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INTERAGENCY REPORT: ASTROGEOLOGY 30
A TEST OF SELF-STATIONARITY
By
Robert D. Regan
Prepared under NASA Contract H-82013A
92·311
The work presented in chis report was accomplished as part of
a study investigating the analysis of traverse geophysics data.
The study was directed toward developing methods of extracting the
maximum amount of information from geophysical measurements ob
tained during proposed automated lunar vehicle traverses.
However, it became apparent that a computer program for test
ing stationarity would be of use in many fields of investigation.
Thus it was dec i ded to publish this report separately.
1
CONTENTS
ABSTRACT -----------------------------------------------
INTRODUCTION --------------------------------------------
BRIEF THEORETICAL SUMMARY -------------------------------Time Series Stochastic Process -----------------
Stochastic Process Functions ---------------------
Statlonarity ------------------------------------
Ergodicity ---------------------------------------
STATIONARITY OF A SINGLE TIME SERIES --------------------
TEST FOR SELF-STATIONARITY -----------------------------
Bendat and Piersol Test --------------------------
Bryan Test ------ -------------- - ---------------
Test Modifications ------------------------------
SELF-STATIONARITY TEST ---------------------------------
COMPUTER PROGRAM STEST ----------------------------------
Program Input -----------------------------------
Program Output ----------------------------------
TEST CASES ---------------------------------------------
APPENDIX ------------------------------------------------
Computer Program STEST ---------------------------
Sample Input ------------------------------------
Sample Output ------------------------------------
REFERENCES ------------------------------------------·---
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31
A TEST OF SELF-STATIONARITY
by
Robert D. Regan
ABSTRACT
A method for testing the stationarity of a single time series has been devised. The method utilizes two pre-existing tests of stationarity. A computer program that permits routine testing of any time series has been developed, and the test has been successfully applied to several known series. In certain cases the self· stationarity determined can be equated to stationarity.
INTRODUCTION
In time-series analysis of any body of observational data,
the fundamental assumption is that of stationarity. Very little
work has been d ~ne with nonstationary time series, and the effects
of nonstationarity upon time-series analysis are lit t le known. In
the prattical application of time-series techniques the assumption
of stationarity is routinely invoked without adequate tests for
the validity of such an assumption. Yet the utilization of some
of the more essential techniques of time-series analysis, such as
the Wiener-Khintchine theorem in the indirect method of power
spectra computat ~ on, is valid on J·· for stationary series.
This situation stems from the f act that, to date, no rapid,
accurate, routine method has been available to test the station
arity of a time series. Such a test is clearly required so that
stationarity or nonstationarity of a series can be routinely de
termined and the validity of the application of time-series anal
ysis be established. The method and computer program outlined in
this report is designed to fulfill this requirement.
Basically, the method outlined and programmed is a combina
tion of tests proposed by Bendat and Piersol (1966) and Bryan
(1967). Their tests and assumptions have been combined with some
additional logic to produce a computer test of the stationarity
of a single time series . The method has proved successful in sev
eral test cases.
1
BRIEF THEORETICAL SUMMARY
Before the test method and rationale are explained it is
essential to discuss some of the concepts and terms pertinent to
stationarity.
Time Series-Stochastic Process
A stochastic process can be considered as being composed of
a family of random variables, and viewed as a function of two
var iables t and w. The sample space associated with this stoch
astic process is doubly infinite and the set of time functions
which can be defined on this space is called an ensemble.
A strict definition of a time series is that it is one reali
zation (outcome) of a stochastic (random) process. A more popular
definition is that it is a set of observations of a parameter
arranged sequentially.
The label "time-series" is perhaps a misnomer, and is directly
applicable only when the observations are made chronologically.
However, for the independent variable, time, there may be substi
tuted any other parameter -e.g., distance.
The basic idea of the statistical theory of time series
analysis is to regard the time series as a set of ob~ervations
made on a family of random variables, i.e., for each tinT,
x (t) is an observed value of a random variable. The set of
observations [ x (t); t ( T } is called a time series.
Stochastic Process Functions
The basic process functions are
a) mean of the process co
\.lx(t) S E (x(t)} • L J x f(x,t) dx
where:
LJ f(x,t)
f
-=
• Lebesgue Integral
• Probability density function
= Expectation operator
2
In the general case the means are different at different
times and must be calculated for each t.
b) autocorrelation function of the process
Rx ( t 1
, t2
) ~ E ( x ( t 1
) , x ( t 2
)}
where:
* x(t2) = complex conjugate of x(t2)
c) autocovariance function of the process
d * Cx(t1 ,t2 ) = E ([ x(t1 ) - ~x(t 1 )] , [ x(t2 ) - ~x(t2 ) J} In the general case these functions must be calculated for
each t1
, t 2 combination.
Stationarity
In general the properties of a stochastic process will be
time dependent. A simplifying assumption which is often made is
that the series has reached some form of steady state in the sense
that the statistical properties of the series are independent of
absolute time. Stationarity can be pictured as the absence of any
time-varying change in the ensemble of member functions as a whole.
A sufficient degree of stationarity for most time-series
analysis is wide-sense stationarity. A process has wide-sense
stationarity if its expected value is a constant and its auto
correlation depends only on the time difference, t1
- t 2 ~ T, and
not on the absolute value of the respective times.
i.e.
and
E (x(t)} a ~ (t) = constant X
* E {x(t+ T), x(t)} = R (T) X
Wide-sense stationarity is also termed stationarity of the second
order, i.e., the series is stationary through its second order
statistical moments. Stationarity of order n implies that all
statistical moments less than n depend only on the time differences.
3
Ergodicity
Ergodicity relates to the problem of determining the statis
tics of the stochastic process from the statistics of one time
series. A stochastic process for which the statistics are thus
determinable is said to be ergodic, and the single time secies is
representative of the ensemble. The ensemble moments can then be
equated to the time moments.
For example the time average equals the ensemble average of
an ergodic process
T J x(t) dt ~ E(x(t)}
-T
If a time series is ergodic we need only to measure the time
averages which are available rather than the postulated ensemble
averages. Since ergodicity is a subclass of stationarity, a time
series must be shown to be stationary before the question of
e~godicity can be considered.
STATIONARITY OF A SINGLE TIME SERIES
In the transformation from the theoretical to the empiri~al,
strict adherence to theory must be tempered with reasonable prac
tical considerations. In most practical applications the single
observed time series is the only information available on the
parent process. Hence ergodicity must be assumed and time-domain
statistics utilized. Also all analyses are performed on the
sample series and thus the stationarity of this series is of prime
interest. Bendat and Piersol (1966) have termed the concept of
stationarity of this one time series as self-stationarity. Thus
we speak of the series being stationary rather tha. the process
being stationary.
However, the concept of self-stationarity is not restrictive.
A necessary condition to extend this self-stationarity to station
arity is that the process be ergodic, i.e. that the series is
representative of the process. Ergodicity is impossible to prove
(except in special instances) when the entire process is not known.
4
Bendat and Piersol (1966, p. 12) state that "in actual practice,
random data representing stationary physical phenomena are gen
erally ergodic".
If the assumption of ergodicit~r is justified then self
stationarity becomes equivalent to stationarity, since the single
time series is representative of the ensemble·.
TEST FOR SELF-STATIONARITY
The test for self-stationarity is based on the methods pro
posed by Bendat and Piersol (1966) and Bryan (1967). The basis
of these methods is that in a stationary series certain statistical
properties of the time series are considered invarian~ with time.
The tests are for second order self-stationarity.
Bendat and Piersol Test
Bendat and Piersol (1966) suggest that the series be divided
into n equal time intervals (either contiguous or non-contiguous)
and that the mean and variance of these intervals be calculated.
The two series thus formed, composed of means and variances, are
then tested for underlying trends or variations by the Run test
and the Trend test, Bendat and Piersol (1966, p. 156). If no
trends or variations are suggested by the application of these
tests, the original series is assumed to be stationary.
Bryan Test
Basically, Bryan's test (1967) is quite similar in that he
tests the invariance of the means and variances obtained from in
dependent, equal time segments. Rather, than using the sample
mean and sample variance he constructs two combi.nations of the
data to serve as estimates of the population mean and population
variance. Those two estimates are independent, and independently
distributed.
Using these two variates, m, a linear function of the data
and an unbiased estimate of the population mean, and Q, a quad
tatic function of the data, he develops a test for the hypothesis
that the time series is stationary, and two test variables, L1
for
5
the Neyman-Pearson L test, and F for the F-distribution test.
Test modifications
The tables for the Run and Trend tests (Bendat and Piersol
1966, p. 170) were extrapolated to include the range n = 1 to
n = 200 . With values of n less than 12 the results proved unre
liable. Hence a low limit c•tt off at n = 12 is utilized. The
Acceptance region was extended to include the lower bound.
The Bryan test was extended to include the 97.5 percent con
fidence interval.
Self-Stationarity Test
In the proposed test both methods are combined and used with
some restrictions and modifications.
First the sampling interval for independent samples is deter
mined. If independent samples cannot be determined, i.e., the
autocorrelation :unction does not damp to zero, the test is aborted
and the series can be considered non-stationary if a reasonable
number of points has been used.
Once the sampling interval has been determined, the series
is segmented into independent samples of length N. Initially the
series is tested with N = 5, then N is increased to 10, and the
final test, if there are enough data points, is for N = 15.
:he minimum test is for KK = N samples of length N. If there
are not enough data points for this number of samples, the re
quirement for independent samples is relaxed slightly (i.e. the
sample separation interval is steadily decreased to a limiting
value equal to the number of lags necessary for the autocorrelation
function to damp to .100). If at this sample interval there are
not N samples, the test is aborted. If this happens at N = 5, it
may be an indication of nonstationarity.
If we have KK independent samples of length N the series is
tested for stationarity at three confidence intervals (95%, 97.5%,
99%) in the following manner
a) if KK is greater than or equal to 12, the Bendat
6
and Piersol test statistics and the Bryan test
statistics are both utilized.
b) if KK is less than 1~ only the Bryan test statistics
are utilized.
The series is considered stationary if both the Run and Trend
tests show no trends or variations for the mean and variance series
and if the two test statistics in the Bryan test indicate station
arity.
In the case where only one method is utilized (i.e. KK < 12)
stationarity is tested on the merits of the Bryan test statistics
alone.
It should be noted that the 95 percent confidence interval is
the most restrictive (i.e. the smallest acceptance region) and the
other confidence intervals progressively less restrictive.
Also results at the largest N used are preferable since more
data points are used in each sample to determine the test statis
tic and the assumption of normality in the Bryan test statistics
is more closely approximated.
COMPUTER PROGRAM STEST
The stationarity test has been programmed on an IBM 360/30 as
computer program STEST. A copy of the computer program is con
tained in the Appendix.
The program requires approximately 53,000 bytes of storage
and to test a series of 400 data points for all values of N re
quires approximately ~ ~inutes of computer time.
Pt·ogram Input
The input to the program is simply the number of data points
in the series and the values of the data points. An example is
contained in the Appendix.
Program Output
The program output has several forms. Initially the lag re
quired for the autocorrelatiot function to damp to zero is indi
cated along with the value of the autocorrelation function at
7
that lag.
If the autocorrelation function does not damp to zero, a
statement is printed stating that this occurred and that it may be
indicative of non-stationarity.
The stationarity te3t is then cc-·.ducted for values of N = 5,
10, 15. If at any value there are not at least N samples available
for testing, the sampling interval i& progressively decreased to a
limiting value to obtain N values. If N values are obtained in
this manner a statement indicating that correlated samples are
being used along with the autocorrelation lag and value at this
sample interval is printed out. If N samples cannot be obtained
the test is aborted.
If there are KK samples (KK ~ N) of length N, the test re
sults are printed out for the 95%, 97.5%, and 99% confidence inter
vals.
A sample output is contained in the Appendix.
TEST CASES
Seven series that have been tested are seri~ ~ A-F as given
in Box and Jenkins (1970) and a second order auto regressive pro
cess as given in Jenkins (1968). The results are shown in Table 1.
In all cases except series A the test accurately indicated the
stationarity or non-stationarity of the series. The discrepancy
in series A may be attributable to the fact that correlated samples
were used.
It is interesting to note that the series generated from the
second order AR process is stationary and the process itself is
stationary. Thus, in this case self-stationarity is indicative
of stationarity.
8
Table 1
Series A - Non stationary
STEST Results
- Correlated samples used -
A) N =- 5
95% confidence interval
97.5% confidence interval
99% confidence interval
B) N :;;; 10
non stationary
stationary
stationary
Not enough data points for independent or
correlated samples.
Series B - Non stationary
STEST Results
A) N ~ 5
There are not enough data points for independent
or correlated samples. Since this occurred for
a sample of length 5 and the length of the in
put series is 369, this may be indicative of
non-stationarity.
Ser4ss C - Non stationary
STEST Results
A) N a 5
95% confidence interval
97.5% confidence interval
99% confidence interval
B) N -= 10
non stationary
non stationary
non stationary
Not enough data points for independent or cor
related samples.
9
Series D - Non stationary
STEST Results
A) N =- 5
For N =- 5 there are not enough data points for
independent or correlated samples. Since this
occurred for a sample of length 5 and the length
of the input series is 310 this may be indica
tive of non-stationarity.
Series E - Stationary
STEST Results
A) N = 5
95% confidence interval
97.5% confidence interval
99% confidence interval
B) N = 10
non stationary
stationary
stationary
Not enough data points for independent or cor
related samples.
Series F - Stationary
STEST Results
A) N = 5
95% confidence interval
97.5% confidence interval
99% confidence interval
B) N • 10
stationary
stationary
stationary
Not enough data points for independent or cor
related samples.
10
Second Order A. R. Proces ~ - Stationary
STEST Results
A) N :-.: 5
95% confidence interval
97.5% confidence interval
99% confidence interval
B) N = 10
95~ confidence interval
97.5% confidence interval
99% confidence interval
C) N :: 15
95% confidence interval
97.5% confidence interval
99% confidence interval
11
stationary
stationary
stationary
stationary
stationary
stationary
stationary
stationary
stationary
Appendix
12
Computer Program STEST
13
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104 FOR~ATC///,tOx,•FOA N • 1 ,ll,?I,•JHEAf A~f. ""' ENOUGH DATA POINTS F lOR INOf.PfNOF~T OA tnRRElATEO ~AMPlf.S-----TfST A80ATE0. 1 1
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s..,t. Input
27
. SAMPLE IIPUT
(Seriea C - lOx aad Jeaklaa, 1970, p. 521)
0226 26.6 27.0 27.1 27.1 27.1 27.1 26.9 26.1 26.7 26.4
• • • 20.2 19.7 19.3 19.1 19.0 18.1
21
s..,t. Output
29
w 0
AAAAA LAG IIIPCallltiOII AAA*A
IMDIPBIIDIIT TAU • 20 I( 20) • -0.0033
AAAAA ITATtOMAIITY 118T fOl K • S AAAAA
95.0 PaCIII! CCIIFIDDICB tlft'IIYAL. Till DAtA !18!8 ... TAUCIMIY
97. 5 PDCIIft' COMPIDIIICI IMTIIYAL.
99.0 PDCIIIT CCIIFIDIMCI INIIRYAL.
twl DATA Tl8!8 ... !AftCIIAIY
'ftll DAft Ultl ... TAftOIMIY
AAAAA STATIOMAIITY !1ST POl I • 10 AAAAA
VITM BU • 20 All) II • 10 'ftllll All lUI' IIIOOOH DATA POIIJS rca IJI)IPIII)IIft IMIPLIS
rca II • 10 'niDI ARB NOT IIIOUGH DATA POIIft'S rca IMDiftii)UI' OR C«aULATID 8AMPLI8••••!18T DCaftD.
lndat, J. s., Mea-101, A. G., 1966, MIUua-.. t ad aa&1JI11 of
rud• data: ._ Yoa-k, 1111.,.
lox, G. 1. '·• J.akiaa, G. t.L, 1970, !~ ..riel aa&lJiil, fo~e
cutiaa aad coat .. o1: laa haaciaco, Ro1•ea-DaJ.
lr,aa, J~ G., 1967, Statiatica1 teat of tbe b,oth•aia that a tiM Mttiea ia atatioaHJ: GeophJiica, v. 22, p. 499-511.
Jeakiaa, G. M. , Vatta 1 D. G. 1 1961, SpecUal aaalJiil aDd ita
appllcatioat: Baa ~aaciaco, HoldeD•DaJ.
31
