Comparison of two exploratory data analysis methods for fmri: fuzzy clustering vs. principal component analysis

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1 Magnetic Resonance Imaging 18 (2000) Technical Note Comparison of two exploratory data analysis methods for fmri: fuzzy clustering vs. principal component analysis R. Baumgartner, L. Ryner, W. Richter, R. Summers, M. Jarmasz, R. Somorjai* Institute for Biodiagnostics, National Research Council Canada, 435 Ellice Ave., Winnipeg Manitoba, Canada, R3B 1Y6 Received 8 March 1999; accepted 7 August 1999 Abstract Exploratory data-driven methods such as Fuzzy clustering analysis (FCA) and Principal component analysis (PCA) may be considered as hypothesis-generating procedures that are complementary to the hypothesis-led statistical inferential methods in functional magnetic resonance imaging (fmri). Here, a comparison between FCA and PCA is presented in a systematic fmri study, with MR data acquired under the null condition, i.e., no activation, with different noise contributions and simulated, varying activation. The contrast-to-noise (CNR) ratio ranged between We found that if fmri data are corrupted by scanner noise only, FCA and PCA show comparable performance. In the presence of other sources of signal variation (e.g., physiological noise), FCA outperforms PCA in the entire CNR range of interest in fmri, particularly for low CNR values. The comparison method that we introduced may be used to assess other exploratory approaches such as independent component analysis or neural network-based techniques. Crown Copyright Published by Elsevier Science Inc. Keywords: Functional MR imaging; Principal component analysis; Fuzzy clustering analysis 1. Introduction In the recent functional magnetic resonance imaging (fmri) literature, two paradigm-free (data-driven) methods have been used, fuzzy clustering analysis (FCA) and principal component analysis (PCA). FCA has been successfully applied in a variety of fmri experiments [1 5]. Principal component analysis has been found useful in investigating functional connectivity, i.e., identification of regions with correlated signal responses in human brain mapping [6]. Here, we compare FCA with PCA in a systematic simulated fmri study with two types of data: 1) a water phantom with scanner noise contributions only and 2) in vivo data acquired under null hypothesis conditions (with parameters typically used in fmri, but with a variety of scanner and physiological noise contributions [7], [8]). The activated foci were simulated as localized regions in the motor cortex. * Corresponding author. Tel.: ; fax: Materials and methods Water phantom data: Three single slice data sets were acquired on a GE Signa Horizon LX MRI System (1.5T), with a homogeneous birdoage coil, by single shot EPI (FA/ TE 90 /50, matrix size , 120 volumes) with TRs of 3500, 2500, and 1250 ms, respectively. A homogeneous region (3898 pixels) (with contribution from scanner noise only [9]) was excised from the mean image of the water phantom, and a focal activated region (n 46) was superimposed on it. The time-course (TC) of this activated region was defined as an off/on function with two on cycles, where the on (30 volumes each) cycles were simulated as a combination of two gamma functions [10]. For each TR, ten simulated data sets were generated, with contrast-to-noise ratio (CNR) ranging between 1 10, (CNR S/ n, where S is the signal enhancement and n is the noise standard deviation [11]) in the activation foci. The water phantom with the overlaid activation focus is shown in Fig. 1. In vivo fmri data under null hypothesis condition: Three in vivo (2D) data sets (with both scanner and physiological noise contributions) were acquired with identical parame X/00/$ see front matter Crown Copyright Published by Elsevier Science Inc. PII: S X(99)

2 90 R. Baumgartner et al. / Magnetic Resonance Imaging 18 (2000) Fig. 1. Water phantom, with the activation region overlaid. ters. Simulated activation: An activated focal area (n 46 pixels, as above) was defined in the motor cortex. In each pixel of this region, a hemodynamic response (as a combination of two gamma functions) was simulated, with two 30 volumes on cycles. Again, from each in vivo data set ten simulated data sets were generated by varying the CNR between 1 10 in the activation focus. The in vivo anatomy (mean image of the time series), with the overlaid activation focus is shown in Fig Methods of data analysis Fig. 2. In vivo data acquired under null hypothesis conditions: Anatomy (TR 3500 ms), with the activation region overlaid. The methods used have already been extensively described (PCA, see references 12, 13; FCA, see 14, 15), hence only short descriptions are given in the Appendix. PCA was coded in Matlab and FCA was used as implemented in the software package EvIdent TM v [3]. The FC algorithm in EvIdent TM is initialized with a large number (typically 30) of clusters and the final number is determined by a data-driven, novel cluster merging algorithm [3]. Prior to FCA, the median was subtracted from each TC. This eliminates intensity dependence, hence clustering is based only on TC shape similarity. Both PCA and FCA were applied directly to the water phantom data. Before exploratory analysis of the in vivo data, noisy pixels were removed by preprocessing. For all pixel TCs we first calculated their 1-lag-shifted autocorrelation function (p 0.01) [16]. Thus, only potentially interesting TCs survive for the actual FCA. The survivors include pixels with activation and pixels corrupted by other sources of signal variation, such as physiological (colored) noise, artifacts, etc. FCA applied to the in vivo data results in typically 8 clusters after merging [3]. The methods performance was assessed by determining the maximum Pearson correlation coefficient (MCC) between the simulated (known) activation time-course and all representative time-courses obtained by PCA and FCA, respectively. 3. Results In Fig. 3, the results of applying FCA and PCA to the water phantom are shown (TR 3500 ms, CNR 4.0). Figure 3a shows the activation map extracted by FCA overlaid onto the mean image of the time series acquired. Figure 3b shows the activation map extracted by PCA, overlaid on the mean image for the same data set. Note the good agreement between the maps calculated by FCA and PCA. Figure 3c shows the corresponding time-courses. Time-courses obtained by both FCA and PCA correlate well with the simulated time-course (cc 0.98). In Figure 4, the results of applying FCA and PCA to in vivo data are shown (TR 3500 ms, CNR 4.0). Figure 4a shows the activation map extracted by FCA overlaid on the T 2 * anatomy, i.e., the mean image of the time series. The simulated activation focus was correctly recognized. Figure 4b displays the activation map extracted by PCA, overlaid on the T 2 * anatomy. PCA failed to identify the activation focus, and produced well-scattered false positives. Figure 4c shows the corresponding time-courses. The time-course (cluster centroid) obtained by FCA correlates well with the simulated time-course (cc 0.98). In contrast to FCA, the time-course extracted by PCA is corrupted and the correlation coefficient with the simulated time-course is In Figure 5, a quantitative comparison of applying PCA and FCA to the water phantom is shown. Since the direction of an eigenvector obtained by PCA is arbitrary, PCA cannot distinguish globally between positively and negatively correlated time-courses, reflected by the flipping sign of the corresponding eigenvector (time-course). Therefore, we

3 R. Baumgartner et al. / Magnetic Resonance Imaging 18 (2000) Fig. 3. (a), FCA: Water phantom (TR 3500 ms, CNR 4.0), with the region extracted overlaid. (b), PCA: Water phantom (TR 3500 ms, CNR 4.0), with the extracted region overlaid. (c), FCA, PCA applied to water phantom: time-courses corresponding to FCA (solid line, a) and PCA (dotted and dashed line, b) extracted. Note the good agreement obtained by both exploratory methods with the reference time-course (dashed line REF). compared MCC obtained from FCA with MCC obtained from PCA. For the entire CNR range simulated, both methods are comparable for all TRs (for CNR 1, MCC 0.95 for FCA and MCC 0.95 for PCA, Fig. 5). Figure 6 shows a quantitative comparison of applying PCA and FCA to the in vivo data. Again, using PCA, the positively and negatively correlated time-courses were not discriminated and we compared MCC obtained by FCA with MCC obtained by PCA. PCA was able to identify simulated activation for CNR 6 for all TRs (Fig. 6). In Fig. 4. (a), FCA: In vivo anatomy (TR 3500 ms, CNR 4.0), with the region extracted by FCA overlaid. (b, PCA: Anatomy (TR 3500 ms, CNR 4.0), with the extracted region by PCA overlaid. (c), FCA, PCA applied to in vivo data: time-courses corresponding to Figure 4. FCA (solid line, a); PCA (dotted and dashed line, b) extracted. Note the good agreement obtained by FCA with the reference time course (dashed line. REF), and, in contrast, the failure of PCA to detect the simulated activation. contrast to PCA, FCA consistently and unambiguously identified the simulated hemodynamic response in all data sets with CNR 3(MCC 0.98) and recognized the simulated focal region. Other sources of signal variation in in vivo data, especially for lower CNR values, contaminate the activation map obtained by PCA. Consequently, the localized region is missed and the scattered pixels identified are false positives (Fig. 4c).

4 92 R. Baumgartner et al. / Magnetic Resonance Imaging 18 (2000) Fig. 5. FCA and PCA: Plot of MCC vs. CNR shows that with increasing CNR the simulated hemodynamic response was properly detected (for CNR 3 in all three TRs almost perfect match, MCC 0.98). To compare FCA and PCA directly, MCC vs. CNR for PCA is displayed. 4. Discussion We have tested the performance of two exploratory, data-driven methods (FCA, PCA) when applied to fmri data analysis. In a systematic simulated study, MR time series were combined with varying simulated activation with CNR in the range Two types of data with different noise characteristics obtained from an MR experiment were used: 1) water phantom data with scanner noise only and 2) MR time series acquired under null condition with both scanner and physiological noise present. Both methods have been tested with the same data sets under identical conditions. The results suggest that 1. If the time series are corrupted with scanner noise only, both methods show comparable performance. 2. If other sources of signal variation (e.g., physiological Fig. 6. FCA: MCC vs. CNR shows that with increasing CNR the simulated hemodynamic response was properly identified (for CNR 3 in all three TRs almost perfect match, MCC 0.98) in contrast to PCA. To compare FCA and PCA directly, MCC vs. CNR for PCA is displayed.

5 R. Baumgartner et al. / Magnetic Resonance Imaging 18 (2000) noise) are present, PCA fails to identify activation at lower CNRs, which may be critical in fmri. FCA outperforms PCA in the entire CNR range simulated. 3. Due to the arbitrary sign of the eigenvectors obtained from the correlation matrix decomposition, PCA cannot immediately (globally) distinguish between positively and negatively correlated time-courses, i.e., activation and deactivation Local projection of individual TCs onto the various PCs are needed to determine sign and amplitude. This works unambiguously if only a few scores dominate. FCA preserves the original shapes of the time-courses and as a consequence yields immediately interpretable results. PCA used in time series analysis may be considered as a cluster identification method [11], for which the PCs extracted represent simple data structures (a term frequently used in factor analysis and PCA-related methods [12]). From this point of view, we have in fact compared two clustering procedures. The exploratory methods are complementary techniques to the hypothesis-led inferential methods [17], which use a priori model assumptions (usually the time-courses are modeled [18]) to calculate the spatial distribution of activation, i.e., activation maps on which then the statistical inference is carried out. On the other hand, exploratory analysis is data-driven, and it aims to decompose the data in space and time and thus generate multiple hypotheses for inferential methods. Furthermore, it may be used in experiments when no a priori knowledge about the signal variation is available, e.g., during MR system stability testing, single trial fmri, clinical studies, etc. Realistic phantoms to test analysis methods are used in nuclear medicine [19], where the in vivo data are mixed with simulated time-series and evaluated in multicenter studies. In this preliminary investigation we showed a possible approach for fmri. Of course, for more detailed comparisons a more sophisticated set of phantoms should be designed and evaluated at several sites. The comparison method proposed in this study may also be used when evaluating other exploratory approaches, such as hard k-means clustering [20 22], Independent Component Analysis [23], or other neural network based methods, e.g., Self Organizing Maps [24,25]. normalized eigenvectors from an eigenvector-eigenvalue decomposition of the correlation matrix. Correlation matrix R 1/(n 1)ZZ T, where Z is the matrix X normalized to zero mean and unit standard deviation. (Note that PCA, based on the covariance matrix, gave identical results). The i-th principal component is calculated as PC i EV i i, where EV i is the i-th eigenvector and i is the i-th eigenvalue of the matrix R. The principal component images are then obtained by a least squares projection of the PCs onto the observed matrix Z. Fuzzy Clustering Analysis In fuzzy clustering the time-courses are considered as points in tdim-dimensional space. They are to be assigned to one of the C cluster centers (representative time-courses) which are defined by a matrix V(C,tdim). Furthermore, the C-partition of X is defined by the matrix U(C,n). The members of U(C,n), u ik are the membership values of the k-th pixel to the i-th centroid C i 1 n u ik 1, 0 u ik 1, u ik n. k 1 The matrices U and V are determined by an enhanced version of the fuzzy C means algorithm proposed by Bezdek [13], which minimizes the functional J n : n J n k 1 C u m ik d 2 ik, i 1 where d ik is the Euclidean distance of the k-th data point from the i-th cluster centroid. The solution of minimizing J n is found by a two stage iteration, n n C v ij u m ik x kj / u m ik, u ik 1/ d ik /d jk 2/ m 1, k 1 k 1 i 1 v ij is the element of the matrix V and m 1 is a parameter which controls the fuzziness of the clusters (we used m 1.1). The iterations stop when the algorithm satisfies predetermined convergence criteria. Note, that in EvIdent the final number of clusters is determined by a data-driven, novel cluster merging algorithm [3]. Appendix 1 Let us define the observed data matrix X(n,tdim), with dimensions n and tdim, where n is the number of pixels and tdim is the number of time instances. Principal component analysis The principal components (some of which may approximate the centroids of the time-courses of interest) are the References [1] Scarth G, Somorjai R. Fuzzy clustering versus principal component analysis of fmri. In: Book of abstracts: Fourth scientific meeting of the Society for Magnetic Resonance in Medicine, Vol. 3. New York: Butterworth-Heinemann, p [2] Scarth G, McIntyre M, Wowk B, Somorjai R. Detection of novelty in functional images using fuzzy clustering. In: Book of abstracts: Twelfth annual meeting of the Society of Magnetic Resonance. Vol. 1. Nice: Butterworth-Heinemann, p. 238.

6 94 R. Baumgartner et al. / Magnetic Resonance Imaging 18 (2000) [3] Jarmasz M, Somorjai R. Time to join! Cluster-merging in unsupervised fuzzy clustering of functional MRI data. In: Book of abstracts: Sixth scientific meeting of the Society for Magnetic Resonance in Medicine, Vol. 3. Sydney: Butterworth-Heinemann, p [4] Baumgartner R, Windischberger C, Moser E. Quantification in functional magnetic resonance imaging: fuzzy clustering vs. correlation analysis. Magn Reson Imaging 1998;16: [5] Golay X, Kollias S, Stoll G, Meier D, Valavanis A, Boesiger P. A new correlation-based fuzzy logic clustering algorithm for fmri. Magn Reson Med 1998;40: [6] Friston K, Frith C, Liddle P, Frackowiak R. Functional connectivity: the principal component analysis of large (PET) data sets. J Cereb Blood Flow Metab 1993;13:5 14. [7] Zarahn E, Aguirre G, D Esposito M. Empirical analysis of BOLD fmri statistics. I. Spatially unsmoothed data collected under null hypothesis conditions. NeuroImage 1997;5: [8] Constable R, Skudlarski P, Gore J. An ROC approach for evaluating functional brain MR imaging and postprocessing protocols. Magn Reson Med 1995;34: [9] Sijbers A. Signal and noise estimation from magnetic resonance images. PhD thesis University Antwerp, visielab/theses.html. [10] Lange N, Zeger S. Non-linear Fourier time series analysis for human brain mapping by functional magnetic resonance imaging. Appl Statist 1997;46:1 29. [11] Lange N. Tutorial in biostatistics: statistical approaches to human brain mapping by functional magnetic resonance imaging. Statistics Med 1996;15: [12] Basilevski A. Statistical Factor Analysis and Related Methods. New York: John Wiley & Sons, Inc., [13] Sychra J, Bandettini P, Bhattacharya N, Lin Q. Synthetic images by subspace transforms. I. Principal components images and related filters. Med Phys 1994;21: [14] Bezdek J. Pattern recognition with fuzzy objective function algorithms. New York: Plenum Press; [15] Gordon H, Somorjai R. Fuzzy cluster analysis of molecular dynamics trajectories. Proteins 1992;14: [16] Somorjai R, Jarmasz M, Baumgartner R, Richter W. Exploratory Analysis of fmr Images by Fuzzy Clustering: Voxel Preselection via Self-Similarity Seventh scientific meeting of the Society for Magnetic Resonance in Medicine, Philadelphia: p [17] Ford I. Some nonontological and functionally unconnected views on current issues in the analysis of PET datasets. J Cereb Blood Flow Metab 1994;15: [18] Friston K, Holmes A, Poline J, Grasby P, Williams S, Frackowiak R, Turner R. Analysis of fmri time-series revisited. NeuroImage 1995; 2: [19] Samal M, Bergmann H. Hybrid phantoms for testing the measurement of regional dynamics in dynamic renal scintigraphy. Nucl Med Commun 1998;19: [20] Baune A, Sommer FT, Erb M, Wildgruber D, Kardatzki B, Palm G, Grodd W. Dynamical cluster analysis of cortical fmri activation. NeuroImage 1999;9: [21] Goutte C, Toft P, Rostrup E, Nielsen F, Hansen L. On clustering fmri time series. NeuroImage 1999;9: [22] Filzmoser P, Baumgartner R, Moser E. Hierarchical clustering in fmri. Magn Reson Imaging 1999;17: [23] McKeown M, Makeig S, Brown G, Jung T, Kindermann S, Bell A, Sejnowski T. Analysis of fmri data by blind separation into independent spatial components. Hum Brain Mapp 1998;3: [24] Fischer H, Hennig J. Neural network based analysis of MR time series. Magn Reson Med 1999;41: [25] Kohonen T. Self-organizing maps. New York: Springer, 1995.