{fcall}=\mathrm{Nag\_FALSE}$, n must be unchanged since previous call.

On entry, $\mathbf{ns}=〈\mathit{\text{value}}〉$.
On entry at previous call, $\mathbf{ns}=〈\mathit{\text{value}}〉$.
Constraint: if $\mathbf{fcall}=\mathrm{Nag\_FALSE}$, ns must be unchanged since previous call.

On entry, $\mathbf{shi}=〈\mathit{\text{value}}〉$.
On exit from previous call, $\mathbf{shi}=〈\mathit{\text{value}}〉$.
Constraint: if $\mathbf{fcall}=\mathrm{Nag\_FALSE}$, shi must be unchanged since previous call.

On entry, $\mathbf{slo}=〈\mathit{\text{value}}〉$.
On exit from previous call, $\mathbf{slo}=〈\mathit{\text{value}}〉$.
Constraint: if $\mathbf{fcall}=\mathrm{Nag\_FALSE}$, slo must be unchanged since previous call.

NE_PRIME_FACTOR

On entry, $\mathbf{ns}=〈\mathit{\text{value}}〉$.
Constraint: Largest prime factor of ns must not exceed $19$.

On entry, $\mathbf{ns}=〈\mathit{\text{value}}〉$.
Constraint: Total number of prime factors of ns must not exceed $20$.

NE_REAL

On entry, $\mathbf{window}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{window}>0.0$.

NW_POTENTIAL_PROBLEM

On entry, $\mathbf{slo}=〈\mathit{\text{value}}〉$ and $\mathbf{shi}=〈\mathit{\text{value}}〉$.
On entry, $\min \left(\mathbf{x}\right)=〈\mathit{\text{value}}〉$ and $\max \left(\mathbf{x}\right)=〈\mathit{\text{value}}〉$.
Expected values of at least $〈\mathit{\text{value}}〉$ and $〈\mathit{\text{value}}〉$ for slo and shi.
All output values have been returned.

7  Accuracy

8  Parallelism and Performance

9  Further Comments

10  Example

10.1  Program Text

Program Text (g10bbce.c)

10.2  Program Data

Program Data (g10bbce.d)

10.3  Program Results

Program Results (g10bbce.r)