7. The Fatal Error of Lin's Rotation Theory

7.1 Description of the main results

C.D.Lin et al. extends the algorithm of Steinberg & Lin (2006)[Biometrika,93,2, 279-288,(2006)], and the main results are as follows,

"Partition" A_j= OA(n²,2f,n,2) as A_j1,...,A_jf， where each of A_j1,..., A_jf has two columns." ∀ A'_jA_s=0，(j,s=1,2,...,f)，where 0 is a full 0 matrix。 The method of Lin grouping transformation is the same as Steinberg transformation for OA(n²,2f,n,2).

M'_jM_s only if A'_jA_s=0 =0, Then M is OLH(N, 2pf), and theorem 1(b) can be established.

7.2 Is OA orthogonal in the sense that the inner product is 0?

The original OA is not defined in the real Euclidean space. When OA is defined, the level is not a number, and the numerical form of the level is just a substitute for symbols such as Latin-Greek. Hadayat et al. (1999) explicitly note that orthogonality does not imply that the inner product is zero. I don't know who unconditionally equate OA with orthogonality in real Euclidean space. When considering the numbers in the Taguchi orthogonal arrays in place of Latin symbols as numerical values, I have checked these tables and they are all zero-correlated. In He (2009), I boldly conjectured that moving these OAs into real Euclidean space is zero-correlated, I firmly believe it can be proven but I haven't tried to prove it yet. In 1973, when I used these tables for experimental design, when quantitative factors and qualitative factors coexisted, I didn't know how to handle regression analysis, and I hesitated until I heard about Bayes' method. I still don't think the two orthogonal concepts are equivalent. However, as we have seen, C.D.Lin mixed up the two. She digitized Latin-Greek notation, defined distance, mean, variance, and inner product, and unconditionally moved OA into real Euclidean space. Needless to say, C.D.Lin is a very intelligent scholar. Her mind is very active and bold, she has a lot of conjectures. She made a program to calculate the correlation matrix of a matrix. I also designed one in 2005. There is also a standard module in the R system. The report correlation matrix is only a dozen sentences. It is an essential tool for constructing weak correlation designs. She used this program to test the matrix she guessed to be orthogonal, If it is orthogonal, she assumes that this class of matrices is orthogonal without proving it, not even saying it is provable. She grouped these results into separate items called propositions. It is natural to use these conjectures as theorems.

Under the Proposition entry should be some logical judgments that are simpler than theorems, obvious, do not need to be proved or are easy to prove. C.D.Lin's propositions did not this property. For example, Proposition 1 in Lin (2008, 2010) says that (L,U) is orthogonal，which is equivalent to the proposition "L, U are independent of each other". This judgment is not obvious, and its proof is quite difficult and cannot be omitted. I still have doubts. The author deliberately avoided proof.

7.3 Who defines and names the coupling operation?

If step 1 in Section 2 is coupling, why not give the definition explicitly and name it coupled for easy call. If it was first defined by you, you must prove that the result of OA being coupled is still OA. The author neither defines coupling nor indicates who did so. The coupling is not named but this name is used repeatedly throughout the text to refer to this algorithm. I have to suspect that such a coupled algorithm was not discovered by this author.

Whether the result of OA being coupled is OA has not yet been proved, what can be used to ensure that the result of Step 2 is OLH? A conjecture is thus used as a theorem.

7.4 The n×p matrix B did not orthogonal and cannot derive Theorem 1(b)

The first line of Figure 7.1 did not require B to be orthogonal, the premise does not match Theorem 1(b), and Theorem 1(b) cannot be deduced from the premise. In fact, the author did not succeed in proving the theorem.

The key to the proof of the theorem is A'_jA_s=0. The key passages of the author's proof are as follows.

Equations (4) and (5) in Figure 7.2 do not exist naturally, and the author flew over the entire reasoning process. We will not discuss the flaws of these proof now. The author said: "where r_js is the (j,s)th element of R." r_js=0 only if B is an orthogonal matrix; However, The preconditions did not specify that B is an orthogonal matrix, then, r_js≠0., the (L.7) annotated by me is not 0, (L.8) does not hold. The correlation matrix of M consists of p² non-0 blocks, like (2) or (3) in Figure 7.3, not (1).

The conclusion required by Theorem 1(b) cannot be obtained. However, the proof process was over. The author failed to prove his Theorem 1 before the end of the proof process.

7.5 The author got two concepts wrong.

(i)."The maximum absolute correlation" is a misconception, which is the result of the author deliberately removing the core condition of the mcc concept in order to avoid accountability for stole mcc of He (2009)， for details, please refer to the third section of this report "C.D.Lin and Boxin Tang Theoretical System Missing Critical Values".

(ii). "The maximum absolute correlation in R^~ is the same as that in R.” The author did not make basic preparations for the subject research and did not establish a critical value calculation program. In order to avoid the critical value calculation, no statistical test critical value is introduced. There is absolutely no trace of critical values in the author's theoretical system, NOLH is completely independent of correlation, except for orthogonal it is NOLH. In Lin (2009), a constant 0.05 is used as the correlation threshold, and there is no threshold at all in other papers. When n=9, r=0.05 corresponding correlation confidence probability is about P=0.102, which is weak correlation.The coupling magnifies the system by a factor of n. When N=81, the correlation coefficient does not change, but P=0.342, which is by no means weakly correlated. The mcc values of the 7th, 8th, and 9th columns of NOLH(11,9) in the third section are all 0.0364. When n=11, P=0.0847, the correlation is weaker, when coupled to N=121, P=0.3398, which is outside the correlation tolerance range. The tolerable weak correlation for NOLH(121,x) is 84 columns instead of 108. The author claims that OLH(13,12) coupled with OA(169,14,13,2) yields NOLH(169,168), The absolute value of the maximum correlation coefficient on its non-main diagonal line is mcc=0.0495. It seems that the correlation coefficient is small enough, The degree of freedom has changed due to coupling, and its correlation confidence probability exceeds the weak correlation tolerance limit.

The following gives the correlation confidence probability changes for the three points of NOLH(13,12).

Column 8， mcc=0.0275, P=0.07145; if N=169, P=0.277343;

Column 10， mcc=0.0385, P=0.0994; if N=169, P=0.38086;

Column 11， mcc=0.0495, P=0.1277; if N=169, P=0.4754.

It can be seen from these simulations that the correlation confidence probability p=0.277343 corresponding to the correlation coefficient of 0.0275 in column 8, which has exceeded the usual weak correlation range. If the coupled results are weaker correlated, only the first 7 columns can be used, forming OLH(13,7) instead of OLH(13,12). Coupling and transforming yields NOLH(169,98), not NOLH(169,168). The benefits of NOLH coupled OA are far less than the authors claim.

7.6 Afterword

The preconditions in Figure 7.1 match only propositions: mcc in R is the same as in R. This is the author's "Remark 1(ii)." Accordingly, the last line of the proof process in Figure 7.2 (It is a wrong conclusion) should be deleted. Its Theorem 1(b) is actually "Remark 1(i)." "If B is an orthogonal Latin hypercube, then so is M." This strange way of writing fully proves that the writing of this article is hasty and sloppy. The reason for so mach haste is the rush to publish the results of Tables 1 and 2 before my manuscript is published.