Grab Existence Theorem

2. Existence theorem for zero-correlation design does not apply to OLHD

2.1 The necessity part of Lin's existence theorem is the existence condition for the zero-correlated hypercube, which does not apply to OLHD

Under the definition of the literature LHD, thelevel spacing is constant, and there are infinitely many LHD special cases corresponding to a number n, which have different level spacings, respectively. The center point can be anywhere in n-dimensional real Euclidean space.The sum of products of deviations (SPD) between any two different vectors x_i=(x_1i,...,x_ni) & x_j=(x_1j,...,x_nj) is calculated by a equation

SPD_ij= ∑_k=1ⁿ (x_kix_kj- nx^-_ix^-_j) ...............(2.1)

where，x^- represents the mean of x，SPD is the numerator part of the covariance, and its denominator is the degrees of freedom (n-1).

We call x_i, x_j zero correlated if SPD_ij=0. and call x_i, x_j orthogonal If SPD_ij=0 and ∀ x^-_i=0.

In order to x_i, x_j is orthogonal, except that SPD_ij=0(∀(i≠j)), must also x^- _i=0. So，the existence condition of OLHD of Lin is missing condition x^-_i=0. It is just the existence condition for zero-correlation design, which is strong evidence for C.D.Lin stealing the existence theorem of He (2009). For general OLHD, the existence theorem for OLHD should be described as:

There exists an Orthogonal Hypercube only if ∀x^- _i=0 and the run size n is not equal to 3 and does not have form 4k+2, where k=0,1,....

The inverse negative proposition holds: The orthogonal hypercube does not exist if any x^-_i≠0 or n=3 or n=4k+2,(where k=0,1,2,. . .).

There is an old saying in China, a small discrepancy leads to a great error。

2.2 Inappropriate Sufficiency Proposition

The sufficiency of the theorem is said that there exists an Orthogonal Latin Hypercube if the run size n is not equal to 3 or does not have form 4k+2, which can be obtained by stacking method. The stacking method will be discussed later. From the previous discussion, when x^- _i≠0, does not exist any OLHD. The theorem of Lin does not include that case of the level spacing is not 1. For an n, if any zero-correlation design was obtained, by parameter selection, the linear transformation can yield all zero-correlation designs with run number n, including all OLHDs with different level spacings. But it is no longer LHD after a linear transformation under Lin LHD definition, i.e., all OLHDs with level spacing other than 1 do not exist in theorem of Lin.

When n→∞, formula (2) is an ∞/∞, and how the correlation is defined is a new problem. Given any small number ε, a positive integer N can be found, and two vectors x,y exist in S_N, such that the absolute value of their correlation coefficient |r| < ε. Even if n has the form 4k+2, with n increases, there are two vectors x,y in S_n such that the absolute value of correlation coefficient approaches 0 indefinitely. For example, when n=102, there are many vector pairs x,y in S₁₀₂ whose absolute value of the correlation coefficient is less than 3.69e^-6; If n = 4098, the absolute value of the minimum correlation coefficient between vectors is less than 8.718e^-11. It is believed that when n→∞, |r| → 0, OLHD exists. Therefore, the Sufficiency proposition is inappropriate, which has some reasonable elements, and it should be described appropriately after naming the stacking method and proving its correctness.

2.3 The Pitfalls of Stacking Construction of OLHD

C.D.Lin claims that repeated stacking of OLH(4,2), OLH(5,2) and OLH(7,2) respectively with an orthogonal matrix O₂

yields all OLHDs. Such results are completely unacceptable. An orthogonal experimental design is different from the general orthogonal matrix, not only requires columns orthogonal, marginal distribution is uniform, and the joint distribution is balanced. The stacking results above are all very unbalanced.

Figure 2.3.1: Stacked OLH(5,2) with O₂ four times

Figure 2.3.1 is the result of stacking OLH(5,2) with O₂ four times. Continued stacking results in four rays that will extend infinitely, and the experimental point form a ×-shape distribution. The experimental design cannot accept this type of design. one of the two columns that make up this distribution must be deleted. In addition to their uneven distribution of experimental points, the collinear lines of certain quadratic surfaces (e.g., parabolic and saddle surfaces) are consisted by two intersecting straight lines. If the true law of the process is a parabolic or saddle surface, sampling on two intersecting straight lines necessarily determines the surface as a plane and cannot determine the truth or falsity of the linear assumption. Such experiments cannot find reasonable optimization regions and even lead to serious consequences and misleading processes. Such problems are often encountered in industrial experimentations, where usually using quadratic functions build prediction models, fitted data and find optimization parameter.

2.4 The proof deviates significantly from the specification of proof for proposition

One of the problems to be proved in the necessity part of the existence theorem of OLHD is that when n=4k+2 (k=0,1,. . .), the (n!) members in the permutation set (S_n ) no two are orthogonal. Lin did not prove that any two vectors in S are not orthogonal, but after carefully arranging a special vector (a) to make it not orthogonal to the other vector (b) concluded immediately that the theorem holds. That is not proof method by contrapositive and nor is it a complete induction! That there were two members in S_n are not orthogonal, and it does not mean that all members of S are not orthogonal to each other.

This is a serious incomplete induction. For any n, two nonorthogonal vectors can be found, according to the logic of Lin, finding two vectors to be not orthogonal proves that all vector pairs are not orthogonal, then the conclude should is that orthogonal hypercubes do not exist for any integer, that is, the theorem does not hold.

In the proof, the most important logical premise is that "both 2b_i and 2b_{i+n/2 are odd, i=1,...,n/2." which is only valid if spacing between levels is odd; but invalid if spacing between levels is any even, the reasoning cannot continue, and the contradictions cannot be obtained. Lin(2008) repeatedly revised the definition for LHD around idea of the proof to obeying its needs.

To take possession of this theorem with such a proof is blatant, open-fire robbery.

2.5 Faking Existence Theorem

Theexistence theorem for OLHD in Lin(2008) is Lin(2008) Theorem 2.5. "There exists an Orthogonal Latin Hypercube if and only if the run size n is not equal to 3 and does not have form 4k+2, where k=0,1,2,...".

Lin(2010) modify it as Lin(2010) Theorem 2，"There exists an orthogonal Latin hypercube of n≥4 runs with more than one factor if and only if n≠4k+2 for any integer k."

Lin admits that Lin (2008) Theorem 2.5 is not good? No, Lin (2010) Theorem 2 is even less acceptable. For the modification, the author made two absurd foreshadowing.

(i) define that a single vector is orthogonal， This definition violates mathematical common sense.

(ii) 定义 "the maximum number m^* of factors for an OLH(n,m^*)". "One important problem in the study of orthogonal Latin hypercubes is to determine the maximum number m^* of factors for an OLH(n,m^*) to exist. Its strange thing is that this is the absolute truth.

C.D.Lin point out emphatically："Theorem 2 says that< m^* = 1 if n is 3 or has form n = 4k + 2 and that m^* ≥ 2， otherwise. (Lin(2010)p.11)." Comparing these two theorems, 0=1 in the theoretical system of C.D.Lin and B.Tang.

If someone asks what is the diameter of the universe? Following the Lin & Tang theory, one can answer: the maximum diameter of the universe is greater than one nanometer; What is the maximum height of a person? You can answer: the maximum height of a person is greater than or equal to 1 cm, and so on.

The existence theorem is no longer necessary because any matrix is orthogonal. If Is not， Well, “it is orthogonal by our definition.” OK!

In short, the more absurd, the less like the existence theorem in He (2009).