nag_rand_exp_smooth (g05pmc) : NAG Library, Mark 23

2 Specification

#include <nag.h>

#include <nagg05.h>

void	nag_rand_exp_smooth (Nag_InitialValues mode, Integer n, Nag_ExpSmoothType itype, Integer p, const double param[], const double init[], double var, double r[], Integer state[], const double e[], Integer en, double x[], NagError *fail)

3 Description

nag_rand_exp_smooth (g05pmc) returns

\{x_{t} : t = 1, 2, \dots, n\}

, a realization of a time series from an exponential smoothing model defined by one of five smoothing functions:

Single Exponential Smoothing
$\begin{matrix} x_{t} & = & m_{t - 1} + ε_{t} \\ m_{t} & = & α x_{t} + (1 - α) m_{t - 1} \end{matrix}$

Brown Double Exponential Smoothing

\begin{matrix} x_{t} & = & m_{t - 1} + \frac{r_{t - 1}}{α} + ε_{t} \\ m_{t} & = & α x_{t} + (1 - α) m_{t - 1} \\ r_{t} & = & α (m_{t} - m_{t - 1}) + (1 - α) r_{t - 1} \end{matrix}

Linear Holt Exponential Smoothing

\begin{matrix} x_{t} & = & m_{t - 1} + ϕ r_{t - 1} + ε_{t} \\ m_{t} & = & α x_{t} + (1 - α) (m_{t - 1} + ϕ r_{t - 1}) \\ r_{t} & = & γ (m_{t} - m_{t - 1}) + (1 - γ) ϕ r_{t - 1} \end{matrix}

Additive Holt–Winters Smoothing

\begin{matrix} x_{t} & = & m_{t - 1} + ϕ r_{t - 1} + s_{t - 1 - p} + ε_{t} \\ m_{t} & = & α (x_{t} - s_{t - p}) + (1 - α) (m_{t - 1} + ϕ r_{t - 1}) \\ r_{t} & = & γ (m_{t} - m_{t - 1}) + (1 - γ) ϕ r_{t - 1} \\ s_{t} & = & β (x_{t} - m_{t}) + (1 - β) s_{t - p} \end{matrix}

Multiplicative Holt–Winters Smoothing

\begin{matrix} x_{t} & = & (m_{t - 1} + ϕ r_{t - 1}) \times s_{t - 1 - p} + ε_{t} \\ m_{t} & = & α x_{t} / s_{t - p} + (1 - α) (m_{t - 1} + ϕ r_{t - 1}) \\ r_{t} & = & γ (m_{t} - m_{t - 1}) + (1 - γ) ϕ r_{t - 1} \\ s_{t} & = & β x_{t} / m_{t} + (1 - β) s_{t - p} \end{matrix}

where

m_{t}

is the mean,

r_{t}

is the trend and

s_{t}

is the seasonal component at time

t

with

p

being the seasonal order. The errors,

ε_{t}

are either drawn from a normal distribution with mean zero and variance

σ^{2}

or randomly sampled, with replacement, from a user-supplied vector.

5 Arguments

1: mode – Nag_InitialValuesInput

On entry: indicates if nag_rand_exp_smooth (g05pmc) is continuing from a previous call or, if not, how the initial values are computed.

$mode = Nag_InitialValuesSupplied$: Values for $m_{0}$ , $r_{0}$ and $s_{- j}$ , for $j = 0, 1, \dots, p - 1$ , are supplied in init.
$mode = Nag_ContinueNoUpdate$: nag_rand_exp_smooth (g05pmc) continues from a previous call using values that are supplied in r. r is not updated.
$mode = Nag_ContinueAndUpdate$: nag_rand_exp_smooth (g05pmc) continues from a previous call using values that are supplied in r. r is updated.

Constraint:

mode = Nag_InitialValuesSupplied

Nag_ContinueNoUpdate

Nag_ContinueAndUpdate

2: n – IntegerInput

On entry: the number of terms of the time series being generated.

Constraint:

n \geq 0

3: itype – Nag_ExpSmoothTypeInput

On entry: the smoothing function.

$itype = Nag_SingleExponential$: Single exponential.
$itype = Nag_BrownsExponential$: Brown's double exponential.
$itype = Nag_LinearHolt$: Linear Holt.
$itype = Nag_AdditiveHoltWinters$: Additive Holt–Winters.
$itype = Nag_MultiplicativeHoltWinters$: Multiplicative Holt–Winters.

Constraint:

itype = Nag_SingleExponential

Nag_BrownsExponential

Nag_LinearHolt

Nag_AdditiveHoltWinters

Nag_MultiplicativeHoltWinters

4: p – IntegerInput

On entry: if

itype = Nag_AdditiveHoltWinters

Nag_MultiplicativeHoltWinters

, the seasonal order,

p

, otherwise p is not referenced.

Constraint: if

itype = Nag_AdditiveHoltWinters

Nag_MultiplicativeHoltWinters

p > 1

5: param[

\dim

] – const doubleInput

Note: the dimension, dim, of the array param must be at least

$1$ when $itype = Nag_SingleExponential$ or $Nag_BrownsExponential$ ;
$3$ when $itype = Nag_LinearHolt$ ;
$4$ when $itype = Nag_AdditiveHoltWinters$ or $Nag_MultiplicativeHoltWinters$ .

On entry: the smoothing parameters.

itype = Nag_SingleExponential

Nag_BrownsExponential

param [0] = α

and any remaining elements of param are not referenced.

itype = Nag_LinearHolt

param [0] = α

param [1] = γ

param [2] = ϕ

and any remaining elements of param are not referenced.

itype = Nag_AdditiveHoltWinters

Nag_MultiplicativeHoltWinters

param [0] = α

param [1] = γ

param [2] = β

and

param [3] = ϕ

and any remaining elements of param are not referenced.

Constraints:

if $itype = Nag_SingleExponential$ , $0.0 \leq α \leq 1.0$ ;
if $itype = Nag_BrownsExponential$ , $0.0 < α \leq 1.0$ ;
if $itype = Nag_LinearHolt$ , $0.0 \leq α \leq 1.0$ and $0.0 \leq γ \leq 1.0$ and $ϕ \geq 0.0$ ;
if $itype = Nag_AdditiveHoltWinters$ or $Nag_MultiplicativeHoltWinters$ , $0.0 \leq α \leq 1.0$ and $0.0 \leq γ \leq 1.0$ and $0.0 \leq β \leq 1.0$ and $ϕ \geq 0.0$ .

6: init[

\dim

] – const doubleInput

Note: the dimension, dim, of the array init must be at least

$1$ when $itype = Nag_SingleExponential$ ;
$2$ when $itype = Nag_BrownsExponential$ or $Nag_LinearHolt$ ;
$2 + p$ when $itype = Nag_AdditiveHoltWinters$ or $Nag_MultiplicativeHoltWinters$ .

On entry: if

mode = Nag_InitialValuesSupplied

, the initial values for

m_{0}

r_{0}

and

s_{- j}

, for

j = 0, 1, \dots, p - 1

, used to initialize the smoothing.

itype = Nag_SingleExponential

init [0] = m_{0}

and any remaining elements of init are not referenced.

itype = Nag_BrownsExponential

Nag_LinearHolt

init [0] = m_{0}

and

init [1] = r_{0}

and any remaining elements of init are not referenced.

itype = Nag_AdditiveHoltWinters

Nag_MultiplicativeHoltWinters

init [0] = m_{0}

init [1] = r_{0}

and

init [2]

init [2 + p - 1]

hold the values for

s_{- j}

, for

j = 0, 1, \dots, p - 1

. Any remaining elements of init are not referenced.

7: var – doubleInput

On entry: the variance,

σ^{2}

of the Normal distribution used to generate the errors

ε_{i}

. If

var \leq 0.0

then Normally distributed errors are not used.

8: r[

\dim

] – doubleInput/Output

Note: the dimension, dim, of the array r must be at least

$13$ when $itype = Nag_SingleExponential$ , $Nag_BrownsExponential$ or $Nag_LinearHolt$ ;
$13 + p$ when $itype = Nag_AdditiveHoltWinters$ or $Nag_MultiplicativeHoltWinters$ .

On entry: if

mode = Nag_ContinueNoUpdate

Nag_ContinueAndUpdate

, r must contain the values as returned by a previous call to nag_rand_exp_smooth (g05pmc), r need not be set otherwise.

On exit: if

mode = Nag_ContinueNoUpdate

, r is unchanged. Otherwise, r contains the information on the current state of smoothing.

Constraint: if

mode = Nag_ContinueNoUpdate

Nag_ContinueAndUpdate

, r must have been initialized by at least one call to nag_rand_exp_smooth (g05pmc) or nag_tsa_exp_smooth (g13amc) with

mode \neq Nag_ContinueNoUpdate

, and r must not have been changed since that call.

9: state[

\dim

] – IntegerCommunication Array

Note: the actual argument supplied must be the array state supplied to the initialization functions nag_rand_init_repeatable (g05kfc) or nag_rand_init_nonrepeatable (g05kgc).

On entry: contains information on the selected base generator and its current state.

On exit: contains updated information on the state of the generator.

10: e[en] – const doubleInput

On entry: if

en > 0

and

var \leq 0.0

, a vector from which the errors,

ε_{t}

are randomly drawn, with replacement.

en \leq 0

, e is not referenced.

11: en – IntegerInput

On entry: if

en > 0

, then the length of the vector e.

If both

var \leq 0.0

and

en \leq 0

then

ε_{t} = 0.0

, for

t = 1, 2, \dots, n

12: x[n] – doubleOutput

On exit: the generated time series,

x_{t}

, for

t = 1, 2, \dots, n

13: 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_ENUM_INT

On entry,

itype = 〈value〉

and

p = 〈value〉

.
Constraint: if

itype = Nag_AdditiveHoltWinters

Nag_MultiplicativeHoltWinters

p > 1

NE_INT

On entry,

n = 〈value〉

.
Constraint:

n \geq 0

NE_INT_2

On entry,

p = 〈value〉

.
Constraint: if

itype = Nag_AdditiveHoltWinters

Nag_MultiplicativeHoltWinters

p \geq 2

NE_INT_ARRAY

On entry, some of the elements of the array r have been corrupted or have not been initialized.

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_INVALID_STATE

On entry, state vector has been corrupted or not initialized.

NE_REAL_ARRAY

Model unsuitable for multiplicative Holt–Winter, try a different set of parameters.

On entry,

param [〈value〉] = 〈value〉

.
Constraint:

0 \leq param [i] \leq 1

On entry,

param [〈value〉] = 〈value〉

.
Constraint: if

itype = Nag_BrownsExponential

0 < param [i] \leq 1

On entry,

param [〈value〉] = 〈value〉

.
Constraint:

param [i] \geq 0

NAG Library Function Document

nag_rand_exp_smooth (g05pmc)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

NAG Library Function Documentnag_rand_exp_smooth (g05pmc)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

NAG Library Function Document

nag_rand_exp_smooth (g05pmc)