The Time Series module contains a wide range of descriptive, modeling, decomposition, and
forecasting methods for both time and frequency domain models. These procedures are integrated,
that is, the results of one analysis (e.g., ARIMA residuals) can be used directly in subsequent
analysis (e.g., to compute the autocorrelation of the residuals). Also, numerous flexible options
are provided to review and plot single or multiple series. Analyses can be performed on even very
long series. Multiple series can be maintained in the active work area of the program (e.g.,
multiple raw input data series or series resulting from different stages of the analysis); the
series can be reviewed and compared.
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The program will automatically keep track of successive analyses, and maintain a log of
transformations and other results (e.g., ARIMA residuals, seasonal components, etc.). Thus, the
user can always return to prior transformations or compare (plot) the original series together
with its transformations. Information about the consecutive transformations is maintained in the
form of long variable labels, so if you save the newly created variables into a dataset, the
"history" of each of the series will be permanently preserved. The specific Time Series procedures
are described in the following subsections.
Transformations, Modeling, Plots, Autocorrelations
The available time series transformations allow the user to fully explore patterns in the input
series, and to perform all common time series transformations, including: de-trending, removal of
autocorrelation, moving average smoothing (unweighted and weighted, with user-defined or Daniell,
Tukey, Hamming, Parzen, or Bartlett weights), moving median smoothing, simple exponential smoothing
, differencing, integrating, residualizing, shifting, 4253H smoothing, tapering, Fourier
(and inverse) transformations, and others. Autocorrelation, partial autocorrelation, and
crosscorrelation analyses can also be performed.
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ARIMA and Interrupted Time Series (Intervention) Analysis
The Time Series module offers a complete implementation of ARIMA. Models may include a constant, and
the series can be transformed prior to the analysis; these transformations will automatically be
"undone" when ARIMA forecasts are computed, so that the forecasts and their standard errors are
expressed in terms of the values of the original input series. Approximate and exact maximum-
likelihood conditional sums of squares can be computed, and the ARIMA implementation in the Time
Series module is uniquely suited to fitting models with long seasonal periods (e.g., periods of
30 days). Standard results include the parameter estimates and their standard errors and the parameter correlations.
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Forecasts and their standard errors can be computed and plotted, and appended to the input series.
In addition, numerous options for examining the ARIMA residuals (for model adequacy) are available,
including a large selection of graphs. The implementation of ARIMA in the Time Series module also
allows the user to perform interrupted time series (intervention) analysis. Several simultaneous
interventions may be modeled, which can either be single-parameter abrupt-permanent interventions,
or two-parameter gradual or temporary interventions (graphs of different impact patterns can be
reviewed). Forecasts can be computed for all intervention models, which can be plotted (together
with the input series) as well as appended to the original series.
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Seasonal and Non-Seasonal Exponential Smoothing
The Time Series module contains a complete implementation of all 12 common exponential smoothing
models. Models can be specified to contain an additive or multiplicative seasonal component and/or
linear, exponential, or damped trend; thus, available models include the popular Holt-Winter linear
trend models. The user may specify the initial value for the smoothing transformation, initial trend
value, and seasonal factors (if appropriate). Separate smoothing parameters can be specified for the
trend and seasonal components.The user can also perform a grid search of the parameter space in order
to identify the best parameters.
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The respective results spreadsheet will report for all combinations of parameter values the mean
error, mean absolute error, sum of squares error, mean square error, mean percentage error, and
mean absolute percentage error. The smallest value for these fit indices will be highlighted in the
spreadsheet. In addition, the user can also request an automatic search for the best parameters with
regard to the mean square error, mean absolute error, or mean absolute percentage error (a general
function minimization procedure is used for this purpose). The results of the respective exponential
smoothing transformation, the residuals, as well as the requested number of forecasts, are available
for further analyses and plots. A summary plot is also available to assess the adequacy of the
respective exponential smoothing model; that plot will show the original series together with the
smoothed values and forecasts, as well as the smoothing residuals plotted separately against the
right-Y axis.
Classical Seasonal Decomposition (Census Method I).
The user may specify the length of the seasonal period, and choose either the additive or
multiplicative seasonal model. The program will compute the moving averages, ratios or differences,
seasonal factors, the seasonally adjusted series, the smoothed trend-cycle component, and the
irregular component. Those components are available for further analysis; for example, the user
may compute histograms, normal probability plots, etc. for any or all of these components (e.g.,
to test model adequacy).
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X-11 Monthly and Quarterly Seasonal Decomposition and Seasonal Adjustment (Census Method II).
The Time Series module contains a full-featured implementation of the US Bureau of the Census
X-11 variant of the Census Method II seasonal adjustment procedure. While the original X-11
algorithms were not year-2000 compatible (only data prior to January 2000 could be analyzed),
the STATISTICA implementation of X11 can handle data containing dates prior to January 1, 2000,
after that date, or series that will start prior to that date but terminate in or after the
year 2000. The arrangement of options and dialogs closely follows the definitions and conventions
described in the Bureau of the Census documentation.
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Additive and multiplicative seasonal models may be specified. The user may also specify prior
trading-day factors and seasonal adjustment factors. Trading-day variation can be estimated via
regression (controlling for extreme observations), and used to adjust the series (conditionally
if requested). The standard options are provided for graduating extreme observations, for computing
the seasonal factors, and for computing the trend-cycle component (the user can choose between
various types of weighted moving averages; optimal lengths and types of moving averages can also
automatically be chosen by the program). The final components (seasonal, trend-cycle, irregular)
and the seasonally adjusted series are automatically available for further analyses and plots;
those components can also be saved for further analyses with other programs. The program will
produce the plots of the different components, including categorized plots by months
(or quarters)
Polynomial Distributed Lag Models
The implementation of the polynomial distributed lag methods in the Time Series module will estimate
models with unconstrained lags as well as (constrained) Almon distributed lags models. A selection
of graphs are available to examine the distributions of the model variables.
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Spectrum (Fourier) and Cross-Spectrum Analysis.
The Time Series module includes a full implementation of spectrum (Fourier decomposition) analysis
and cross-spectrum analysis techniques. The program is particularly suited for the analysis of
unusually long time series (e.g., with over 250,000 observations), and it will not impose any
constraints on the length of the series (i.e., the length of input series does not have to be a
multiple of 2). However, the user may also choose to pad or truncate the series prior to the
analysis. Standard pre-analysis transformations include tapering, subtraction of the mean, and
detrending. For single spectrum analysis, the standard results include the frequency, period,
sine and cosine coefficients, periodogram values, and spectral density estimates.
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The density estimates can be computed using Daniell, Hamming, Bartlett, Tukey, Parzen, or user-
defined weights and user-defined window widths. An option that is particularly useful for long
input series is to display only a user-defined number of the largest periodogram or density
values in descending order; thus, the most salient periodogram or density peaks can be easily
identified in long series. The user can compute the Kolmogorov-Smirnov d test for the periodogram
values to test whether they follow an exponential distribution . Numerous plots are available to
summarize the results; the user can plot the sine and cosine coefficients, periodogram values,
log-periodogram values, spectral density values, and log-density values against the frequencies,
period, or log-period. For long input series, the user can choose the segment (period) for which
to plot the respective periodogram or density values, thus enhancing the "resolution" of the
periodogram or density plot. For cross-spectrum analysis, in addition to the single spectrum
results for each series, the program computes the cross-periodogram (real and imaginary part),
co-spectral density, quadrature spectrum, cross-amplitude, coherency values, gain values, and the
phase spectrum. All of these can also be plotted against the frequency, period, or log-period,
either for all periods (frequencies) or only for a user-defined segment. A user-defined number of
the largest cross- periodogram values (real or imaginary) can also be displayed in a spreadsheet
in descending order of magnitude to facilitate the identification of salient peaks when analyzing
long input series. As with all other procedures in the Time Series module, all of these result
series can be appended to the active work area, and will be available for further analyses with
other time series methods or other Statistica modules.
Regression-Based Forecasting Techniques.
Finally, Statistica offers regression-based time series techniques for lagged or non-lagged
variables (including regression through the origin, nonlinear regression, and interactive what-if
forecasting).
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