seasonal

Chapter 8: Time Series and Forecasting

seasonal_fit

Estimates the optimum seasonality parameters for a time series using an autoregressive model , AR(p), to represent the time series.

Synopsis

#include <imsls.h>

float * imsls_f_seasonal_fit(int n_obs, float z[], int maxlag,
int n_differences, int n_s_initial, int s_initial[],…,0)

The type double function is imsls_d_seasonal_fit.

Required Arguments

int n_obs (Input)
Number of observations in the time series.

float z[] (Input)
An array of length n_obs containing the time series. No missing values in the series are allowed.

int maxlag (Input)
The maximum lag allowed when fitting an AR(p) model.

int n_differences (Input)
The number of differences to perform. Argument n_differences must be greater than or equal to one.

int n_s_initial (Input)
The number of rows of the array containing the seasonal differences.

int s_initial[] (Input)
Array of dimension n_s_initial by n_differences containing the seasonal differences to test. All values of s_initial must be greater than or equal to one.

Return Value

Pointer to an array of length n_obs or n_obs-n_lost containing the optimum seasonally adjusted, autoregressive series. The first n_lost observations in this series are set to NaN, missing values. The seasonal adjustment is done by selecting optimum values for , (m=n_differences) and in the AR model:

where is the original time series,is the backward shift operator defined by , , is Gaussian white noise with and ,
,
with , and is a centering parameter for the differenced series.

NOTE that , the identity operator, i.e. .

If an error occurred, then NULL is returned.

Synopsis with Optional Arguments

#include <imsls.h>

float * imsls_f_seasonal_fit (int n_obs, float z[], int maxlag,
int n_differences, int n_s_initial, int s_initial[],
IMSLS_RETURN_USER, float w[],
IMSLS_D_INITIAL, int n_d_initial, int d_initial[],
IMSLS_SET_FIRST_TO_NAN, or IMSLS_EXCLUDE_FIRST,
IMSLS_CENTER, int n_center,
IMSLS_LOST, int *n_lost,
IMSLS_BEST_PERIODS, int **s,
IMSLS_BEST_PERIODS_USER, int s[],
IMSLS_BEST_ORDERS, int **d,
IMSLS_BEST_ORDERS_USER, int d[],
IMSLS_AR_ORDER, int *p,
IMSLS_AIC, float *aic,
0)

Optional Arguments

IMSLS_RETURN_USER, float w[] (Output)
An array of length n_obs supplied by the user to hold the seasonally adjusted series returned by imsls_f_seasonal_fit.

IMSLS_D_INITIAL, int n_d_initial, int d_initial[] (Input)
An array of dimension n_d_initial by n_differences containing the candidate values for d[], from which the optimum is being selected. All candidate values in d_initial[] must be non-negative and
n_d_initial ³ 1.
Default: n_d_initial=1, d_initial an array of length n_differences filled with ones.

IMSLS_SET_FIRST_TO_NAN, or IMSLS_EXCLUDE_FIRST (Input)
If IMSLS_EXCLUDE_FIRST is specified, the first n_lost values are excluded from w due to differencing. The differenced series w is of length
n_obs–n_lost. If IMSLS_SET_FIRST_TO_NAN is specified, the first n_lost observations are set to NaN (Not a Number).
Default: IMSLS_SET_FIRST_TO_NAN.

IMSLS_CENTER, int n_center (Input)
If supplied, IMSLS_CENTER controls the method used to center the differenced series. If n_center=0 then the series is not centered. If n_center=1, the mean of the series is used to center the data, and if n_center=2, the median is used.
Default: n_center=1.

IMSLS_LOST, int *n_lost (Output)
The number of observations lost due to differencing the time series. This is also equal to the number of NaN values that appear in the first n_lost locations of the returned seasonally adjusted series.

IMSLS_BEST_PERIODS, int **s (Output)
Address of a pointer to an internally allocated array of length m=n_differences containing the optimum values for the seasonal adjustment parameters selected from the list of candidates contained in s_initial[].

IMSLS_BEST_PERIODS_USER, int s[] (Output)
A user supplied array of length n_differences for storage of the array s.

IMSLS_BEST_ORDERS, int **d (Output)
Address of a pointer to an internally allocated array of length m=n_differences containing the optimum values for the seasonal adjustment parameters selected from the list of candidates contained in d_initial[].

IMSLS_BEST_ORDERS_USER, int d[] (Output)
A user supplied array of length n_differences for storage of the array d.

IMSLS_AR_ORDER, int *p (Output)
The optimum value for the autoregressive lag.

IMSLS_AIC, float *aic (Output)
Akaike’s Information Criterion (AIC) for the optimum seasonally adjusted model.

Description

Many time series contain seasonal trends and cycles that can be modeled by first differencing the series. For example, if the correlation is strong from one period to the next, the series might be differenced by a lag of 1. Instead of fitting a model to the series , the model is fitted to the transformed series: . Higher order lags or differences are warranted if the series has a cycle every 4 or 13 weeks.

Function imsls_f_seasonal_fit does not center the original series. If IMSLS_CENTER is specified with either n_center =1 or n_center =2, then the differenced series, , is centered before determination of minimum AIC and optimum lag. For every combination of rows in s_initial and d_initial, the series is converted to the seasonally adjusted series using the following computation

.

where , represent specific rows of arrays s_initial and d_initial respectively, and =n_differences.

This transformation of the series to is accomplished using function imsls_f_difference(). After this transformation,

is (optionally) centered and a call is made to imsls_f_auto_uni_ar to automatically determine the optimum lag for an AR(p) representation for . This procedure is repeated for every possible combination of rows of s_initial and d_initial. The series with the minimum AIC is identified as the optimum representation and returned.

Example

Consider the Airline Data (Box, Jenkins and Reinsel 1994, p. 547) consisting of the monthly total number of international airline passengers from January 1949 through December 1960. Function imsls_f_seasonal_fit is used to compute the optimum seasonality representation of the adjusted series

where

and

As differenced series with minimum AIC,

is identified.

#include <imsls.h>

#include <stdlib.h>

#include <stdio.h>

void main()

{

int i;

int maxlag = 10;

int nobs = 144;

int n_differences = 2;

int n_s_initial = 2;

int nlost;

int npar;

float aic;

int s_init[] = { 1, 1,

1, 12};

int *s = NULL;

int *d = NULL;

float *z = NULL;

float *difference = NULL;

z = imsls_f_data_sets(4, 0);

difference = imsls_f_seasonal_fit(nobs, z, maxlag, n_differences,

n_s_initial, s_init,

IMSLS_LOST, &nlost,

IMSLS_BEST_PERIODS, &s,

IMSLS_BEST_ORDERS, &d,

IMSLS_AIC, &aic,

IMSLS_AR_ORDER, &npar,

0);

printf("\nnlost = %d\n", nlost);

printf("s = (%d, %d)\n", s[0], s[1]);

printf("d = (%d, %d)\n", d[0], d[1]);

printf("Order of optimum AR process: %d\n", npar);

printf("aic = %lf\n", aic);

printf("\ni\tz[i]\tdifference[i]\n");

for (i=0; i<nobs; i++)

printf("%d\t%f\t%f\n", i, z[i], difference[i]);

if (s)

{

free(s);

s = NULL;

}

if (d)

{

free(d);

d = NULL;

}

if (z)

{

free(z);

z = NULL;

}

if (difference)

{

free(difference);

difference = NULL;

}

return;

}

Output

nlost = 13

s = (1, 12)

d = (1, 1)

Order of optimum AR process: 1

aic = 829.780334

i z[i] difference[i]

0 112.000000 NaN

1 118.000000 NaN

2 132.000000 NaN

3 129.000000 NaN

4 121.000000 NaN

5 135.000000 NaN

6 148.000000 NaN

7 148.000000 NaN

8 136.000000 NaN

9 119.000000 NaN

10 104.000000 NaN

11 118.000000 NaN

12 115.000000 NaN

13 126.000000 5.000000

14 141.000000 1.000000

15 135.000000 -3.000000

16 125.000000 -2.000000

17 149.000000 10.000000

18 170.000000 8.000000

19 170.000000 0.000000

20 158.000000 0.000000

21 133.000000 -8.000000

22 114.000000 -4.000000

23 140.000000 12.000000

24 145.000000 8.000000

25 150.000000 -6.000000

26 178.000000 13.000000

27 163.000000 -9.000000

28 172.000000 19.000000

29 178.000000 -18.000000

30 199.000000 0.000000

31 199.000000 0.000000

32 184.000000 -3.000000

33 162.000000 3.000000

34 146.000000 3.000000

35 166.000000 -6.000000

36 171.000000 0.000000

37 180.000000 4.000000

38 193.000000 -15.000000

39 181.000000 3.000000

40 183.000000 -7.000000

41 218.000000 29.000000

42 230.000000 -9.000000

43 242.000000 12.000000

44 209.000000 -18.000000

45 191.000000 4.000000

46 172.000000 -3.000000

47 194.000000 2.000000

48 196.000000 -3.000000

49 196.000000 -9.000000

50 236.000000 27.000000

51 235.000000 11.000000

52 229.000000 -8.000000

53 243.000000 -21.000000

54 264.000000 9.000000

55 272.000000 -4.000000

56 237.000000 -2.000000

57 211.000000 -8.000000

58 180.000000 -12.000000

59 201.000000 -1.000000

60 204.000000 1.000000

61 188.000000 -16.000000

62 235.000000 7.000000

63 227.000000 -7.000000

64 234.000000 13.000000

65 264.000000 16.000000

66 302.000000 17.000000

67 293.000000 -17.000000

68 259.000000 1.000000

69 229.000000 -4.000000

70 203.000000 5.000000

71 229.000000 5.000000

72 242.000000 10.000000

73 233.000000 7.000000

74 267.000000 -13.000000

75 269.000000 10.000000

76 270.000000 -6.000000

77 315.000000 15.000000

78 364.000000 11.000000

79 347.000000 -8.000000

80 312.000000 -1.000000

81 274.000000 -8.000000

82 237.000000 -11.000000

83 278.000000 15.000000

84 284.000000 -7.000000

85 277.000000 2.000000

86 317.000000 6.000000

87 313.000000 -6.000000

88 318.000000 4.000000

89 374.000000 11.000000

90 413.000000 -10.000000

91 405.000000 9.000000

92 355.000000 -15.000000

93 306.000000 -11.000000

94 271.000000 2.000000

95 306.000000 -6.000000

96 315.000000 3.000000

97 301.000000 -7.000000

98 356.000000 15.000000

99 348.000000 -4.000000

100 355.000000 2.000000

101 422.000000 11.000000

102 465.000000 4.000000

103 467.000000 10.000000

104 404.000000 -13.000000

105 347.000000 -8.000000

106 305.000000 -7.000000

107 336.000000 -4.000000

108 340.000000 -5.000000

109 318.000000 -8.000000

110 362.000000 -11.000000

111 348.000000 -6.000000

112 363.000000 8.000000

113 435.000000 5.000000

114 491.000000 13.000000

115 505.000000 12.000000

116 404.000000 -38.000000

117 359.000000 12.000000

118 310.000000 -7.000000

119 337.000000 -4.000000

120 360.000000 19.000000

121 342.000000 4.000000

122 406.000000 20.000000

123 396.000000 4.000000

124 420.000000 9.000000

125 472.000000 -20.000000

126 548.000000 20.000000

127 559.000000 -3.000000

128 463.000000 5.000000

129 407.000000 -11.000000

130 362.000000 4.000000

131 405.000000 16.000000

132 417.000000 -11.000000

133 391.000000 -8.000000

134 419.000000 -36.000000

135 461.000000 52.000000

136 472.000000 -13.000000

137 535.000000 11.000000

138 622.000000 11.000000

139 606.000000 -27.000000

140 508.000000 -2.000000

141 461.000000 9.000000

142 390.000000 -26.000000

143 432.000000 -1.000000

Visual Numerics, Inc.
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http://www.vni.com/
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