findstysols which, given two time series,
tsx and tsy, and a given complexity Ncoefs,
finds the set (number dependent and equal to Nsims)
of solutions that solve the optimization problem ``find the coefficient
vectors that cause the time-varying linear combination of tsx
and tsy to be as most stationary as possible''.
findstysols is capable of running one optimization run,
or several. The former can be useful when running a single optimization
job as part of several on a multiprocessor machine, however the function
can run all your optimization jobs a single call (it'll just take longer).Typically, one should start with Ncoefs small (it must be one less
than a dyadic number, e.g. 3, 7, 15, for Haar wavelets), 3 say, run
several optimizations to see how likely the linear combinations are to
be stationary. If the time series themselves are fairly stationary
(but not exactly, if they were both stationary then it is trivial
to find a stationary linear combination) then it will be easy to find
many stationary combinations. If the time series are highly non-stationary,
evolving significantly over time, then more complex time-varying combination
vectors will be required to get stationarity more often.
Note that since the algorithm is based on numerical optimization, one is
never guaranteed to find an optimal solution even if one exists.
Our methodology adopts the time-honoured strategy of multiple random
starts to find stationary solutions. If many solutions indicate
stationarity has been reached then you have found the types and complexity
of solutions that give stationarity. If many solutions indicate that
stationarity has not been reached (look at the p-value output of
link{findstysols}) then you should increase Ncoefs to the
next dyadic number (less one) and then try again.
Once you have discovered a set of solutions which appear to contain
a number of stationary ones then the functions LCTSres
and COEFbothscale can be used to interrogate the solution
set to find clusters of solutions. Since findstysols uses
multiple numerical optimizations the solution set tends to cluster around
a number of ``true'' linear combination vectors and the interrogation
functions are useful for identifying the number of these, and also what
they represent.
The function mergexy can merge together the result sets
of multiple findstysols runs with the resultant object
being as if one run of findstysols had been used.
This again can be useful to concatenate result sets from multiple
runs on, say, a multiprocessor/core machine.
The test of stationarity is contained in the myTOS
function, which can take a time series and uses a bootstrap method
to test for stationarity. The exact test statistic is contained
in TOSts which is called by myTOS.
In principle, another test of stationarity could be substitued for
the wavelet based bootstrap test, a Fourier test, or something else
could potentially be used. The costationarity idea is independent
of the particular test statistic used. However, the judgement of
a particular linear combination of two time series and whether it is
stationary does, of course, depend on the particular test used.
This package also contains a number of other functions of use in
analysing single and pairs of locally stationary time series. For example,
lacv computes the time-localized autocovariance of
any time series (including stationary ones!) using a wavelet method (although
the quantity itself only depends on time and lag), crosslacv
performs a similar job, but computes the cross time-localized covariance
between two series, ewcrossspec the time-localized cross
spectrum, localvar computes the time-localized variance of
the series, again computed via a wavelet method, but not, in the end,
dependent on it, just time.
The functions tstosscan and plottstosscan
take a single time series, apply a test of stationarity at multiple
locations on multiple length windows of a series, and the latter function
produces a plot that displays the results of the tests. These functions
are very useful in discovering the degree of non-stationarity of a time series.
I.e. not only whether a series, or bits of it, are non-stationary, but where
and also over what periods. E.g. a series might be stationary over a window
of 32 or 64 observations, but decidedly non-stationary over a longer
time period (plus the statistical power of detection is greater for
longer series as there is generally more information).
#
# See comprehensive examples in the help pages for the major functions
# outlined above.
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