New Statistical Test for Quality Control in High Dimension Data Set

Transcription

1 International Journal of Applie Engineering Research ISSN Volume, Number 6 (7) pp New Statistical Test for Quality Control in High Dimension Data Set Shamshuritawati Sharif, Suzilah Ismail an Zurni Omar School of Quantitative Sciences, UUM-College of Arts an Sciences, Universiti Utara Malaysia, 6 UUM Sintok, Keah, Malaysia Abstract The existing statistical test for quality control such as Box s M test caters for number of sample size n larger than number of variables or imensions p (n>p). However, in real worl application of Small Meium Enterprise (SME), the number of variables or imensions p can be larger than sample size n (p>n) which is known as high imension ata set. This is ue to small number of aily prouctions which lea to small number of sample size (n) but high imensions proucts (p). One of the examples is rubber gloves which rely on machine capacity or latex supply that limiting the aily prouctions (n) but involves many imensions (p) of measurement such as the size of five ifferent fingers, the with of the palm an wrist; the strength, number of holes an etc. Another rawback, once the samples are teste for quality control, they are iscare which is very wasteful if uses large sample size. Therefore, in this stuy we have evelope a new statistical test known as S* test to accommoate high imension ata set (p>n). A simulation stuy was conucte in comparing the performance of Box s M test an S* test using power of test. Base on, replications an 5% significance level, the power of test inicate that S* test outperforme the Box s M test. Interestingly, when n is smaller than p, S* test still can be compute which proven it can be use for quality testing in high imension ata set. INTRODUCTION Quality control is a crucial approach in riving an inustry such as Small Meium Enterprise (SME) to be more effective an competitive. The existing statistical test for quality control such as Box s M statistics focuses on number of sample size n larger than number of variables or imensions p (n p) (Djauhari, 5; Djauhari, 9; Wan Yusoff, 3; Sharif, 3). However, in real worl application of Small Meium Enterprise (SME), the number of variables or imensions p can be larger than sample size n (p>n) which is known as high imension ata set. The main reason is because of small number of aily prouctions which lea to selecting small sample size (n) for quality testing but involves high imensions prouct (p). An example is the quality testing of rubber gloves which requires many imensions (p) of measurement such as the size of five ifferent fingers, the with of the palm an wrist; the strength, number of holes an etc.; but limite number of aily prouctions in SME perhaps ue to machine capacity or the shortage of latex supply. Thus, lea to selecting small sample size (n) for quality testing. Another rawback, once the samples are teste for quality control, they are not saleable an iscare which is very wasteful if uses large sample size. Therefore, in this stuy we have evelope a new statistical test known as S* test to accommoate high imension ata set (p>n). The following sections outline the evelopment of the test, valiation using simulation stuy base on power of test an conclusions. DEVELOPMENT OF NEW STATISTICAL TEST (S* Test) The asymptotic istribution of S is investigate for p using the proposition erive by Kollo an Rosen (5) an theorem establishe by Anerson (3). Now, let X, X,, X n be a ranom sample of size n rawn from a p-variate normal istribution with covariance matrix Σ. The sample mean vector an sample covariance matrix are given by X = n X n i= i an S = n (X n i= i X )(X i X ) t respectively. Proposition (Kollo an Rosen, 5). n {vec (S) vec(σ)} N p (, Π) Π = (I p + K)(Σ Σ), p p t K = i= j= H ij H ij is the commutation matrix of size (p p ) an H ij is a matrix of size (p p) its (i, j)-th element is equal to an else. Proof : V(S) = For p =, let (I n p + K)(Σ Σ) 64

2 International Journal of Applie Engineering Research ISSN Volume, Number 6 (7) pp Σ = [ σ σ σ σ, S = [ s s s s, vec(s) = [s s s s t, V(S) = Var(vec(S)) = s s s s s s s s s s s s s s s s [ s s s s s s s s, s s s s s s s s K = [ with the size of p p, an I p = [. Then, Π = (I p + K)Σ Σ = [ [ σ σ σ σ [ σ σ σ σ Theorem Anerson (3, Theorem 4..3., p. 3) : Let {U(n)} be a sequence of p-component ranom vectors an b a fixe vector such that n [U (n) b N (, γ) as n. Let f(u) be a vector-value function of u such that each component f j (u) satisfies f j (u) u=b. If f j (u) u=b is the ( i, j ) th component of. Then n [f(u(n)) f(b) (, t γ ). The new proposition when Σ is unknown is formulate on the basis of removing the uplication matrix from vec (S) to vec (S L ). Covariance matrix Σ is a p p symmetric matrix contains uplication or reunant elements since σ ij = σ ji, for all (i j). Let vec(σ L ) efine as a vector of size r = p(p + ) representing the lower triangular elements of Σ. The uplication matrix T p is the matrix of size p p(p + ) will be use to transform vec(σ) into vec(σ L ). Next, we efine u(vec (S)) = T p (vec (S)) = vec (S L ) to erive the asymptotic istribution of vec (S L ).Thus, the new proposition is, u i u i σ [ σ σ σ = [ [ σ σ [ σ σ σ σ σ [ σ σ σ σ σ [ σ σ σ σ Proposition. n (vec (S L ) vec(σ L )) N p (, φ) σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ = [ [ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ + σ σ σ σ + σ σ σ σ + σ σ σ σ + σ σ σ = [ σ + σ σ σ σ + σ σ σ σ + σ σ σ σ + σ σ σ σ + σ σ σ σ + σ σ σ σ + σ σ σ σ + σ σ σ σ + σ σ σ σ + σ σ σ σ + σ σ σ σ + σ σ φ = T p t (I p + K)(Σ Σ)T m, an T p = (a ij ) is a matrix zero-one with p blocks as erive in Appenix. To put into practice, using Theorem 3.4. in Maria et al. (979) an the asymptotic istribution of vec(s L ) given in Proposition, we present two principal results of erivation into Proposition 3 an Proposition 4. These proposition are actually the Mahalanobis square istance between v(s L ) an v(σ L ). V(s s ) Cov(s s ) Cov(s s ) Cov(s s ) Cov(s s ) V(s s ) Cov(s s ) Cov(s s ) = Cov(s s ) Cov(s s ) V(s s ) Cov(s s ) [ Cov(s s ) Cov(s s ) Cov(s s ) V(s s ) V(S) = n (I p + K)(Σ Σ) Accoring to asymptotic istribution of vec (S) in Proposition an the following theorem, Proposition 3. [vec(s L) vec(σ L ) t φ [vec(s L ) vec(σ L) χ r the egrees of freeom r = p(p + ). Proposition 4. [vec(s k,l ) vec(σ,l ) t φ [vec(s k,l ) vec(σ,l ) χ r the egrees of freeom r = p(p + ). Let S k be the covariance matrix relate to the k-th inepenent sample of size rawn from N p (μ, Σ k ); k =,,, g. Consier 64

3 International Journal of Applie Engineering Research ISSN Volume, Number 6 (7) pp the hypothesis testing H : Σ = Σ =...=Σ g (=Σ, say). Uner H, if Σ is known, then Proposition 5. [vec(s k,l ) vec(σ,l ) t vec(σ,l ) χ r φ [vec(s k,l ) S k,l an Σ,L are lower elements of S k an Σ, respectively, φ =T p t (I p + K)(Σ Σ )T p, is a iagonal matrix of Σ, an r = p(p + ). Now, we consier the hypothesis H : Σ = Σ =...=Σ g (=Σ, say) versus H : Σ i Σ j for at least one pair (i, j). Testing that hypothesis is equivalent to the repeate tests (Montgomery, 5). Therefore the hypothesis testing can be written as, H : Σ k = Σ for all k =,,, g versus H : Σ k Σ for at least one k; k =,,, g. Therefore, propose statistic, S is as follows S = [vec(s L ) vec(σ L ) t φ [vec(s L ) vec(σ L ) χ r, S L an Σ L are lower elements of S k an Σ, respectively φ =T p t (I p + K)(Σ Σ )T p r = p(p + ) The null hypothesis is rejecte if S > χ r. In real case stuy, when is Σ unknown Σ = Σ. Matrix Σ is estimate reference sample of covariance matrix, i.e poole sample covariance matrix. Next, a simulation stuy was conucte in comparing the performance of S* test an Box s M test using power of test. The Box s M test is as follows g () M = Nln S n i ln S i S = n N i= is i is the poole sample variance-covariance matrix, S i is the variance-covariance matrix calculate from the sample i, g is the number of subgroup the stability of matrices is hypothesize, an g i= N = n + n + + n g ; n i = i-th sample size. The power of test is use to compare between ifferent statistical testing proceures the most powerful statistical will contain the higher number of rejection of null hypothesis (Mittelhammer, 996). Accoring to Yue et al. (), the power can be estimate by, Power = R n N R n is the number of experiments that fall in the rejection region an N is the total number of repetition in simulation experiments. The simulation stuy was execute base on 5 replications to obtain the accuracy of power approximation. In this stuy, the simulation is base on, replications, 5% significance level, several conitions which are ifferent number of variables (p =,3), ifferent number of sample sizes(n =,, 3, 4,5),ifferent covariance shift (k =.,.5,.,.5,.) an ifferent correlations (.,.5,.7,. 9 ). Let,,, ) represent the population stanar ( p eviation vector of the p variables for i =,,, p, an ii is corresponing variance of the i-th variable. We consier in each ata set, it consist of ifferent the number of variables, as well as sample size, n. At the same time, for all pairs of variables an sample size will have the same correlations. These values are representatives of low, moerate, high an very high correlation. As an example, size of shift, k =.5 then, the hypothesis testing is, H : Σ m = Σ versus H : Σ m = Σ Σ = I p an Σ are as follows,.5.5*.5.5*.5.5*.5.5.5*.5.5*.5.5*.5 ;.5 i ii.5 Generally, the power of a test or known as β is the probability of correctly rejecting the null hypothesis when it is false. It actually refer to the sensitivity of the statistical test the ability to etect a true. From the power of test, the sensitivity level of the S* test an Box s M test to the shift in covariance structure can be etermine. RESULTS AND DISCUSSIONS Table an isplay the simulations results base on several conitions as liste previously. When p =, n =, ρ =.7 an k =., S* test has reach the maximum power of test but M test reveal the least power of test an only obtain the highest power of test when p =, n = 3, ρ =.9 an k = 643

4 International Journal of Applie Engineering Research ISSN Volume, Number 6 (7) pp These inicates S* test outperform M test at lower correlation (ρ) an smaller sample size (n). Interesting results obtain when p = 3an n =, no power of test can be compute for M test. It is very obvious S* test performs well than M test when n < p but even better when n > p. S* test also shows sensitivity when there is a small shift from k =. to.5 as compare to k =. to.. Although S* test has low power of test when ρ =. but still outperform M test. Overall S* test has high power of test when ρ =.5 an above for all p an sample sizes (n). This fining aligns with real worl application regaring the inter-relationships of variables (p) the variables are correlate. One example is regaring the rubber gloves the measurements of the five fingers shoul be correlate as to reflect the proportionate measurements of human fingers. CONCLUSIONS In this stuy, we successfully evelope a new test for quality control name as S* test for high imension ata set. The power of test reveals S* test outperforme Box s M test when sample size (n) less than number of variables or imensions (p) which fulfille the conition of high imensions ata set. This S* test can be use in real worl settings when there is a nee to select small sample size for quality testing especially in SME. Table : Power of test when p = k n = n = n = 3 n = 4 n = 5 S* test M test S* test M test S* test M test S* test M test S* test M test

5 International Journal of Applie Engineering Research ISSN Volume, Number 6 (7) pp Table : Power of test when p = 3 k n = n = n = 3 n = 4 n = 5 S* test M test S* test M S* test M S* test M S* test M test test test test

6 International Journal of Applie Engineering Research ISSN Volume, Number 6 (7) pp REFERENCES [ Anerson, T. W. (3). Introuction to Multivariate Statistical Analysis. New York: John Wiley & Sons, Inc. [ Djauhari, M. A. (5). Improve Monitoring of Multivariate Process Variability. Journal of Quality Technology, 37, pp [3 Djauhari, M.A. (9). Asymptotic Distribution of Sample Covariance Determinant. Journal MATEMATIKA, 5 (). [4 Djauhari, M.A. (b). A Multivariate Process Variability Monitoring Base on Iniviual Observations. Journal of Moern Applie Science, 4(). [5 Djauhari, M.A. (). Geometric Interpretation of Vector variance. Journal MATEMATIKA, 5(). [6 Kollo, T. & von Rosen, D. (5). Avance Multivariate Statistics with Matrices. The Netherlans: Springer. [7 Maria, K. V., Kent, J. T. an Bibby, J. M. (979). Multivariate Analysis. Lonon: Acaemic Press. [8 Mittelhammer, R. C., & Mittelhammer, R. C. (996). Mathematical statistics for economics an business (Vol. 78). New York, NY, USA: Springer. [9 Montgomery, D.C. (5). Introuction to statistical quality control, 5 th Eition. John Wiley & Sons, Inc., New York. [ Sharif, S. (3). A new statistic to the theory of correlation stability testing in financial market. (Unpublishe PhD Thesis), Universiti Teknologi Malaysia, Johor, Malaysia [ Wan Yusoff, W.N.S. (3). A new covariance structure stability test base on vector variance. (Unpublishe PhD Thesis), Universiti Teknologi Malaysia, Johor, Malaysia [ Yue, Shang. Paul Pilon & George Cavaias. (). Power of the Mann-Kenall an Spearman s rho tests for etecting monotonic trens in hyrological series. Journal of Hyrology, 59, APPENDIX Construction of Generalise Transformation Covariance Matrices Case p = s s S = [ s s s s, V(S) = [ s R 4=. s Choosing the lower triangular elements of S, we construct the covariance vector V(S L ) as follows s V(S L ) = [ s R 3= (+). s Now, T V(S) = V(S L ) Since the sizes of V(S) an V(S L ) are 4 an 3 respectively, T must be of the size 3 4,i.e s s s [ [ s = [ s s s We can further partition T into two blocks as follows [ The location of entries with element for each block can be presente as below: First block: (, p ), (, p) Secon block: (3,p) 646

7 International Journal of Applie Engineering Research ISSN Volume, Number 6 (7) pp Case p = 3 s s s 3 S = [ s s s 3, V(S) = s 3 s 3 s s s 3 s s s 3 s 3 s 3 R 9=3. We then construct covariance V(S L ) by choosing the lower triangular elements of S, i.e V(S L ) = s s s 3 s s 3 R 6=3(3+). Since T 3 V(S) = V(S L ) an the the sizes of V(S) an V(S L ) are now 9 an 6 respectively, T 3 must be of the size 6 9,i.e s s s s 3 s s s s 3 = s s 3 s s 3 [ 3 s 3 We partition T 3 into three blocks as epicte below [ whose the location of entries with element is given by First block: (, p ), (, p ), (3, p) Secon block: (4,p ), (5,p) Thir block: Case p = 4 (6,3p) s s s 3 s 4 s S = [ s 3 s s 3 s 3 s 4 s 4 s 4 s 34 s 44, V(S) = s s s 3 s 4 s s s 3 s 4 s 3 s 3 s 4 s 4 s 34 [ s 44 R 6=4. V(S L ) is constructe using the same strategy as before, i.e by choosing the lower triangular elements of S V(S L ) = s s s 3 s 4 s s 3 s 4 R =4(4+). Since T 4 V(S) = V(S L ), T 4 has the size of 6 as the sizes of V(S) an V(S L ) are 6 an respectively [ Partition T 4 into four blocks as follows s s s 3 s 4 s s s 3 s 4 s 3 s 3 s 4 s 4 s 34 [ s 44 [ The inices of entries with element is given by = s s s 3 s 4 s s 3 s 4 First block: (, p 3), (, p ), (3, p ), (4, p) 647

8 International Journal of Applie Engineering Research ISSN Volume, Number 6 (7) pp Secon block: (5,p ), (6,p ), (7,p) Thir block: (8,3p ), (9,3p) Fourth block: (,4p) Using the same proceures as escribe previous we manage to obtain the inices of entries with element for p = 5, p = 6 an p = 7 as follows Case p = 5 First block: (, p 4), (, p 3), (3, p ), (4, p ), (5, p) Secon block: (6,p 3), (7,p ), (8,p ), (9,p) Thir block: (,3p ), (,3p ), (,3p) Fourth block: (3,4p ), (4,4p) First block: (, p 6), (, p 5), (3, p 4), (4, p 3), (5, p ), (6, p ), (7, p) Secon block: (8,p 5), (9,p 4), (,p 3), (,p ), (,p ), (3,p) Thir block: (4,3p 4), (5,3p 3), (6,3p ), (7,3p ), (8,3p) Fourth block: (9,4p 3), (,4p ), (,4p ), (,4p) Fifth block: (3,5p ), (3,5p ), (4,5p ), (5,5p) Sixth block: (6,6p ), (7,6p) Seventh block: (8,7p) The entries of T p consist of an only. In general, for any p, the inices of entries of T p with element is given by Fifth block: Case p = 6 (5,5p) First block ((p) elements): (, p (p )), (, p (p )), (3, p (p 3)), (4, p (p 4)), (5, p (p 5)),, (p, p (p p)) = (p, p) First block: (, p 5), (, p 4), (3, p 3), (4, p ), (5, p ), (6, p) Secon block: (7,p 4), (8,p 3), (9,p ), (,p ), (,p) Thir block: (,3p 3), (3,3p ), (4,3p ), (5,3p) Fourth block: (6,4p ), (7,4p ), (8,4p) Fifth block: (9,5p ), (,5p) Sixth block: (,6p) Secon block ((p ) elements): (p +,p (p )), (p +,p (p 3)), (p + 3,p (p 4)), (p + 4,p (p 5)),, (p + + (p ), p (p p)) = (p, p) Thir block ((p ) elements): (p +,3p (p 3)), (p +,3p (p 4)), (p + 3,3p (p 5)),, (p + (p ), 3p (p p)) = (3p,3p) Case p = 7 648

9 International Journal of Applie Engineering Research ISSN Volume, Number 6 (7) pp Fourth block ((p 3) elements): ((3p ) +,4p (p 4)), ((3p ) +,4p (p 5)),, ((3p ) + (p 3), 4p (p p)) = (4p 5,4p)... (p )th block ( elements): (3,5p ) (3,5p ) (4,5p ) (5,5p) (p)th block ( element): ( p(p+), pp + (p p)) = ( p(p+), p ) Total number of entries having as the element is p + (p ) + (p ) + (p 3) + + (p (p )) + (p (p )) which is an arithmetic series with a = m an =. The sum of this series that represents the total number of entries having is SUM p = p [p + (p )( ) = p (p p + ) = p (p + ) which proves that for size of covariance vector V(S L ) is inee p (p + ). 649