Re: some basic questions on time series analysis

The problem you seem to be talking about is related to sand piles
and their application to earthquake theory.
Per Bak's classic book :
http://www.amazon.com/exec/obidos/tg/detail/-/0387947914/qid=1114528945/sr=1-1/ref=sr_1_1/104-6211140-5094309?v=glance&s=books
Editorial Reviews
Reason, Steven Postrel
. . . In print, at least, what might seem arrogant comes across as a kind of innocent, childlike enthusiasm, a lack of concern
for anything but the sheer joy of figuring things out. His ruthless simplifications of geology, evolution, and neurology pay off
because, as Bak notes, his models describe behavior that is common across these domains. This universality means that trampling across others' turf is not only acceptable, but almost mandatory,
if the underlying principles are to be exposed. Finally, for the most part, Bak wants the reader
to grasp the basic logic of his arguments; only rarely does he try to persuade with flights of poetic language or brute intellectual authority. 

paul wrote:

Hi,

Based on physical understanding of the stress-strain relationship for
soils, some persons feel that there exists an attractor for the
stress-strain curve.  I just came across time series analysis and it
struck me that if there does exist an attractor, then the methods of
time-series analysis should indicate whether this is the case or not.

So, I did a bit of reading and applied the basic methods to the data
that I have available.  I have requested more advanced books from the
local library, but in the meantime, have the following basic questions,
that I thought I would ask this group.

1. Can the time series analysis methods be applied to stress-strain
data.  I have readings of stress vs. strain taken at uniform strain
increments.  The test is run at a constant strain rate.  I think the
answer to my question is yes, just want to confirm.

2. As the test proceeds, the soil sample deforms, bringing in a strain
(time) dependant error, and so, I assume, making time a relelvant
variable i.e. the coupled non-linear system of equations that describe
this test are probably non-autonomous.  Question:  does this impact my
Time Series Analysis in any different way?

3.I have data for six tests.  For each test, I have a time series
analysis for the hydrostatic stress and the octahedral stress q.  I
treat each independantly as a time series and find that the dimension
of the attractor saturates at a value between 0.9 and 1.1.  From my
reading I know that a 1D attractor implies a line.  What does an
attractor that is a non-integer whose value is close to one mean?

4. My phase space portraits typically can be approximated as two
straight lines, connected by a curve, the first line at an angle of say
70 degrees to the horizontal and the second at say 30 degrees to the
horizontal, and heading out at this angle, as a straight line.  What
does this mean if anything?

5. The embedded dimension seems to be between 1 and 5 depending on how
one defines saturation at the resulting curve.  So typically, I get the
following points on the graph of Correlation Dimension vs. Embedded
Dimension (0.94,1) (1.01, 2) (1.04, 3) (1.07, 4) (1.1, 5) (1.1, 6)
(1.1, 7) From other theory, it should be two (if time is not a
variable) and three if time is a varaible.  When is saturation reached
for data such as the one shown above?

6. Related to the question above, I suppose that the embedded
dimension, being that it deals with the number of relevant variables,
has to be an integer?  Given that the sample deforms with time and that
this may be bringing in "error" and impacting the values, is time one
of these dimensions?

7. Also, from the data, I get that the first AMI minimum is found at a
time lag of between 5 to 19, and usually is around 7.  I use this value
of the time lag to get the embedded dimension and correlated dimension.
 My question is, what is the physical significance of the time lag?  My
results (fortuneately) don't seem to be too sensitive to the actual
value chosen.

8. I have also come across a number of papers with titles such as "Does
the Length of Day have a Strange Attractor" and "Strange Attractor of
the Geomagnetic Field Variation" etc.  Usually they calculate the
Correlation Dimension and the Embedded Dimension and show some phase
portaits.  My question is this:  In a classic paper of this type, what
needs to be shown in addition to these to prove that there is a
Attractor (strange or otherwise) in the physical process being
observed.

Thank you very much!
Paul



--
Roger L. Bagula email: rlbagula@xxxxxxxxxxxxx or rlbagulatftn@xxxxxxxxx
11759 Waterhill Road,
Lakeside, Ca. 92040 telephone: 619-561-0814}
.

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