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Stationarity/Nonstationarity Identification
http://vered.rose.utoronto.ca/people/moman/Stationarity/stationarity.
Maurice R. Masliah
Stationarity/Nonstationarity Identification *
Purpose *
Stationarity *
Testing for Stationarity *
The Runs Test (nonparametric) *
Stationarity of Tracking Data *
Parametric Approaches *
Conclusions *
References *
Purpose
The purpose of using formal time series analysis methods on sequential data is to learn
"something" about the nature of the system generating the data. In our case, the system we
are interested in is the human operator interacting with a multiple degree of freedom input
device. The "something" we are trying to learn is the nature of human coordination. Our
definition of coordinated movement is the simultaneous motion along multiple degrees of
freedom resulting in an efficient trajectory.
This write-up is part of an exploration of time series analysis for the purpose of learning
something about the nature of human coordination. The first step in all time analysis
approaches is to check for stationarity, because if the data is stationary then many
simplifying assumptions can be made.
Stationarity
The following definition of stationarity is taken from (Challis and Kitney November 1991).
Stationarity, is defined as a quality of a process in which the statistical parameters (mean
and standard deviation) of the process do not change with time.
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The most important property of a stationary process is that the auto-correlation
function (acf) depends on lag alone and does not change with the time at which the
function was calculated.
A weakly stationary process has a constant mean and acf (and therefore variance)
A truly stationary (or strongly stationary) process has all higher-order moments
constant including the variance and mean.
The previous definition of stationarity is typical of what can be found in the literature. What
is usually not explained in the literature is that strongly stationary processes are never seen
in practice and are discussed only for their mathematical properties. Weakly stationary
processes, are sometimes observed in the real world and are usually assumed to be "close
enough" to stationarity in the strict sense (strong stationarity) to be treated as such. In
addition, stationarity is really a relative term, rather than an absolute as the definition above
may lead one to believe. Any process that "really" is stationary, can only be seen as
stationary if the sampled data from the process is very long compared to the lowest
frequency component in the data. In other words, if one collects data for only a short time,
short compared to the length of wavelength of the data, then even a stationary process will
appear to be nonstationary. Finally, no research exists which discusses what effect
deviations, large or small, from stationarity may have on analysis techniques which require
stationarity.
For the purpose of analysis, the stationarity property is a very good thing to have in one’s
data, since it leads to many simplifying assumptions. Again, the first step in using any
methodology for time series analysis is to check if one’s data is stationary.
Testing for Stationarity
There are two general approaches to testing for stationarity, parametric and nonparametric.
Reviews of the literature seem to indicate that parametric approaches are those usually used
by researchers working in the time domain, such as economists, who are making certain
assumptions about the nature of their data. Nonparametric approaches are more commonly
seen by researchers working in the frequency domain, such as electrical engineers, who
often treat the system as a "black box" and can not make any basic assumptions about the
nature of the system. Nonparametric tests are not based on the knowledge or assumption
that the population is normally distributed (Bethea and Rhinehart 1991). By making no
assumptions about the nature of the data, nonparametric tests are more widely applicable
than parametric tests which often require normality in the data. While more widely
applicable, the trade-off is that nonparametric tests are also less powerful than parametric
tests. To arrive at the same statistical conclusion with the same confidence level,
nonparametric tests require anywhere from 5% to 35% more data than parametric tests
(Bethea and Rhinehart 1991).
The Runs Test (nonparametric)
A run is defined as "a succession of one or more identical symbols, which are followed and
preceded by a different symbol or no symbol at all" (Gibbons 1985). So a series of identical
flips of a coin is a run, where H represents heads and T for tails, such that
…THHHHHHTTHTT…
In the example above, the long succession of H is a counted as a run of heads. Too few or
too many runs is evidence of dependency between the observations, and therefore,
nonstationarity. A runs test is a counting of the number of runs in a series, and comparing
the number found to what one would expect if the observations were independent of one
another. The stationarity of data can be determined by using a runs test (Bendat and Piersol
1986) as follows:
1. Divide the series into time intervals of equal lengths.
2. Compute a mean value (or other, see below) for each interval.
3. Count the number of runs of mean values above and below the median value
of the series.
4. Compare the number of counts found to known probabilities of runs for
random data.
Note that the runs test works equally well on mean values, mean square values, variance,
standard deviation, or any other parameter estimate (Bendat and Piersol 1986). Known
probabilities of runs distributions can be found in (Bendat and Piersol 1986), (Bethea and
Rhinehart 1991), and (Gibbons 1985).
However, the previous references on the runs test all deal with applying the runs test to a
single observation series. This is fine if one only has one sequence of data. What about
experiments where one can obtain multiple realizations of the same process? It does not
seem appropriate to simply select one of the series and apply a runs test only to that series.
The answer proposed here is to apply the runs test to all the data available and then to
compare the distribution of runs found to the distribution of runs for a random series. For
example, let’s say data from 100 realizations of the same process are collected, where
nothing is known about the frequency components of the process generating the data. To
test for stationarity, divide each data sample into 18 equal sized segments (the number 18
has been arbitrarily chosen, any even division should equally well). Then, count the number
of runs above and below the median value (of the particular sample) in each data sample. In
theory, the number of runs can range from 2 to 18 per sample. A truly random process will
expect 90 of the 100 samples to have counts of at least 7 and not more than 14, where 7 and
14 are the 0.05% left and right tail cut-offs (from Table L: Number of Runs Distribution
(Gibbons 1985)).
Note that using 0.05% left and right tail cut-off in the previous example is a more stringent
requirement than using a 0.01% tail probability (which is in the opposite direction of the
probability values from a standard ANOVA test). To illustrate this, imagine using a
0.0001% cut-off. A runs distribution table will give corresponding runs count between 2
and 18, which covers all the possibilities. A 0.0001% criteria is not stringent enough and
will result in any data set passing a test for stationarity. For the purposes of the work here
on human coordination, a 0.05% tail cut-off is considered to be a sufficient criteria.
Stationarity of Tracking Data
Is the error data collected using the MITS software (Zhai 1995) stationary? To test for
stationary, one subject was run through 38 trials of a 40 second six degree-of-freedom
tracking task. The tracking error for each degree of freedom is computed as the difference
between the user’s cursor position and the required target position. Data was sampled at
0.05 seconds during the task. Using the methodology from (Bendat and Piersol 1986) runs
above and below the median were counted for 228 series (38 trials x 6 degrees of freedom),
for different size segments.
The results of the runs count for the 40 second trials divided into 16, 14, 12, and 10
segments (corresponding to segments of 2.5, 2.9, 3.3, and 4.0 seconds in length) are
presented in Figure 1. The 0.05% tail cut-offs are shown for the different size segments. In
order for a process to be considered stationary, 90% of the number of runs distribution
should be between the left and right tails. The results indicate that segments of 3.3 seconds
and greater may be considered stationary, while segments shorter than 3.3 seconds are
nonstationary.
Figure 1. (NOTE: For WEB viewing, it may be necessary to open this image separately in order to view it
properly.) Runs distribution of tracking data from 228 series, divided into 2.5, 2.9, 3.3, and 4.0 second
segments. The 0.05% tails are shown for the different tests. In order for a process to be considered stationary,
90% of the number of runs should be between the left and right tails. Results indicate that segments of 3.3
seconds and greater may be considered stationary, while segments shorter than 3.3 seconds are
nonstationary.
Parametric Approaches
According to (Bowerman and O'Connell 1979) and [Box, 1976 #24], that if a time series is
nonstationary, then the sample auto-correlation function will neither cut off nor die down
quickly, but rather will die down extremely slowly. The next question then becomes, exactly
what is considered quick or slow? Unfortunately, there does not seem to be a quantifiable
answer to this question in the literature. Clearly, the rate at which a function dies down
depends upon the frequency of the signal compared to the sampling rate. Basically,
parametric approaches assume a certain level of experience with the data, and with that
experience one can then tell by looking whether data may be considered stationary or nonstationary.
Figure 2. The auto-correlation function of the Y translation error from a single 40 second tracking trial.
The auto-correlation function of the error from a tracking trial for one of the degrees of
freedom is shown in Figure 2. Does this auto-correlation function agree with the results
from the runs test? The runs test is essentially a test of independence, are different
observations independent of one another or are they correlatated? Results from the runs test
indicates that observations 3.3 seconds and farther apart from each other may be considered
independent. The auto-correlation function shown in Figure 2 drops to near zero correlation
at lags greater than about 3 to 5 seconds. The auto-correlation does seem to agree with
results of the runs test.
Conclusions
A conclusion of stationarity is reasonable when one considers the nature of the tracking
process that is occurring. Stationarity exists if the mean and variance of the data remains
constant. In the MITS task, the target is restricted to a fixed volume area to move about in.
If one assumes that the subject is at all times attempting to track the target with equal effort,
then it reasonable to expect that the subject’s error (and the variance of that error) will
remain relatively constant. The only question remains is what is the minimum time period
which must be considered to achieve a "relatively constant" value. This question has been
answered using the Runs Test. Based on the results from the Runs Test, segments of 3.3
seconds and larger of tracking data from the MITS tracking task may be considered
independent and stationary. Segments smaller than around 3.3 seconds capture only the
higher frequency components of the error process of a human tracking in six degrees of
freedom.
References
Bendat, J. S., and Piersol, A. G. (1986). Random Data: Analysis and
Measurement Procedures, John Wiley & Sons, Inc.
Bethea, R. M., and Rhinehart, R. R. (1991). Applied Engineering Statistics,
Marcel Dekker, Inc., New York, NY.
Bowerman, B. L., and O'Connell, R. T. (1979). Time Series and
Forecasting, Duxbury Press, North Scituate, Massachusetts.
Challis, R. E., and Kitney, R. I. (November 1991). "Biomedical signal
processing (in four parts). Part 1 Time-domain methods." Medical &
Biological Engineering & Computing, 28, 509-524.
Gibbons, J. D. (1985). Nonparametric Methods for Quantitative Analysis,
American Sciences Press, Inc., Columbus, Ohio.
Zhai, S. (1995). "Human Performance in Six Degree of Freedom Input
Control," Ph.D., University of Toronto, Toronto.