nag_rand_exp_smooth (g05pmc) : NAG Library, Mark 24

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 dimension,

\dim

, of this array is dictated by the requirements of associated functions that must have been previously called. This array MUST be the same array passed as argument state in the previous call to 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

On entry,

p = ⟨value⟩

.
Constraint: if

itype = Nag_AdditiveHoltWinters

Nag_MultiplicativeHoltWinters

p \geq 2

NE_INT

On entry,

n = ⟨value⟩

.
Constraint:

n \geq 0

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

10 Example

This example reads

11

observations from a time series relating to the rate of the earth's rotation about its polar axis and fits an exponential smoothing model using nag_tsa_exp_smooth (g13amc).

nag_rand_exp_smooth (g05pmc) is then called multiple times to obtain simulated forecast confidence intervals.

10.1 Program Text

10.2 Program Data

10.3 Program Results

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 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 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 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_rand_exp_smooth (g05pmc)