Friends, I want to write my new posts about 2D Z-series in a programming context so you could use some main functions for powers and products of 2D Z-series and could also take their expectations in order to get jump-started with programming. But I will write detailed posts tomorrow.
Here are some notes copied from past but more detailed information tomorrow.
Two Dimensional Z-Series/Hermite-Series:
Case 1: Random variable X indepenent and Random Variable Y dependent and Conditional on X
Suppose we have two non-Gaussian continuous random variables, X and Y and we can make observations of their joint realizations. We suppose that X is the independent random variable and Y is the dependent random variable. This dependence might follow from causality but would usually be arbitrary for the analytics and we could possibly reverse the dependent and independent random variables and successfully re-do the analytics.
We can calculate the univariate Z-series/Hermite-series of independent random variable X as
\[\,X(Z_x)\,=\, ah_0 \, +\, ah_1 \,H_1(Z_x) \,+\, ah_2 \,H_2(Z_x) \,+\, ah_3 \,H_3(Z_x) \,+\,\ldots\]
Even though we can make a univariate Z-series/Hermite-Series of Y, this would not be helpful from two dimensional perspective. We just define a conditional version of Y as
\begin{align}
\left[Y|X\right](Z_y \, , \, Z_x)\, \,&=\, \Big[\, cH_{0,0} \, + \, cH_{0,1} \,H_1(Z_x) \,+\, cH_{0,2} \, H_2(Z_x) \,\, + \, cH_{0,3} \,H_3(Z_x)\,\,+\,\ldots\,\Big] \\
&+\, \Big[ \, cH_{1,0} \, + \, cH_{1,1} \,H_1(Z_x) \,+\, cH_{1,2} \, H_2(Z_x) \, + \, cH_{1,3} \,H_3(Z_x)\,\,+\,\ldots\, \Big] \, H_1(Z_y) \, \\
&+\, \Big[ \, cH_{2,0} \, + \, cH_{2,1} \,H_1(Z_x) \,+\, cH_{2,2} \, H_2(Z_x) \, \, + \, cH_{2,3} \,H_3(Z_x)\,\,+\,\ldots\, \Big] \, H_2(Z_y) \, \\
&+\,\,\,\ldots\, \\
\end{align}
Bivariate Joint Hermite-Series of X and Y is denoted by a random variable W in the form of a product (of joint realizations of X and Y) and is given as (please note that \(aH_{m,n}\) are merely numerical coefficients while \(H_n(.)\) are hermite polynomials)
\begin{align}
W(Z_y \, , \, Z_x)\, = \, X\, Y\,&=\, \Big[\, aH_{0,0} \, + \, aH_{0,1} \,H_1(Z_x) \,+\, aH_{0,2} \, H_2(Z_x) \,\, + \, aH_{0,3} \,H_3(Z_x)\,\,+\,\ldots\,\Big] \\
&+\, \Big[ \, aH_{1,0} \, + \, aH_{1,1} \,H_1(Z_x) \,+\, aH_{1,2} \, H_2(Z_x) \, + \, aH_{1,3} \,H_3(Z_x)\,\,+\,\ldots\, \Big] \, H_1(Z_y) \, \\
&+\, \Big[ \, aH_{2,0} \, + \, aH_{2,1} \,H_1(Z_x) \,+\, aH_{2,2} \, H_2(Z_x) \, \, + \, aH_{2,3} \,H_3(Z_x)\,\,+\,\ldots\, \Big] \, H_2(Z_y) \, \\
&+\,\,\,\ldots\, \\
\end{align}
The following version of conditional Hermite-Series would hold
\[ \,W(Z_y \, , \, Z_x)\, =\,\left[Y|X\right](Z_y \, , \, Z_x)\,\,X(Z_x)\,\]
Now, we want to do analysis of \(\left[Y|X\right](Z_y \, , \, Z_x)\, \). It is composed of three parts. The first part of Y is directly correlated with X and appears on first row in the formula of \(\left[Y|X\right](Z_y \, , \, Z_x)\, \). The second part of Y is that has variance independent of X and that appears in the first column in the formula of \(\left[Y|X\right](Z_y \, , \, Z_x)\, \). The third part of Y has variance that is dependent on X and this part appears below the first row and towards left of the first column. rewriting the formula of \(\left[Y|X\right](Z_y \, , \, Z_x)\, \), we have
\begin{align}
\left[Y|X\right](Z_x \, , \, Z_y)\, \,&=\, \Big[\, cH_{0,0} \, + \, cH_{0,1} \,H_1(Z_x) \,+\, cH_{0,2} \, H_2(Z_x) \,\, + \, cH_{0,3} \,H_3(Z_x)\,\,+\,\ldots\,\Big] \\
&+\, \Big[ \, cH_{1,0} \, + \, cH_{1,1} \,H_1(Z_x) \,+\, cH_{1,2} \, H_2(Z_x) \, + \, cH_{1,3} \,H_3(Z_x)\,\,+\,\ldots\, \Big] \, H_1(Z_y) \, \\
&+\, \Big[ \, cH_{2,0} \, + \, cH_{2,1} \,H_1(Z_x) \,+\, cH_{2,2} \, H_2(Z_x) \, \, + \, cH_{2,3} \,H_3(Z_x)\,\,+\,\ldots\, \Big] \, H_2(Z_y) \, \\
&+\,\,\,\ldots\, \\
\end{align}
The first row \( \Big[\, cH_{0,0} \, + \, cH_{0,1} \,H_1(Z_x) \,+\, cH_{0,2} \, H_2(Z_x) \,\, + \, cH_{0,3} \,H_3(Z_x)\,\,+\,\ldots\,\Big]\) has part of Y that is directly correlated with X.
And first column given as
\begin{align}
&+\, \Big[ \, cH_{1,0} \, \Big] \, H_1(Z_y) \, \\
&+\, \Big[ \, cH_{2,0} \, \Big] \, H_2(Z_y) \, \\
&+\, \Big[ \, cH_{3,0} \, \Big] \, H_3(Z_y) \, \\
&+\,\,\,\ldots\, \\
\end{align}
is independent of X while the remaining body given as
\begin{align}
&+\, \Big[ \, cH_{1,1} \,H_1(Z_x) \,+\, cH_{1,2} \, H_2(Z_x) \, + \, cH_{1,3} \,H_3(Z_x)\,\,+\,\ldots\, \Big] \, H_1(Z_y) \, \\
&+\, \Big[ \, cH_{2,1} \,H_1(Z_x) \,+\, cH_{2,2} \, H_2(Z_x) \, \, + \, cH_{2,3} \,H_3(Z_x)\,\,+\,\ldots\, \Big] \, H_2(Z_y) \, \\
&+\, \Big[ \, cH_{3,1} \,H_1(Z_x) \,+\, cH_{3,2} \, H_2(Z_x) \, \, + \, cH_{3,3} \,H_3(Z_x)\,\,+\,\ldots\, \Big] \, H_3(Z_y) \, \\
&+\,\,\,\ldots\, \\
\end{align}
is the part whose variance systematically depends upon X.
Friends, I had forgotten to take my tablets(kemadrin) that offset some of the side-effects of antipsychotic injections and therefore had a rather bad evening. I just took the tablets but I am feeling very tired. I want to explain things in the programming context so many friends start writing their own simple functions with 2D Z-series. I am going to sleep now but will write a new post tomorrow.
Serving the Quantitative Finance Community
Re: Breakthrough in the theory of stochastic differential equations and their simulation
You think life is a secret, Life is only love of flying, It has seen many ups and downs, But it likes travel more than the destination. Allama Iqbal
Re: Breakthrough in the theory of stochastic differential equations and their simulation
Friends, In our 2D Framework, there are two basic building blocks/variables. The first variable is univariate Z-series of independent variable X that is mapped on a single independent Gaussian \(Z_x\) . And the second variable is Y which is dependent on both \(Z_x\) and \(Z_y\) and has a 2D Z-series.
The univariate Z-series of X is given as (equivalently, we could have specified Hermite-series)
\[\,X(Z_x)\,=\, a_0 \, +\, a_1 \,Z_x \,+\, a_2 \,{Z_x}^2 \,+\, a_3 \,{Z_x}^3+\, \ldots \]
This can be written more briefly as
\[\,X(Z_x)\,\,=\, \sum\limits_{m=0}^{M}\, a_{m}\,{Z_{x}}^m \]
The bivariate conditional Hermite-Series of X conditional on Y is given as (please note that \(b_{m,n}\) are numerical coefficients). In matlab indices of arrays start at one so in actual programming the indices below are offset by one.
By construction different underlying Gaussians that appear in 2D Z-series are orthogonal and independent.
\begin{align}
\left[Y|X\right](Z_y \, , \, Z_x)\, \,&=\, \Big[\, b_{0,0} \, + \, b_{0,1} \,Z_x \,+\, b_{0,2} \, {Z_x}^2 \,\, + \, b_{0,3} \,{Z_x}^3\,\,+\,\ldots\,\Big] \\
&+\, \Big[ \, b_{1,0} \, + \, b_{1,1} \,Z_x \,+\, b_{1,2} \, {Z_x}^2 \, + \, b_{1,3} \,{Z_x}^3\,\,+\,\ldots\, \Big] \, Z_y \, \\
&+\, \Big[ \, b_{2,0} \, + \, b_{2,1} \,{Z_x} \,+\, b_{2,2} \, {Z_x}^2 \, \, + \, b_{2,3} \,{Z_x}^3\,\,+\,\ldots\, \Big] \, {Z_y}^2 \, \\
&+\,\,\,\ldots\, \\
\end{align}
\[\,\left[Y|X\right](Z_y \, , \, Z_x)\, \,=\, \sum\limits_{n=0}^{N}\, \sum\limits_{m=0}^{M}\, b_{n,m}\,{Z_{y}}^n\,{Z_{x}}^m\,\]
Expected Value of a 2D Z-series is given as
\[\,E\left[Y|X\right](Z_y \, , \, Z_x)\, \,=\, \sum\limits_{n=0}^{N}\, \sum\limits_{m=0}^{M}\, b_{n,m}\,E[{Z_{y}}^n]\,E[{Z_{x}}^m]\,\]
For example pth moment of 2D Z-series [Y|X\right](Z_y \, , \, Z_x) is found by repeated series multiplication of Y by itself and then taking expectation over all Gaussians at the final series.
\[\,E\left[Y|X\right]^p(Z_y \, , \, Z_x)\, \,=\, E\left[ {\left(\sum\limits_{n=0}^{N}\, \sum\limits_{m=0}^{M}\, b_{n,m}\,{Z_{Y}}^n\,{Z_{X}}^m\,\right)}^p\right] \]
Cross moments of X and Y are calculated as repeated series products with themselves to find appropriate powers of X and Y as series and then a final cross product between these series followed by taking expectation over powers of two Gaussians in the 2D Series.
\[\,E\left[Y^p\,X^q\, \right](Z_y \, , \, Z_x)\, \,=\, E\left[ {\left(\sum\limits_{n=0}^{N}\, \sum\limits_{m=0}^{M}\, b_{n,m}\,{Z_{Y}}^n\,{Z_{X}}^m\,\right)}^p \, {\left( \sum\limits_{l=0}^{L}\, a_{l}\,{Z_{x}}^l \, \right)}^q \right] \]
Since all of these calculations are series multiplications on which we later take expectations when necessary, I will take a brief stop to explain the matlab function that handles 2D Series multiplications. I have attached this program at the end.
Programming Notes:
The name of this function that handles multiplications of two 2D Z-series and takes expectation of the product is function "SeriesProduct2DAlgorithmicNew".
Here is the header of the function:
function [cc,EX] =SeriesProduct2DAlgorithmicNew(aa,bb,SeriesOrder10,SeriesOrder01,SeriesOrder20,SeriesOrder02)
It multiplies two 2D or 1D Z-series. aa and bb are both two dimensional arrays that carry the Coefficients of each of the two 2D Z-series that are being multiplied. Let us have two Z-series.
\[\,Y(Z_y \, , \, Z_x)\, \,=\, \sum\limits_{n=0}^{5}\, \sum\limits_{m=0}^{7}\, aa_{n,m}\,{Z_{y}}^n\,{Z_{x}}^m\,\]
The above series has OrderY=5 and OrderX=7. So SeriesOrder10=OrderY+1=6 And SeriesOrder01=OrderX+1=8
We want to multiply it with another Z-series that is just univariate and is given as: \[\,X(Z_x)\,\,=\, \sum\limits_{m=0}^{5}\, bb_{m}\,{Z_{x}}^m \]
This univariate Z-series has SeriesOrder20= OrderY+1 = 0+ 1=1 And SeriesOrder02= OrderX+1 = 5+ 1=6
We input these arguments \(aa_{n,m}\), \(bb_{m}\), SeriesOrder10, SeriesOrder01, SeriesOrder20, SeriesOrder02 into the above function to get the output 2D Z-series and its expectation.
The Series with coefficients cc that would be output will have SeriesOrder30=SeriesOrder10 + SeriesOrder20 -1 And SeriesOrder03=SeriesOrder01 + SeriesOrder02 -1 . Its coefficients range then would be in matlab notation as cc(1:SeriesOrder30, 1:SeriesOrder03)
The output variable cc contains coefficients of the resulting product 2D Z-series and output variable EX will give expectation of the resulting product 2D Z-series.
.
.
.
.
I will follow with more on graphing and calculation of 2D Z-series Density, Calculating conditional Density of dependent variable given a certain realization of Independent variable and also how to take Jacobian derivatives of the cross moments of the two independent 1D and dependent 2D Z-series for optimization purposes.
The univariate Z-series of X is given as (equivalently, we could have specified Hermite-series)
\[\,X(Z_x)\,=\, a_0 \, +\, a_1 \,Z_x \,+\, a_2 \,{Z_x}^2 \,+\, a_3 \,{Z_x}^3+\, \ldots \]
This can be written more briefly as
\[\,X(Z_x)\,\,=\, \sum\limits_{m=0}^{M}\, a_{m}\,{Z_{x}}^m \]
The bivariate conditional Hermite-Series of X conditional on Y is given as (please note that \(b_{m,n}\) are numerical coefficients). In matlab indices of arrays start at one so in actual programming the indices below are offset by one.
By construction different underlying Gaussians that appear in 2D Z-series are orthogonal and independent.
\begin{align}
\left[Y|X\right](Z_y \, , \, Z_x)\, \,&=\, \Big[\, b_{0,0} \, + \, b_{0,1} \,Z_x \,+\, b_{0,2} \, {Z_x}^2 \,\, + \, b_{0,3} \,{Z_x}^3\,\,+\,\ldots\,\Big] \\
&+\, \Big[ \, b_{1,0} \, + \, b_{1,1} \,Z_x \,+\, b_{1,2} \, {Z_x}^2 \, + \, b_{1,3} \,{Z_x}^3\,\,+\,\ldots\, \Big] \, Z_y \, \\
&+\, \Big[ \, b_{2,0} \, + \, b_{2,1} \,{Z_x} \,+\, b_{2,2} \, {Z_x}^2 \, \, + \, b_{2,3} \,{Z_x}^3\,\,+\,\ldots\, \Big] \, {Z_y}^2 \, \\
&+\,\,\,\ldots\, \\
\end{align}
\[\,\left[Y|X\right](Z_y \, , \, Z_x)\, \,=\, \sum\limits_{n=0}^{N}\, \sum\limits_{m=0}^{M}\, b_{n,m}\,{Z_{y}}^n\,{Z_{x}}^m\,\]
Expected Value of a 2D Z-series is given as
\[\,E\left[Y|X\right](Z_y \, , \, Z_x)\, \,=\, \sum\limits_{n=0}^{N}\, \sum\limits_{m=0}^{M}\, b_{n,m}\,E[{Z_{y}}^n]\,E[{Z_{x}}^m]\,\]
For example pth moment of 2D Z-series [Y|X\right](Z_y \, , \, Z_x) is found by repeated series multiplication of Y by itself and then taking expectation over all Gaussians at the final series.
\[\,E\left[Y|X\right]^p(Z_y \, , \, Z_x)\, \,=\, E\left[ {\left(\sum\limits_{n=0}^{N}\, \sum\limits_{m=0}^{M}\, b_{n,m}\,{Z_{Y}}^n\,{Z_{X}}^m\,\right)}^p\right] \]
Cross moments of X and Y are calculated as repeated series products with themselves to find appropriate powers of X and Y as series and then a final cross product between these series followed by taking expectation over powers of two Gaussians in the 2D Series.
\[\,E\left[Y^p\,X^q\, \right](Z_y \, , \, Z_x)\, \,=\, E\left[ {\left(\sum\limits_{n=0}^{N}\, \sum\limits_{m=0}^{M}\, b_{n,m}\,{Z_{Y}}^n\,{Z_{X}}^m\,\right)}^p \, {\left( \sum\limits_{l=0}^{L}\, a_{l}\,{Z_{x}}^l \, \right)}^q \right] \]
Since all of these calculations are series multiplications on which we later take expectations when necessary, I will take a brief stop to explain the matlab function that handles 2D Series multiplications. I have attached this program at the end.
Programming Notes:
The name of this function that handles multiplications of two 2D Z-series and takes expectation of the product is function "SeriesProduct2DAlgorithmicNew".
Here is the header of the function:
function [cc,EX] =SeriesProduct2DAlgorithmicNew(aa,bb,SeriesOrder10,SeriesOrder01,SeriesOrder20,SeriesOrder02)
It multiplies two 2D or 1D Z-series. aa and bb are both two dimensional arrays that carry the Coefficients of each of the two 2D Z-series that are being multiplied. Let us have two Z-series.
\[\,Y(Z_y \, , \, Z_x)\, \,=\, \sum\limits_{n=0}^{5}\, \sum\limits_{m=0}^{7}\, aa_{n,m}\,{Z_{y}}^n\,{Z_{x}}^m\,\]
The above series has OrderY=5 and OrderX=7. So SeriesOrder10=OrderY+1=6 And SeriesOrder01=OrderX+1=8
We want to multiply it with another Z-series that is just univariate and is given as: \[\,X(Z_x)\,\,=\, \sum\limits_{m=0}^{5}\, bb_{m}\,{Z_{x}}^m \]
This univariate Z-series has SeriesOrder20= OrderY+1 = 0+ 1=1 And SeriesOrder02= OrderX+1 = 5+ 1=6
We input these arguments \(aa_{n,m}\), \(bb_{m}\), SeriesOrder10, SeriesOrder01, SeriesOrder20, SeriesOrder02 into the above function to get the output 2D Z-series and its expectation.
The Series with coefficients cc that would be output will have SeriesOrder30=SeriesOrder10 + SeriesOrder20 -1 And SeriesOrder03=SeriesOrder01 + SeriesOrder02 -1 . Its coefficients range then would be in matlab notation as cc(1:SeriesOrder30, 1:SeriesOrder03)
The output variable cc contains coefficients of the resulting product 2D Z-series and output variable EX will give expectation of the resulting product 2D Z-series.
Code: Select all
function [cc,EX] =SeriesProduct2DAlgorithmicNew(aa,bb,SeriesOrder10,SeriesOrder01,SeriesOrder20,SeriesOrder02)
SeriesOrder30=SeriesOrder10+SeriesOrder20-1;
SeriesOrder03=SeriesOrder01+SeriesOrder02-1;
cc(1:SeriesOrder30,1:SeriesOrder03)=0;
for nn1=1:SeriesOrder30
if(nn1<=SeriesOrder10)
nn3=nn1;
else
nn3=SeriesOrder10;
end
for nn2=1:SeriesOrder03
if(nn2<=SeriesOrder01)
nn4=nn2;
else
nn4=SeriesOrder01;
end
for jj1=1:nn3
if(nn1-jj1+1 <=SeriesOrder20)
for jj2=1:nn4
if(nn2-jj2+1 <=SeriesOrder02)
cc(nn1,nn2)=cc(nn1,nn2)+aa(jj1,jj2).*bb(nn1-jj1+1,nn2-jj2+1);
end
end
end
end
end
end
[EZ] = EZZ2();
EX=0;
for nn1=1:SeriesOrder30
for nn2=1:SeriesOrder03
EX=EX+cc(nn1,nn2).*EZ(nn1).*EZ(nn2);
end
end
end.
.
.
Code: Select all
function [EZ] = EZZ2()
EZ(1)=1;
EZ(2)=0;
EZ(3)=1;
for nn=4:128 %Increase for larger products.
if rem(nn,2)==0
EZ(nn)=0;
else
EZ(nn)=EZ(nn-2)*(nn-2);
EZ(nn);
end
end
% EZ(64)
% EZ(64).*EZ(64)
endYou think life is a secret, Life is only love of flying, It has seen many ups and downs, But it likes travel more than the destination. Allama Iqbal
Re: Breakthrough in the theory of stochastic differential equations and their simulation
Friends, I could not do any work today. My mother probably had a minor stroke. She could move her left hand and fingers but the hand was numb. She was mostly talking fine but her speech would become slurry from time to time. She had just told us that her hand was numb but could be moved normally so in the morning we did not properly understand anything. Then I noticed that she was not perfectly saying some words from time to time. And then I noticed that at one point she lost balance while walking but could manage to not fall. At that point I was very worried and took her to hospital right away. We went to several hospitals but neurologists were mostly not available for appointments for several days. I took her to Jinnah hospital and they did her CT Scan and doctor told that no clot was visible and their might be some fleeting clot that did not last for long. In the evening I noticed that she was better but she dropped a glass from her left hand while eating at night and told me that she bit her lips with her teeth twice while eating food and I noticed that at one point very small amount of food came out of her mouth from one side and I was again a bit worried. I think a full recovery from stroke will take a few days or weeks. I will take her to hospital tomorrow again and hope she would be fine in a few days.
Coming back to my research, I have some very different ideas for calibration of the 2D dependent Z-series and I hope to start working on that tomorrow evening after I come back from hospital. I am very optimistic that there could be other ways to analytically improve the calibration to cross-moments. It may take me another 2-3 days after tomorrow to finish the work. Even though I cannot be sure if the idea would work perfectly, I am very very hopeful that it would work and look forward to making it work. I had continued to think of the ideas for past two days and was able to think more clearly today to fit everything together. I wanted to start working today but now would start hopefully tomorrow.
Coming back to my research, I have some very different ideas for calibration of the 2D dependent Z-series and I hope to start working on that tomorrow evening after I come back from hospital. I am very optimistic that there could be other ways to analytically improve the calibration to cross-moments. It may take me another 2-3 days after tomorrow to finish the work. Even though I cannot be sure if the idea would work perfectly, I am very very hopeful that it would work and look forward to making it work. I had continued to think of the ideas for past two days and was able to think more clearly today to fit everything together. I wanted to start working today but now would start hopefully tomorrow.
You think life is a secret, Life is only love of flying, It has seen many ups and downs, But it likes travel more than the destination. Allama Iqbal
Re: Breakthrough in the theory of stochastic differential equations and their simulation
Friends, my mind control is increasing like anything. It is becoming extremely difficult to get good water. They continue to drug underground water in more and more remote areas as I try to go far to get good water. Bread in the city is also being drugged and it is very difficult to get good bread for me. I have been taking bran bread by Dawn but I Was hit twice in past three days by drugged bran bread and then I bought a bread by Dory bakers and it was also drugged. Yesterday, I bought sandwich bread by Sunny breads and it was also drugged. They consider my research as bane of everything against them and think that I was able to get exposure only due to my research and want to stop my research and think once I would not be able to do my research, all good will gained and simultaneously all bad will against their crimes would erode in a few weeks. Rich and evil jews have no remorse about their crimes and more I expose their crimes, the more headstrong they become to stop me at all cost.
True stories of machinations by evil jews against many Americans are considered heresy by general American public who believes strongly in protecting the Jews at all cost and it is a long held tradition in United States to call even true criticism against Jews as anti-semitic talk. These ungrateful evil and cruel jews, on the other hand, respond to this great benevolence of Americans by retarding thousands of innocent American folks through mind control because American defense is a puppet in their hands. Knowledge about mind-control is well known in some jewish circles and many of them they know who to approach in the community to start anybody's mind control and a large number of simple good-natured American folks are placed under mind control when some jew is displeased with them.
Even though American society keeps jews in very high regard, they are not gold-plated. Just like any other people or race, some of them are extremely good human beings, some of them are extremely evil and some of them simply do not care. They tend to be more intelligent and when this intelligence shows itself in the form of Einstein we see one of the greatest human beings ever born but when this intelligence turns evil and machinatory we have exactly opposite form of humanity like evil jewish billionaire godfather/goldfather and his rich jewish cohort who has taken complete control of American army to advance their nefarious designs and have retarded thousands of innocent and intelligent people they did not like. So respect has to be shown in abundance to good people but very carefully to evil people. When great respect is shown to evil people, they become even more emboldened to advance their agendas based on hatred, venom, poison and malice.
I want friends to please protest to American defense that has become a puppet in the hands of rich jews. Black sheep in American defense behind mind control are given very prestigious jobs paying millions of dollars in Jewish companies after they retire and rich jews want to block talented muslims from being successful in United States with an iron fist, they would start wielding some influence and start opposing American policies in Israel and they want to nip this supposed evil in the bud. And American defense has no dearth of wannabees who desperately want to join the ranks of so many others who have gotten extremely high paying jobs with great social status in jewish companies after retirement from American defense. Please do not blindly believe me and find out somebody in your social circle(or well known to somebody in your social circle) you know well and who is in mind control agency of US defense and ask them about my allegations and your eyes would open forever about role of rich jews in United States and how they openly use mind control on anyone they wish especially on a large number of even white Americans whose families have been living in US for hundreds of years. If your eyes do not open forever, I am ready to take any blame you wish. Mind control is basis of symbiotic relationship between American defense and rich and evil Jewish cohort.
Especially very talented American people in science and technology are forced to work in Jewish owned companies due to fear of mind control. Many such people who want to leave are coldly told by administration to leave but their mind control is initiated by American defense on insistence of jews right after they leave the jewish company and once they know the reason behind mind control was that they tried to leave the jewish company against the will of the company owners. And due to large number of such precedents well known to many professionals in many jewish companies, most of the very talented people do not dare to leave the jewish companies against the will of administration since they know very well that they would be instantly put on mind control by American defense who dances on the tunes of rich jews since they want extremely high paying prestigious jobs with jewish companies having great social status. Please do not believe a single word I say blindly but try to find out the actual truth on your own from authentic sources you know well and you will find the truth that would open your eyes forever.
Many friends who are used to formal language all the time might be disturbed by my use of slang expletives for bad people but after reading Einstein's letter to Marie Curie, I consider myself perfectly validated to use the word of scum for American army and for bad jews who have no ethics and are ready to fall to every low level of humanity for some material gain or due to their hatred, venom, poison or malice as in case of bad jews.
Over past thirty years, lot of bad jews have developed a network in academia(with absolute apology to all the great jewish people in academia) and when a victim has to be retarded most of the jews in the network start orchestrating totally complete lies about the victim so that good people think that retarding the victim is indeed is a very good thing.
And I want to tell people that this network of evil jews has greatly perfected itself over past thirty years and not only muslims are retarded, everyone they do not like including europeans, blacks and a large number of white talented Americans are retarded at whim by this network of emboldened evil jews.
I want to request all good people in American society and in european countries to end and thwart this gang of evil jews who are bent on retarding intelligent people of every nation and creed they do not like.
Not everybody is the same, there are at the same time a large number of jews who have nothing to do with any of this and many good jewish companies who do not follow any inhuman practice. But there is a close group of rich jews who are actually involved in large scale mind control with American defsnse working as their proxy. Please try to know the truth on your own.
I want to advise American defense to watch a lot of American movies and try to behave like heroes in those movies. Instead people in American defense are short-tempered scumbags who became belligerent when anybody criticizes them. Contrary to American and Western spirit of freedom of criticism and speech that we all cherish, these short-tempered scumbags are unwilling to tolerate anything other than eulogies based on complete lies. Just like Henry Ford used to say that you can buy any color of car as long as it is black, to these scumbags of American defense the freedom of speech means that you can say anything you like as long it is a eulogy otherwise they become belligerent and start moving thousands of people to silence someone who speaks the truth about them. Let us all of us start cursing Erdogan together now.
When mature and deep people face some criticism, they respond by totally neglecting it if it is wrong. Immature, shallow and short-tempered people respond to criticism by getting angry and deeply offended and start jerking like psychos how to respond meanly to this criticism. American defense is full of such immature blokes and I am very sure that intolerant and arrogant scum of American army will again resort to torture and violence against me after reading my criticism.
I want to request(and suggest to good jews as a friend) to disavow any relationship with jewish scumbag billionaire and his rich jewish cohort who are involved with mind control of innocent and intelligent people of so many nations for their nefarious designs.
I also want to explain very clearly that I have been very severely criticizing bad jews and American defense but this is limited to criticism in writing only. I write only on Wilmott forum where I also contribute my research and I do not write anywhere else(at least for past 10-15 years). I do not want anyone including myself to take the law in our own hands. We have to respect the human dignity of every jew and every human being including those involved in bad practices so even though I openly talk about bad people, it is not an invitation to anyone to take the law in their hands or hurt any any human being. I will continue to fight for my cause of ending mind control of innocent and intelligent people but I want people to completely respect all good jews and good people in American army(even though I wrongly generalized about all of American defense as being bad, if you know any good and patriotic people there, please give them your utmost respect) and make a political struggle for the law or government actions to take their course against corrupt and evil people. I absolutely do not suggest any violence or anybody taking law in their hands.
True stories of machinations by evil jews against many Americans are considered heresy by general American public who believes strongly in protecting the Jews at all cost and it is a long held tradition in United States to call even true criticism against Jews as anti-semitic talk. These ungrateful evil and cruel jews, on the other hand, respond to this great benevolence of Americans by retarding thousands of innocent American folks through mind control because American defense is a puppet in their hands. Knowledge about mind-control is well known in some jewish circles and many of them they know who to approach in the community to start anybody's mind control and a large number of simple good-natured American folks are placed under mind control when some jew is displeased with them.
Even though American society keeps jews in very high regard, they are not gold-plated. Just like any other people or race, some of them are extremely good human beings, some of them are extremely evil and some of them simply do not care. They tend to be more intelligent and when this intelligence shows itself in the form of Einstein we see one of the greatest human beings ever born but when this intelligence turns evil and machinatory we have exactly opposite form of humanity like evil jewish billionaire godfather/goldfather and his rich jewish cohort who has taken complete control of American army to advance their nefarious designs and have retarded thousands of innocent and intelligent people they did not like. So respect has to be shown in abundance to good people but very carefully to evil people. When great respect is shown to evil people, they become even more emboldened to advance their agendas based on hatred, venom, poison and malice.
I want friends to please protest to American defense that has become a puppet in the hands of rich jews. Black sheep in American defense behind mind control are given very prestigious jobs paying millions of dollars in Jewish companies after they retire and rich jews want to block talented muslims from being successful in United States with an iron fist, they would start wielding some influence and start opposing American policies in Israel and they want to nip this supposed evil in the bud. And American defense has no dearth of wannabees who desperately want to join the ranks of so many others who have gotten extremely high paying jobs with great social status in jewish companies after retirement from American defense. Please do not blindly believe me and find out somebody in your social circle(or well known to somebody in your social circle) you know well and who is in mind control agency of US defense and ask them about my allegations and your eyes would open forever about role of rich jews in United States and how they openly use mind control on anyone they wish especially on a large number of even white Americans whose families have been living in US for hundreds of years. If your eyes do not open forever, I am ready to take any blame you wish. Mind control is basis of symbiotic relationship between American defense and rich and evil Jewish cohort.
Especially very talented American people in science and technology are forced to work in Jewish owned companies due to fear of mind control. Many such people who want to leave are coldly told by administration to leave but their mind control is initiated by American defense on insistence of jews right after they leave the jewish company and once they know the reason behind mind control was that they tried to leave the jewish company against the will of the company owners. And due to large number of such precedents well known to many professionals in many jewish companies, most of the very talented people do not dare to leave the jewish companies against the will of administration since they know very well that they would be instantly put on mind control by American defense who dances on the tunes of rich jews since they want extremely high paying prestigious jobs with jewish companies having great social status. Please do not believe a single word I say blindly but try to find out the actual truth on your own from authentic sources you know well and you will find the truth that would open your eyes forever.
Many friends who are used to formal language all the time might be disturbed by my use of slang expletives for bad people but after reading Einstein's letter to Marie Curie, I consider myself perfectly validated to use the word of scum for American army and for bad jews who have no ethics and are ready to fall to every low level of humanity for some material gain or due to their hatred, venom, poison or malice as in case of bad jews.
Over past thirty years, lot of bad jews have developed a network in academia(with absolute apology to all the great jewish people in academia) and when a victim has to be retarded most of the jews in the network start orchestrating totally complete lies about the victim so that good people think that retarding the victim is indeed is a very good thing.
And I want to tell people that this network of evil jews has greatly perfected itself over past thirty years and not only muslims are retarded, everyone they do not like including europeans, blacks and a large number of white talented Americans are retarded at whim by this network of emboldened evil jews.
I want to request all good people in American society and in european countries to end and thwart this gang of evil jews who are bent on retarding intelligent people of every nation and creed they do not like.
Not everybody is the same, there are at the same time a large number of jews who have nothing to do with any of this and many good jewish companies who do not follow any inhuman practice. But there is a close group of rich jews who are actually involved in large scale mind control with American defsnse working as their proxy. Please try to know the truth on your own.
I want to advise American defense to watch a lot of American movies and try to behave like heroes in those movies. Instead people in American defense are short-tempered scumbags who became belligerent when anybody criticizes them. Contrary to American and Western spirit of freedom of criticism and speech that we all cherish, these short-tempered scumbags are unwilling to tolerate anything other than eulogies based on complete lies. Just like Henry Ford used to say that you can buy any color of car as long as it is black, to these scumbags of American defense the freedom of speech means that you can say anything you like as long it is a eulogy otherwise they become belligerent and start moving thousands of people to silence someone who speaks the truth about them. Let us all of us start cursing Erdogan together now.
When mature and deep people face some criticism, they respond by totally neglecting it if it is wrong. Immature, shallow and short-tempered people respond to criticism by getting angry and deeply offended and start jerking like psychos how to respond meanly to this criticism. American defense is full of such immature blokes and I am very sure that intolerant and arrogant scum of American army will again resort to torture and violence against me after reading my criticism.
I want to request(and suggest to good jews as a friend) to disavow any relationship with jewish scumbag billionaire and his rich jewish cohort who are involved with mind control of innocent and intelligent people of so many nations for their nefarious designs.
I also want to explain very clearly that I have been very severely criticizing bad jews and American defense but this is limited to criticism in writing only. I write only on Wilmott forum where I also contribute my research and I do not write anywhere else(at least for past 10-15 years). I do not want anyone including myself to take the law in our own hands. We have to respect the human dignity of every jew and every human being including those involved in bad practices so even though I openly talk about bad people, it is not an invitation to anyone to take the law in their hands or hurt any any human being. I will continue to fight for my cause of ending mind control of innocent and intelligent people but I want people to completely respect all good jews and good people in American army(even though I wrongly generalized about all of American defense as being bad, if you know any good and patriotic people there, please give them your utmost respect) and make a political struggle for the law or government actions to take their course against corrupt and evil people. I absolutely do not suggest any violence or anybody taking law in their hands.
Last edited by Amin on September 14th, 2025, 6:09 am, edited 5 times in total.
You think life is a secret, Life is only love of flying, It has seen many ups and downs, But it likes travel more than the destination. Allama Iqbal
Re: Breakthrough in the theory of stochastic differential equations and their simulation
Friends, I am determined to take my research to conclusion even though I could not work yesterday night due to bad bread again and again. I need some strength in my brain to work but they sapped all my energy after I had bad food and I was unable to work properly. But I hope that I would work better in coming days (and today).
I have some very good ideas and hope that it will lead to far more simpler and more robust calibration to cross-moments. I just want to concentrate on my work but I am stopped again and again by all the difficulties and altered brain due to widespread drugging of food and water with mind control chemicals.
I have some very good ideas and hope that it will lead to far more simpler and more robust calibration to cross-moments. I just want to concentrate on my work but I am stopped again and again by all the difficulties and altered brain due to widespread drugging of food and water with mind control chemicals.
You think life is a secret, Life is only love of flying, It has seen many ups and downs, But it likes travel more than the destination. Allama Iqbal
Re: Breakthrough in the theory of stochastic differential equations and their simulation
Friends, when we take a regression of dependent variable Y (OrderY=5,OrderX=5) on X (!D regression using data), it results in merely first six hermite polynomials. These six hermite polynomials give a break down of the first cross-moment \( \mu_{1,1} \) in terms of component hermite polynomials. Similarly when Y^2 is regressed on X (again 1D Hermite regression using data), the first twelve hermite polynomials give a break down of the cross moment \( \mu_{2,1} \). Similarly all cross moments have their break down given in terms of hermite polynomials (found by 1D hermite regressions) and this number of hermite polynomials can be sometimes very large. I did a regression of Y^2 and Y^3 on X and found this regression to be well-behaved with final coefficients (for example tenth, elventh and twelvth coefficients in case of Y^2) extremely small or miniscule. Even though hermite-regression for variance, we have previously done are sometimes not extremely faithful, these 1D Hermite regressions of "Y on X" I am talking about here are expectations and we could improve any shortcoming in these least square regressions by replacing them with ridge-regressions. We can find the analytical values of these hermite-series coefficients in terms of Z-series coefficients by multiplying 2D Y and 1D X Densities analytically since these hermite coefficients appear in the first row of the 2D product hermite series. We can match these analytical values of hermite coefficients found from (some manipulation of) series products with coefficients found from numerical regressions. There are several hundred or possibly more than a thousand of these hermite polynomial coefficients from all the pairs of cross-moments and there are only 36 (or 64) coefficients of dependent variable Y to be found. This is a very highly over-constrained problem with very large number of equations and only a few tens of coefficients/variables to be backed out. If some regression estimates are supposed to be bad, we can freely remove them from our calibration without making a big deal. We can do Newton for this over-constrained problem and I suppose that dealing with coefficients of hermite polynomials in terms of Z-series coefficients would be far more well-behaved than dealing with exponential type increasing moments since higher hermite coefficients are extremely fast decreasing as opposed to doing Newton with moments where higher moments are very fast increasing. And if there is any lapse in the method, it can give us an extremely good guess that can be perfected by other methods. There can be a lot of ways in which any weakness of the method could be improved.
I have figured out all the details of the new method but I have to deal with mind control drugs everyday that make it difficult for me to work otherwise I would have already done most of the work. I am determined to complete the work in next few days and thought I would share the basic ideas with friends.
I have figured out all the details of the new method but I have to deal with mind control drugs everyday that make it difficult for me to work otherwise I would have already done most of the work. I am determined to complete the work in next few days and thought I would share the basic ideas with friends.
You think life is a secret, Life is only love of flying, It has seen many ups and downs, But it likes travel more than the destination. Allama Iqbal
Re: Breakthrough in the theory of stochastic differential equations and their simulation
Sorry friends for not writing on the thread for so many days. My elder sister was visiting Lahore and I had to spend some time with her in the home and outside. And then, earlier today, I had to take my mother to Jinnah Hospital for her carotid artery color doppler test. As it turned out, a lot of her carotid artery is blocked. We were asked by other doctors to have CT-Angio or MRI-Angio. My brother is also in Lahore and he is going to take my mother for tests tomorrow. My mother would possibly need angiography to clean her blocked artery.
I have been slowly working for the algorithm that tries to find the value of coefficients of 2D Z-series of Y by matching analytical values of first few hermite polynomial coefficients of cross products between Y^m and X^n (to be matched) with numerical values of these first few hermites from data directly. We had earlier tried to match cross moments which are coefficients of zeroth hermites of cross products between Y^m and X^n. I hope the program would be ready in a few days. I have completed part of the program that calculates derivatives of the hermite coefficients of cross products between powers of X and Y.
I hope to mention about progress on my work for next few days.
I have been slowly working for the algorithm that tries to find the value of coefficients of 2D Z-series of Y by matching analytical values of first few hermite polynomial coefficients of cross products between Y^m and X^n (to be matched) with numerical values of these first few hermites from data directly. We had earlier tried to match cross moments which are coefficients of zeroth hermites of cross products between Y^m and X^n. I hope the program would be ready in a few days. I have completed part of the program that calculates derivatives of the hermite coefficients of cross products between powers of X and Y.
I hope to mention about progress on my work for next few days.
You think life is a secret, Life is only love of flying, It has seen many ups and downs, But it likes travel more than the destination. Allama Iqbal
Re: Breakthrough in the theory of stochastic differential equations and their simulation
Friends, I do not know what this is but I have become totally listless due to some new mind control drugs they are trying on me. I eat something at home and mostly it is always lightly drugged. So some mind control drugs continue to go into my body. In order to stop me from any further research, they are using some new drugs and, as a result, I have become completely listless and devoid of energy. I always work with enthusiasm and passion but for past few days, I have lost all enthusiasm and passion for my research and I sit on my computer and lethargically continue to waste time. I am feeling very bad about it but this listlessness still takes over me. Please protest to US defense and mind control group to please stop trying new mind control drugs and chemicals on me and let me do my work and research.
You think life is a secret, Life is only love of flying, It has seen many ups and downs, But it likes travel more than the destination. Allama Iqbal