SEASONAL_FIT Function

PV-WAVE Advantage > IMSL Statistics Reference Guide > Time Series and Forecasting > 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