Identifying Relevant Group of mirnas in Cancer using Fuzzy Mutual Information

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1 Noname manuscrip No. (will be insered by he edior) Idenifying Relevan Group of mirnas in Cancer using Fuzzy Muual Informaion (Supplemenary Maerial) Jayana Kumar Pal Shubhra Sankar Ray Sankar K Pal 1 Fuzzy muual informaion Here he concep of fuzzy muual informaion (FMI) [1] is described. Consider, A as a Fuzzy aribue se in a finie se U, d as he number of fuzzy equivalence classes and as he oal number of objecs in U. Now, he fuzzy equivalen pariion marix (M A ) is denoed as µ A 11 µ A µ A 1 M A = µ A 21 µ A µ A (1) µ A d1 µa d2... µa d where, M A is a d marix, d µ A uv = 1, v, µ A uv [0, 1], represens he u=1 membership of he vh objec (say, x v ) in he uh fuzzy equivalence class F u. Each row of he marix M A represens a fuzzy equivalence class in U and each column represens an objec in U. Therefore, F u can be represened as F u = µa u1 x 1 + µa u2 x µa u x (2) where, he + sign represens he union operaor. Now, he cardinaliy of F u can be calculaed as Fu = µ A uv. (3) v=1 The enropy of he Fuzzy aribue se A is given as H(A) = d λ Fu logλ Fu (4) u=1

2 2 J. K. Pal, S. S. Ray and S. K. Pal where, he fuzzy relaive frequency is represened by λ Fu and i is defined as From Eq. 3 and Eq. 4, H(A) can be rewrien as H(A) = Fu λ Fu =. (5) d [ 1 u=1 v=1 µ A uv ] [ 1 log v=1 µ A uv ]. (6) Consider, P and Q are wo subses of A. The numbers of fuzzy equivalence classes generaed by P and Q are p and q, respecively. The ah and bh fuzzy equivalence classes of P and Q are represened as P a and Q b, respecively. The join relaive frequency of P a and Q b is given as ( P a Q b )/, where, Pa Q b = (µ P ac µ Q bc ). (7) Here, µ P ac represens he membership value of he ch objec in he ah fuzzy equivalence class, µ Q bc represens he membership value of he ch objec in he bh fuzzy equivalence class and is he oal number of objecs in boh of he fuzzy aribue ses P and Q. Therefore join enropy H(P, Q) can be wrien as p q [ 1 H(P, Q) = (µ P ac µq )] [ 1 log (µ P bc ac µq )]. (8) bc a=1 b=1 The Fuzzy muual informaion (FMI) beween he wo fuzzy ses P and Q is given as I(P, Q) = H(P ) + H(Q) H(P, Q). (9) Using Eqs. 6 and 8, Eq. 9 can be rewrien as I(P, Q) = p a=1 q b=1 + [1 [1 p q a=1 b=1 µ P ] [1 ac log µ Q ] [1 bc log [1 log [ 1 µ P ] ac µ Q ] bc (µ P ac µ Q bc )] (µ P ac µ Q bc )]. (10)

3 FMIMS 3 Fig. 1 Block Diagram of he Proposed Mehod 2 Block diagram The block diagram of FMIMS is shown in Fig. 1. I has hree seps. In he firs sep, he mirnas are iniially grouped by SVM. Noe ha, he number of groups is auomaically deermined by he grouping algorihm. In he second sep, he relevance of each mirna (wih respec o a decision), for a given group, is calculaed by FMI. The mos imporan group is hen seleced according o he highes average relevance value of he mirnas in a group. In he final sep, for each mirna, he redundancies wih respec o oher mirnas (in he mos relevan group) are calculaed separaely and he average of all he redundancies is considered as he redundancy of ha paricular mirna. This procedure is repeaed for all mirnas in he mos relevan group and he mirnas are ranked according o heir redundancy. Highly redundan mirnas can be removed according o he user defined crierion. Noe ha he user can skip he redundancy removal process and proceed for furher experimens wih all relevan mirnas.

4 4 J. K. Pal, S. S. Ray and S. K. Pal 3 Descripion of he daa ses In his invesigaion six daa ses, viz., breas [2], renal [3], colorecal [4], lung [5], melanoma [6] and prosae [7] cancer daa ses, are used o es he performance of he proposed mehodology, FMIMS. The breas, renal, colorecal, lung, melanoma and prosae cancer daa ses consis of 98 (5 normal and 93 cancer), 24 (12 normal and 12 cancer), 66 (8 normal and 58 cancer), 36 (19 normal and 17 cancer), 57 (22 normal and 35 cancer) and 24 (12 normal and 12 cancer) samples and 309, 12033, 352, 866, 866 and mirnas, respecively. The summary of he used daa ses is presened in Table 1. Table 1 Summary of he daa ses Cancer T oal No. of No. of T ype No. of Normal Cancer mirnas P aiens P aiens Breas cancer Renal cancer Colorecal cancer Lung cancer Melanoma cancer Prosae cancer Measures for performance evaluaion In his secion we will define sensiiviy, specificiy, F score and accuracy. These are defined as Sensiiviy = rue posiives rue posiives+false negaives, (11) Specificiy = Accuracy(%) = F = rue negaives and (12) rue negaives+false posiives 2 Sensiiviy Specificiy. (13) Sensiiviy+Specificiy No. of correcly classified samples T oal samples 100, (14) Here, he rue posiive refers o he number of correcly deeced cancer mirna expressions and false negaive refers o he number of undeeced cancer mirna expressions, for a cancer sample. True negaive implies he number of correcly deeced normal mirna expressions and false posiive implies he wrongly deeced cancer mirna expressions.

5 FMIMS 5 5 Calculaion of membership values Here we describe he way of calculaing membership values used in fuzzy muual informaion (FMI). The membership values can be calculaed by any funcion, such ha, he membership value of a daa poin wih respec o a class, is non-decreasing wih is closeness (in erms of disance) o ha class cener. Consider, Xh k as he hh expression value (normal or cancer, anyhing) of he kh mirna and 1 h (N + M), c k 1 and c k 2 as he average expression values of normal and cancer classes (i.e., he ceners of he wo classes) of he kh mirna, respecively. Here, he membership value (µ k eh ) of he hh sample of he kh mirna in eh class is calculaed as µ k eh = 1/[ (c k e Xk h )2 /(c k 1 Xk h )2 + (c k e Xk h )2 /(c k 2 Xk h )2] (15) where, e = 1, 2 (as here are wo classes, normal and cancer). Some properies of Eq. 15 are (i) µ k eh [0, 1], (ii) 2 µ k eh = 1, h, (iii) he value of µk eh increases e=1 as he closeness of Xh k o eh group cener increases, i.e., ck e Xh k decreases and (iv) if he hh sample coincides wih any class cener (i.e., he disance of he sample from ha class cener is 0) hen he value of µ k eh for he corresponding class will be 1, and he value will be 0 for he oher class. Noe ha in Eq. 15, 0 n 0 is considered as 1 as lim n 0 n = 1. 6 Comparison wih variable se of mirnas In his secion we compared he performance of FMIMS wih relaed algorihms by varying he number of seleced mirnas. Here he percenage of op mirnas is varied from 10 o 70, in seps of 10, for various mehods. The experimenal resuls using k-nn are shown in Figs. 2(a)-2(f). In hese figures, he performance of FMIMS is shown by a sraigh line parallel o he x axis as i is consan for a paricular daa se where he mirnas are auomaically seleced hrough he mos relevan group. I is observed ha our mehod achieved he bes F score for all he daa ses, as compared o oher mehods, using variable number of op mirnas. I is also observed ha all oher mehods show heir bes performance wihin op 10%-30% of mirnas. Similar curves are also obained by using SVM as a classifier, hough no repored.

6 6 J. K. Pal, S. S. Ray and S. K. Pal (a) Comparison using Breas cancer (b) Comparison using Renal cancer

7 FMIMS 7 (c) Comparison using Colorecal cancer (d) Comparison using Lung cancer

8 8 J. K. Pal, S. S. Ray and S. K. Pal (e) Comparison using Melanoma cancer (f) Comparison using Prosae cancer Fig. 2 Comparison using various se of mirnas

9 FMIMS 9 7 Oher Comparisons So far, he mos relevan group of mirnas, obained by FMIMS, is considered as he seleced se of mirnas. The selecion process can also be performed using wo oher ways. In he firs approach, groups can be obained using SVM in a similar fashion and he represenaive of each group can consiue ogeher he se of resuling mirnas. Le his mehod be called grouping and selecing group represenaives (GR). This mehod esablishes he imporance of he mos relevan group. In he second approach, he oupu of he firs block (shaded in Fig. 1) can be considered o rank he mirnas simply based on similariy measure and a percenage of he op ranked mirnas can be seleced. Le his mehod be referred as normalized disance based ranking (NDR). The mehod also demonsrae he effeciveness of grouping. These wo mehods are described in he following paragraphs. Table 2 F score of GR and NDR mehod Cancer ype No. of mirnas F score F score Obained by SVM Obained by k-nn FMIMS GR NDR FMIMS GR NDR FMIMS GR NDR Breas Renal Colorecal Lung Melanoma Prosae GR: The way of selecing he group represenaives in GR is as follows. Le, G 1, G 2,..., G f,..., G s be he groups of mirnas. The average expression profile of mirnas in he group G f (say Ḡf ) is calculaed as [ 1 Ḡ f = L L k=1 X k 1,..., 1 L L k=1 X k h,..., 1 L L XN+M k where, 1 h (N + M), N and M are he oal number of normal and cancer samples, respecively, Xh k is he expression value (i.e., normal or cancer) corresponding o he hh sample of he kh mirna and L is he oal number of mirnas in he group G f. The mirna neares o he Ḡf, is considered as he represenaive of he group G f. The process is followed for all he groups. NDR: In our experimen 10% of he op ranked mirnas, as obained from he oupu of shadded region of Fig. 1 on he basis of highes value of α k (See Eq. 13 in main aricle), is seleced. Table 2 shows he classificaion performance of he mirnas, seleced by FMIMS, GR, and NDR as obained by boh SVM and k-nn. I is observed ha he mirnas, seleced by he FMIMS, perform beer han oher wo mehods in erms of F score. For example, while he F score of he FMIMS k=1 ] (16)

10 10 J. K. Pal, S. S. Ray and S. K. Pal varies from 0.72 o 0.86 (using SVM as classifier) and 0.66 o 0.83 (using k- NN classifier), i varies from 0.46 o 0.72 (using SVM as classifier) and 0.33 o 0.56 (using k-nn as classifier) for he GR mehod. For NDR mehod, he F scores vary from 0.40 o 0.60 for SVM and 0.33 o 0.53 for k-nn, depending on various daa ses. Therefore i can be said ha our mehod provides beer resuls han oher wo mehods. 8 Overall Performance In our experimen, i is seen ha he average F scores, over all mehods, achieved by FMIMS (0.79 and 0.74 for SVM and k-nn classifiers, respecively) are highes among all oher average F scores obained wih differen mehods. The second highes F scores (0.69 and 0.64) are achieved by FRSIM [1] algorihm for boh he classifiers. The ime requiremens o execue he selecion process by differen mehods are also compared and are shown in Fig. 3. All he mehods are implemened in Malab 2012Ra and execued on a compuer having Inel core i7 2nd generaion processor wih 16 GB of RAM. The ime scale shown in he figure is he log value wih base 2 of he acual execuion ime calculaed in seconds. For example FMIMS akes seconds o Fig. 3 Execuion ime for differen mehods complee he selecion process for renal cancer daa se, and he corresponding log 2 value (7.14) is shown in he figure. I is observed ha, alhough SVMRFE and SVMRFE wih MRMR ake much higher ime han FMIMS, he seleced mirnas by FMIMS provides beer classificaion performance. The mehods MRMR, FRSIM and GR ake less or marginally higher ime as compared o

11 FMIMS 11 FMIMS, bu FMIMS performs bes for mos of he daa ses. As expeced, he NDR mehod akes he lowes compuaional ime among all he mehods as i is based on only inerclass and inraclass disances of individual mirna, bu is performance is considerably inferior o he proposed mehod. References 1. P. Maji, S.K. Pal, IEEE Trans. Sysems, Man and Cyberneics-Par B: Cyberneics 40, 741 (2010) 2. C. Blenkiron, e al., Genome Biology 8, R214.1 (2007) 3. M. Jung, e al., Journal of Cellular and Molecular Medicine 13, 3918 (2009) 4. G.M. Arnd, e al., BMC Cancer 9, 374 (2009) 5. A. Keller, e al., BMC Cancer 9, 353 (2009) 6. P. Leidinger, e al., BMC Cancer 10, 262 (2010) 7. A. Schaefer, e al., Inernaional Journal of Cancer 126, 1166 (2010)