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Efficient statistical significance approximation for local association analysis of high-throughput time series data

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Efficient statistical significance approximation for local association analysis of high-throughput time series data

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Content EFFICIENT STATISTICAL SIGNIFICANCE APPROXIMATION FOR LOCAL ASSOCIATION ANALYSIS OF HIGH-THROUGHPUT TIME SERIES DATA by Li Charlie Xia A Dissertation Presented to the FACULTY OF THE USC GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulllment of the Requirements for the Degree MASTER OF SCIENCE (STATISTICS) August 2012 Copyright 2012 Li Charlie Xia Dedication To my wife Ge, my parents Songjuan and Xianhui, and to my professors and friends. ii Acknowledgements I am most grateful to my advisor, Prof. Fengzhu Sun. Without his insightful vision and continuous support, I would not have come this far. I am also deeply impressed and in uenced by his rigorous academic attitude, for instance, careful examination of every line of my manuscripts, which I will carry on these merits I learned from him to my future research and life. I also would like to thank Prof. Allan Schumitzky and Prof. Jianfeng Zhang, who served as my MS committee. With their mathematical expertise, they have been valu- able resource to resort to and particular helpful during my research. The questions and comments raised by them directed me to further improve my works. I want to express my thanks to Profs. Sun and Chen's current and previous group members: Lin, Xiting, Joyce, Jing, Quan, Wangshu, Xuemei, Yang-ho, Kjong, Sungjie, Xiaolin, Tade and others, and Prof. Fuhrman group, Joshua, Jacob, Cheryl, Rohan and others I cannot enumerate. Thank you for your company through my PhD journey. Finally, I want to express my thanks to my family, other USC faculty members and friends for sharing with me a period of wonderful time in Los Angeles. Thank you all! iii Table of Contents Dedication ii Acknowledgements iii List of Tables vi List of Figures viii Abstract ix Chapter 1: Introduction 1 1.1 Local association analysis approaches . . . . . . . . . . . . . . . . . . . . . 1 1.2 Current limitations and our developments . . . . . . . . . . . . . . . . . . 3 Chapter 2: Methods 4 2.1 Previous works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 Maximum absolute partial sums of i.i.d and Markovian random variables 6 2.3 Statistical signicance for local similarity scores . . . . . . . . . . . . . . . 9 2.4 Statistical signicance for local shape analysis . . . . . . . . . . . . . . . . 11 2.5 Simulation studies and application to real datasets . . . . . . . . . . . . . 12 Chapter 3: Results 14 3.1 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 3.2 The CDC dataset . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 3.3 The SPOT dataset . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.4 The MPH dataset . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 3.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 Chapter 4: Conclusions 26 4.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 Bibliography 28 Appendix A Supplementary Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 A.1 Dealing with replicates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 iv A.2 Data normalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 A.3 Dealing with multiple zeros . . . . . . . . . . . . . . . . . . . . . . . . . . 33 Appendix B Supplementary Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 v List of Tables 3.1 Theoretical approximation for local similarity analysis p-values versus the simulated probability P (LS(D)= p n x). The theoretical approximate probability based on equation 2.3 with = 1 is given in the 2nd column and the simulated probability that LS(D)= p n x is given in the 3rd to the 9th columns. D = 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 3.2 Theoretical approximation for local shape analysis p-values versus the sim- ulated probabilityP (LS(D)= p 1:25nx). The theoretical approximate probability based on equation 2.3 with = 1 is given in the 2nd column and the simulated probability that LS(D)= p 1:25nx is given in the 3rd to the 9th columns. D = 0. . . . . . . . . . . . . . . . . . . . . . . . . . . 25 B.1 Theoretical approximation for local similarity analysis p-values versus the simulated probability P (LS(D)= p n x). The theoretical approximate probability based on equation 2.3 with = 1 is given in the 2nd column and the simulated probability that LS(D)= p n x is given in the 3rd to the 14th columns. D = 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 B.2 Theoretical approximation for local similarity analysis p-values versus Sim- ulated probabilityP (LS(D)= p nx). The theoretical approximate prob- ability based on equation 2.3 with = 1 is given in the 2nd column and the simulated probability that LS(D)= p n x is given in the 3rd to the 14th columns. D = 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 B.3 Theoretical approximation for local similarity analysis p-values versus the simulated probability P (LS(D)= p n x). The theoretical approximate probability based on equation 2.3 with = 1 is given in the 2nd column and the simulated probability that LS(D)= p n x is given in the 3rd to the 14th columns. D = 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 B.4 Theoretical approximation for local similarity analysis p-values versus the simulated probability P (LS(D)= p n x). The theoretical approximate probability based on equation 2.3 with = 1 is given in the 2nd column and the simulated probability that LS(D)= p n x is given in the 3rd to the 14th columns. D = 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 vi B.5 Theoretical approximation for local shape analysis p-values versus the sim- ulated probability P (LS(D)= p 1:25n x). The theoretical approximate probability based on equation 2.3 with = 1 is given in the 2nd column and the simulated probability that LS(D)= p 1:25nx is given in the 3rd to the 14th columns. D = 0. . . . . . . . . . . . . . . . . . . . . . . . . . 43 B.6 Theoretical approximation for local shape analysis p-values versus the sim- ulated probability P (LS(D)= p 1:25n x). The theoretical approximate probability based on equation 2.3 with = 1 is given in the 2nd column and the simulated probability that LS(D)= p 1:25nx is given in the 3rd to the 14th columns. D = 1. . . . . . . . . . . . . . . . . . . . . . . . . . 44 B.7 Theoretical approximation for local shape analysis p-values versus the sim- ulated probability P (LS(D)= p 1:25n x). The theoretical approximate probability based on equation 2.3 with = 1 is given in the 2nd column and the simulated probability that LS(D)= p 1:25nx is given in the 3rd to the 14th columns. D = 2. . . . . . . . . . . . . . . . . . . . . . . . . . 45 B.8 Theoretical approximation for local shape analysis p-values versus the sim- ulated probability P (LS(D)= p 1:25n x). The theoretical approximate probability based on equation 2.3 with = 1 is given in the 2nd column and the simulated probability that LS(D)= p 1:25nx is given in the 3rd to the 14th columns. D = 3. . . . . . . . . . . . . . . . . . . . . . . . . . 46 B.9 The comparison of signicant gene pairs usingP theo andP perm given type-I error 0.05 for local similarity (`original') and shape (`trendBy0') analysis of 25 randomly selected factors from the CDC dataset: (a-d), local similarity scores; (e-h), local shape scores; (a,e) D=0, (b,f) D=1, (c,g) D=2, (d,h) D=3. The total number of comparisons is 300. . . . . . . . . . . . . . . . 46 B.10 The comparison of signicant OTU pairs usingP theo andP perm given type- I error 0.05 for local similarity (`original') and shape (`trendBy0') analysis of 40 selected OTUs from SPOT dataset: (a-d), local similarity scores; (e- h), local shape scores; (a,e) D=0, (b,f) D=1, (c,g) D=2, (d,h) D=3. The total number of OTU pairs is 780. . . . . . . . . . . . . . . . . . . . . . . 47 vii List of Figures 3.1 The histogram of local similarity scores LS(D)= p n for n = 200 and D = 0; 1; 2; 3 together with the theoretical approximate density function given in Equation 2.6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3.2 The histogram of local shape scores LS(D)= p 1:25n for n = 200 and D = 0; 1; 2; 3 together with the theoretical approximate density function given in Equation 2.6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3.3 P theo and P perm comparison for all-to-all pairwise local similarity (`orig- inal') and shape (`trendBy0') analysis of 25 selected factors from CDC dataset. Columns D0 to D3 are for D = 0; 1; 2; 3, respectively. . . . . . . 19 B.1 Local similarity analysis P theo vs P perm for 10,000 pairs simulated data. Columns D0 to D3 are for D = 0; 1; 2; 3. Rows n10 to n100 are for n = 10; 20; 30; 40; 60; 80; 100. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 B.2 Local shape analysisP theo vsP perm for 10,000 pairs simulated data. Columns D0 to D3 are forD = 0; 1; 2; 3. Rows n10 to n100 are forn = 10; 20; 30; 40; 60; 80; 100. 40 B.3 P theo and P perm comparison for all-to-all pairwise local similarity (`origi- nal') and shape (`trendBy0') analysis of 40 abundant OTUs from SPOT dataset. Columns D0 to D3 are for D = 0; 1; 2; 3, respectively. . . . . . . 41 B.4 Examples of signicant local associations from `F4' feces sample of MPH dataset. Proles are shifted to synchronize the co-occurrence according to local similarity analysis. (left) Coprococcus and Escherichia (LS=-0.3179, P=0.0002; r=-0.3314, P=0.0001); (right) Eubacterium and Oscillospira (LS=0.3862, P=0.0001; r=0.3525, P=0.0001) . . . . . . . . . . . . . . . . 42 B.5 Running time comparison for example real dataset computation. Note that the constant computation time using the theoretical approach that is independent of sample size as compared to sample-size and precision dependent computation time of permutation approaches. . . . . . . . . . . 42 viii Abstract Local association analysis, such as local similarity analysis and local shape analysis, of biological time series data helps elucidate the varying dynamics of biological systems. However, their applications to large scale high-throughput data are limited by slow per- mutation procedures for statistical signicance evaluation. We developed a theoretical approach to approximate the statistical signicance of local similarity and local shape analysis based on the approximate tail distribution of the maximum partial sum of in- dependent identically distributed (i.i.d) and Markovian random variables. Simulations show that the derived formula approximates the tail distribution reasonably well (starting at time points > 10 with no delay and > 20 with delay) and provides p-values compa- rable to those from permutations. The new approach enables ecient calculation of statistical signicance for pairwise local association analysis, making possible all-to-all association studies otherwise prohibitive. As a demonstration, local association analy- sis of human microbiome time series shows that core OTUs are highly synergetic and some of the associations are body-site specic across samples. The new approach is im- plemented in our eLSA package, which now provides pipelines for faster local similarity and shape analysis of time series data. The tool is freely available from eLSA's website: http://meta.usc.edu/softs/lsa. ix Chapter 1 Introduction 1.1 Local association analysis approaches Understanding how genes regulate each other and when the regulations are active is an important problem in molecular biological research. Similarly, in ecological studies, it is important to understand how dierent organisms and environmental factors, such as food resources, temperature, etc. regulate each other to aect the whole community. For generality, we will refer to either genes in gene regulation studies or organisms or environmental factors in ecological studies asfactors. Time series data can give signicant insights about the regulatory relationships among dierent factors. Many computational or statistical approaches have been developed to cluster the genes into dierent groups so that the expression proles of genes in each cluster are highly correlated, see reviews [1, 3]. Most of these methods consider the correlation of expression patterns across the entire time interval of interest. For many gene regulation relationships, the regulation may be active in certain subintervals. Methods based on the global relationships of the gene expression proles may fail to detect these relationships. 1 Several methods have been developed to address this problem. Borrowing the idea from local alignment for molecular sequences, [18] proposed to identify local and potential time-delayed (-lagged) associations between gene expression proles. Here, local indicates the two factors are only associated within some time subinterval, and time-delayed in- dicates there is time shift in the associated proles. The strength of local association is measured by local similarity (LS) score and the statistical signicance of LS score is evaluated by a large number of permutations. The authors showed that such analysis can identify associated pairs that are not detectable through global analysis. [20] used a similar approach to study local associations of microbial organisms in the ocean over a four year period and this approach has been used in several other recent ecological studies [4, 6, 10, 21, 23]. [25] recently extended the approach to deal with replicated time series where not only statistical signicance of LS score can be evaluated, but also a bootstrap condence interval can be obtained. Several investigators have extended the basic local similarity approach for the gene expression proles to local shape analysis [2, 12, 14], where in [14], the time delay is xed and in [12], the time delay is not pre-dened but is estimated from the data. In local shape analysis, expression proles of n consecutive time points are changed to a n 1 time point series corresponding to decrease, no change, and increase in expression levels. The Smith-Waterman algorithm is then used to locate subintervals I andJ in the pair of obtained series, respectively, so that the product is maximized. The statistical signicance is evaluated by permutation procedures similar to that of local similarity analysis. However, this approach is problematic since the shape data are not independent even under the assumption that the real observed data are independent. 2 1.2 Current limitations and our developments One of the major limitations of the local similarity/shape analysis is the time consuming permutation procedure used to evaluate the statistical signicance (p-value) of the LS score. When a large number of G genes are considered, G(G 1)=2 gene pairs need to be evaluated. For a type I error , in order to adjust for multiple testing, the Bonferroni corrected threshold is 2=(G(G 1)). For G = 5000, the threshold is 4 10 9 when = 0:05, which will need over 2:510 8 permutations that is prohibitive. While in practice false discovery rate (q-value) is used to mitigate the multiple comparison problem, still, fast and ecient theoretical approximation for the statistical signicance of the LS score is urgently needed to accurately estimate the p-values. In this thesis work, we provide a theoretical approximation for the distribution of local similarity analysis LS scores as well as that based on local shape analysis. In the \Methods" chapter, we provide the theoretical bases for deriving the approximate tail probability that the LS score is above a threshold. In the \Results" chapter, we use simulations to study the number of data pointsn needed for the theoretical approximation to be valid. We also use the theoretical formula to study three real datasets arising from dierent high-throughput experiments: microarray, molecular nger printing, and NGS tag-sequencing. The thesis concludes with some discussion on further applications and future research directions. 3 Chapter 2 Methods 2.1 Previous works Consider time series data for two factors with levels X 1 ;X 2 ; ;X n and Y 1 ;Y 2 ; ;Y n , respectively. The rst step is to normalize the expression levels of each time series so that they can be regarded as normally distributed. Without loss of generality, we assume that they are already normally distributed. Second, dynamic programming algorithm is used to nd intervals I = [i;i +l 1] and J = [j;j +l 1] of the same length l such that the absolute value ofS = P l1 k=0 X i+k Y j+k is maximized, which is referred to as local similarity (LS) score [18, 20, 25]. Here the starting positions of the subintervals i and j, and the length of the intervalsl are not pre-specied and are all derived from the data. In most practical problems, investigators may only be interested in local associations with short delays, for example, the starting positions of the intervals in the two time series i and j are at most D apart, i.e.jijjD. We denote the LS score with time delay at most D by LS(D). 4 In the third step, statistical signicance for the LS score corresponding to the null hypothesis that the two factors are not associated is approximated by permuting one of the time series data many times and calculating the fraction of times that the LS score for the permuted data is higher than that for the real data [18, 20]. With such a permutation approach for p-value, the authors implicitly assumed that the observations for the samples at the dierent time points are independent under the null model. However, in many practical problems, in particular, time series data, the observations for each factor may depend on each other and the permutation based approach may not work well. Another drawback of the permutation based approach is computational time scales linearly with the inverse of the p-value precision and is computationally expensive for large dataset of long series. Here, we provide theoretical formulas to approximate the p-value overcoming both problems. The local shape analysis [2, 12, 14] is similar to local similarity analysis given above. The only dierence is to change the vectors of expression levels to a vector off-1, 0, 1g corresponding to decrease, no signicant change, or increase of the expression levels at consecutive time points. Under this situation, the permutation approach for the obtained vectors is even more problematic because the adjacent terms are highly correlated and the permutation approach destroys such dependency resulting in incorrect estimation of p-values. We will also provide new theoretical approaches to calculate the p-value corresponding to local shape analysis. 5 2.2 Maximum absolute partial sums of i.i.d and Markovian random variables In order to derive theoretical formulas to calculate the p-value related to the local sim- ilarity score, we resort to classical theoretical studies on the range of partial sums for independent identically distributed (i.i.d) random variables with zero mean [9] and the extensions to Markovian random variables [7]. The results from such studies when the expectation of the random variables is negative played key roles in the derivation of sta- tistical signicance for local sequence alignment (e.g. BLAST) which forms a milestone in the eld of computational biology [15]. On the other hand, the theoretical results on the approximate distributions when the mean is zero is not widely used in the computational biology community. Based on these previous theoretical studies, we present some theoretical results re- garding the range of partial sums for either i.i.d. or Markovian random variables. [9] studied the approximate distribution of the range of the sum of n random variables with mean 0. Let Z i be i.i.d. random variables such that E(Z i ) = 0 and Var(Z i ) = 2 . Let S n =Z 1 +Z 2 ++Z n ,M n = maxf0;S 1 ;S 2 ; ;S n g, andm n = minf0;S 1 ;S 2 ; ;S n g. The range is dened as R n = M n m n . It is shown in [9]: E(R n =) = 2 p 2n=, Var(R n =) = 4n(log(2) 2=). Using the theory of Bachelier-Wiener processes, [9] approximated the density function of R n = by (equations 3.7 and 3.8 in that paper) (n;r), (n;r) = r 2 r 1 L 0 (r=(2 p n)); (2.1) 6 where, L(z) = p 2z 1 1 X k=0 exp (2k + 1) 2 2 =8z 2 : Thus, P (R n =( p n)x) = Z 1 p nx r 2 r 1 L 0 r 2 p n dr = 1 8 1 X k=0 1 x 2 + 1 (2k + 1) 2 2 exp (2k + 1) 2 2 2x 2 (2.2) Next we give an upper bound for the tail in equation 2. This upper bound can be used to determine when we stop the summation in equation 2 for practical calculations. Note exp (2k+1) 2 2 2x 2 < exp (2k+1) 2 2x 2 , for k > 0. Thus, for any K > 0 such that (2K + 1)>x, we have, 1 X k=K 1 x 2 + 1 (2k + 1) 2 2 exp (2k + 1) 2 2 2x 2 < 2 x 2 1 X k=K exp (2k + 1) 2 2x 2 = 2 exp (2K+1) 2 2x 2 x 2 (1 exp 2 2 2x 2 ) Thus, for an approximation error threshold , we can choose K so that 16 exp (2K+1) 2 2x 2 x 2 1 exp 2 2 2x 2 : 7 Then we just approximate P (R n =( p n)x) by 1 8 K1 X k=0 1 x 2 + 1 (2k + 1) 2 2 exp (2k + 1) 2 2 2x 2 : (2.3) [7] studied the distribution of the maximum partial sum of an aperiodic Markov chain taken values on a nite subset of the real line, i.e H n = max 0i<jn (S j S i ). Let v be the stationary distribution of the Markov chain Z n , n = 0; 1; 2; with E v (Z 1 ) = 0 and 2 =E v (Z 2 1 ) + 2 P 1 k=2 E v (Z 1 Z k ). It was shown in [7] that lim n!1 P H n p n x = 4 1 X k=0 (1) k 2k + 1 exp (2k + 1) 2 2 8x 2 : (2.4) [8] provided an upper bound of C p log(n)=n for the approximation in equation 2.4. Similarly, we can dene L n = min 0i<jn (S j S i ). Since E(Z i ) = 0, L n has the same limiting distribution as H n . It can also be seen easily that R n = max(H n ;L n ). When x is large, the probability offH n >xg T fL n >xg will be small and P (R n =( p n)x) = P (max(H n ;L n )=( p n)x) = P (fH n =( p n)xg [ fL n =( p n)xg) P (H n =( p n)x) +P (L n =( p n)x) 2P (H n =( p n)x) 2 1 4 1 X k=0 (1) k 2k + 1 exp (2k + 1) 2 2 8x 2 ! ; x 0: (2.5) 8 The approximation works very well when x 2. However, when x is small, the approxi- mation does not work well and actually the above quantity can be larger than 1. 2.3 Statistical signicance for local similarity scores We next use the theory outlined in section 2.2 to approximate the statistical signi- cance in local similarity and local shape analyses. For time series data of two factors X 1 ;X 2 ; ;X n and Y 1 ;Y 2 ; ;Y n , we use the dynamic programming algorithm to cal- culate the LS score with maximum delay D denoted as s D . Corresponding to the null hypothesis that the two time series data are not related, the statistical signicance is given by p-value = P (LS(D) s D ), where LS(D) = max i;j;l;jijjD j P l1 k=0 X i+k Y j+k j: First consider the case thatD = 0. LetZ i =X i Y i ;i = 1; 2; ;n. Assuming that bothX i and Y i are independent standard normally distributed, we have E(Z i ) = 0 and E(Z 2 i ) = 1. Therefore, we can directly use the theory developed above in equations 2.3 to calculate the p-value. Next let us assume D > 0. Let S (d) n be the LS score with no time delay for the pair of series (d = 0;1;2; ;D) x 1 x 2 x 3 x n2 x n1 x n y 1+d y 2+d y 3+d x n2+d x n1+d x n+d where we consider the data as missing when the subscript is outside the range [1;n] and the pair is not considered when the LS score is calculated. When n is suciently large, S (d) n ford = 0;1;2; ;D can be considered as approximately identically distributed because S (d) n is the LS score for nd pairs of i.i.d. normal random variables. The tail distribution function ofS (d) n =( p n) can be approximated by equation 2.3. NoteLS(D) = 9 max D d=D S (d) n : In order to derive an approximate cumulative distribution function of LS(D), we pretend that S (d) n ;d = 0;1;2; ;D are independent although they are not. Then P (LS(D)=( p n)x) = D Y d=D P (S (d) n =( p n)x) (use independence assumption) = 8 2D+1 1 X k=1 1 x 2 + 1 (2k 1) 2 2 exp (2k 1) 2 2 2x 2 ! 2D+1 Thus, the tail probability of LS(D) can be approximated by L(x) = P(LS(D)=( p n)x) 18 2D+1 0 @ 1 X k=1 1 x 2 + 1 (2k1) 2 2 ! exp (2k1) 2 2 2x 2 ! 1 A 2D+1 : From Equation 2.6, we can obtain the approximate density function of R (D) n =( p n) by f D (x) = (2D+1) x 3 8 2D+1 0 @ 1 X k=1 1 x 2 + 1 (2k1) 2 2 ! exp (2k1) 2 2 2x 2 ! 1 A 2D 1 X k=1 (2k1) 2 2 x 2 1 ! exp (2k1) 2 2 2x 2 ! : (2.6) Finally, we can model time series data with replicate data as well as with large fraction of zero values based on equation 2.6, using additional scaling in , and the detail of the method is given in the Supplementary Methods. 10 2.4 Statistical signicance for local shape analysis Next, we use the theory developed in section 2.2 to approximate the p-value corresponding to local shape (also LS) score. Note that in local shape analysis, we change the n- dimensional vectorX to an 1 dimensional vector (d X i ;i = 1; 2; ;n 1), whered X i = sign(X i+1 X i ), sign(x) = 1 whenx> 0 and sign(x) =1 whenx< 0 . Note that even if the components ofX are independent, the random variablesd X i ;i = 1; 2; ;n1 are not independent because both d X i and d X i+1 depend on X i+1 . Assuming that X 1 ;X 2 ; ;X n are i.i.d continuous random variables such that the probability of taking a xed value to be 0, we haveP [(d X i ;d X i+1 ) = (1; 1)] =P [(d X i ;d X i+1 ) = (1;1)] = 1=6 andP [(d X i ;d X i+1 ) = (1;1)] =P [(d X i ;d X i+1 ) = (1; 1)] = 1=3. Note that P (d X i = 1) = P (d X i =1) = 1=2 if the X's are exchangible. Therefore d X i ; i = 1; 2; ;n1 are not independent. Actuallyd X 1 ;d X 2 ; ;d X n1 do not even form a Markov chain. However, let us pretend that they form a Markov chaind X i with transition matrix T = 1 -1 1 1/3 2/3 -1 2/3 1/3 : Then it can be shown by spectral expansion that T k = 1 2 0 B B @ 1 + (1) k =3 k 1 (1) k =3 k 1 (1) k =3 k 1 + (1) k =3 k 1 C C A : 11 For k > 1, we have P (d X 1 d X k+1 = 1) = (1 + (1) k =3 k )=2 and P (d X 1 d X k+1 = 1) = (1 (1) k =3 k )=2. Thus, E(d X 1 d X k+1 ) = (1) k =3 k : In local shape analysis, we com- pare d X 1 ;d X 2 ; ;d X n1 with d Y 1 ;d Y 2 ; ;d Y n1 . The score for the local shape analysis is the range of P i d X i d Y i when D = 0. Using the results of [7], we have 2 XY = E((d X 1 ) 2 )E((d Y 1 ) 2 ) + 2 P 1 k=1 E(d X 1 d X k+1 )E(d Y 1 d Y k+1 ) = 1 + 2 P 1 k=1 1=3 2k = 1 + 1=4 = 1:25. Thus, for the local shape score, P (LS(D)s D ) =P LS(D) XY p n s D XY p n =L s D p 1:25n ; where the functionL is dened in equation 2.6. 2.5 Simulation studies and application to real datasets In deriving the approximate p-values for local similarity and local shape analysis in sec- tions 2.3 and 2.4, we made several simplifying assumptions, whose eects on the accuracy of the approximations were evaluated by simulations. We rst study the accuracy of the approximation for the tail probability of R n =( p n) in equation 2.6 using simulations for both local similarity and shape analysis. Firstly, for given number of time points n, we generaten pairs of i.i.d standard normal random variables (X i ;Y i ),i = 1; 2; ;n andX i and Y i are independent. Secondly, the dynamic programming algorithm is then used to calculate the local similarity score with at most time delayD,LS(D). Thirdly, we repeat the rst two steps 10,000 times and obtain the empirical distribution of LS(D)=( p n). We compare the empirical distributions with the theoretical approximation given in equa- tion 2.6 for local similarity scores ( = 1). We did similar simulation studies for local 12 shape scores, using the transformed series off1; 0; 1g from the original standard normal series ( = p 1:25). We then apply our method to analyze three real datasets. The rst one is a mi- croarray yeast gene expression dataset, synchronized by cdc-15 gene, from [22] (referred to as `CDC'). The second one is an ARISA molecular nger printing microbial ecology dataset from San Pedro Ocean Time Series in [23] (referred to as `SPOT') . The third one is a 16S RNA tag-sequencing dataset from Moving Pictures of Human microbiota sampling of human symbiotic microbial communities from [5] (referred to as `MPH'). We applied both local similarity and shape analysis to re-analyze the rst two datasets and compared the theoretical and permutation approaches. We are the rst to analyze the third dataset for local similarity and shape analyses. For the local similarity analysis, datasets are normalized by the procedures described in Supplementary Methods (referred to as `original'). For the local shape analysis, datasets are converted to local trend series off1; 0; 1g and analyzed without normalization (referred to as `trendBy0'). 13 Chapter 3 Results 3.1 Simulations The approximate p-values for the local similarity and local shape analysis given in sections 2.3 and 2.4 are only applicable when the p-value is small and the number of time points is large. Thus, it is important to know the range of applicability for the approximation. Table 3.1 gives the theoretical tail probability based on equation 2.6 (2nd column) and the simulated probabilityP (LS(0)= p nx) (3rd to 9th columns) for dierent number of time points when D = 0. It can be seen that the theoretical tail probability is very close to the simulated probability when x 3 corresponding to the theoretical p-value less than 0.011. The approximation is even reasonable when the number of time points is just 10. In general, the theoretical tail probability is slightly larger than the simulated values when D = 0. Similar results are observed for local shape analysis (Table 3.2) when n is as small as 20. When D> 0, the theoretical approximation is close to the simulated tail probability when n 20 and n 30 for local similarity and shape analysis, respectively 14 (see Tables SB.1-B.8 in Supplementary Results). Thus, if we use the theoretical approx- imate distribution to calculate the p-value, we will be slightly conservative in declaring signicant associations. For relatively small value of x, the theoretical approximation can be much larger than the simulated tail probability. One potential explanation is that the R (D) n =( p n) is stochastically increasing and that asn increases the theoretical approximation become closer to the simulated distribution ofLS(D)=( p n). We also tested if theP (LS(D)=( p n) x) = 1 (1P (R (0) n =( p n) x)) 2D+1 is generally true using the simulated tail prob- abilities and it can be clearly seen from Tables SB.1-B.8 that this relationship is indeed reasonable indicating that S (d) n ; d = 0;1;D can indeed be considered as indepen- dent. In equation 2.6, we derive the approximate density function of R (D) n =( p n). We superimpose this approximate density function to the histograms of the simulated values of LS(D)=( p n) at n = 200 and D = 0; 1; 2; 3 as in Figures 3.1-3.2 for local similarity and local shape scores, respectively. Several observations can be made from the gure. First, the values of LS(D)=( p n) increases as a function of D as expected. Second, the approximate theoretical density function is slightly lower than the simulated frequency when x is lower than the mode of the theoretical distribution and is slightly higher than the simulated frequency when x is larger than the model of the theoretical distribution, thus the tail probability based on the theoretical approximation is slightly higher than the simulated value. We next see how p-values (P theo ) derived from theoretical approximation compare to that of permutation (P perm ) given the same data, see Figures SB.1-B.2. For local 15 Number of time points n x Theory 10 20 30 40 60 80 100 2 0.1815 0.0848 0.0987 0.1062 0.1122 0.1201 0.1235 0.1290 2.2 0.1111 0.0541 0.0621 0.0645 0.0665 0.0699 0.0771 0.0767 2.4 0.0656 0.0341 0.0367 0.0392 0.0416 0.0411 0.0435 0.0457 2.6 0.0373 0.0223 0.0221 0.0252 0.0235 0.0232 0.0249 0.0261 2.8 0.0204 0.0147 0.0128 0.0154 0.0131 0.0129 0.0138 0.0163 3.0 0.0108 0.0093 0.0082 0.0088 0.0074 0.0069 0.0071 0.0090 3.2 0.0055 0.0056 0.0051 0.0038 0.0036 0.0030 0.0035 0.0054 3.4 0.0027 0.0033 0.0031 0.0017 0.0022 0.0009 0.0016 0.0027 3.6 0.0013 0.0019 0.0020 0.0011 0.0014 0.0004 0.0006 0.0012 3.8 0.0006 0.0007 0.0008 0.0006 0.0010 0.0002 0.0004 0.0009 4.0 0.0003 0.0004 0.0005 0.0003 0.0005 0.0000 0.0003 0.0004 4.2 0.0001 0.0002 0.0004 0.0002 0.0005 0.0000 0.0001 0.0002 4.4 0.0000 0.0001 0.0003 0.0001 0.0002 0.0000 0.0000 0.0001 4.6 0.0000 0.0000 0.0003 0.0001 0.0000 0.0000 0.0000 0.0001 4.8 0.0000 0.0000 0.0002 0.0001 0.0000 0.0000 0.0000 0.0000 5.0 0.0000 0.0000 0.0001 0.0001 0.0000 0.0000 0.0000 0.0000 Table 3.1: Theoretical approximation for local similarity analysis p-values versus the simulated probabilityP (LS(D)= p nx). The theoretical approximate probability based on equation 2.3 with = 1 is given in the 2nd column and the simulated probability that LS(D)= p nx is given in the 3rd to the 9th columns. D = 0. similarity case, starting from D = 0 and n = 20, points in scatter plots become con- centrated on the diagonal line (where P perm =P theo ) and they become more aligned as n increases. Similar trend is observed for local shape analysis. This indicates an increasing rate of agreement between the theoretical and permutation p-values, representing their reasonable approximation to the null distribution despite of inherent variance associated with the permutation procedures. The same is true with D > 0 for both analysis and the theoretical approximation become signicantly closer to the permutation one as n increase. Though, whenD> 0, the variation seems more substantial and close alignment only starts at n = 30. In summary, we can see that if we are interested in statistical sig- nicance given some type I error threshold, the theoretical approach shall provide results comparable to that from permutation starting from n = 20. 16 Figure 3.1: The histogram of local similarity scores LS(D)= p n for n = 200 and D = 0; 1; 2; 3 together with the theoretical approximate density function given in Equation 2.6. Figure 3.2: The histogram of local shape scores LS(D)= p 1:25n for n = 200 and D = 0; 1; 2; 3 together with the theoretical approximate density function given in Equation 2.6. 17 3.2 The CDC dataset The CDC dataset consists of the expression proles of 6177 genes at 24 time points. It is extremely time consuming to approximate the p-values for all the gene pairs using permutations. Thus, we only randomly selected 25 genes and estimated the p-value for each of the 300 gene pairs by permuting the original data 1000 times. We then compared the p-values by our theoretical approximation denoted by P theo with the p-values by the permutation approach denoted as P perm . The results are given in Figure 3.3. It can be seen from the gure that P theo is highly positively associated with P perm , but P theo is slightly higher than P perm indicating that it is conservative when we declare statistical signicance usingP theo . For a specic example, Table SB.9 in the supplementary materials compares the gene pairs declared as signicant by eitherP theo orP perm for the type-I error threshold 0.05. For all the situations considered, none of P theo is less than 0.05 when P perm > 0:05. Among the gene pairs with P perm 0:05, over half of them are declared as signicant byP theo . For the local similarity analysis (`original'), using D = 0, we have 233 (78%) out of 300 found to be non-signicant by both theoretical approximation and permutations. Among the remaining, 48 (16%) are found signicant by both methods, and in total 281 (94%) are in agreement. The results are similar with D=1,2,3, with 262 (88%), 262 (88%) and 262 (88%) in agreement, respectively. For local shape analysis (`trendBy0'), the value of P theo is more closer to the value of P perm . With D=0 and type-I error 0.05, we have 53 (18%) out of 300 found signicant while 241 (80%) non-signicant by both approaches, and in total 296 (98%) are in agree- ment. Among the gene pairs withP perm > 0:05, none of them are signicant usingP theo . 18 - log 10 (P theo ) - log 10 (P perm ) 1 2 3 1 2 3 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● D0 trendBy0 ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ●● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● D1 trendBy0 1 2 3 ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ●● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●●● ● ● ● ●● ● ● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● D2 trendBy0 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● D3 trendBy0 ●●● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● D0 original 1 2 3 ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● D1 original ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ●● ● ● ● ● ● ● ● ●● ●● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●● ● ● ●● ● ● ●● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● D2 original 1 2 3 1 2 3 ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ●● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● D3 original Figure 3.3: P theo and P perm comparison for all-to-all pairwise local similarity (`original') and shape (`trendBy0') analysis of 25 selected factors from CDC dataset. Columns D0 to D3 are for D = 0; 1; 2; 3, respectively. Among the gene pairs declared as signicant by P perm , about 53/59 (90%) are declared as signicant by P theo . Similarly, with D=1,2,3, there are 291 (97%), 281 (94%) and 278 (93%) p-value pairs in agreement by bothP perm andP theo , respectively. In fact, all-to-all pairwise analysis of the whole CDC dataset with D=3 and permutation 1000 times cannot be completed in 100 hours on a \Dell, PE1950, Xeon E5420, 2.5GHz, 12010MB RAM" computing node, while, using the theoretical approach, the analysis nishes in 10 hours on the same node. 3.3 The SPOT dataset The SPOT dataset consists of ten-year monthly (114 time points) sampled operational taxonomic unit (OTU) abundance data. A major numerical dierence between the CDC and the SPOT dataset is that the later has a large number of zeros, where the OTU's abundance drops below the measurement limit. We adjusted our model for this fact as described in Supplementary Methods. As in the above section, we selected 40 abundant 19 OTUs from the SPOT dataset with the criteria \the OTU occurs at least 20 times with minimum relative abundance 1% and has less than 10 missing values". In the selected set of OTUs, we observe the fraction of zeros ranging from 0% to 46% with median at 12%, while there is almost none zeros in CDC. We summarize p-value comparison for local similarity and shape analysis in Table SB.10 and Figure SB.3 in the supplementary materials. For local similarity analysis (`original'), with D = 0 and type-I error 0.05, we have 488 (63%) out of 780 found non-signicant and 261 (33%) signicant by both methods. In total, 685 (96%) are in agreement. All of the remaining 31 (4%) pairs are signicant by P theo but non-signicant byP perm . The results are similar with D=1,2,3: 733 (94%), 723 (93%) and 727 (93%) in concordance, respectively. There are about 6-7% associations signicant byP perm but non-signicant byP theo , showing thatP theo is more conservative. For local shape analysis (`trendBy0'), the result show even better concordance. With D = 0 and type-I error 0.05, we have 578 (74%) and 182 (23%) out of 780 found signicant and non-signicant, respectively, by both methods. In total 760 (97%) are in agreement. Among the 20 (3%) OTU pairs with discordant signicance by P theo and P perm , all of them are signicant by P perm but non-signicant by P theo . The results are similar with D=1,2,3, with 754 (97%), 757 (97%) and 749 (96%) in concordance, respectively, and about 3-4% incidences signicant byP perm but non-signicant byP theo . The concordance for the signicant results between P theo and P perm for the SPOT dataset is better than that for the CDC dataset. This can be explained by the fact that the number of time points for the SPOT data (114) is much higher than that for the CDC dataset (24) and the approximation is better when the number of time points is large. 20 3.4 The MPH dataset The MPH dataset consists of 130, 133 and 135 daily sequenced samples from feces, palm and tongue sites of a female (`F4') and 332, 357, 372 samples from a male (`M3') [5]. The genus level OTU abundance is used for our analysis. There are 335, 1295, 373 unique OTUs from feces, palm and tongue sites of `F4' and `M3', respectively. With D = 3, we analyzed the MPH dataset with local similarity and shape analyses. Because of the intra-person variability (both time and site) of the human microbiota, one important step in analyzing human microbiota datasets is to identify the core (persisting) group of microbes for a specic body site of a person. Based on the discussion in [5], we consider core OTUs as those showing in at least 60% of samples from the same body site of a specic person. Using this criteria, we identied 45, 252 and 41 core OTUs for the feces, palm and tongue sites of `F4', and 59, 269 and 56 core OTUs for the corresponding sites of `M3'. Subsequent analysis show these symbiotic core microbes are highly synergetic. We use local shape analysis as our main approach and report signicant local associations with p-value<=0.05, Q-value<=0.05 and Aligned Length>= 80% time points [25]. We found 159 signicant associated pairs within the subset formed by the 45 core OTUs of the `F4' feces samples and the intra association rate (the average degree divided by number of OTUs minus one in an association subnetwork) is 16%. The same rates are 26% and 25% for `F4' tongue and palm samples, whereas 44%, 56% and 34% for `M3' feces, tongue and palm samples. These percentages translate into a picture of high connectivity between these OTUs, supporting their role as crucial players in the corresponding microbiota. 21 However, do these site-specic signicant local associations persist across individu- als? In fact, we found some of the body-site specic dynamics are shared across human subjects. We take the intersection (shared) of all signicant association pairs between two samples and calculate the shared percentage dividing by the union between the two. We found `F4' and `M3' share 83 (9%) in their feces microbiota. These are mostly from the Ruminococcaceae, Lachnospiraceae, Clostridiaceae, Clostridiaceae Family XI Incertae Sedis, Prevotellaceae, Coriobacteriaceae families. In tongue and palm sites, the fractions are 17% and 21%, respectively. In contrast, signicant associations are generally not shared between dierent body sites even within one person. For example, only 1 ( 0%) pair shows signicance in both feces and tongue sites of `F4'. Similar low fractions were observed for other sites. On the other hand, using local similarity analysis, we found some time-delayed lo- cal associations. For example, in `F4' feces sample, the global proles Coprococcus and Escherichia are not signicantly correlated by PCC (r=-0.1708, P=0.0521) while signif- icantly by eLSA (LS=-0.3179, P=0.0002). The association is signicantly negative for 126 consecutive time points, where Coprococcus leads Escherichia 3 time points in the co-occurrence. Hinted by this, shifting the Coprococcus prole 3 time units backward, we see their global prole are signicantly negatively correlated (r=-0.3314, P=0.0001), see Figure SB.4. As another example in `F4' feces sample, Eubacterium and Oscillospira is not signicantly correlated by PCC (r=0.1313, P=0.1364), however, signicantly associ- ated for 122 consecutive time points by eLSA (LS=0.3862, P=0.0001), where the former 22 leads the later 2 time points in the co-occurrence. Hinted by this, shifting the Eubac- terium prole 2 time units backward, we see they actually are also globally signicantly correlated (r=0.3525, P=0.0001), see Figure SB.4. 3.5 Discussion In this paper, we provide theoretical formulas to approximate the statistical signicance of local similarity and local shape analyses for time series data. The theoretical approx- imations make it possible to evaluate the statistical signicance of comparisons of time series data for a large number of factors such as genes in gene expression analysis or OTUs in metagenomic studies, which is impossible to carry out using the original per- mutation based approach. The theoretical approximation is more mathematically sound with specied assumptions of data distribution veriable before applying the analysis. The permutation test, however, heavily depends on data-specic empirical distributions and can be biased by the numerical properties of specic data as well as its intrinsic vari- ability. In particular, in the local shape analysis, permuting the sequence independently may be misleading because the trend sequence is generated with dependency. In addition, if we are interested in the tail distribution (in most applications), the two methods are mostly in agreement with each other in predictions given the same type- I error threshold. We have results from setting threshold to lower values (0.01, 0.005, 0.001, etc.) showing high overall agreement rate. Therefore, from the practical point of view, we can substitute permutations with the theoretical method in such applications. Moreover, from the simulations and our real analysis, P theo is more conservative P perm 23 { a property particularly useful in biological applications prone to substantial number of false positives, such as the microarray analysis [17]. The most important reason for us to embrace the theoretical method is computational eciency. As shown in [25], for a given type-I error, , the time complexity of P perm is O(DMN=), where D is the delay limit, N is the sample number and M the replicate number. With P theo , we may compute and store them into a hash table, before any pairwise comparison. Then, for each p-value calculation, it only costs constant time O(1) to read out P theo and is independent of D, M, N and Q, a strongly desired feature for large scale analysis. The superiority of eciency is evident from Figure SB.5, in which, the time cost of analyzing 40 factors of 113 and 114 time points are compared. The per- pair time cost is about 40 seconds forP perm while negligible forP theo and independent of sample size, which is a big saver of computing resource, energy, and research time. 24 The number of time points n x Theory 10 20 30 40 60 80 100 2 0.1815 0.0491 0.1242 0.0921 0.0986 0.1092 0.1439 0.1308 2.2 0.1111 0.0491 0.0677 0.0613 0.0680 0.0599 0.0832 0.0801 2.4 0.0656 0.0132 0.0367 0.0353 0.0430 0.0420 0.0492 0.0487 2.6 0.0373 0.0053 0.0168 0.0211 0.0158 0.0207 0.0266 0.0207 2.8 0.0204 0.0053 0.0094 0.0057 0.0106 0.0098 0.0137 0.0125 3.0 0.0108 0.0000 0.0039 0.0027 0.0034 0.0057 0.0069 0.0080 3.2 0.0055 0.0000 0.0016 0.0013 0.0021 0.0026 0.0032 0.0044 3.4 0.0027 0.0000 0.0002 0.0007 0.0009 0.0009 0.0008 0.0020 3.6 0.0013 0.0000 0.0002 0.0000 0.0004 0.0005 0.0004 0.0009 3.8 0.0006 0.0000 0.0000 0.0000 0.0001 0.0003 0.0001 0.0004 4.0 0.0003 0.0000 0.0000 0.0000 0.0000 0.0000 0.0001 0.0002 4.2 0.0001 0.0000 0.0000 0.0000 0.0000 0.0000 0.0001 0.0000 4.4 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 4.6 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 4.8 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 5.0 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 Table 3.2: Theoretical approximation for local shape analysis p-values versus the sim- ulated probability P (LS(D)= p 1:25n x). The theoretical approximate probability based on equation 2.3 with = 1 is given in the 2nd column and the simulated probability that LS(D)= p 1:25nx is given in the 3rd to the 9th columns. D = 0. 25 Chapter 4 Conclusions 4.1 Conclusions The recent advent of high-throughput technologies made possible large scale time-resolved omics studies (proteomics, transcriptomics, metagenomics), tracking hundreds, thou- sands, or even tens of thousands molecules simultaneously. Time-series generated from these studies provide an invaluable opportunity to investigate the varying dynamics of biological systems. However, to make full use of huge datasets, accurate and ecient statistical and computational methods are urgently needed in all levels of analysis, from accurate estimation of abundance and expression levels, to pairwise association and net- work analysis. The theoretical statistical signicance approximation we proposed in this work can serve as an ecient alternative for calculating p-values in local similarity and shape anal- ysis. Its time cost is always constant, which reduces the computational burden in a large scale pairwise analysis. For example, in metagenomics, after short read assignment 26 and abundance estimation [13, 24], proles of thousands of microbial OTUs are avail- able. Before this work, pairwise local association analysis with this number of factors are hardly tractable using permutation procedures, if not impossible. Parallel computation and hardware acceleration or some pre-clustering and ltering approaches are required, in- creasing the diculty of analysis. With the new method, researchers can quickly compute the statistical signicance for all OTU pairs on desktop computers, allowing on-the- y network mining and analysis. Analyzing the MPH dataset with the new method, we found body-site specic human microbiota core OTUs are highly coordinated. There exist robust site-specic associations across persons. We implemented the new method in the eLSA package [25], now providing faster pipelines for local similarity and shape analysis. The methodological part remain true for other types of local associations that use the maximum range of partial sum of i.i.d and markovian zero-mean and nite variance random variables as their metrics. It may be further developed for application in more complex analysis, such as local shape with non-zero thresholds and local liquid association analysis. 27 Bibliography [1] I. P. Androulakis, E. Yang, and R. R. Almon. Analysis of time-series gene expression data: methods, challenges, and opportunities. Annu Rev Biomed Eng, 9:205{28, 2007. [2] R. Balasubramaniyan, E. Hullermeier, N. Weskamp, and J. Kamper. Clustering of gene expression data using a local shape-based similarity measure. Bioinformatics, 21(7):1069{77, 2005. [3] Z. Bar-Joseph. Analyzing time series gene expression data. Bioinformatics, 20(16):2493{503, 2004. [4] J. M. Beman, J. A. Steele, and J. A. Fuhrman. Co-occurrence patterns for abundant marine archaeal and bacterial lineages in the deep chlorophyll maximum of coastal california. ISME J, 5(7):1077{85, 2011. [5] J. G. Caporaso, C. L. Lauber, E. K. Costello, D. Berg-Lyons, A. Gonzalez, J. Stombaugh, D. Knights, P. Gajer, J. Ravel, N. Fierer, et al. Moving pictures of the human microbiome. Genome Biol, 12(5):R50, 2011. [6] S. Charon, H. Rehrauer, J. Pernthaler, and C. von Mering. A global network of coexisting microbes from environmental and whole-genome sequence data. Genome Res, 20(7):947{59, 2010. [7] J. J. Daudin, M. P. Etienne, and P. Vallois. Asymptotic behavior of the local score of independent and identically distributed random sequences. Stoch Proc Appl, 107(1):1{28, 2003. [8] M. P. Etienne and P. Vallois. Approximation of the distribution of the supremum of a centered random walk. application to the local score. Methodol Comput Appl, 6(3):255{275, 2004. [9] W. Feller. The asymptotic distribution of the range of sums of independent random variables. Ann Math Stat, 22(3):427{432, 1951. [10] J. A. Gilbert, J. A. Steele, J. G. Caporaso, L. Steinbruck, J. Reeder, B. Temperton, S. Huse, A. C. McHardy, R. Knight, I. Joint, et al. Dening seasonal marine microbial community dynamics. ISME J, 6(2):298{308, 2011. 28 [11] M. S. Gilthorpe, M. Frydenberg, Y. Cheng, and V. Baelum. Modelling count data with excessive zeros: the need for class prediction in zero-in ated models and the issue of data generation in choosing between zero-in ated and generic mixture models for dental caries data. Statistics in medicine, 28(28):3539{53, 2009. [12] F. He and A. P. Zeng. In search of functional association from time-series microarray data based on the change trend and level of gene expression. BMC Bioinformatics, 7:69, 2006. [13] P. He and L. Xia. Oligonucleotide proling for discriminating bacteria in bacterial communities. Comb Chem High T Scr, 10(4):247{255, 2007. [14] L. Ji and K. L. Tan. Identifying time-lagged gene clusters using gene expression data. Bioinformatics, 21(4):509{16, 2005. [15] S. Karlin, A. Dembo, and T. Kawabata. Statistical composition of high-scoring segments from molecular sequences. Ann Stat, 18(2):571{581, 1990. [16] K. C. Li. Genome-wide coexpression dynamics: theory and application. Proc Natl Acad Sci U S A, 99(26):16875{80, 2002. [17] Y. Pawitan, S. Michiels, S. Koscielny, A. Gusnanto, and A. Ploner. False discovery rate, sensitivity and sample size for microarray studies. Bioinformatics, 21(13):3017{ 24, 2005. [18] J. Qian, M. Dolled-Filhart, J. Lin, H. Yu, and M. Gerstein. Beyond synexpression relationships: local clustering of time-shifted and inverted gene expression proles identies new, biologically relevant interactions. J Mol Biol, 314(5):1053{66, 2001. [19] G. P. Quinn and M. J. Keough. Experimental design and data analysis for biologists. Cambridge University Press, Cambridge, UK ; New York, 2002. [20] Q. Ruan, D. Dutta, M. S. Schwalbach, J. A. Steele, J. A. Fuhrman, and F. Sun. Local similarity analysis reveals unique associations among marine bacterioplankton species and environmental factors. Bioinformatics, 22(20):2532{8, 2006. [21] A. Shade, C. Y. Chiu, and K. D. McMahon. Dierential bacterial dynamics pro- mote emergent community robustness to lake mixing: an epilimnion to hypolimnion transplant experiment. Environ Microbiol, 12(2):455{66, 2010. [22] P. T. Spellman, G. Sherlock, M. Q. Zhang, V. R. Iyer, K. Anders, M. B. Eisen, P. O. Brown, D. Botstein, and B. Futcher. Comprehensive identication of cell cycle- regulated genes of the yeast saccharomyces cerevisiae by microarray hybridization. Mol Biol Cell, 9(12):3273{97, 1998. [23] J. A. Steele, P. D. Countway, L. Xia, P. D. Vigil, J. M. Beman, D. Y. Kim, C. E. Chow, R. Sachdeva, A. C. Jones, M. S. Schwalbach, et al. Marine bacterial, archaeal and protistan association networks reveal ecological linkages. ISME J, 5(9):1414{25, 2011. 29 [24] L. Xia, J. Cram, T. Chen, J. Fuhrman, and F. Sun. Accurate genome relative abun- dance estimation based on shotgun metagenomic reads. PLoS ONE, 6(12):e27992, 2011. [25] L. Xia, J. Steele, J. Cram, Z. Cardon, S. Simmons, J. Vallino, J. Fuhrman, and F. Sun. Extended local similarity analysis (elsa) of microbial community and other time series data with replicates. BMC Syst Biol, 5(Suppl 2):S15, 2011. 30 Appendix A Supplementary Methods A.1 Dealing with replicates. To reduce the eect of biological and/or technical variation on the LSA results, replicate experiments are frequently carried out [19]. An extended LSA (eLSA) [25] approach was developed for time series data with replicates. First, for each sample the replicate data at each time point was summarized by a function, for example, the average over the replicates. Second, the local similarity score is calculated using the averages for the sample pairs. Third, statistical signicance for testing the hypothesis that the two sequences are related by randomly shuing the data along the dierent time points. Finally, the bootstrap condence interval for the LS score is obtained by bootstrapping the data at each time point by sampling from the observed data with replacement. With the theory developed above, we can signicantly speed up the process of evaluat- ing the statistical signicance of the LS score in the third step. LetX (m) = (X (m) 1 ; ;X (m) n ) and Y (m) = (Y (m) 1 ; ;Y (m) n ) be the m-th replicate for the time series data, m = 1; 2; ;M. The essence of eLSA is to calculate the local similarity score of U i = 31 F (X (1) i ; ;X (M) i ) andV i =F (Y (1) i ; ;Y (M) i ), whereF () is the summarizing function. Then by replacing X and Y in non-replicated case with U and V , respectively, similar approaches can be used to obtain the p-value. In particular, ifU i =F (X (1) i ; ;X (M) i ) = X i = P M m=1 X (m) i =M and all the X (m) i are standard normal, then Var(U i ) = 1=M. Sim- ilarly, we have Var(V i ) = 1=M assuming that each of the X (m) i is already normalized to be standard normal. Thus, 2 = Var(U i V i ) = 1=M 2 . Let the local similarity score with time delay at most D for the real data be s D . Then the p-value is calculated by P (LS(D)s D ) =P MLS(D) p n Ms D = p n =L(Ms D = p n); where the functionL is dened in equation 2.6 in the Method section. A.2 Data normalization eLSA requires the input series of factors X and Y to be standard normally distributed, which may not be satised by the raw data. Through normalization, the normality of the data can be ensured for subsequent analysis. To accommodate possible nonlinear associations and the variation of scales within the raw data, we apply the following approach to normalize the raw data before any LS score calculations [16]. We use x i to denote the original raw data of the i-th time spot of a factor X. First, we take r k = rank ofx k infx 1 ;x 2 ;:::;x n g: 32 Then, we take s k = 1 r k n + 1 ; where is the cumulative distribution function of the standard normal distribution. In case of smalln, we nd that the above transformed dataS =s [1:n] do not necessarily follow a standard normal distribution closely. When the variance is not 1 and mean is not zero, it will cause the LS scores calculated to be smaller than that expected from the theory and can lead to unexpected high p-values. To overcome this diculty, we further scale and shift S =s [1:n] using the Z-score transformation, such that z i = s i S S : We will take Z =z [1:n] as the standardized normalization of X. A.3 Dealing with multiple zeros When normalizing the time series data, we rank the input data by their numerical value and then take the standard normal percentile which equals to their rank divided by the total number of data points plus one. In some real situations, we may encounter a number of indistinguishable zeros (beyond measurement limit) from the input, which violates our assumption that the data comes from a normal distribution. To accommodate these indistinguishable zeros, we adjust our theoretical model using mixture ideas similar to zero-in ated models [11]. Let the proportion of zeros in the input sequencesX andY be 1 and 1, respectively. In the normalization step, we bypass 33 these zeros, while the other non-zeros are normalized as usual. The normalized series Z X and Z Y thus can be modeled as i.i.d. sampled from the following mixture distributions: Z X (1)f0g +W X ; Z Y (1)f0g +W Y ; where W X and W Y are independent and standard normal N(0; 1). Consequently, Z X Z Y (1)f0g +W X W Y : Therefore, EfZ X Z Y g = 0 and Var(Z X Z Y ) = . Fortunately, the standard theory we have developed still applies here, with a shrink in variance corresponding to the remaining non-zero portion of the product mixture distribution. Then the p-value can be calculated by: P (LS(D)s D ) =P LS(D) p n s D = p n) =L s D = p n ; where the functionL is dened in equation 2.6 in the Method section. Similar results can be obtained for replicated series and local trend series with multiple zeros. 34 Appendix B Supplementary Results The number of time points n x Theory 10 20 30 40 60 80 100 120 140 160 180 200 2 0.1815 0.0848 0.0987 0.1062 0.1122 0.1201 0.1235 0.1290 0.1294 0.1355 0.1375 0.1430 0.1379 2.2 0.1111 0.0541 0.0621 0.0645 0.0665 0.0699 0.0771 0.0767 0.0783 0.0798 0.0798 0.0861 0.0831 2.4 0.0656 0.0341 0.0367 0.0392 0.0416 0.0411 0.0435 0.0457 0.0451 0.0477 0.0483 0.0526 0.0498 2.6 0.0373 0.0223 0.0221 0.0252 0.0235 0.0232 0.0249 0.0261 0.0253 0.0261 0.0276 0.0301 0.0275 2.8 0.0204 0.0147 0.0128 0.0154 0.0131 0.0129 0.0138 0.0163 0.0129 0.0141 0.0159 0.0159 0.0152 3.0 0.0108 0.0093 0.0082 0.0088 0.0074 0.0069 0.0071 0.0090 0.0072 0.0066 0.0087 0.0083 0.0074 3.2 0.0055 0.0056 0.0051 0.0038 0.0036 0.0030 0.0035 0.0054 0.0040 0.0042 0.0043 0.0043 0.0038 3.4 0.0027 0.0033 0.0031 0.0017 0.0022 0.0009 0.0016 0.0027 0.0019 0.0018 0.0027 0.0028 0.0017 3.6 0.0013 0.0019 0.0020 0.0011 0.0014 0.0004 0.0006 0.0012 0.0011 0.0007 0.0012 0.0015 0.0008 3.8 0.0006 0.0007 0.0008 0.0006 0.0010 0.0002 0.0004 0.0009 0.0007 0.0002 0.0008 0.0008 0.0004 4.0 0.0003 0.0004 0.0005 0.0003 0.0005 0.0000 0.0003 0.0004 0.0004 0.0001 0.0002 0.0002 0.0001 4.2 0.0001 0.0002 0.0004 0.0002 0.0005 0.0000 0.0001 0.0002 0.0003 0.0000 0.0001 0.0001 0.0001 4.4 0.0000 0.0001 0.0003 0.0001 0.0002 0.0000 0.0000 0.0001 0.0000 0.0000 0.0000 0.0000 0.0000 4.6 0.0000 0.0000 0.0003 0.0001 0.0000 0.0000 0.0000 0.0001 0.0000 0.0000 0.0000 0.0000 0.0000 4.8 0.0000 0.0000 0.0002 0.0001 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 5.0 0.0000 0.0000 0.0001 0.0001 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 Table B.1: Theoretical approximation for local similarity analysis p-values versus the simulated probabilityP (LS(D)= p nx). The theoretical approximate probability based on equation 2.3 with = 1 is given in the 2nd column and the simulated probability that LS(D)= p nx is given in the 3rd to the 14th columns. D = 0. 35 The number of time points n x Theory 10 20 30 40 60 80 100 120 140 160 180 200 2 0.4516 0.1692 0.2289 0.2464 0.2698 0.2972 0.3084 0.3220 0.3354 0.3393 0.3410 0.3531 0.3491 2.2 0.2977 0.1139 0.1485 0.1636 0.1708 0.1871 0.1978 0.2043 0.2169 0.2150 0.2119 0.2300 0.2254 2.4 0.1841 0.0757 0.0976 0.0988 0.1050 0.1123 0.1176 0.1261 0.1352 0.1313 0.1292 0.1401 0.1393 2.6 0.1077 0.0513 0.0606 0.0608 0.0626 0.0650 0.0692 0.0748 0.0813 0.0776 0.0745 0.0768 0.0832 2.8 0.0601 0.0318 0.0385 0.0357 0.0379 0.0374 0.0398 0.0464 0.0460 0.0426 0.0447 0.0434 0.0470 3.0 0.0320 0.0186 0.0224 0.0187 0.0233 0.0208 0.0222 0.0276 0.0245 0.0220 0.0246 0.0249 0.0248 3.2 0.0164 0.0127 0.0124 0.0113 0.0133 0.0117 0.0115 0.0139 0.0122 0.0129 0.0128 0.0127 0.0133 3.4 0.0081 0.0078 0.0074 0.0062 0.0073 0.0058 0.0063 0.0078 0.0065 0.0066 0.0066 0.0060 0.0066 3.6 0.0038 0.0044 0.0039 0.0030 0.0045 0.0025 0.0032 0.0043 0.0039 0.0035 0.0035 0.0034 0.0034 3.8 0.0017 0.0032 0.0023 0.0015 0.0023 0.0012 0.0019 0.0023 0.0017 0.0015 0.0018 0.0015 0.0014 4.0 0.0008 0.0013 0.0013 0.0005 0.0011 0.0007 0.0010 0.0009 0.0004 0.0005 0.0007 0.0006 0.0005 4.2 0.0003 0.0010 0.0009 0.0003 0.0003 0.0002 0.0004 0.0004 0.0003 0.0005 0.0003 0.0003 0.0003 4.4 0.0001 0.0007 0.0003 0.0003 0.0001 0.0001 0.0004 0.0000 0.0002 0.0004 0.0002 0.0002 0.0002 4.6 0.0001 0.0004 0.0002 0.0003 0.0000 0.0001 0.0002 0.0000 0.0001 0.0002 0.0002 0.0001 0.0000 4.8 0.0000 0.0002 0.0002 0.0001 0.0000 0.0000 0.0002 0.0000 0.0000 0.0000 0.0002 0.0000 0.0000 5.0 0.0000 0.0002 0.0001 0.0000 0.0000 0.0000 0.0002 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 Table B.2: Theoretical approximation for local similarity analysis p-values versus Simu- lated probability P (LS(D)= p nx). The theoretical approximate probability based on equation 2.3 with = 1 is given in the 2nd column and the simulated probability that LS(D)= p nx is given in the 3rd to the 14th columns. D = 1. 36 The number of time points n x Theory 10 20 30 40 60 80 100 120 140 160 180 200 2 0.6326 0.2199 0.2914 0.3429 0.3704 0.4167 0.4447 0.4709 0.4792 0.4855 0.4935 0.5072 0.5178 2.2 0.4452 0.1514 0.1947 0.2316 0.2437 0.2759 0.3004 0.3132 0.3245 0.3264 0.3305 0.3429 0.3456 2.4 0.2876 0.1053 0.1228 0.1478 0.1524 0.1738 0.1884 0.1964 0.2032 0.2020 0.2043 0.2145 0.2141 2.6 0.1730 0.0713 0.0765 0.0887 0.0949 0.1036 0.1119 0.1146 0.1194 0.1203 0.1211 0.1226 0.1301 2.8 0.0981 0.0447 0.0453 0.0534 0.0558 0.0602 0.0660 0.0681 0.0657 0.0704 0.0690 0.0721 0.0732 3.0 0.0528 0.0288 0.0256 0.0303 0.0316 0.0347 0.0356 0.0385 0.0352 0.0370 0.0360 0.0421 0.0412 3.2 0.0272 0.0185 0.0139 0.0172 0.0170 0.0188 0.0199 0.0187 0.0179 0.0213 0.0185 0.0227 0.0192 3.4 0.0134 0.0121 0.0084 0.0100 0.0099 0.0097 0.0099 0.0089 0.0089 0.0092 0.0103 0.0120 0.0091 3.6 0.0063 0.0077 0.0046 0.0060 0.0051 0.0048 0.0059 0.0053 0.0044 0.0048 0.0045 0.0064 0.0053 3.8 0.0029 0.0053 0.0021 0.0037 0.0026 0.0021 0.0021 0.0031 0.0030 0.0023 0.0024 0.0035 0.0024 4.0 0.0013 0.0031 0.0011 0.0018 0.0019 0.0013 0.0007 0.0017 0.0017 0.0011 0.0014 0.0015 0.0010 4.2 0.0005 0.0020 0.0003 0.0009 0.0006 0.0006 0.0003 0.0011 0.0012 0.0003 0.0004 0.0006 0.0005 4.4 0.0002 0.0008 0.0003 0.0005 0.0005 0.0001 0.0001 0.0006 0.0006 0.0002 0.0003 0.0003 0.0001 4.6 0.0001 0.0006 0.0001 0.0004 0.0004 0.0000 0.0001 0.0003 0.0002 0.0000 0.0001 0.0001 0.0000 4.8 0.0000 0.0002 0.0000 0.0002 0.0001 0.0000 0.0001 0.0002 0.0001 0.0000 0.0000 0.0000 0.0000 5.0 0.0000 0.0001 0.0000 0.0002 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 Table B.3: Theoretical approximation for local similarity analysis p-values versus the simulated probabilityP (LS(D)= p nx). The theoretical approximate probability based on equation 2.3 with = 1 is given in the 2nd column and the simulated probability that LS(D)= p nx is given in the 3rd to the 14th columns. D = 2. 37 The number of time points n x Theory 10 20 30 40 60 80 100 120 140 160 180 200 2 0.7539 0.2509 0.3331 0.4103 0.4544 0.5120 0.5402 0.5571 0.5804 0.5972 0.6128 0.6059 0.6259 2.2 0.5616 0.1731 0.2301 0.2772 0.3124 0.3533 0.3707 0.3917 0.4077 0.4109 0.4368 0.4256 0.4406 2.4 0.3779 0.1177 0.1486 0.1772 0.2000 0.2293 0.2344 0.2539 0.2613 0.2679 0.2819 0.2722 0.2805 2.6 0.2336 0.0785 0.0952 0.1122 0.1236 0.1366 0.1401 0.1578 0.1562 0.1573 0.1756 0.1678 0.1669 2.8 0.1346 0.0513 0.0583 0.0679 0.0736 0.0767 0.0813 0.0918 0.0875 0.0894 0.0977 0.0998 0.0947 3.0 0.0732 0.0320 0.0343 0.0379 0.0416 0.0443 0.0453 0.0534 0.0469 0.0481 0.0523 0.0517 0.0509 3.2 0.0379 0.0199 0.0205 0.0207 0.0210 0.0237 0.0264 0.0278 0.0246 0.0261 0.0259 0.0275 0.0271 3.4 0.0187 0.0123 0.0129 0.0116 0.0110 0.0124 0.0122 0.0142 0.0107 0.0133 0.0145 0.0145 0.0128 3.6 0.0089 0.0083 0.0073 0.0055 0.0059 0.0058 0.0061 0.0076 0.0052 0.0067 0.0068 0.0069 0.0058 3.8 0.0040 0.0047 0.0046 0.0030 0.0029 0.0029 0.0036 0.0048 0.0023 0.0031 0.0028 0.0034 0.0027 4.0 0.0018 0.0028 0.0032 0.0013 0.0015 0.0015 0.0013 0.0024 0.0005 0.0014 0.0011 0.0010 0.0011 4.2 0.0007 0.0019 0.0021 0.0004 0.0007 0.0010 0.0005 0.0008 0.0001 0.0005 0.0008 0.0003 0.0002 4.4 0.0003 0.0009 0.0015 0.0002 0.0004 0.0007 0.0002 0.0002 0.0000 0.0000 0.0003 0.0003 0.0001 4.6 0.0001 0.0006 0.0009 0.0002 0.0003 0.0003 0.0000 0.0001 0.0000 0.0000 0.0000 0.0001 0.0001 4.8 0.0000 0.0003 0.0002 0.0000 0.0002 0.0000 0.0000 0.0001 0.0000 0.0000 0.0000 0.0000 0.0000 5.0 0.0000 0.0000 0.0000 0.0000 0.0002 0.0000 0.0000 0.0001 0.0000 0.0000 0.0000 0.0000 0.0000 Table B.4: Theoretical approximation for local similarity analysis p-values versus the simulated probabilityP (LS(D)= p nx). The theoretical approximate probability based on equation 2.3 with = 1 is given in the 2nd column and the simulated probability that LS(D)= p nx is given in the 3rd to the 14th columns. D = 3. 38 Figure B.1: Local similarity analysis P theo vs P perm for 10,000 pairs simulated data. Columns D0 to D3 are for D = 0; 1; 2; 3. Rows n10 to n100 are for n = 10; 20; 30; 40; 60; 80; 100. 39 Figure B.2: Local shape analysisP theo vsP perm for 10,000 pairs simulated data. Columns D0 to D3 are for D = 0; 1; 2; 3. Rows n10 to n100 are for n = 10; 20; 30; 40; 60; 80; 100. 40 - log 10 (P theo ) - log 10 (P perm ) 1 2 3 1 2 3 ● ● ● ● ● ● ●●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● D0 trendBy0 ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ●● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● D1 trendBy0 1 2 3 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ●●● ●● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● D2 trendBy0 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ● ● ●● ● ● ● ● ● ● ● ● ●●● ●● ● ● ● ● ● ●● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● D3 trendBy0 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ●● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 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●●●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● D1 original ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 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● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● D2 original 1 2 3 1 2 3 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ●●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● D3 original Figure B.3: P theo andP perm comparison for all-to-all pairwise local similarity (`original') and shape (`trendBy0') analysis of 40 abundant OTUs from SPOT dataset. Columns D0 to D3 are for D = 0; 1; 2; 3, respectively. 41 Figure B.4: Examples of signicant local associations from `F4' feces sample of MPH dataset. Proles are shifted to synchronize the co-occurrence according to local simi- larity analysis. (left) Coprococcus and Escherichia (LS=-0.3179, P=0.0002; r=-0.3314, P=0.0001); (right) Eubacterium and Oscillospira (LS=0.3862, P=0.0001; r=0.3525, P=0.0001) Figure B.5: Running time comparison for example real dataset computation. Note that the constant computation time using the theoretical approach that is independent of sample size as compared to sample-size and precision dependent computation time of permutation approaches. 42 The number of time points n x Theory 10 20 30 40 60 80 100 120 140 160 180 200 2 0.1815 0.0491 0.1242 0.0921 0.0986 0.1092 0.1439 0.1308 0.1314 0.1371 0.1325 0.1511 0.1422 2.2 0.1111 0.0491 0.0677 0.0613 0.0680 0.0599 0.0832 0.0801 0.0859 0.0735 0.0770 0.0871 0.0888 2.4 0.0656 0.0132 0.0367 0.0353 0.0430 0.0420 0.0492 0.0487 0.0442 0.0486 0.0517 0.0509 0.0508 2.6 0.0373 0.0053 0.0168 0.0211 0.0158 0.0207 0.0266 0.0207 0.0288 0.0252 0.0283 0.0275 0.0234 2.8 0.0204 0.0053 0.0094 0.0057 0.0106 0.0098 0.0137 0.0125 0.0134 0.0112 0.0141 0.0140 0.0122 3.0 0.0108 0.0000 0.0039 0.0027 0.0034 0.0057 0.0069 0.0080 0.0075 0.0067 0.0066 0.0074 0.0065 3.2 0.0055 0.0000 0.0016 0.0013 0.0021 0.0026 0.0032 0.0044 0.0030 0.0032 0.0035 0.0041 0.0033 3.4 0.0027 0.0000 0.0002 0.0007 0.0009 0.0009 0.0008 0.0020 0.0016 0.0016 0.0013 0.0013 0.0016 3.6 0.0013 0.0000 0.0002 0.0000 0.0004 0.0005 0.0004 0.0009 0.0008 0.0005 0.0008 0.0007 0.0010 3.8 0.0006 0.0000 0.0000 0.0000 0.0001 0.0003 0.0001 0.0004 0.0001 0.0001 0.0005 0.0004 0.0002 4.0 0.0003 0.0000 0.0000 0.0000 0.0000 0.0000 0.0001 0.0002 0.0001 0.0001 0.0002 0.0002 0.0002 4.2 0.0001 0.0000 0.0000 0.0000 0.0000 0.0000 0.0001 0.0000 0.0000 0.0001 0.0000 0.0002 0.0001 4.4 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0002 0.0000 4.6 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0002 0.0000 4.8 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 5.0 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 Table B.5: Theoretical approximation for local shape analysis p-values versus the simu- lated probabilityP (LS(D)= p 1:25nx). The theoretical approximate probability based on equation 2.3 with = 1 is given in the 2nd column and the simulated probability that LS(D)= p 1:25nx is given in the 3rd to the 14th columns. D = 0. 43 The number of time points n x Theory 10 20 30 40 60 80 100 120 140 160 180 200 2 0.4516 0.0833 0.2633 0.2173 0.2282 0.2753 0.3391 0.2938 0.3116 0.3099 0.3124 0.3556 0.3352 2.2 0.2977 0.0833 0.1596 0.1415 0.1557 0.1512 0.2142 0.1929 0.2122 0.1799 0.1889 0.2263 0.2154 2.4 0.1841 0.0261 0.0837 0.0874 0.1073 0.1081 0.1255 0.1211 0.1111 0.1209 0.1295 0.1361 0.1327 2.6 0.1077 0.0038 0.0463 0.0496 0.0461 0.0549 0.0688 0.0570 0.0681 0.0633 0.0690 0.0795 0.0629 2.8 0.0601 0.0038 0.0209 0.0144 0.0293 0.0252 0.0368 0.0321 0.0317 0.0285 0.0353 0.0445 0.0322 3.0 0.0320 0.0000 0.0081 0.0073 0.0106 0.0169 0.0208 0.0181 0.0180 0.0183 0.0178 0.0231 0.0173 3.2 0.0164 0.0000 0.0031 0.0034 0.0059 0.0052 0.0105 0.0095 0.0084 0.0073 0.0087 0.0108 0.0079 3.4 0.0081 0.0000 0.0004 0.0015 0.0013 0.0018 0.0027 0.0022 0.0051 0.0046 0.0036 0.0039 0.0036 3.6 0.0038 0.0000 0.0004 0.0003 0.0007 0.0007 0.0021 0.0012 0.0024 0.0017 0.0021 0.0018 0.0017 3.8 0.0017 0.0000 0.0000 0.0000 0.0003 0.0000 0.0008 0.0004 0.0013 0.0002 0.0008 0.0005 0.0007 4.0 0.0008 0.0000 0.0000 0.0000 0.0000 0.0000 0.0003 0.0001 0.0006 0.0000 0.0004 0.0002 0.0001 4.2 0.0003 0.0000 0.0000 0.0000 0.0000 0.0000 0.0002 0.0001 0.0001 0.0000 0.0003 0.0001 0.0000 4.4 0.0001 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0003 0.0000 0.0000 4.6 0.0001 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0001 0.0000 0.0000 4.8 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 5.0 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 Table B.6: Theoretical approximation for local shape analysis p-values versus the simu- lated probabilityP (LS(D)= p 1:25nx). The theoretical approximate probability based on equation 2.3 with = 1 is given in the 2nd column and the simulated probability that LS(D)= p 1:25nx is given in the 3rd to the 14th columns. D = 1. 44 The number of time points n x Theory 10 20 30 40 60 80 100 120 140 160 180 200 2 0.6326 0.1180 0.3762 0.3057 0.3305 0.3924 0.4665 0.4317 0.4601 0.4528 0.4621 0.5079 0.4778 2.2 0.4452 0.1180 0.2275 0.1984 0.2337 0.2262 0.3020 0.2930 0.3275 0.2747 0.2984 0.3396 0.3221 2.4 0.2876 0.0308 0.1268 0.1236 0.1575 0.1684 0.1814 0.1897 0.1850 0.1866 0.2142 0.2080 0.2038 2.6 0.1730 0.0054 0.0631 0.0765 0.0671 0.0866 0.0993 0.0819 0.1153 0.0967 0.1201 0.1232 0.1050 2.8 0.0981 0.0054 0.0301 0.0239 0.0422 0.0398 0.0530 0.0471 0.0552 0.0461 0.0628 0.0667 0.0584 3.0 0.0528 0.0000 0.0120 0.0134 0.0138 0.0262 0.0283 0.0258 0.0304 0.0281 0.0297 0.0350 0.0314 3.2 0.0272 0.0000 0.0044 0.0070 0.0074 0.0108 0.0137 0.0135 0.0123 0.0124 0.0148 0.0166 0.0171 3.4 0.0134 0.0000 0.0002 0.0031 0.0025 0.0033 0.0043 0.0049 0.0061 0.0063 0.0070 0.0060 0.0090 3.6 0.0063 0.0000 0.0002 0.0005 0.0013 0.0011 0.0026 0.0027 0.0019 0.0024 0.0040 0.0030 0.0046 3.8 0.0029 0.0000 0.0000 0.0002 0.0005 0.0005 0.0008 0.0013 0.0011 0.0008 0.0016 0.0018 0.0016 4.0 0.0013 0.0000 0.0000 0.0000 0.0002 0.0001 0.0003 0.0006 0.0004 0.0005 0.0007 0.0007 0.0007 4.2 0.0005 0.0000 0.0000 0.0000 0.0002 0.0000 0.0001 0.0002 0.0001 0.0000 0.0003 0.0004 0.0002 4.4 0.0002 0.0000 0.0000 0.0000 0.0000 0.0000 0.0001 0.0001 0.0000 0.0000 0.0002 0.0004 0.0002 4.6 0.0001 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0003 0.0001 4.8 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0002 0.0000 5.0 0.0000 0.0001 0.0000 0.0002 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 Table B.7: Theoretical approximation for local shape analysis p-values versus the simu- lated probabilityP (LS(D)= p 1:25nx). The theoretical approximate probability based on equation 2.3 with = 1 is given in the 2nd column and the simulated probability that LS(D)= p 1:25nx is given in the 3rd to the 14th columns. D = 2. 45 The number of time points n x Theory 10 20 30 40 60 80 100 120 140 160 180 200 2 0.7539 0.1128 0.4443 0.3969 0.4194 0.4821 0.5824 0.5342 0.5639 0.5610 0.5598 0.6095 0.5956 2.2 0.5616 0.1128 0.2694 0.2690 0.3001 0.2906 0.3933 0.3684 0.4121 0.3551 0.3669 0.4284 0.4173 2.4 0.3779 0.0308 0.1480 0.1650 0.2068 0.2163 0.2490 0.2423 0.2356 0.2496 0.2633 0.2737 0.2732 2.6 0.2336 0.0044 0.0739 0.0960 0.0847 0.1126 0.1441 0.1178 0.1550 0.1362 0.1525 0.1643 0.1405 2.8 0.1346 0.0044 0.0351 0.0325 0.0528 0.0557 0.0744 0.0688 0.0708 0.0664 0.0837 0.0938 0.0806 3.0 0.0732 0.0000 0.0147 0.0161 0.0184 0.0359 0.0379 0.0379 0.0413 0.0408 0.0410 0.0493 0.0438 3.2 0.0379 0.0000 0.0060 0.0073 0.0105 0.0155 0.0183 0.0193 0.0166 0.0184 0.0190 0.0250 0.0237 3.4 0.0187 0.0000 0.0006 0.0038 0.0030 0.0064 0.0049 0.0072 0.0090 0.0102 0.0082 0.0083 0.0108 3.6 0.0089 0.0000 0.0006 0.0009 0.0013 0.0020 0.0038 0.0035 0.0039 0.0043 0.0043 0.0036 0.0055 3.8 0.0040 0.0000 0.0002 0.0003 0.0006 0.0009 0.0015 0.0016 0.0018 0.0022 0.0018 0.0020 0.0022 4.0 0.0018 0.0000 0.0001 0.0002 0.0002 0.0001 0.0004 0.0002 0.0006 0.0008 0.0009 0.0009 0.0006 4.2 0.0007 0.0000 0.0000 0.0001 0.0001 0.0001 0.0001 0.0000 0.0004 0.0003 0.0002 0.0005 0.0003 4.4 0.0003 0.0000 0.0000 0.0000 0.0000 0.0001 0.0001 0.0000 0.0002 0.0000 0.0001 0.0003 0.0002 4.6 0.0001 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0001 0.0000 0.0000 0.0002 0.0001 4.8 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 5.0 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 Table B.8: Theoretical approximation for local shape analysis p-values versus the simu- lated probabilityP (LS(D)= p 1:25nx). The theoretical approximate probability based on equation 2.3 with = 1 is given in the 2nd column and the simulated probability that LS(D)= p 1:25nx is given in the 3rd to the 14th columns. D = 3. (a) original, D=0 P theo >0.05 P theo 0.05 Pperm >0.05 233 0 Pperm0.05 19 48 (b) original, D=1 P theo >0.05 P theo 0.05 Pperm >0.05 225 0 Pperm0.05 37 38 (c) original, D=2 P theo >0.05 P theo 0.05 Pperm >0.05 228 0 Pperm0.05 36 36 (d) original, D=3 P theo >0.05 P theo 0.05 Pperm >0.05 228 0 Pperm0.05 36 36 (e) trendBy0, D=0 P theo >0.05 P theo 0.05 Pperm >0.05 241 0 Pperm0.05 6 53 (f) trendBy0, D=1 P theo >0.05 P theo 0.05 Pperm >0.05 244 0 Pperm0.05 9 47 (g) trendBy0, D=2 P theo >0.05 P theo 0.05 Pperm >0.05 243 0 Pperm0.05 19 38 (h) trendBy0, D=3 P theo >0.05 P theo 0.05 Pperm >0.05 240 0 Pperm0.05 22 38 Table B.9: The comparison of signicant gene pairs using P theo and P perm given type-I error 0.05 for local similarity (`original') and shape (`trendBy0') analysis of 25 randomly selected factors from the CDC dataset: (a-d), local similarity scores; (e-h), local shape scores; (a,e) D=0, (b,f) D=1, (c,g) D=2, (d,h) D=3. The total number of comparisons is 300. 46 (a) real, D=0 P theo >0.05 P theo 0.05 Pperm >0.05 488 0 Pperm0.05 31 261 (b) real, D=1 P theo >0.05 P theo 0.05 Pperm >0.05 492 0 Pperm0.05 47 241 (c) real, D=2 P theo >0.05 P theo 0.05 Pperm >0.05 505 0 Pperm0.05 57 218 (d) real, D=3 P theo >0.05 P theo 0.05 Pperm >0.05 516 0 Pperm0.05 53 211 (e) trendBy0, D=0 P theo >0.05 P theo 0.05 Pperm >0.05 578 0 Pperm0.05 20 182 (f) trendBy0, D=1 P theo >0.05 P theo 0.05 Pperm >0.05 606 0 Pperm0.05 26 148 (g) trendBy0, D=2 P theo >0.05 P theo 0.05 Pperm >0.05 622 0 Pperm0.05 23 135 (h) trendBy0, D=3 P theo >0.05 P theo 0.05 Pperm >0.05 627 0 Pperm0.05 31 122 Table B.10: The comparison of signicant OTU pairs using P theo andP perm given type-I error 0.05 for local similarity (`original') and shape (`trendBy0') analysis of 40 selected OTUs from SPOT dataset: (a-d), local similarity scores; (e-h), local shape scores; (a,e) D=0, (b,f) D=1, (c,g) D=2, (d,h) D=3. The total number of OTU pairs is 780. 47

Abstract (if available)

Abstract Local association analysis, such as local similarity analysis and local shape analysis, of biological time series data helps elucidate the varying dynamics of biological systems. However, their applications to large scale high-throughput data are limited by slow permutation procedures for statistical significance evaluation. We developed a theoretical approach to approximate the statistical significance of local similarity and local shape analysis based on the approximate tail distribution of the maximum partial sum of independent identically distributed (i.i.d) and Markovian random variables. Simulations show that the derived formula approximates the tail distribution reasonably well (starting at time points > 10 with no delay and > 20 with delay) and provides p-values comparable to those from permutations. The new approach enables efficient calculation of statistical significance for pairwise local association analysis, making possible all-to-all association studies otherwise prohibitive. As a demonstration, local association analysis of human microbiome time series shows that core OTUs are highly synergetic and some of the associations are body-site specific across samples. The new approach is implemented in our eLSA package, which now provides pipelines for faster local similarity and shape analysis of time series data. The tool is freely available from eLSA's website: https://meta.usc.edu/softs/lsa.

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Asset Metadata

Creator Xia, Li Charlie (author)

Core Title Efficient statistical significance approximation for local association analysis of high-throughput time series data

School College of Letters, Arts and Sciences

Degree Master of Science

Degree Program Statistics

Publication Date 08/06/2012

Defense Date 08/03/2012

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag high-throughput data,local association,local shape analysis,local similarity analysis,microbial community,OAI-PMH Harvest,time-series data

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Sun, Fengzhu Z. (committee chair), Schumitzky, Alan (committee member), Zhang, Jianfeng (committee member)

Creator Email lxia@usc.edu

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c3-87579

Unique identifier UC11288900

Identifier usctheses-c3-87579 (legacy record id)

Legacy Identifier etd-XiaLiCharl-1140.pdf

Dmrecord 87579

Document Type Thesis

Rights Xia, Li Charlie

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Access Conditions The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the a...

Repository Name University of Southern California Digital Library

Repository Location USC Digital Library, University of Southern California, University Park Campus MC 2810, 3434 South Grand Avenue, 2nd Floor, Los Angeles, California 90089-2810, USA