NAG Library Routine Document

G13AMF

Exponential smoothing is a relatively simple method of short term forecasting for a time series. G13AMF provides five types of exponential smoothing; single exponential, Brown's double exponential, linear Holt (also called double exponential smoothing in some references), additive Holt–Winters and multiplicative Holt–Winters. The choice of smoothing method used depends on the characteristics of the time series. If the mean of the series is only slowly changing then single exponential smoothing may be suitable. If there is a trend in the time series, which itself may be slowly changing, then double exponential smoothing may be suitable. If there is a seasonal component to the time series, e.g., daily or monthly data, then one of the two Holt–Winters methods may be suitable.

For a time series

y_{t}

, for

t = 1, 2, \dots, n

, the five smoothing functions are defined by the following:

Single Exponential Smoothing
$\begin{matrix} m_{t} & = & α y_{t} + (1 - α) m_{t - 1} \\ {\hat{y}}_{t + f} & = & m_{t} \\ var ({\hat{y}}_{t + f}) & = & var (ε_{t}) (1 + (f - 1) α^{2}) \end{matrix}$

Brown Double Exponential Smoothing

\begin{matrix} m_{t} & = & α y_{t} + (1 - α) m_{t - 1} \\ r_{t} & = & α (m_{t} - m_{t - 1}) + (1 - α) r_{t - 1} \\ {\hat{y}}_{t + f} & = & m_{t} + ((f - 1) + 1 / α) r_{t} \\ var ({\hat{y}}_{t + f}) & = & var (ε_{t}) (1 + \sum_{i = 1}^{f - 1} {(2 α + (i - 1) α^{2})}^{2}) \end{matrix}

Linear Holt Smoothing

\begin{matrix} m_{t} & = & α y_{t} + (1 - α) (m_{t - 1} + ϕ r_{t - 1}) \\ r_{t} & = & γ (m_{t} - m_{t - 1}) + (1 - γ) ϕ r_{t - 1} \\ {\hat{y}}_{t + f} & = & m_{t} + \sum_{i = 1}^{f} ϕ^{i} r_{t} \\ var ({\hat{y}}_{t + f}) & = & var (ε_{t}) (1 + \sum_{i = 1}^{f - 1} {(α + \frac{α γ ϕ (ϕ^{i} - 1)}{(ϕ - 1)})}^{2}) \end{matrix}

Additive Holt–Winters Smoothing

\begin{matrix} m_{t} & = & α (y_{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 (y_{t} - m_{t}) + (1 - β) s_{t - p} \\ {\hat{y}}_{t + f} & = & m_{t} + (\sum_{i = 1}^{f} ϕ^{i} r_{t}) + s_{t - p} \\ var ({\hat{y}}_{t + f}) & = & var (ε_{t}) (1 + \sum_{i = 1}^{f - 1} ψ_{i}^{2}) \\ ψ_{i} & = & \{\begin{matrix} 0 & if ​ i \geq f \\ α + \frac{α γ ϕ (ϕ^{i} - 1)}{(ϕ - 1)} & if ​ i mod p \neq 0 \\ α + \frac{α γ ϕ (ϕ^{i} - 1)}{(ϕ - 1)} + β (1 - α) & otherwise \end{matrix} \end{matrix}

Multiplicative Holt–Winters Smoothing

\begin{matrix} m_{t} & = & α y_{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 y_{t} / m_{t} + (1 - β) s_{t - p} \\ {\hat{y}}_{t + f} & = & (m_{t} + \sum_{i = 1}^{f} ϕ^{i} r_{t}) \times s_{t - p} \\ var ({\hat{y}}_{t + f}) & = & var (ε_{t}) (\sum_{i = 0}^{\infty} \sum_{j = 0}^{p - 1} {(ψ_{j + i p} \frac{s_{t + f}}{s_{t + f - j}})}^{2}) \end{matrix}

and

ψ

is defined as in the additive Holt–Winters smoothing,

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

f

-step ahead forecasts are given by

{\hat{y}}_{t + f}

and their variances by

var ({\hat{y}}_{t + f})

. The term

var (ε_{t})

is estimated as the mean deviation.

The parameters,

α

β

and

γ

control the amount of smoothing. The nearer these parameters are to one, the greater the emphasis on the current data point. Generally these parameters take values in the range

0.1

0.3

. The linear Holt and two Holt–Winters smoothers include an additional parameter,

ϕ

, which acts as a trend dampener. For

0.0 < ϕ < 1.0

the trend is dampened and for

ϕ > 1.0

the forecast function has an exponential trend,

ϕ = 0.0

removes the trend term from the forecast function and

ϕ = 1.0

does not dampen the trend.

For all methods, values for

α

β

γ

and

ψ

can be chosen by trying different values and then visually comparing the results by plotting the fitted values along side the original data. Alternatively, for single exponential smoothing a suitable value for

α

can be obtained by fitting an

ARIMA (0, 1, 1)

model (see G13BEF). For Brown's double exponential smoothing and linear Holt smoothing with no dampening, (i.e.,

ϕ = 1.0

), suitable values for

α

and

γ

can be obtained by fitting an

ARIMA (0, 2, 2)

model. Similarly, the linear Holt method, with

ϕ \neq 1.0

, can be expressed as an

ARIMA (1, 2, 2)

model and the additive Holt–Winters, with no dampening, (

ϕ = 1.0

), can be expressed as a seasonal ARIMA model with order

p

of the form

ARIMA (0, 1, p + 1) (0, 1, 0)

. There is no similar procedure for obtaining parameter values for the multiplicative Holt–Winters method, or the additive Holt–Winters method with

ϕ \neq 1.0

. In these cases parameters could be selected by minimizing a measure of fit using one of the nonlinear optimization routines in Chapter E04.

In addition to values for

α

β

γ

and

ψ

, initial values,

m_{0}

r_{0}

and

s_{- j}

, for

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

, are required to start the smoothing process. You can either supply these or they can be calculated by G13AMF from the first

k

observations. For single exponential smoothing the mean of the observations is used to estimate

m_{0}

. For Brown double exponential smoothing and linear Holt smoothing, a simple linear regression is carried out with the series as the dependent variable and the sequence

1, 2, \dots, k

as the independent variable. The intercept is then used to estimate

m_{0}

and the slope to estimate

r_{0}

. In the case of the additive Holt–Winters method, the same regression is carried out, but a separate intercept is used for each of the

p

seasonal groupings. The slope gives an estimate for

r_{0}

and the mean of the

p

intercepts is used as the estimate of

m_{0}

. The seasonal parameters

s_{- j}

, for

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

, are estimated as the

p

intercepts –

m_{0}

. A similar approach is adopted for the multiplicative Holt–Winter's method.

One step ahead forecasts,

{\hat{y}}_{t + 1}

are supplied along with the residuals computed as

(y_{t + 1} - {\hat{y}}_{t + 1})

. In addition, two measures of fit are provided. The mean absolute deviation,

\frac{1}{n} \sum_{t = 1}^{n} |y_{t} - {\hat{y}}_{t}|

and the square root of the mean deviation

\sqrt{\frac{1}{n} \sum_{t = 1}^{n} {(y_{t} - {\hat{y}}_{t})}^{2}} .

4 References

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

5 Parameters

1: MODE – INTEGERInput

On entry: indicates if G13AMF is continuing from a previous call or, if not, how the initial values are computed.

$MODE = 0$: Required values for $m_{0}$ , $r_{0}$ and $s_{- j}$ , for $j = 0, 1, \dots, p - 1$ , are supplied in INIT.
$MODE = 1$: G13AMF continues from a previous call using values that are supplied in R.
$MODE = 2$: Required values for $m_{0}$ , $r_{0}$ and $s_{- j}$ , for $j = 0, 1, \dots, p - 1$ , are estimated using the first $k$ observations.

Constraint:

MODE = 0

1

2

2: ITYPE – INTEGERInput

On entry: the smoothing function.

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

Constraint:

ITYPE = 1

2

3

4

5

3: P – INTEGERInput

On entry: if

ITYPE = 4

5

, the seasonal order,

p

, otherwise P is not referenced.

Constraint: if

ITYPE = 4

5

P > 1

4: 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

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 $ϕ \geq 0.0$ .

5: N – INTEGERInput

On entry: the number of observations in the series.

Constraint:

N \geq 0

6: Y(N) – REAL (KIND=nag_wp) arrayInput

On entry: the time series.

7: K – INTEGERInput

On entry: if

MODE = 2

, the number of observations used to initialize the smoothing.

MODE \neq 2

, K is not referenced.

Constraints:

if $MODE = 2$ and $ITYPE = 4$ or $5$ , $2 \times P \leq K \leq N$ ;
if $MODE = 2$ and $ITYPE = 1$ , $2$ or $3$ , $1 \leq K \leq N$ .

8: INIT( $*$ ) – REAL (KIND=nag_wp) arrayInput/Output

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 the remaining elements of INIT are not referenced.

ITYPE = 2

3

INIT (1) = m_{0}

and

INIT (2) = r_{0}

and the remaining elements of INIT are not referenced.

ITYPE = 4

5

INIT (1) = m_{0}

INIT (2) = r_{0}

and

INIT (3)

INIT (p + 2)

hold the values for

s_{- j}

, for

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

. The remaining elements of INIT are not referenced.

On exit: if

MODE \neq 1

, the values used to initialize the smoothing. These are in the same order as described above.

9: NF – INTEGERInput

On entry: the number of forecasts required beyond the end of the series. Note, the one step ahead forecast is always produced.

Constraint:

NF \geq 0

10: FV(NF) – REAL (KIND=nag_wp) arrayOutput

On exit:

{\hat{y}}_{t + f}

, for

f = 1, 2, \dots, NF

, the next NF step forecasts. Where

t = N

, if

MODE \neq 1

, else

t

is the total number of smoothed and forecast values already produced.

11: FSE(NF) – REAL (KIND=nag_wp) arrayOutput

On exit: the forecast standard errors for the values given in FV.

12: YHAT(N) – REAL (KIND=nag_wp) arrayOutput

On exit:

{\hat{y}}_{t + 1}

, for

t = 1, 2, \dots, N

, the one step ahead forecast values, with

YHAT (i)

being the one step ahead forecast of

Y (i - 1)

13: RES(N) – REAL (KIND=nag_wp) arrayOutput

On exit: the residuals,

(y_{t + 1} - {\hat{y}}_{t + 1})

, for

t = 1, 2, \dots, N

14: DV – REAL (KIND=nag_wp)Output

On exit: the square root of the mean deviation.

15: AD – REAL (KIND=nag_wp)Output

On exit: the mean absolute deviation.

16: 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

, R must contain the values as returned by a previous call to G05PMF or G13AMF, R need not be set otherwise.

ITYPE = 1

2

3

, only the first

13

elements of R are referenced, otherwise the first

13 + p

elements are referenced.

On exit: the information on the current state of the smoothing.

Constraint: if

MODE = 1

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

MODE \neq 1

, and R should not have been changed since the last call to G05PMF or G13AMF.

17: 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,	$MODE < 0$ ,
or	$MODE > 2$ .

$IFAIL = 2$

On entry,	$ITYPE < 1$ ,
or	$ITYPE > 5$ .

$IFAIL = 3$

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

$IFAIL = 6$: A multiplicative Holt–Winters model cannot be used with the supplied data.

$IFAIL = 7$: On entry, $MODE = 2$ and $K < 1$ or $K > N$ .

On entry, $MODE = 2$ , $ITYPE = 4$ or $5$ and $K < 2 \times P$ .

$IFAIL = 9$

On entry,

NF < 0

$IFAIL = 16$: On entry, $MODE = 1$ and the array R has not been initialized correctly.

7 Accuracy

Not applicable.

8 Further Comments

Single exponential, Brown's double exponential and linear Holt smoothing methods are stable, whereas the two Holt–Winters methods can be affected by poor initial values for the seasonal components.

See also the routine document for G05PMF.

9 Example

This example smooths a time series relating to the rate of the earth's rotation about its polar axis.

NAG Library Routine DocumentG13AMF

+− 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

G13AMF