Decomposes a time series into trend, seasonal, and an error component.

The type double function is imsls_d_bayesian_seasonal_adj.

The average Akaike Bayesian Information Criterion for the estimated model. NaN is returned on error.

Function imsls_f_bayesian_seasonal_adj is based upon the algorithm published by Akaike (1980). This algorithm uses a Bayesian approach to the problem of fitting the following autoregressive model for a time series Wt decomposed into a trend and a seasonal component.

Adopting the notation described earlier in the Usage Notes section of this chapter, if

then a seasonal autoregressive model can be represented by the following relationship:

where Wt is the stationary time series with mean μ, Tt denotes an underlying trend, St denotes a seasonal component and At denotes a noise or irregular component.

A non-Bayesian approach to this problem would be to estimate the trend and seasonal components by minimizing

where the difference operator is denoted by ∇ and defined as ∇Tt = (Tt - Tt-1), for k ≥ 1, ∇kTt = ∇(∇k-1Tt), with ∇0Tt =  Tt. Similarly, ∇pSt = (St -  St-p) and ∇lpSt = ∇p(∇pl-1St), l ≥ 1, with ∇0pSt = St.

The period of the seasonal component, p, the trend order, k, and the seasonal order l correspond to parameter options IMSLS_PERIOD, IMSLS_TREND_ORDER, and IMSLS_SEASONAL_ORDER respectively. d, z, and r are constants determined as follows.

In imsls_f_bayesian_seasonal_adj, the approach is to select the parameter d, which controls the smoothness of the trend and seasonality estimates, using Bayesian methods. The prior distribution controls the smoothness of the trend and seasonal components by assuming low-order Gaussian autoregressive models for some differences of these components. The choice of the variance of the Gaussian distribution is realized by maximizing the log likelihood of the Bayesian model.

The other smoothing parameters, r and z, are determined by the value of rigidity. The default value for rigidity is 1. Larger values of rigidity produce a more rigid seasonal pattern. Normally, a series is first fit using the default value for rigidity. The smoothness of the trend and seasonality estimates are examined and then rigidity is either increased or decreased depending upon whether more or less seasonal smoothing is needed.

Additionally, imsls_f_bayesian_seasonal_adj selects the optimum autoregressive model as the model that minimizes the Akaike Bayesian Information Criterion (ABIC).

where the likelihood in this case is the mixed Bayesian maximum likelihood. Smaller values of ABIC represent a better fit. The basic minimization procedure is applied to blocks of data of length span*period. The final return value of the criterion is averaged over these blocks. By default, the data is treated in one block.

This example uses unadjusted unemployment for women over 20 years of age in the U.S. for 1991-2001, as reported by the U.S. Bureau of Labor Statistics (www.bls.gov).