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RE: Your comments on m-exponential equation (ii)
Hi, Zheng,
Thank you very much for your comments and corrections dated on Jan 28 to my abstract of " M-Exponential equation and vegetation dynamic analysis". As I promised, I am responding after I have red your mail and equipped my self well. I am sorry responding so late.
The main difference between you and me is that you are a well, too well, trained statistician and always judge things by statistics. But MDSM is not a statistic model. Instead, it is a vector analysis model. I attached a citation about Vector Analysis from The 1995 Grolier Multimedia Encyclopedia. Hope you will enjoy reading it.
Thank you very much for your interest.
Jay Bai
Attachment
Citation about Vector Analysis
from The 1995 Grolier Multimedia Encyclopedia
A vector, in mathematics, is a quantity stating both a magnitude and a direction. Force, velocity (speed in a particular direction), acceleration, angular momentum, and torque, for example, are quantities that are vectors. By contrast, a quantity with a magnitude but not a direction-mass and volume, for example-is called a scalar. Because physical quantities such as velocity and force are vectors, vector analysis-the mathematical manipulation of vectors-plays an important role in physics and engineering. Whereas scalars follow the ordinary arithmetical laws of addition, subtraction, and so on, vectors must be dealt with by geometrical techniques because they involve direction as well. The most simple example of this is the addition of two vectors, such as two forces acting on a body in two different directions. Vectors are commonly represented by directed line segments-whose direction and length represent the direction and magnitude of the quantity-and are written in boldface. Subtraction of vectors can be performed in the same manners simply by reversing the direction of the representative line segment to be subtracted and constructing the appropriate parallelogram. No limit exists to the number of vectors that can be handled graphically in this way. More-complex problems involving vectors, however, require the use of the calculus and other advanced mathematical operations.
Bibliography:
Campbell, Hugh G., Introduction to Matrices, Vectors and Linear Programming, 2d ed. (1977);
Crowe, Michael J., History of Vector Analysis (1967);
Davis, Harry F., and Snider, Arthur D., Introduction to Vector Analysis, 4th ed. (1979);
Thrall, R. M., and Tornheim, Leonard, Vector Spaces and Matrices (1957; repr. 1970).
>
>
> Hi, Dr. Jay Bai,
>
> Could you please give more explanation about the following
> equation you used. I have a hard time to understand it.
>
> y_i,k+x_=y_i,k_*t_i,k_^x^ + d_i,k+x_, I=1,2,...n.
>
> My questions are related to the relationships between
> y_i,k+x_ and d_i,k+x_.
>
> 1. Did you notice that alpha plus beta is equal to 2
> instead 1 in this equation. In the following part of your post,
> you mentioned alpha + beta = 1. If alpha + beta = 2, can you
provide your rationale for that.
Thank you for your comment and correction. Since now on, I am going to use alpha and 1-alpha to represent the weighing factors for sampled data and prediction. The equation then is looking like this:
Y_i,k+x_= (1-alph)*Y_i,k_*T_i,k_^x + alpha*D_i,k+x_
>
> 2. Is your purpose to estimate the population? Not really. My purpose is NOT to estimate the magnitude of the vector but the DIRECTION of the vector, or composition of a community.
> guess it is, because you used current sample information
> in the model. Otherwise, I will claim that I can do any
> prediction if you tell me the truth :-).
In classical statistics,
As I mentioned several times, MDSM is not a statistic model. It is a vector analysis model. It mainly works on vector directions. It adopted vector standardization, centralization, vector division, but not to variances, nor possibilities, yet.d_i,k+x_ is the unbiased
> estimate for the mean of y_i,k+x_ at time t+x, which means
> E(d_i,k+x_)=y_i,k+x_. When the sample error is
> approximating zero(e.g., by increasing the sample size), How about, if we could not increase the sample size. There are many limitations that limited our sample size, time, labor, money, etc.
> y_i,k+x_=d_i,k+x_ (alpha=0). Actually, for the unbiased
> estimate under any sample size, the alpha in your equation
> should be zero! The adjustion for the estimate of the mean
> (not the variance) is only needed for the biased sampling
> process.
>
> 3. Is d_i,k+x_ a positive number? If one always
> puts a positive number in his equation, it means the equation
> would under-predict the dependent variable if the positive
> number is removed. Is there any evidence that y_i,k_*t_i,k_^x^
> will under-predict y_i,k+x_? Why you did not use minus
> instead plus sign before d_i,k+x_ term?
> (the alpha + beta =2 causes such problem). As I said, I am going to use alpha and (1-alpha) instead of alpha and beta.
>
> 4. I would like to say that the use of the past
> information, and current limited sample information
> would be great idea for estimating the current
> population (both mean and variance), under the condition
> that large sample size is impossible to obtain.That is almost always the true situation.
> I understand your idea in the following equation.
>
> y_i,k+1_=alpha*p_i,k+1_+beta*d_i,k+1_
> alpha+beta=1
>
> Another good point you made is the alpha and beta
> values are related to the variance of the sampling and
> projection. However, the question is how to determine
> the alpha or the beta value. In your deterministic model,
> you only focus about the mean, it seems it is not
> necessary to adjust it for unbiased sample, and very
> difficult to adjust it for the biased sample.
>
> In more comprehensive model, the adjustion works for
> the mean and variance under biased and unbiased sample.
> The effect from the adjustion will yield more accurate
> and tight estimate of the distribution of the true
> population density. This is what exactly Bayesian
> statistics is doing. In classical Bayesian, there
> is some question to determine influences from the prior.
> It is very similar to the problem you met.
> The basic difference between your model and Bayesian is
> that you only use the mean value of all variables, and
> Bayesian uses the probability density distributions.
This is just the beginning of the model. It is still in a very basic stage.>
> I hope some day I can see some applications in this research
> direction. The dificulty is that there is no software available
If you are interested, you can down load a DOS executable software, SMM52.exe, from http://lamar.colostate.edu/~jbai. It is a step by step system monitoring program handling up to 52 variables and up to 30 time intervals.
> to do such analysis. One of my professor gave a Baysian
> question to me in my comprehensive test, if you have Mathcad
I don't have Mathcad.
> program and would like to play it, I can send one copy to
> you (the adjustion is carried out for both variance and mean).
>
> Your work is very interesting.
>
> I would be wrong in any point above, please not take
> it too seriously.
>
> Thank you for your attention, and good luck with your paper.
>
>
> Zheng Zeng
>
Thank you. Our first paper in English was published, Ecological Modelling, 97/1,2, pp.75-86. "Multi_dimensional sphere model and vegetation instantaneous trend analysis">