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Discussion on a temporal dynamics model (fwd)
Hi, friends,
I am forwarding the following message to this list. The
Multi-Dimensional Sphere Model (MDSM) is a new data synthesis
and temporal dynamics analysis tool for multi-variable, multi-
sample, and multi-time intervals data, D(i,j,k). It is still in
the research phase and needs your inputs to be improved.
I am going to present my research report on Session 155, on Aug.
14. I hope this my help me and my audience warmed up. If you
feel this is not for you, then please excuse me posting this
message here.
Thank you for your attention.
Jay Bai
Attachment:
Hi, Zeng,
Thank you for your important comments about "two observations",
"long time trend", "correlation of variables", and "linear
model". Let me summarize my responses:
a) One of the big advantage of using the MDSM is that it uses all
the information from D(i,j,k). Each state is generated from
j+j*(k-1)=j*k observations, instead of two. Every observation has
i values. See detail below.
b) The definition of instantaneous trend included the long time
trend, as time interval can be any length, year, decade, or
century, as long as data supplies. Instantaneous trend is a
function of time, it can be visualized as the tangent in n-space.
c) The correlation between one and all is considered, and this is
reflected in the standardization formula,
e(i)/SQRT(SUM(e(i)^2)),
instead of correlation between each others. The standardization
is the key of this model. It filters out the annual fluctuation,
reflects the relations of the species with all others, and
visualizes the dynamics by projecting the vegetation onto the
hypersphere.
d) MDSM is not a linear model, as it uses an alternative
definitions of vector multiplication and division. However, It IS
a multivariate space model. It expects "linear in the
neighborhood" (Jameson, 1986). This means it is an exponential
model, but can be treated as linear in a very short segment.
I was telling people that this model is basic, simple, and easy.
I hope this discussion would not scare potential users away.
Thank you.
Jay Bai
Your mail is attached with my respond with ** under related line.
P.S. I realized you have another post. I am working on it after
this. Looks the most of the comments on that post have been
discussed here.
=======================
Date: Sun, 28 Jul 1996 22:24:46 -0600 (MDT)
To: Tugjayzhab Bai <jbai@lamar.ColoState.EDU>
Cc: sino-ecol@biomod.fw.vt.edu, mdsm@gpsrv1.gpsr.colostate.edu
Subject: Re: Your comments on MDSM
On Sun, 28 Jul 1996, Tugjayzhab Bai wrote:
> Hi, Zeng,
>
> Glad to hear your comments on the MDSM again. I appreciate your
> time and effort to be help. So I tried my best to respond to
your
> mail. I copied your mail and add my comments with ** under
> elated context.
Hi, Dr. Jai Bai.
Thank you for your response!
** Again, my name is Jay Bai.
>
> There are a few things I have to point out that the t, x are
> n-vector so they should be in lower case, conventionally. Thus
the
> subscript of time should be k, instead of t, in sake of our
> discussion.
That's fine.
> As I mentioned in the last mail to you and QiYe. This model is
in
> its beginning phase. Don't treat it too fancy, too complex. It
is
> very basic and simple, now. Just like your salary increased
from
> 100k to 110k, then the trend is 11/10=1.1000. As t is greater
than
> one, it is increasing, and the increasing rate is 110%... The
MDSM
> is the extensions of division and percentage from one dimension
to
> n-dimension. That is all about this model.
This is the key in your model! However, the value you get, I
think, is the fact of increase instead of the trend. You
considered
as the trend, and then used it for the prediction in a stochastic
world.
** You may consider the world is stochastic, not me, nor this
model.
In my view, you can define it as the trend, but you cannot use it
to
do the prediction and infer any changes in the future. This is
our
major difference. Based on my understanding, nobody can detect
the
trend based on two observations. Please see more in the
followings.
** It is not based on two observations. It is based on two
** sequential states, and every states included j observations
plus
** a prediction that is from previous j*(k-1) observations, total
**of j*k observation.
** e(k+1)=p(k+1)+d(k+1), but
** d(k+1)=avg(d(j,k+1)), and
** p(k+1)=d(k)*t(k)
** =d(k)*[e(k)/e(k-1)]
** =d(k)*[d(k)+p(k)]/[d(k-1)+p(k-1)]
** =d(k)*[d(k)+d(k-1)*t(k-1)]/[d(k-1)+d(k-2)*t(k-2)]
** ...
>
> In this note we can not discuss much. If you are interested, I
can
> send you my new paper or proposal in ASCII form. As this is a
new
> model, it changes every month. What you were comment on are
mostly
> be solved in email discussion, e.g. "what we can discover
> from 1-2-3 to 2-3-4?".
I read "from 1-2-4 to 2-3-4". I will be happy if you can send me
your new paper. Now, it may be not appropriate for me to read
your proposal, because one of my past graduate committee members
is
going to submit a proposal for temporal and spatial data
analysis.
My mail address is
Dept. of Entomology
Montana State University
Bozeman, MT 59717
** 59717 or 59715?
>
> Thank you for your attention.
>
> Jay Bai
>
> ===================
> I read Dr. Jai Bai's MDSM paper last year as well as some
of
> ** Jay Bai
>
>
> I admire Dr. Jai Bai's spirit in enhancing this new
methodology,
> which from my view, totally differ from many ecological
> literatures I have read.
> ** Thank you.
>
> 1. I personally think that MDSM method is nothing to do with
the
> trend analysis, and the method itself is untestable.
> ** Try to be nice and positive.
I wish I could. The formal trend analysis needs a very long
time
** how long is very long? **
series. MDSM caimed that it is a trend analysis method and could
detect the trend based on two observations. This is not true.
Nobody can detect that based on two observations in a stochastic
world. One may be interested in Manly Bryan's randomization book
or
Harvey's structural time series book. The later one includes
many
ad hoc approaches in modeling the trend.
I believe, if you get some ideas from any standard approaches,
you
may not call current MDSM as a trend analysis method. By the
way,
Bryan will be in this year ESA meeting too.
** Could you introduce me to him?
>
> 1a. Prediction by MDSM.
> Based on Dr. Jai Bai, his MDSM model can be formulated as
> X(t-1)*T(t)=X(t), where X is the standardized community state
> vector based on the formula in (2). T(t) is a ??trend matrix??
> ** t(k) is multivariate-instantaneous-trend, an n-vector,
How about multivariate-instantaneous change rate. It is a
n-vector only if you force the species to be independent with
each
other. However, this is not true. At least your transformation
will
make them correlated.
** Do you really think all the samples in statistic are true
**independent? No! Not in any meaning, except having the equal
**chance to be sampled. In another words, they are only be
treated
**like independent, under the condition of being randomly
sampled.
**In an extremely example, all people in the world are related,
some
**how. If classified by gender, there are only two groups of
people.
**But this fact never invalidated any statistical analysis
performed
**by social scientists.
**And I suppose the relation among the plant species are much
less
**important, strong, than that of between the people. When
**performing vegetation dynamics analysis, we should treat the
**change of cholla cactus independent from the change of blue
grama.
**Don't we?
>
> and can be solved as T(t)=X(t)/X(t-1). In some case, Dr. Jai
> ** t(k-1)=x(k)/x(k-1), there is a time lag there.
Sorry for the error.
> considered it as the constant; while in some case, Dr. Jai Bai
> used it as the time-varying parameter. It is better to
consider
> the trend matrix T as the constant matrix in the following.
> ** t is short for multivariate instantaneous trend, it is an
> ** n-vector, not matrix, it has a subscript k, so it is a
function
> ** of time.
You may need to consider it as the matrix in the model.
>
> Here is the problem in MDSM. It is not a valid method to
> estimate the trend index by letting T = X(t)/X(t-1)! Based on
> **Don't treat this model too complex. It is very basic and
simple:
> ** if your hourly wage is 11 dollar per hour, but was 10 dollar
> ** per hour last months, then the trend of your wage is going
up,
> ** and the trend value is 11/10=1.1000, a 110% increase!
This is not the trend. But I like it if it were true, then I
would be rich some day. :)
** You may name it changing ratio, but that is not a big deal, is
** it?.
>
> Dr. Jai Bai, an more appropriate model expression is:
>
> X(t) = X(t-1)*T+E(t).
>
> E(t) can be assumed to be a normally, independent random
vector.
> ** This is for prediction, but the previous one is for
> ** calculation of trend. Definition of trend and prediction are
>
> ** related, but they are different.
> ** p(k+1)=d(k)*t(k)+E(k+1)
I suggest using following dynamic model without considering
the
observation error,
d(k+1)=d(k)*t(k)+E(k+1)
The conditional mean for this model is E(d(k+1))=d(k)*t(k)
The prediction variance is
var(d(k+1))=var(d(k)*t(k))+
var(E(k+1))=t(k)*d(k)*t(k)^t+var(E(k+1))
The trend is determined by the value in t(k)(it is a matrix)
This is the real prediction instead of yours.
** All right. If I am not worry about the var, then there is no
big
**difference.
>
> If one wants to do prediction based on the model above, one
needs
> a stochastic term, because all variation in X(t) cannot be
> explained by X(t-1)! This model is actually called as a
> multivariate first order autoregressive model. There are a few
> *** Thank you. Any citation?
You may find it in many time series textbooks.
** I knew you would say something like this.
Because you do not consider the correlation between
different species, you can estimate one species by one species
using
univariate time series analysis method.(But, this assumption is
not
true).
** That is what MDSM doing, species by species.
> methods to estimate the parameters in T and E.
> But,?? T is not equal to X(t)/X(t-1)! ??
> ** Why not? It is the definition of the t: t(k-1)=x(k)/x(k-1)
To compute the multivariate-instantaneous change rate between k
and k-1, you can do it.
In order to estimate a constant t based on information between
time
1 to 2, 2 to 3, ..., n-1 to n, you cannot solve t as
t(k-1)=x(k)/x(k-1).
** Why a constant t? MDSM is not estimating a constant t,
instead,
**it estimate the instantaneous trend, which is a function of
time.
** This is a exponential model, but can be simulated as linear.
This is almost identical to a linear regression.
** Almost.
> **
> 1b. Stochastic disturbance is considered as the trend in
> MDSM method in the paper I read. This is inappropriate because
> ??it fails to separate the noise from trends. ??
As the example given in another post, MDSM will trace the
noise
in a stochastic world.
>
> 1c. Model X(t) = X(t-1)*T+E(t) model does not necessarily
> mean any trends and instability of the model. There are some
> criteria for ??matrix T?? to judge the stability of the model.
Yes.
>
> 1d. Nobody can test any hypothesis based on original MDSM
> model, because it is impossible to form a null and alternative
> hypotheses there. Nobody knows how large is "large" for the
> element in matrix T without knowing the variation of the
element.
> For example, if one gets 0.5 in one element of the "trend
> matrix", this value does not mean any trend.
> ** The prediction was judged by actual samples from next year.
> ** e(k+1)=p(k+1)*d(k+1), instead of hypothesis.
> ** The expectation of the trend value is one, t>1 increasing;
> ** t<1, decreasing. People don't need to know the variance
> ** distribution to find out 1 is greater than 0.9
No, unless you have a zero variance. In a stochastic world,
one
can be equal to 1000 under a limited number of sample size.
** Why you suppose MDSM is a statistic model? Statistics happens
to
** be used in multivariate analysis does not mean every model
**dealing with multi-variable is statistics. As indicated in the
**name this is a multivariable space model.
>
> 1e. Trend analysis can always be tested by simulations.
I
> wonder if any one can do such simulation study related to the
> power of the MDSM method in testing the trend? I guess that
one
> is not able to setup a simulation, because MDSM is not real
> dynamic model.
>
> 2. Is MDSM a good index for monitoring community change?
> Look at the following example before making your conclusion.
>
> Generally, we do not suggest doing a calculation such as a
a
> apples + b oranges = ?
> **a+b= fruit, in this case.
>
> then what does a apples/(a apples + b oranges) mean?
> ** apple/fruit mean how much share is the apple in the fruit.
> ** Then MDSM compares the sequential apple shares in fruit
> ** to determine if the apple-share increased or decreased
> ** in the total fruit production.
This is interesting. I am still confused. Because, the
proportion
will be controlled by high density species. And all data may no
longer independent with each other. My question: what is the
advantage to transform the data and what is the disadvantage in
using
the original abundance data in dynamic modeling?
> How about a^m apples /(a^m apples + b^m oranges)^(1/m)?
In
> MDSM, m is equal to 2. What is the advantage letting m=2
instead
> of letting m=1.
> ** Advantage of m=2 is that every variable is orthogonal to
each
> ** other, independent. For example, 3+4=7 is in one dimension.
> **
> ** But 3^2+4^2=5^2, is in two dimension (Shang Gao). The 3,4,5
> ** makes a triangle. In this triangle, 3/5=0.6
> ** is the cosine of angle between 3 and 5, 4/5=0.8 is the
> ** cosine between 4 and 5. Thus these two cosines determined
> ** the direction of the vector(3,4) in two dimension space.
> ** When the 0.6 and 0.8 changes over time, the vector rotates.
> ** And the composition of the system changes, i.e., fruit
production
> ** in the above case.
I understand how it works. But I fails to find the advantages
for
doing this.
** The advantages: It simplifies, visualizes, and fully uses
**information.
>
> I am curious about such an additive algorithm regarding
elements
> in different entities.
>
> Summary: The basic task in trend monitoring is to
separate
> the trend from noise. A good trend analysis should help
> ** MDSM filter out the random sampling error by centralization,
No. MDSM cannot do that unless you use some complex modeling
approach.
** Are you saying you don't want to accept MDSM, because it is
not
** complex?
> ** or say averaging matrix to centroid n-vector;
You may remove the intercept in the model by doing this.
> ** and filter out the annual fluctuation by standardization, or
It may be true. But you may remove the trend either. Can you
remove the annual fluctuation only without removing the trend?
Cay
you prove this?
** use MDSM to analyze 1-2-3 to 2-3-4, you can prove it yourself.
> ** divided by vector sum. These two are the main noise factors
> ** in vegetation data.
> ** D(k)= t(k)+Y(k)+S(k), our data included: trend, yearly
fluctuation,
> ** sampling error,... mainly.
>
Yes.
> prediction of future and explain things happened in the past.
> Trend monitoring is more complex than we thought. We need to
> know if there are any trends exist; what is the type of trends
> (e.g., circles, linear trends); what are the potential
mechanisms
> ** It is not necessary to be complex. or say we start with
simple.
> ** all the multivariate-instantaneous-trend is straight line,
> ** the tangent line, as the instantaneous speed in two
dimension.
This only works in deterministic and linear world.
** And deterministic and linear world is the basics. If there is
no
** special reason, let start with the basic. Or until someone
prove
** the simple one does not work, lets start with the simple one,
and
**improve it to higher level.
>
> to generate such trend and how to test trend hypothesis.
Current
> version MDSM seems nothing to do with these topics.
>
> I hope to see Dr. Jai Bai's new version of MDSM. I agree with
Dr.
> ** If you are serious, I can email you my proposal in ASCII
form.
Paper is fine for me. Thanks!
> With a data set fewer than 10 observations, it is less
> likely for one to find any trend in the data. Based on my
> experience in modeling population dynamics. If the data set is
> fewer than 20 observations, one may not detect complex behavior
> built in the time series data.
>
> *P.S. Hi, Zeng, what is your email address? I would like to
send
you
> *some new development on MDSM. To me, your comments mostly
based
on
> *last year' discussion.
>
I am very happy that I can see more changes in your research.
From
your this response, I found that you paid some attention about
the
seasonality and observation error (sampling error).
** Yes. The seasonality and observation error are filtered out by
** standardization and centralization. As most of us agree that
** these two are the main noise for the vegetation data, this
model
** should work.
Now you considered your model might be some kind of stochastic
model. I think it is a complex model too. However, there are
some
statistical problems in your model formulation, parameter
estimation
and prediction. I think it may prevent you from doing
simulations
and testing your approaches. If you follow the common approaches,
I
guess you will end without enough data.
If everybody accepts your transformation method, I think the
model
part is a very weak part. If you solve the modeling approach,
you
may find you do not need to use the transformation, because the
dynamic model is the natural way in monitoring the community. If
you solve these problems, you may get some wonderful temporal and
spatial community dynamic model, though now it is still far from
that.
** Sorry, I did not get these. What are transformation and
modeling
** approach ? (P.S. I saw you another post)
All of my views are serious, and based on my modeling experience
in
** I thought you are for fun.
temporal and spatial analysis. I hope that you can falsify all
my
views. If you like, we can discuss more about MDSM during ESA
meeting.
Thank you!
Zheng Zeng