Performs a harmonic analysis of a univariate time series (G. Tunnicliffe Wilson & R.P. Littlejohn).
|Controls printed output (
||What to plot (
||What harmonic regression model to fit (
||Groups for plot of means|
||Order for time series model; default
||Colour for each level of
||Factor giving levels of the short cycle|
||Number of nested cycles, must be 0, 1, or 2; default 0|
||Length of the short cycle; default 24|
||Length of the long cycle; default 365.225|
||Label for the short cycle; default
||Label for the long cycle; default
||Number of harmonics for the short cycle; default 5|
||Number of harmonics for the long cycle; default 3|
||Variate with two values, defining the frequency range within [0,0.5] to draw a portion of the periodogram|
||Saves the periodogram of
||Saves the frequencies at which the periodogram is calculated|
||Saves the table of means of the fitted model for each value of
||Saves the residuals from the fitted model|
||Saves the fitted values from the model|
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
SHORTCYCLE options, respectively. The defaults 365.225 and 24, correspond to hourly measurement of annual and daily cycles. The
LABSHORTCYCLE options supply labels for these cycles for the plots, with defaults of
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.
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
tsm setting of the
TRANSFERFUNCTION in a time series analysis of
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
RESIDUALS parameters. The
residuals setting of
PLOT produces time-series plots of the residuals, from the
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.
There must not be any restrictions.
Commands for: Time series.
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