NAG Library Routine Document

G05PMF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

+− 9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

1 Purpose

G05PMF simulates from an exponential smoothing model, where the model uses either single exponential, double exponential or a Holt–Winters method.

2 Specification

SUBROUTINE G05PMF (

MODE, N, ITYPE, P, PARAM, INIT, VAR, R, STATE, E, EN, X, IFAIL)

INTEGER	MODE, N, ITYPE, P, STATE(*), EN, IFAIL
REAL (KIND=nag_wp)	PARAM(), INIT(), VAR, R(*), E(EN), X(N)

3 Description

G05PMF returns

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

, a realisation 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 Parameters

1: MODE – INTEGERInput

On entry: indicates if G05PMF is continuing from a previous call or, if not, how the initial values are computed.

$MODE = 0$: Values for $m_{0}$ , $r_{0}$ and $s_{- j}$ , for $j = 0, 1, \dots, p - 1$ , are supplied in INIT.
$MODE = 1$: G05PMF continues from a previous call using values that are supplied in R. R is not updated.
$MODE = 2$: G05PMF continues from a previous call using values that are supplied in R. R is updated.

Constraint:

MODE = 0

1

2

2: N – INTEGERInput

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

Constraint:

N \geq 0

3: ITYPE – INTEGERInput

On entry: the smoothing function.

$ITYPE = 1$: Single exponential.
$ITYPE = 2$: Brown's double exponential.
$ITYPE = 3$: Linear Holt.
$ITYPE = 4$: Additive Holt–Winters.
$ITYPE = 5$: Multiplicative Holt–Winters.

Constraint:

ITYPE = 1

2

3

4

5

4: P – INTEGERInput

On entry: if

ITYPE = 4

5

, the seasonal order,

p

, otherwise P is not referenced.

Constraint: if

ITYPE = 4

5

P > 1

5: PARAM( $*$ ) – REAL (KIND=nag_wp) arrayInput

Note: the dimension of the array PARAM must be at least

1

ITYPE = 1

2

3

ITYPE = 3

and at least

4

ITYPE = 4

5

On entry: the smoothing parameters.

ITYPE = 1

2

PARAM (1) = α

and any remaining elements of PARAM are not referenced.

ITYPE = 3

PARAM (1) = α

PARAM (2) = γ

PARAM (3) = ϕ

and any remaining elements of PARAM are not referenced.

ITYPE = 4

5

PARAM (1) = α

PARAM (2) = γ

PARAM (3) = β

and

PARAM (4) = ϕ

and any remaining elements of PARAM are not referenced.

Constraints:

if $ITYPE = 1$ , $0.0 \leq α \leq 1.0$ ;
if $ITYPE = 2$ , $0.0 < α \leq 1.0$ ;
if $ITYPE = 3$ , $0.0 \leq α \leq 1.0$ and $0.0 \leq γ \leq 1.0$ and $ϕ \geq 0.0$ ;
if $ITYPE = 4$ or $5$ , $0.0 \leq α \leq 1.0$ and $0.0 \leq γ \leq 1.0$ and $0.0 \leq β \leq 1.0$ and $ϕ \geq 0.0$ .

6: INIT( $*$ ) – REAL (KIND=nag_wp) arrayInput

Note: the dimension of the array INIT must be at least

1

ITYPE = 1

2

ITYPE = 2

3

and at least

2 + P

ITYPE = 4

5

On entry: if

MODE = 0

, the initial values for

m_{0}

r_{0}

and

s_{- j}

, for

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

, used to initialize the smoothing.

ITYPE = 1

INIT (1) = m_{0}

and any remaining elements of INIT are not referenced.

ITYPE = 2

3

INIT (1) = m_{0}

and

INIT (2) = r_{0}

and any remaining elements of INIT are not referenced.

ITYPE = 4

5

INIT (1) = m_{0}

INIT (2) = r_{0}

and

INIT (3)

INIT (2 + p)

hold the values for

s_{- j}

, for

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

. Any remaining elements of INIT are not referenced.

7: VAR – REAL (KIND=nag_wp)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( $*$ ) – REAL (KIND=nag_wp) arrayInput/Output

Note: the dimension of the array R must be at least

13

ITYPE = 1

2

3

and at least

13 + P

ITYPE = 4

5

On entry: if

MODE = 1

2

, R must contain the values as returned by a previous call to G05PMF, R need not be set otherwise.

On exit: if

MODE = 1

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

Constraint: if

MODE = 1

2

, R must have been initialized by at least one call to G05PMF or G13AMF with

MODE \neq 1

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

9: STATE( $*$ ) – INTEGER arrayCommunication Array

Note: the actual argument supplied must be the array STATE supplied to the initialization routines G05KFF or G05KGF.

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) – REAL (KIND=nag_wp) arrayInput

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) – REAL (KIND=nag_wp) arrayOutput

On exit: the generated time series,

x_{t}

, for

t = 1, 2, \dots, n

13: IFAIL – INTEGERInput/Output

On entry: IFAIL must be set to

0

- 1 ​ or ​ 1

. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.

For environments where it might be inappropriate to halt program execution when an error is detected, the value

- 1 ​ or ​ 1

is recommended. If the output of error messages is undesirable, then the value

1

is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is

0

. When the value $- 1 or 1$ is used it is essential to test the value of IFAIL on exit.

On exit:

IFAIL = 0

unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry

IFAIL = 0

- 1

, explanatory error messages are output on the current error message unit (as defined by X04AAF).

Errors or warnings detected by the routine:

$IFAIL = 1$: On entry, $MODE \neq 0$ , $1$ or $2$ .

$IFAIL = 2$: On entry, $ITYPE \neq 1$ , $2$ , $3$ , $4$ or $5$ .

$IFAIL = 3$

On entry,

ITYPE = 4

5

and

P < 2

$IFAIL = 4$: On entry, at least one of $α$ , $β$ or $γ > 0.0$ or $> 1.0$ .

On entry, $ITYPE = 2$ and $α = 0.0$ .

On entry, $ϕ < 0.0$ .

$IFAIL = 5$

On entry,

N < 0

$IFAIL = 8$

On entry,

MODE = 1

2

and the array R has not been initialized correctly.

$IFAIL = 9$: On entry, the array STATE has not been initialized correctly.

$IFAIL = 12$: $ITYPE = 5$ and model is unsuitable for multiplicative Holt–Winter.

7 Accuracy

Not applicable.

8 Further Comments

None.

9 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 G13AMF.

G05PMF is then called multiple times to obtain simulated forecast confidence intervals.

NAG Library Routine DocumentG05PMF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

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 Routine Document

G05PMF