Graphical Display of fmri Data. Visualizing Multidimensional Space.
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1 Gaphical Display of fmri Data. Visualizing Multidimensional Space. R. Baumgatne, R. Somojai+ Institute fo Biodiagnostics, National Reseach Council Canada, 435 Ellice Ave., Winnipeg Manitoba, Canada, R3B 1Y6 Running title: Gaphical Display of fmri Data Submitted to Magnetic Resonance Imaging as a Technical Note + Coesponding autho: R. Somojai Institute fo Biodiagnostics National Reseach Council Canada 435 Ellice Avenue Winnipeg, Manitoba MB R3B 1Y6 Canada Tel: Fax: somojai@ibd.nc.ca
2 Abstact Visualization of multidimensional data is an integal pat of computational statistics and exploatoy data analysis (EDA). We show how visualization of fmri time-couses may be used to eveal the fmri data stuctue. We conside fmri time-couses (TCs) as points in multidimensional space. In simulated and in vivo data, we show that minimum spanning tee (MST) based sequencing of multivaiate time-couses, in combination with a homogeneity map visualization, allows fo effective and useful gaphical display of the goups of coactivated timecouses obtained by tempoal clusteing. This display may seve as a tool fo investigation of bain connectivity. We also suggest a simple oveall display of the entie fmri data set. Keywods: functional MR imaging, gaphical display, exploatoy data analysis, minimum spanning tee, homogeneity map
3 Intoduction In computational statistics, multidimensional data stuctue exploation may be pefomed by data-diven appoaches such as clusteing (had, fuzzy, Kohonen maps) and pojection-pusuit based methods (Pincipal Component Analysis - PCA, Independent Component Analysis ICA o multidimensional scaling - MDS). Application of data-diven methods in fmri data analysis lagged behind the taditional hypothesis-led infeential methods (e.g. Ref. 1); howeve, ecently EDA appoaches have been gaining populaity (see e.g. Refs. 2-12, etc.). Afte stuctue exploation, pasimonious and compact summay / gaphical display of the esults is desiable / useful (13-17). In fact, as was suggested by J. Tukey in 1977 (18), gaphical display of multidimensional data must be an integal pat of exploatoy data analysis (EDA). In paticula, in fmri, coactivation of bain egions is of inteest when these egions show simila tempoal behavio (19). Theefoe, we focus hee on the esults of tempoal cluste analysis, which natually yields coactivated bain egions. In fmri, the esults ae customaily summaized as colo-coded values of the coelation coefficient (cc), o coesponding p-values, supeimposed on the anatomical image. This display in fact educes the multidimensional time-couses (TCs) to a single featue (cc / p-value). Hee we suggest visualization of an entie (single) cluste of TCs, with the TCs in thei oiginal, expeimentally acquied fom, i.e. a tempoal sequence of intensity values. The idea is to show all the featues contained in the cluste of inteest. We show how the combination of sequencing / anking of the fmri TCs with a paticula homogeneity map display may be used to visually epesent the coheence of a cluste of TCs obseved in fmri. We also suggest a visually evealing oveall display of the entie fmri data set. Mateials and Methods Simulated data: To demonstate the efficiency of visually displaying multidimensional TCs, two types of TCs, activation (# of TCs = 10) and noise TCs (# of TCs = 10) wee selected. Hemodynamic esponse was simulated by a two-paamete gamma function (20), with two on phases of 30 instances each. Noise TCs wee simulated by i.i.d. Gaussian noise. Fo the activation case,
4 noise was added to the simulated wavefoms, with the contast-to-noise atio (CNR= S/σ noise, whee S is the signal enhancement and σ noise is the noise standad deviation) 5.0 (20). In vivo data: An activation cluste was obtained fom an in vivo single tial fmri study with a mental otation paadigm (21). The data set was acquied with an EPI sequence on a 4 T whole body imaging system (matix size = 64x64x64 with 88 time instances). The atifact cluste was obtained fom fmri data obtained unde null condition (without stimulation). Data wee acquied with an EPI sequence on a 1.5 T GE Signa Hoizon scanne (matix size = 128 x 128x1 with 120 time instances). To acquie this data set a shot TR value (TR=335 ms) was chosen to avoid aliasing beathing into low fequencies. Thus, the beathing atifact is not undesampled as would be fo highe TRs. Data patition: We used fuzzy C-means clusteing in the tempoal domain, as implemented in the softwae package EvIdent (22,23). Homogeneity map single cluste visualization: In the homogeneity map, the vetical coodinate coesponds to the TC numbe in the cluste; along the hoizontal coodinate the intensity value of a paticula TC is displayed as a function of time, colo-coded such that high intensity values ae yellow, low ones ae blue. (The homogeneity map has been a standad featue in EvIdent ). The TCs in the homogeneity map ae odeed. Hee, we used anking accoding to the minimum spanning tee. Othe ankings ae possible, e.g., accoding to the degee of coelation with the cluste centoid, o the fuzzy membeship values. Sequencing of the TCs, Minimum Spanning Tee (MST): MST is a spanning tee with minimum length (fo a detailed desciption see e.g. Ref. 24). We constucted the MST of a goup of TCs, consideing each TC as a node of the tee. The connections between nodes ae called edges. We calculated the length of an edge as the Euclidean distance (L 2 nom) between two nodes (TCs). MST is a genealization of a onedimensional soted list. It may be used fo sequencing multivaiate obsevations (25).
5 Display of the entie data stuctue: As the epesentative of a cluste, we chose its centoid (the weighted aveage of the TCs of the cluste). Then we calculated the coelation coefficient of the centoid with all TCs. We expect the highest coelation values fo the membes of the cluste the centoid was deived fom. We displayed the values of the coelation coefficient in a two-dimensional block image. The block image is shown schematically in Fig. 1. The hoizontal coodinate denotes the cluste numbe. The dimensions of the image ae (Σ j N j xc), whee C is the numbe of clustes and N j is the numbe of TCs in cluste j. To demonstate the visualization of the entie data set, in the esults section we use equal numbe of TCs fo each cluste, N j = 50, j = 1,,C. Note howeve, that in geneal the numbe of TCs is diffeent in the diffeent clustes. Fo each cluste we display the coelation between all TCs and the cluste centoid. Fo example, the element (k) ij in the block image (Fig. 1) is the coelation coefficient between the ith TC of cluste j and the centoid of cluste k. Within cluste j, the values (k) ij ae in ascending ode. Results The homogeneity map of simulated data, ( noise and activation goups combined in one cluste), is shown in Fig. 2. This example demonstates the application of the homogeneity map in the exteme case of pue noise mixed with activation. Using the MST-sequencing and displaying its esults by the homogeneity map eveals clealy (and visually) the stuctue of the activation goup against the stuctueless noise goup. The white aow indicates the bounday between the two oiginal clustes, with the noise cluste above the aow, and the activation cluste beneath it. The homogeneity map of the in vivo data ( activation cluste ) is shown in Fig. 3. This visualization eveals the quality of fuzzy clusteing. High tempoal concodance among the TCs in the cluste is appaent fom the display. The MST-odeing also shows the gadual boadening of the hemodynamic esponse, fom top down (moe yellow towads the bottom of the map). The homogeneity map of in vivo data obtained unde null hypothesis is shown in Fig.4. The cluste identified epesents a beathing atifact, with 9 off-on cycles (the peaks, 9 vetical
6 yellow bands of appoximately the same width, sepaated by the toughs, blue vetical bands). Again, the homogeneity map displays the high intenal coheence within the cluste. Eight clustes of fifty TCs each (C=8 / N=50) fom in vivo data ae shown in an oveall gaphical display in Fig. 5. The diagonal elements (TCs) of the block image have high coelation values with thei own centoids. The black aow shows a cluste distinctly patitioned fom all the othes. The white aow denotes a cluste that is moe simila to thee othe clustes. Discussion Sequencing (anking) the multivaiate data in combination with the homogeneity map povides a simultaneous visual display of a cluste of TCs. The tempoal stuctue of the multidimensional data is appaent fom this display. Thee ae othe possible methods fo visualization and sequencing of multidimensional data. Bivaiate plots (whee the plot dimension is educed to 2) based on PCA o MDS could be used fo the display (14). Note, howeve, that in contast to MDS/PCA, MST based plots do not defom the Euclidean distances between TCs. Anothe possibility of odeing the TCs is accoding to thei coelation with some (weighted) mean TC, such as the fuzzy membeship-weighted cluste centoid. We have shown that befoe featue eduction, a homogeneity map display, with all featues pesent, is useful in high-dimensional stuctue exploation in fmri, especially when the coactivation o concodance between the TCs is to be visualized. The visual display of goups of TCs may be inteesting and useful not only fo EDA, but also fo infeential methods. Acknowledgements We thank D. Wolfgang Richte and D. Lawence Ryne fo poviding the mental otation fmri data set and the null fmri data set, espectively.
7 Refeences 1. Fiston, K.; Holmens, A.; Wosley, K.; Poline J.; Fith, C. Statistical paametic maps in functional imaging: A geneal linea appoach. Human Bain Mapping 2: , Baumgatne, R.; Windischbege, C.; Mose, E. Quantification in fmri: Fuzzy clusteing vs. coelation analysis. Magnetic Resonance Imaging 16: , Baumgatne, R.; Somojai, R.; Summes, R.; Richte, W.; Ryne, L. Coelato bewae: Coelation has limited selectivity fo fmri data analysis. NeuoImage 12: (2000) 4. Baune, A.; Somme, F.; Eb, M.; Wildgube, D.; Kadatzki, B.; Palm, G.; Godd, W. Dynamical cluste analysis of cotical fmri activation. NeuoImage 9: , Chuang, K.; Chiu M.; Lin, C.; Chen, J. Model fee fmri analysis using Kohonen clusteing neual netwok and fuzzy c-means. IEEE Tansactions on Medical Imaging 18: , Fadili, M.; Ruan, S.; Bloyet, D.; Mazoye, B. A multistep unsupevised fuzzy clusteing analysis of fmri time seies. Human Bain Mapping 10: , Filzmose, P.; Baumgatne, R.; Mose, E. A hieachical clusteing method fo analyzing fmri. Magnetic Resonance Imaging 17: , Fische, H.; Hennig, J. Neual netwok-based analysis of MR time-seies. Magnetic Resonance in Medicine 41: , Golay, X.; Kollias, S.; Stoll, G.; Meie, D; Valavanis, A.; Boesige, P. A new coelation-based fuzzy logic clusteing algoithm fo fmri. Magnetic Resonance in Medicine 40: , McKeown, M.; Jung, T.; Makeig, S.; Bown, G.; Kindemann, S.; Lee, T.; Sejnowski, T. Analysis of fmri data by blind sepaation into independent spatial components. Human Bain Mapping 6: , Mose, E.; Baumgatne, R.; Bath, M.; Windischbege, C. Exploative signal pocessing in fmri. Intenational Jounal of Imaging Systems and Technology 10: , Ngan, S.; Hu, X. Analysis of fmri data using self-oganizing mapping with spatial connectivity. Magnetic Resonance in Medicine 41: , 1999.
8 13. Minotte, M.; West W. The data image: a tool fo exploing high dimensional data sets. Poceedings of the ASA Section on Statistical Gaphics, in pess Pison, G.; Stuyf, A.; Rousseeuw P. Displaying a clusteing with CLUSPLOT. Computational Statistics and Data Analysis 30: , Wegman, E. Hypedimensional data analysis using paallel coodinates. Jounal of the Ameican Statistical Association 85: , Chenoff, H. The use of faces to epesent points in k-dimensional space gaphically. Jounal of the Ameican Statistical Association 68, , Andews, D. Plots of high-dimensional data. Biometics 28: , Tukey, J. Exploatoy data analysis. Addison-Wesley, Capente, P.; Just, M. Modeling the mind: vey-high-field fmri activation duing cognition. Topics in Magnetic Resonance Imaging 10:16-36, Lange, N. Tutoial in biostatistics; statistical appoaches to human bain mapping by fmri. Statistics in Medicine 15: , Richte, W.; Ugubil, K.; Geogopoulos, A.; Kim, S-G. Time esolved fmri of mental otation. Neuoepot 8: , Somojai, R.; Jamasz, M. Exploatoy data analysis of fmri: Philosophy, stategies, tools, implementation. Seventh Annual Meeting ISMRM, Philadelphia, USA, 1714, Somojai, R.; Jamasz, M.; Baumgatne, R. EVIDENT: A two-stage stategy fo the exploatoy data analysis of fmri data by fuzzy clusteing. Institute fo Biodiagnostics, Technical epot #37, EvIdent a 3D analysis softwae package Haay, F. Gaph theoy. Addison-Wesley, Fiedman, J.; Rafsky, L. Multivaiate genealizations of the Wald-Wolfowitz and Sminov twosample tests. Annals of Statistics 7: , 1979.
9 Figue Captions Fig. 1 Schematic figue fo the block image of the entie data set. N j numbe of TCs in the jth cluste, C numbe of cluste displayed. Fig. 2 Simulated data: Homogeneity map of two clustes, one consisting of TCs following the paadigm (two cycles, yellow bands), the othe of pue noise, meged and anked by MST. Aow indicates the bode between activation and noise egions, with the latte on top. Fig. 3 In vivo data: Homogeneity map of the MST-anked activation cluste. Note the high tempoal concodance among the TCs. Fig. 4 In vivo data: Homogeneity map of the MST-anked beathing atifact cluste obtained fom null hypothesis data. Again, note the high tempoal concodance among the TCs. Fig. 5 In vivo data: Entie data display. Black aow points to (a ow which epesents) a cluste distinctly sepaated fom all the othes. White aow shows (a ow which epesents) a cluste that is moe simila to thee othe clustes.
10 ... ( C) N 1,1 ( C) N 1 1,1 ( C) 1,1... ( C ) N C, C ( C ) N C 1, C ( C ) 1, C (1) N 1,1 (1) N 1 1,1 (1) 1,1... (1) N C, C (1) N C 1, C (1) 1, C Cluste Cluste C Fig. 1
11 TC numbe TC numbe Time Fig. 2 Time Fig. 3
12 TC numbe Time instance Fig. 4
13 Cluste membes with odeed coelation coefficients Cluste numbe Fig. 5
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