Gives periodogram-based analyses for replicated time series (R.P. Littlejohn).
Options
PRINT = string token |
What to print (pair, randomization, glm); default * i.e. none |
|---|---|
PLOT = string token |
What graphs to plot (group, mean, logmean, cumulative, cv, pair); default mean, logm |
TITLE = text |
Title for each page of graphs |
REPRESENTATION = string token |
Form of data in SERIES (timeseries, meanperiodogram); default time |
LENGTH = scalar or variate |
Scalar specifying that the first N units of the series are to be used, or a variate specifying the first and last units of the series to be used |
SEED = scalar |
Seed for randomization; default 0 |
NRANDOMIZATIONS = scalar |
Number of randomizations; default 99 |
TREATMENTS = factor |
Contains ordered classification of SERIES |
PAIR = variates |
Treatment pair levels for pairwise comparisons |
COLOUR = text or variate |
Colours for each level of TREATMENTS; default * sets suitable colours automatically |
MEANPERIODOGRAM = pointer |
Saves mean periodograms according if REPRESENTATION=timeseries |
REPLICATION = scalar or variate |
Inputs or saves number of replicate series if REPRESENTATION=timeseries; scalar can be used for equal replication |
Parameter
SERIES = variates |
Specify the time series to be analysed |
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Description
REPPERIODOGRAM gives periodogram-based analyses of replicated time series. The data are supplied in a list of variates using the SERIES parameter, either as the original time series (option REPRESENTATION set to timeseries) with the level for each series given by the factor specified by the TREATMENTS option, or as the mean periodograms for each treatment level (option REPRESENTATION set to meanperiodogram), with levels and labels optionally given by the TREATMENTS factor and the multiplicity of each treatment defined by the REPLICATION option. In the former case the LENGTH option can specify that only part of each series is to be used, using either a scalar N to indicate that the first N values are to be used, or a variate of length two, holding the values of the first and last units of the required subseries. This may be used to eliminate missing values, which are otherwise not permitted. Further, when REPRESENTATION=timeseries, periodogram means and the replication variate can be saved using the MEANPERIODOGRAM and REPLICATION options, respectively.
Graphical output is controlled by the PLOT option. For the group (REPRESENTATION=timeseries only), mean, logmean and cumulative periodogram and cv graphs, the COLOUR option can be used to code for treatments; by default, the standard colours are used in the same order as for pens 2, 3… (see PEN). The cv plot (REPRESENTATION=timeseries only) gives a scatterplot of coefficients of variation for each treatment group at each frequency, together with lines for the means of these cvs at each frequency for those treatments with replication greater than one, and cv=1, the theoretical value if there is no subject-specific variation. For these graphs a title can be supplied using the TITLE option. Graphs are also given for the differences between pairs of log periodograms as defined by PAIR (see below), with 95% confidence intervals on the sample and null (equal periodograms) distribution.
Output of various test statistics for pairwise comparison of treatment levels described by Diggle (1990) and Diggle & Fisher (1991) is controlled by the PRINT and PAIR options. PAIR is a list of 2-unit variates representing treatment levels, e.g.
PAIR=!(1,2),!(3,4)
gives tests comparing treatment levels 1 and 2, followed by tests for levels 3 and 4. With PRINT=pair, the maximum absolute value and range of the difference of log periodograms give (weak) tests against the null hypotheses of equal and proportional spectra, respectively. With PRINT=random, a randomization test is given for the equality of cumulative spectra, which is insensitive to the alternative of proportional spectra. The seed for the ramdomizations can be set using the SEED option, and the number of randomizations is specified by NRANDOMIZATIONS (default 99). This is available only if the treatments in the pair have equal replication.
When PRINT=glm, a generalized linear model is fitted to the mean periodograms for all treatments, adjusting for frequency, and testing for differences with treatment in constant (proportional spectra), linear (power shift) and quadratic (power spread) contrasts with frequency (Diggle 1990). Results are presented in the accumulated analysis of deviance table and tables of parameter estimates, within which the Intercept-Difference, Slope-Difference and Curve-Difference estimates relate to the above hyptheses.
Options: PRINT, PLOT, REPRESENTATION, LENGTH, TREATMENTS, PAIR, SEED, NRANDOMIZATIONS, COLOUR, TITLE, MEANPERIODOGRAM, REPLICATION.
Parameter: SERIES.
Method
The series are mean-corrected, but not trend corrected, before transformation, and are not smoothed. Critical values for the Range test are obtained from tables in Potscher & Reschenhofer (1988) and Coates & Diggle (1986). Random numbers are generated using URAND. The analysis for PRINT=glm is obtained from fitting a generalized linear model with DISTRIBUTION=gamma, LINK=log and DISPERSION=1/nr, where nr is number of replicates of the treatments.
Action with RESTRICT
The SERIES may not be restricted; restriction of the input series to a contiguous set of units may be achieved by use of the LENGTH parameter.
References
Coates, D.S. & Diggle, P.J. (1986) Tests for comparing two estimated spectral densities. Journal of Time Series Analysis, 7, 7-20.
Diggle, P.J. (1990). Time Series: A Biostatistical Introduction. Oxford, Clarendon Press.
Diggle, P.J. & Fisher, N.I. (1991). Nonparametric comparisons of cumulative periodograms. Applied Statistics, 40, 423-434.
Potscher, B.M. & Reschenhofer, E. (1988). Discriminating between two spectral densities in case of replicated observations. Journal of Time Series Analysis, 9, 221-224.
See also
Directive: FOURIER.
Procedures: DFOURIER, MCROSSPECTRUM, PERIODTEST, SMOOTHSPECTRUM.
Commands for: Time series.
Example
CAPTION 'REPPERIODOGRAM example',\
!t('Data from Diggle, P.J. (1990),',\
'Time Series: A Biostatistical Introduction,',\
'Clarendon Press, Oxford, Table A.1 Lutenizing hormone.');\
STYLE=meta,plain
VARIATE [NVALUES=48] Number,lh[1...4]
FACTOR [LEVELS=2; LABELS=!T(early,late); VALUES=(1,2)2] treat
READ Number,lh[1...4]
1 2.2 5.5 2.4 4.3
2 2.2 4.5 2.4 4.6
3 2.3 5.1 2.4 4.7
4 2.0 5.5 2.2 4.1
5 1.6 5.7 2.1 4.1
6 1.4 5.1 1.5 5.2
7 1.8 4.3 2.3 5.0
8 2.2 4.8 2.3 4.4
9 2.9 5.6 2.5 4.2
10 2.6 5.9 2.0 5.1
11 2.4 6.0 1.9 5.1
12 2.1 5.1 1.7 4.7
13 3.0 5.2 2.2 4.4
14 2.5 4.4 1.8 3.9
15 2.7 5.5 3.2 5.4
16 2.2 5.4 3.2 5.9
17 2.4 4.1 2.7 4.2
18 2.7 4.4 2.2 4.1
19 3.1 4.7 2.2 4.1
20 2.5 4.6 1.9 3.6
21 2.4 6.0 1.9 3.1
22 2.3 5.6 1.8 4.8
23 2.4 5.1 2.7 5.1
24 1.9 4.7 3.0 5.1
25 3.3 4.8 2.3 4.5
26 3.8 5.5 2.0 4.6
27 3.7 5.1 2.0 5.8
28 3.5 5.2 2.9 5.0
29 3.1 5.0 2.9 5.1
30 2.7 4.0 2.7 4.5
31 4.1 3.7 2.7 4.2
32 4.0 4.8 2.3 6.0
33 3.4 5.9 2.6 5.6
34 3.2 5.5 2.4 5.4
35 3.7 4.9 1.8 5.0
36 3.6 4.4 1.7 4.4
37 4.1 4.7 1.5 4.6
38 2.0 4.2 1.4 5.7
39 4.6 5.5 2.1 5.2
40 4.1 4.9 3.3 5.0
41 3.2 4.8 3.5 4.4
42 2.9 4.5 3.5 5.7
43 2.7 4.9 3.1 5.7
44 3.0 4.9 2.6 4.8
45 * 4.5 2.1 3.4
46 * 4.2 3.4 5.5
47 * 4.9 3.0 5.5
48 * 5.9 2.9 5.6 :
REPPERIODOGRAM [PRINT=pair,random,glm;\
PLOT=group,mean,logmean,cumulative,cv,pair;\
TITLE='Luteinizing hormone'; REPRESENTATION=timeseries;\
LENGTH=44; TREATMENTS=treat; SEED=376512; PAIR=!(1,2);\
COLOUR=!t(red,limegreen); MEANPERIODOGRAM=mPer;\
REPLICATION=rep] lh[1...4]
PRINT mPer[]
PRINT rep