SEASONAL_FIT Function
Estimates the optimum seasonality parameters for a time series using an autoregressive model, AR(p), to represent the time series.
Usage
result = SEASONAL_FIT(z, maxlag, s_initial)
Input Parameters
z—Array of length n_obs containing the time series, where n_obs is the number of observations in the time series. No missing values in the series are allowed.
maxlag—The maximum lag allowed when fitting an AR(p) model.
s_initial—A scalar, 1D, or 2D array of dimension n_s_initial by n_differences containing the seasonal differences to test. n_s_initial is the number of rows of the array containing the seasonal differences and n_differences is the number of differences to perform. n_differences must be greater than or equal to one. All values of s_initial must be greater than or equal to one. If s_initial is a 1D array, n_differences is assumed to be 1. A scalar s_initial value implies n_s_initial = n_differences = 1.
Returned Value
result—Array of length n_obs, or n_obs – N_lost if the Exclude_first keyword is set, 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 d1, ..., dm, s1, ..., sm (m = n_differences) and p in the AR model:
where:
{
Zt} is the original time series
B is the backward shift operator defined by
, k
≥ 0
at is Gaussian white noise with
E(
at) = 0
VAR(
at)
=
σ 2,
, 0
≤ p ≤ maxlag , with s > 0, d ≥ 0
, and μ is a centering parameter for the differenced series.
Note that
, the identity operator, i.e.
. Input Keywords
Double—If present and nonzero, double precision is used.
D_initial—A scalar, 1D array, or 2D 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 ≥ 1. Default: N_d_initial = 1, D_initial an array of length n_differences filled with ones.
Exclude_first—If Exclude_first is specified, the first N_lost values are excluded from the returned value due to differencing. The differenced series, the returned value, is of length n_obs – N_lost. By default, the first N_lost observations are set to NaN (Not a Number).
Center—If supplied, controls the method used to center the differenced series. If Center = 0 then the series is not centered. If Center = 1, the mean of the series is used to center the data, and if Center = 2, the median is used. Default: Center = 1.
Output Keywords
N_lost—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 when Set_first_to_nan is set or the number of values excluded if Exclude_first is set.
Best_periods—Array of length m = n_differences containing the optimum values for the seasonal adjustment parameters s1, s2, ..., sm selected from the list of candidates contained in s_initial.
Best_orders—Array of length m = n_differences containing the optimum values for the seasonal adjustment parameters d1, d2, ..., dm selected from the list of candidates contained in D_initial.
Ar_order—The optimum value for the autoregressive lag.
Aic—Akaike’s Information Criterion (AIC) for the optimum seasonally adjusted model.
Discussion
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 Zt, the model is fitted to the transformed series: Wt = Zt – Zt–1. Higher order lags or differences are warranted if the series has a cycle every 4 or 13 weeks.
SEASONAL_FIT does not center the original series. If Center = 1 or Center = 2, then the differenced series, Wt, is centered before determination of minimum AIC and optimum lag. For every combination of rows in s_initial and D_initial, the series Zt is converted to the seasonally adjusted series using the following computation:
where
,
represent specific rows of arrays
s_initial and
D_initial respectively, and
m =
n_differences.
This transformation of the series Zt to Wt(s, d) is accomplished using DIFFERENCE. After this transformation:
Wt(s, d)
is (optionally) centered and a call is made to AUTO_UNI_AR to automatically determine the optimum lag for an AR(p) representation for Wt(s, d). 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. SEASONAL_FIT is used to compute the optimum seasonality representation of the adjusted series:
where:
s = (1,1)
or:
s = (1,12)
and:
d = (1,1)
As differenced series with minimum AIC:
is identified.
maxlag = 10
nobs = 144
s_init = TRANSPOSE([[1, 1], [1, 12]])
z = statdata (4)
difference = SEASONAL_FIT(z, maxlag, s_init, $
N_Lost=nlost, $
Best_Periods=s, $
Best_Orders=d, $
Aic=aic, $
Ar_Order=npar)
PRINT, nlost, Format='("nlost = ", I2)'
PRINT, s(0), s(1), Format='("s = (", I1, ", ", I2, ")")'
PRINT, d(0), d(1), Format='("d = (", I1, ", ", I2, ")")'
PRINT, npar, Format='("Order of optimum AR process:", I2)'
PRINT, aic, Format='("aic = ", F7.3)'
PRINT, ''
PRINT, " i z[i] difference[i]"
FOR i=0L, nobs-1 DO $
PRINT, i, z(i), difference(i), Format='(I4, F15.4, F15.4)'
Output
nlost = 13
s = (1, 12)
d = (1, 1)
Order of optimum AR process: 1
aic = 829.780
i z[i] difference[i]
0 112.0000 NaN
1 118.0000 NaN
2 132.0000 NaN
3 129.0000 NaN
4 121.0000 NaN
5 135.0000 NaN
6 148.0000 NaN
7 148.0000 NaN
8 136.0000 NaN
9 119.0000 NaN
10 104.0000 NaN
11 118.0000 NaN
12 115.0000 NaN
13 126.0000 5.0000
14 141.0000 1.0000
15 135.0000 -3.0000
16 125.0000 -2.0000
17 149.0000 10.0000
18 170.0000 8.0000
19 170.0000 0.0000
20 158.0000 0.0000
21 133.0000 -8.0000
22 114.0000 -4.0000
23 140.0000 12.0000
24 145.0000 8.0000
25 150.0000 -6.0000
26 178.0000 13.0000
27 163.0000 -9.0000
28 172.0000 19.0000
29 178.0000 -18.0000
30 199.0000 0.0000
31 199.0000 0.0000
32 184.0000 -3.0000
33 162.0000 3.0000
34 146.0000 3.0000
35 166.0000 -6.0000
36 171.0000 0.0000
37 180.0000 4.0000
38 193.0000 -15.0000
39 181.0000 3.0000
40 183.0000 -7.0000
41 218.0000 29.0000
42 230.0000 -9.0000
43 242.0000 12.0000
44 209.0000 -18.0000
45 191.0000 4.0000
46 172.0000 -3.0000
47 194.0000 2.0000
48 196.0000 -3.0000
49 196.0000 -9.0000
50 236.0000 27.0000
51 235.0000 11.0000
52 229.0000 -8.0000
53 243.0000 -21.0000
54 264.0000 9.0000
55 272.0000 -4.0000
56 237.0000 -2.0000
57 211.0000 -8.0000
58 180.0000 -12.0000
59 201.0000 -1.0000
60 204.0000 1.0000
61 188.0000 -16.0000
62 235.0000 7.0000
63 227.0000 -7.0000
64 234.0000 13.0000
65 264.0000 16.0000
66 302.0000 17.0000
67 293.0000 -17.0000
68 259.0000 1.0000
69 229.0000 -4.0000
70 203.0000 5.0000
71 229.0000 5.0000
72 242.0000 10.0000
73 233.0000 7.0000
74 267.0000 -13.0000
75 269.0000 10.0000
76 270.0000 -6.0000
77 315.0000 15.0000
78 364.0000 11.0000
79 347.0000 -8.0000
80 312.0000 -1.0000
81 274.0000 -8.0000
82 237.0000 -11.0000
83 278.0000 15.0000
84 284.0000 -7.0000
85 277.0000 2.0000
86 317.0000 6.0000
87 313.0000 -6.0000
88 318.0000 4.0000
89 374.0000 11.0000
90 413.0000 -10.0000
91 405.0000 9.0000
92 355.0000 -15.0000
93 306.0000 -11.0000
94 271.0000 2.0000
95 306.0000 -6.0000
96 315.0000 3.0000
97 301.0000 -7.0000
98 356.0000 15.0000
99 348.0000 -4.0000
100 355.0000 2.0000
101 422.0000 11.0000
102 465.0000 4.0000
103 467.0000 10.0000
104 404.0000 -13.0000
105 347.0000 -8.0000
106 305.0000 -7.0000
107 336.0000 -4.0000
108 340.0000 -5.0000
109 318.0000 -8.0000
110 362.0000 -11.0000
111 348.0000 -6.0000
112 363.0000 8.0000
113 435.0000 5.0000
114 491.0000 13.0000
115 505.0000 12.0000
116 404.0000 -38.0000
117 359.0000 12.0000
118 310.0000 -7.0000
119 337.0000 -4.0000
120 360.0000 19.0000
121 342.0000 4.0000
122 406.0000 20.0000
123 396.0000 4.0000
124 420.0000 9.0000
125 472.0000 -20.0000
126 548.0000 20.0000
127 559.0000 -3.0000
128 463.0000 5.0000
129 407.0000 -11.0000
130 362.0000 4.0000
131 405.0000 16.0000
132 417.0000 -11.0000
133 391.0000 -8.0000
134 419.0000 -36.0000
135 461.0000 52.0000
136 472.0000 -13.0000
137 535.0000 11.0000
138 622.0000 11.0000
139 606.0000 -27.0000
140 508.0000 -2.0000
141 461.0000 9.0000
142 390.0000 -26.0000
143 432.0000 -1.0000