A personal annotations of Veach's thesis (9)

Page 50: Stratified

I was not familiar with the word `Stratified.' A Monte-Carlo method usually uses a pseudo random number sequence. A random number sequence usually should have a property, `unexpectedness,' this means a same number sequence like 1, 1, 1, 1, 1, ..., 1 could be also a part of random number sequence. This issue is well explained in Knuth's book (D.E.Knuth, The Art of Computer Programming, Vol.2 Random numbers). The sampling pattern could be the left of Figure 1.

Figure 1. Stratified example. Left example of random 12 samples (clumping problem), Right example of 2x2 stratified random 12 samples

In Figure 1 left, some part is not well sampled, or some part is oversampled -- the sampling pattern is biased. This is clumping, a bad luck in a sense. Usually it is solved when the number of samples is increased, but, it computationally costs, means it takes more time to compute. To solve this clumping problem without losing performance, the sampling region is subdivided --- or stratified. In this way, even we have a bad luck, still we could sample more uniformly. This method is called a stratified method. Readers might wonder, ``then why don't you sample uniformly?'' Uniform sampling can avoid clumping problem, but, the uniform sampling causes another problem, ``Aliasing problem'' by the sampling theorem. My friend said that ``the Quasi-Monte-Carlo method is a good stratified method.''

Acknowledgements

Special thanks to Daniel S. and Leonhard G. who explained me what is the Stratified method.

Comments

Why A^{T}A is invertible? (2) Linear Algebra

Why A^{T}A has the inverse Let me explain why A^{T}A has the inverse, if the columns of A are independent. First, if a matrix is n by n, and all the columns are independent, then this is a square full rank matrix. Therefore, there is the inverse. So, the problem is when A is a m by n, rectangle matrix. Strang's explanation is based on null space. Null space and column space are the fundamental of the linear algebra. This explanation is simple and clear. However, when I was a University student, I did not recall the explanation of the null space in my linear algebra class. Maybe I was careless. I regret that... Explanation based on null space This explanation is based on Strang's book. Column space and null space are the main characters. Let's start with this explanation. Assume x where x is in the null space of A . The matrices ( A^{T} A ) and A share the null space as the following: This means, if x is in the null space of A , x is also in the n...

Gauss's quote for positive, negative, and imaginary number

Recently I watched the following great videos about imaginary numbers by Welch Labs. https://youtu.be/T647CGsuOVU?list=PLiaHhY2iBX9g6KIvZ_703G3KJXapKkNaF I like this article about naming of math by Kalid Azad. https://betterexplained.com/articles/learning-tip-idea-name/ Both articles mentioned about Gauss, who suggested to use other names of positive, negative, and imaginary numbers. Gauss wrote these names are wrong and that is one of the reason people didn't get why negative times negative is positive, or, pure positive imaginary times pure positive imaginary is negative real number. I made a few videos about explaining why -1 * -1 = +1, too. Explanation: why -1 * -1 = +1 by pattern https://youtu.be/uD7JRdAzKP8 Explanation: why -1 * -1 = +1 by climbing a mountain https://youtu.be/uD7JRdAzKP8 But actually Gauss's insight is much powerful. The original is in the Gauß, Werke, Bd. 2, S. 178 . Hätte man +1, -1, √-1) nicht positiv, negative, imaginäre (oder gar um...

Why parallelogram area is |ad-bc|?

Here is my question. The area of parallelogram is the difference of these two rectangles (red rectangle - blue rectangle). This is not intuitive for me. If you also think it is not so intuitive, you might interested in my slides. I try to explain this for hight school students. Slides: A bit intuitive (for me) explanation of area of parallelogram (to my site, external link) .

Lx=d: SundayResearch

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