g05pmc simulates from an exponential smoothing model, where the model uses either single exponential, double exponential or a Holt–Winters method.

2 Specification

#include <nag.h>

void	g05pmc (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)

The function may be called by the names: g05pmc, nag_rand_times_smooth_exp or nag_rand_exp_smooth.

3 Description

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.

4 References

Chatfield C (1980) The Analysis of Time Series Chapman and Hall

5 Arguments

1: $mode$ – Nag_InitialValues Input

On entry: indicates if 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.
$mode = Nag_ContinueNoUpdate$: g05pmc continues from a previous call using values that are supplied in r. r is not updated.
$mode = Nag_ContinueAndUpdate$: 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: $n$ – Integer Input

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

Constraint:

n \geq 0

3: $itype$ – Nag_ExpSmoothType Input

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: $p$ – Integer Input

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 double Input

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 double Input

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: $var$ – double Input

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: $r [\dim]$ – double Communication Array

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 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 g05pmc or g13amc with

mode \neq Nag_ContinueNoUpdate

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

9: $state [\dim]$ – Integer Communication 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 double Input

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: $en$ – Integer Input

On entry: if

en > 0

, 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: $x [n]$ – double Output

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 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
NE_BAD_PARAM: On entry, argument $⟨ value ⟩$ had an illegal value.
NE_ENUM_INT: On entry, $p = ⟨ value ⟩$ .
Constraint: if $itype = Nag_AdditiveHoltWinters$ or $Nag_MultiplicativeHoltWinters$ , $p \geq 2$ .
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.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_INVALID_STATE: On entry, state vector has been corrupted or not initialized.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
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: $param [i] \geq 0$ .

7 Accuracy

Not applicable.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.

g05pmc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9 Further Comments

None.

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 g13amc.

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

g05pm: FL CL CPP AD PY MB

NAG CL Interfaceg05pmc (times_​smooth_​exp)

▸▿ 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 CL Interface
g05pmc (times_smooth_exp)