Performs a harmonic analysis of a univariate time series (G. Tunnicliffe Wilson & R.P. Littlejohn).
Options
PRINT = string tokens |
Controls printed output (accumulated, means, tsm); default * |
|---|---|
PLOT = string tokens |
What to plot (periodogram, harmonics, means, residuals, cumulative, range); default peri, harm, mean, resid |
MODELTYPE = string token |
What harmonic regression model to fit (none, best, full); default none |
GROUPS = factor |
Groups for plot of means |
ORDER = variate |
Order for time series model; default !(1,0,0) |
COLOURS = text or variate |
Colour for each level of GROUPS |
FACSHORTCYCLE = factor |
Factor giving levels of the short cycle |
NCOMPONENTS = scalar |
Number of nested cycles, must be 0, 1, or 2; default 0 |
SHORTCYCLE = scalar |
Length of the short cycle; default 24 |
LONGCYCLE = scalar |
Length of the long cycle; default 365.225 |
LABSHORTCYCLE = text |
Label for the short cycle; default 'daily' |
LABLONGCYCLE = text |
Label for the long cycle; default 'annual' |
NHSHORTCYCLE = scalar |
Number of harmonics for the short cycle; default 5 |
NHLONGCYCLE = scalar |
Number of harmonics for the long cycle; default 3 |
RANGE = variate |
Variate with two values, defining the frequency range within [0,0.5] to draw a portion of the periodogram |
Parameters
DATA = variates |
Time series |
|---|---|
PERIODOGRAM = variates |
Saves the periodogram of DATA |
FREQUENCY = variates |
Saves the frequencies at which the periodogram is calculated |
MEANS = tables |
Saves the table of means of the fitted model for each value of FACSHORTCYCLE by each level of GROUPS |
RESIDUALS = variates |
Saves the residuals from the fitted model |
FITTEDVALUES = variates |
Saves the fitted values from the model |
Description
DFOURIER performs a harmonic analysis for a univariate time series which is supplied, in a variate, by the DATA parameter. In its basic form, it can produce 3 pages of graphs to study the series. These graphs are all controlled by the PLOT option. Setting PLOT=periodogram produces a page of graphs showing the time series, its periodogram and its log periodogram. The frequencies for the periodogram are calculated internally, and noted in the output. These can be saved, in a variate, by the FREQUENCY parameters, and the PERIODOGRAM parameter can save the periodogram. The cumulative setting of PLOT plots the cumulative periodogram (on a separate page), and the range setting plots the periodogram over the range specified by the RANGE option (this must be a value within [0,0.5]).
Other graphs are useful if you anticipate that the series will show some specific components. The number of these components is specified by the NCOMPONENTS option, and may be either 0 (no components, the default), 1 (a “short” cycle) or 2 (a “short” and a “long” cycle). The lengths of the long and short cycles are specified by the LONGCYCLE and SHORTCYCLE options, respectively. The defaults 365.225 and 24, correspond to hourly measurement of annual and daily cycles. The LABLONGCYCLE and LABSHORTCYCLE options supply labels for these cycles for the plots, with defaults of 'annual' and 'daily' respectively.
The components are particularly useful for analysing meterological time series (such as air temperatures) measured hourly over several years, where you want to describe how the diurnal pattern varies throughout the year. A single (non-sinusoidal) periodic component with period p (e.g. p = 24 for hourly observations) produces a main spike in the periodogram at the frequency f = 1/p, followed by a series of diminishing spikes at integer multiples of f known as harmonics.
When there are two periodic components with interacting rhythms, signals are observed in the periodogram not only at harmonics of each frequency, but at integer differences of the lower frequency from the higher. Thus, if hourly and annual frequencies are denoted by fd and fa, spikes may be observed in the periodogram at
fda = n × fd + m × fa,
where n is a non-negative integer, and m is an integer, which must be positive when n is zero.
These spikes generated by the interaction are generally hard to discern in an ordinary graph of the periodogram. The harmonic setting of PLOT produces a trellis plot that zooms in on a narrow range of about n × fd, for integer values of n ranging from 1 up to a value defined by the NHSHORTCYCLE option. This can be set to either 5 (default), 7 or 8, producing respectively a 3 × 2, 4 × 2 or 3 × 3 array of graphs. The NHLONGCYCLE option specifies the number of vertical lines to be drawn, within each graph, at positions corresponding to differences due to the long cycle. This can be set to 0, 1, 2 or 3 (default). It should be set to 0 if there is only one periodicity in the sampling protocol. The y-axes of the plots are scaled individually to a suitable order of magnitude, which is denoted in each graph. The frequency range for each panel is
n × fd +/- 5.1 × fa.
The MODELTYPE option allows a harmonic regression analysis to be conducted on DATA. The setting full fits sine and cosine terms for each frequency indicated in the harmonics graph. Alternatively, the setting best fits the full model and then drops terms that are non-significant at the 5% level. This does not guarantee that all terms remaining in the model are necessarily significant at the 5% level. In practice, however, dropping these additional terms will usually make little difference to the fitted model or residual variance. The accumulated setting of the PRINT option prints the accumulated analysis of deviance table from the fit.
With the tsm setting of the PRINT option, the model fitted as above is then used as the TRANSFERFUNCTION in a time series analysis of DATA. The TSM is defined by the ORDER option; by default this is set to a first-order autoregression (i.e. ORDER=!(1,0,0)). Note that this may take a long time to fit if there are many missing values in the data.
The fitted values and residuals from the final model (tsm is fitted after best, which is fitted after full) can be saved by the FITTEDVALUES and RESIDUALS parameters. The residuals setting of PLOT produces time-series plots of the residuals, from the BJIDENTIFY procedure.
DFOURIER forms tables of means of the fitted values classified by the the short cycle component and another factor, specified by the GROUPS option. You can supply the short cycle factor using the FACSHORTCYCLE option; this must have SHORTCYCLE levels or a fault will be generated. If FACSHORTCYCLE is unset, the required factor will be internally generated with levels 1…SHORTCYCLE. The factor GROUPS may, for example, be month or season. The SHORTCYCLE factor should be nested within GROUPS to provide meaningful output, but no checks are carried out on this.
You can plot the means using the means setting of the PLOT option. The points in each group are plotted in different colours, and you can supply these using the COLOURS option. If COLOURS is unset, the colours are set by default. If GROUPS has 4 levels, it is assumed they correspond to season, and pens 1 to 4 are defined to be red, gold, blue and green, corresponding to summer, autumn, winter and spring. If GROUPS has 12 levels, it is assumed that they correspond to months, and pens 1 to 12 are given decreasing intensities within the seasonal shades in clusters of three. Thus pens 1 to 3 are given crimson, red and salmon for the summer months. Note that this is tuned to a southern hemisphere calendar.
Options: PRINT, PLOT, MODELTYPE, GROUPS, ORDER, COLOURS, FACSHORTCYCLE, NCOMPONENTS, SHORTCYCLE, LONGCYCLE, LABSHORTCYCLE, LABLONGCYCLE, NHSHORTCYCLE, NHLONGCYCLE, RANGE.
Parameters: DATA, PERIODOGRAM, FREQUENCY, MEANS, RESIDUALS, FITTEDVALUES.
Action with RESTRICT
There must not be any restrictions.
See also
Directive: FOURIER.
Procedures: MCROSSPECTRUM, PERIODTEST, PREWHITEN, REPPERIODOGRAM, SMOOTHSPECTRUM.
Commands for: Time series.
Example
CAPTION 'DFOURIER example',\
!t('Data hourly temperatures at Tara Base',\
'courtesy of Alison Rutherford.'); STYLE=meta,plain
IMPORT '%gendir%/examples/DFOU-1.gsh'
DFOURIER [PRINT=accumulated,means; MODELTYPE=best;\
PLOT=periodogram,harmonics,means,residuals,cumulative,range;\
GROUPS=season; FACSHORT=hour; NCOMPONENTS=2; NHSHORTCYCLE=5;\
NHLONGCYCLE=2; RANGE=!(0.13,0.225)] TB