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Thread Subject:
Tests for stationarity, ergodicity

Subject: Tests for stationarity, ergodicity

From: Jerry Malone

Date: 24 Nov, 2002 13:15:00

Message: 1 of 3

I have a series of data in an array (approximately 12,000 samples)
and I would like to know if the data is stationary (and if so, if it
is ergodic).


Is there a simple way to do this in Matlab?


Thanks,


--Jerry

Subject: Tests for stationarity, ergodicity

From: Ken Davis

Date: 24 Nov, 2002 18:35:20

Message: 2 of 3

Hello,

Stationarity and ergodicity are properties of the underlying stochastic
process and not properties of a single realization of that process.
Stationarity is the property that the expected values of the moments of the
process are independent of the temporal index. Ergodicity is the property
that the expected values of the moments of the process are equal to the time
averages of the moments of the process. Since the expectation operator is
the average over all realizations of the process, you can't say anything for
sure with just a single realization of the process. If, however, you assume
in advance that the process is ergodic, then you can draw conclusions about
ensemble averages knowing only the time average, which you can estimate from
a single realization.

Thus, if you have only a single time series AND YOU ASSUME ergodicity, then
you can test for stationarity by seeing whether the statistics (mean,
variance, etc.) as estimated in one part of the time series is close enough
to the statistics as estimated in other parts of the time series.

HTH,

Ken

"Jerry Malone" <malonej@uscolo.edu> wrote in message
news:eeb5724.-1@WebX.raydaftYaTP...
> I have a series of data in an array (approximately 12,000 samples)
> and I would like to know if the data is stationary (and if so, if it
> is ergodic).
>
>
> Is there a simple way to do this in Matlab?
>
>
> Thanks,
>
>
> --Jerry

Subject: Tests for stationarity, ergodicity

From: Jerry Malone

Date: 25 Nov, 2002 10:18:49

Message: 3 of 3

Thanks very much.
You are right.


Actually, what I am trying to do is prove that the underlying process
is in fact not stationary.


Previous researchers (in the 70s) have determined that the process is
partly deterministic and partly random; but in the literature this
fact is often ignored and people take great liberties in analyzing
this type of data, assuming that it is stationary.


The data is distance-, rather than time-based, and it changes very
slowly. It I take one series of measurements, then wait 6 months and
take a second, they will look nearly identical, except for a few
localized differences. In fact, I can do this over many years, and
still see only minor trends.


Therefore, I understand why many would assume it is ergodic. One
sample appears to have the same statistical properties as other
samples.


However, if I take any single sample and divide it into short
segments, then groups of these segments have significantly different
statistical properties than other groups. (One channel, for example,
shows obvious, rapid changes in mean value at certain locations.)


If I take a single, representative sample, break it up into segments,
and and demonstate that these segments are statistically very
different, is that sufficent to show that the data is not stationary?
(Or perhaps only locally stationary?)


Thanks,
--Jerry


Ken Davis wrote:
>
>
> Hello,
>
> Stationarity and ergodicity are properties of the underlying
stochastic
> process and not properties of a single realization of that process.
> Stationarity is the property that the expected values of the
moments of the
> process are independent of the temporal index. Ergodicity is the
property
> that the expected values of the moments of the process are equal to
the
> time
> averages of the moments of the process. Since the expectation
operator is
> the average over all realizations of the process, you can't say
anything
> for
> sure with just a single realization of the process. If, however,
you assume
> in advance that the process is ergodic, then you can draw
conclusions about
> ensemble averages knowing only the time average, which you can
estimate
> from
> a single realization.
>
> Thus, if you have only a single time series AND YOU ASSUME
ergodicity, then
> you can test for stationarity by seeing whether the statistics
(mean,
> variance, etc.) as estimated in one part of the time series is
close enough
> to the statistics as estimated in other parts of the time series.
>
> HTH,
>
> Ken
>
> "Jerry Malone" <malonej@uscolo.edu> wrote in message
> news:eeb5724.-1@WebX.raydaftYaTP...
>> I have a series of data in an array (approximately 12,000
samples)
>> and I would like to know if the data is stationary (and if so,
if it
>> is ergodic).
>>
>>
>> Is there a simple way to do this in Matlab?
>>
>>
>> Thanks,
>>
>>
>> --Jerry
>
>
>

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