1 00:00:00,025 --> 00:00:05,131 [SOUND] So they often use multivariate decomposition 2 00:00:05,131 --> 00:00:09,547 methods to study functional connectivity. 3 00:00:09,547 --> 00:00:13,129 These methods provide a decomposition of the data into separate components. 4 00:00:13,129 --> 00:00:16,038 And they can be used to find coherent brain networks and 5 00:00:16,038 --> 00:00:20,349 provide information on how different brain regions interact with one another. 6 00:00:21,490 --> 00:00:26,410 The most common decomposition methods are principal components analysis, or PCA, and 7 00:00:26,410 --> 00:00:28,450 independent components analysis, or ICA. 8 00:00:30,190 --> 00:00:35,510 So throughout, we're going to organize the fMRI data in an M x N matrix X. 9 00:00:35,510 --> 00:00:38,160 The row dimension is the number of time points and 10 00:00:38,160 --> 00:00:40,520 the column dimension the number of voxels. 11 00:00:40,520 --> 00:00:45,920 So we're just going to put all the data together in a time by voxel matrix. 12 00:00:45,920 --> 00:00:51,960 So here in contrast to say the GLM mode where we analyze 13 00:00:51,960 --> 00:00:56,610 each voxel at a time, here we're going to analyze all the voxels simultaneously. 14 00:00:58,960 --> 00:01:02,780 So principle components analysis Is a mutivariate procedure 15 00:01:02,780 --> 00:01:04,365 concerned with explaining the variance, 16 00:01:04,365 --> 00:01:06,810 co-variance structure of a high dimensional random vector. 17 00:01:07,890 --> 00:01:12,420 So in PCA, a set of correlated variables are transformed into a set of uncorrelated 18 00:01:12,420 --> 00:01:15,900 variables ordered by the amount of variability in the data that they explain. 19 00:01:18,120 --> 00:01:23,802 So in fMRI principal component analysis involves finding the spatial modes, 20 00:01:23,802 --> 00:01:26,046 or eigenimages, in the data. 21 00:01:26,046 --> 00:01:27,576 These are the patterns that account for 22 00:01:27,576 --> 00:01:30,470 most of the various-covariance structure in the data. 23 00:01:30,470 --> 00:01:34,011 And they're ranked in order of the amount of variation that they explain. 24 00:01:34,011 --> 00:01:38,770 The eigenimages images can be contained using singular value decomposition or 25 00:01:38,770 --> 00:01:43,101 SVD, which decomposes the data into two sets of orthogonal vectors that 26 00:01:43,101 --> 00:01:45,670 correspond to patterns in space and time. 27 00:01:47,280 --> 00:01:52,475 So, the single value of decomposition is an operation that decomposes the matrix X, 28 00:01:52,475 --> 00:01:54,300 into three other matrices. 29 00:01:54,300 --> 00:01:58,549 So, we write X is equal to U times S times V transpose, 30 00:01:58,549 --> 00:02:02,799 where V transpose V is equal to the identity matrix and 31 00:02:02,799 --> 00:02:06,878 U transpose U is also equal to the identity matrix. 32 00:02:06,878 --> 00:02:11,680 And S is a diagonal matrix whose elements are called singular values. 33 00:02:13,290 --> 00:02:16,370 So, pictorially, we can write this as follows. 34 00:02:16,370 --> 00:02:21,392 We can take our matrix, X, which again was time by voxels, 35 00:02:21,392 --> 00:02:25,921 and decompose it into three matrices, U, S, and V. 36 00:02:25,921 --> 00:02:30,360 Pictorially we can represent the singular value decomposition as follows. 37 00:02:30,360 --> 00:02:35,273 Here we take the matrix X which is again time by voxels, and 38 00:02:35,273 --> 00:02:39,297 we can separate it into U, S, and V Transpose. 39 00:02:39,297 --> 00:02:43,820 So here what I'm going to claim is that the columns of V or 40 00:02:43,820 --> 00:02:47,768 the rows of V transpose are the Eigenimages, and 41 00:02:47,768 --> 00:02:52,940 the columns of U represent the corresponding time courses. 42 00:02:52,940 --> 00:02:56,369 So these are the time courses that correspond to the respective Eigenimages. 43 00:02:57,860 --> 00:03:03,940 So, we can write this as X is equal to U times S V transpose, but because of 44 00:03:03,940 --> 00:03:10,254 the diagonal nature of S we can also decompose it into each column of U and V. 45 00:03:10,254 --> 00:03:16,139 We can write S1 times U1 V1 transpose, etc for each of the subsequent columns. 46 00:03:17,190 --> 00:03:19,500 Here we see a real data example. 47 00:03:19,500 --> 00:03:22,570 Here we have x is the first image here. 48 00:03:22,570 --> 00:03:26,551 And this can be decomposed into a number of sub-matrices as 49 00:03:26,551 --> 00:03:28,886 indicated on the previous slide. 50 00:03:28,886 --> 00:03:34,050 The first sub-matrix consists of S1 which is a scaler 51 00:03:34,050 --> 00:03:38,140 times U1 which is a time course corresponding to the first Eigenimage. 52 00:03:38,140 --> 00:03:41,671 And v1 transpose which is the first Eigenimage. 53 00:03:41,671 --> 00:03:46,212 Then we have the second sub-matrix which is S2 times U2. 54 00:03:46,212 --> 00:03:50,555 Which is the time course corresponding to the second Eigenimage V2 55 00:03:50,555 --> 00:03:54,160 transpose which is the second Eigenimage etc., etc. 56 00:03:54,160 --> 00:03:59,030 Now, each of these Vs Have a length of the number of voxels, and they can be 57 00:03:59,030 --> 00:04:04,100 sort of reconstructed into images corresponding to the spacial modes here. 58 00:04:04,100 --> 00:04:09,250 So here we see the first Eigenimage which is the V1 transpose here, 59 00:04:09,250 --> 00:04:13,020 and we have U1 transpose which is the corresponding time course. 60 00:04:13,020 --> 00:04:15,990 Similarly, we get the second Eigenimage, and 61 00:04:15,990 --> 00:04:19,300 it's corresponding time of course, etc., etc. 62 00:04:19,300 --> 00:04:22,340 So if we do this, we can get several different 63 00:04:22,340 --> 00:04:26,670 temporal components corresponding to each of the columns of U, 64 00:04:26,670 --> 00:04:31,510 and we get the corresponding Eigenimages below. 65 00:04:31,510 --> 00:04:34,150 And here's an example of a PCA analysis. 66 00:04:34,150 --> 00:04:38,220 Here we see the first four Eigenimage on the bottom, so 67 00:04:38,220 --> 00:04:44,564 in the bottom panel we see four rows, one for each Eigenimage. 68 00:04:44,564 --> 00:04:47,829 And on the top panel we see the corresponding time courses. 69 00:04:49,030 --> 00:04:53,670 And to the right of these time courses, you see percentages. 70 00:04:53,670 --> 00:04:57,940 Those percentages are the percent of variation, explained by each 71 00:04:57,940 --> 00:05:02,710 of the components, and they're related to the values of S in the singular matrix. 72 00:05:04,310 --> 00:05:08,270 >> So independent component analysis, or ICA, is a family of techniques used to 73 00:05:08,270 --> 00:05:11,730 extract independent signals from some source signal. 74 00:05:11,730 --> 00:05:15,440 ICA provides a method to blindly separate the data into spatially independent 75 00:05:15,440 --> 00:05:17,150 components. 76 00:05:17,150 --> 00:05:20,000 Here, the key assumption is that the data set consists of p 77 00:05:20,000 --> 00:05:24,400 spatially independent components which are linearly mixed, but spatially fixed. 78 00:05:25,620 --> 00:05:30,530 The ICA model differs a little bit from what we used in PCA. 79 00:05:30,530 --> 00:05:35,160 Here, the matrix x is decomposed into two matrices, A and S. 80 00:05:36,330 --> 00:05:40,290 Here A is referred to as the mixing matrix and S as the source matrix. 81 00:05:41,380 --> 00:05:46,139 So our goal is ultimately to use this information to find 82 00:05:46,139 --> 00:05:50,288 an un-mixing matrix W such that Y is equal to WX, 83 00:05:50,288 --> 00:05:56,587 provides a good approximation to S which are these independent sources. 84 00:05:56,587 --> 00:06:00,730 If the mixing matrix is known, the problem is straight forward and almost trivial. 85 00:06:00,730 --> 00:06:06,330 However, ICA tries to solve this problem without knowing the mixing parameters. 86 00:06:06,330 --> 00:06:09,480 So instead, what it does is it exploits some key assumptions. 87 00:06:09,480 --> 00:06:13,020 First it assumes there is linear mixing of sources, 88 00:06:13,020 --> 00:06:17,850 then it assumes that the components SI are statistically independent of one another. 89 00:06:17,850 --> 00:06:21,600 And it assumes that the components are non-Gaussian, or most 1. 90 00:06:21,600 --> 00:06:23,930 >> It can be Gaussian. 91 00:06:23,930 --> 00:06:26,250 When applying ICA for fMRIs, 92 00:06:26,250 --> 00:06:30,850 assume that fMRI data can be modeled by identifying sets of voxels whose activity 93 00:06:30,850 --> 00:06:35,400 vary both over time and are different from activity in other sets. 94 00:06:35,400 --> 00:06:39,250 We try to decompose the data into spatially independent component maps 95 00:06:39,250 --> 00:06:41,370 with a set of corresponding time courses. 96 00:06:42,940 --> 00:06:47,228 Here's the kind of cartoon image here we have x, which again is time by voxels, 97 00:06:47,228 --> 00:06:48,254 that's our data. 98 00:06:48,254 --> 00:06:50,754 And we have two matrixes a and s. 99 00:06:50,754 --> 00:06:55,800 So again, S represents the spatially independent components, one for each row. 100 00:06:57,590 --> 00:07:01,731 The columns of A represent the time courses corresponding to the spatially 101 00:07:01,731 --> 00:07:03,253 independent components. 102 00:07:03,253 --> 00:07:08,530 So the first column corresponds to the first row of S. 103 00:07:10,990 --> 00:07:15,659 And what we want to do here is use an ICA algorithm to find both A and S. 104 00:07:17,350 --> 00:07:19,710 Here's an example fitting ICA. 105 00:07:19,710 --> 00:07:21,800 And here's two different components. 106 00:07:21,800 --> 00:07:26,750 So this corresponds to two different spatial components 107 00:07:26,750 --> 00:07:30,450 from the ICA decomposition. 108 00:07:30,450 --> 00:07:33,504 And first we see a task related component. 109 00:07:33,504 --> 00:07:36,980 And in a second, we see a more noise component. 110 00:07:36,980 --> 00:07:41,988 And see here, you can see that this is a noise component by a lot of activation 111 00:07:41,988 --> 00:07:47,713 around the edges of the brain, which is probably due to emotion-related artifacts. 112 00:07:47,713 --> 00:07:50,405 Here's an example of eight of the most common and 113 00:07:50,405 --> 00:07:55,630 consistently identified resting state networks, which are identified by ICA. 114 00:07:55,630 --> 00:08:00,110 And so this week, we looked at the previous lecture on resting state fMRI. 115 00:08:00,110 --> 00:08:02,340 And we showed these results. 116 00:08:02,340 --> 00:08:07,180 And here, we can now come back to this and say that this was obtained using ICA. 117 00:08:08,860 --> 00:08:11,440 So what's the difference between PCA and ICA? 118 00:08:11,440 --> 00:08:14,870 Well PCA assumes an orthonormality constraint. 119 00:08:14,870 --> 00:08:19,240 In contrast ICA assumes statistical independence among a collection of spatial 120 00:08:19,240 --> 00:08:20,520 patterns. 121 00:08:20,520 --> 00:08:24,380 So independence is a stronger requirement than orthonormality. 122 00:08:24,380 --> 00:08:28,990 However in ICA, the spatially independent components are not ranked in order of 123 00:08:28,990 --> 00:08:32,430 importance such as they are when performing PCA. 124 00:08:32,430 --> 00:08:36,160 So it behooves you to go through all the components after the fact and find out 125 00:08:36,160 --> 00:08:39,460 which ones are important and which ones are just related to noise and whatnot. 126 00:08:40,480 --> 00:08:42,530 Okay, so that's the end of this module. 127 00:08:42,530 --> 00:08:44,920 Here we've introduced principle component analysis and 128 00:08:44,920 --> 00:08:46,200 independent component analysis. 129 00:08:46,200 --> 00:08:51,780 So these are two ways of taking the full time by voxel data and 130 00:08:51,780 --> 00:08:54,030 finding interesting patterns of activation in it. 131 00:08:54,030 --> 00:08:58,240 And so, these are commonly used in functional connectivity analysis. 132 00:08:58,240 --> 00:09:02,672 Okay, in the next module, we'll talk a little bit about dynamic connectivity. 133 00:09:02,672 --> 00:09:03,642 See you then, bye. 134 00:09:03,642 --> 00:09:05,087 [SOUND]