1 00:00:00,930 --> 00:00:01,150 Hi. 2 00:00:01,150 --> 00:00:04,210 In the previous module, we started talking about functional connectivity. 3 00:00:04,210 --> 00:00:09,010 So we talked about methods that use correlation and partial correlation. 4 00:00:09,010 --> 00:00:11,370 In this module, we'll talk about a different class of ways 5 00:00:11,370 --> 00:00:13,520 of measuring functional connectivity that are 6 00:00:13,520 --> 00:00:16,220 based on using multivariate decomposition methods. 7 00:00:17,690 --> 00:00:22,450 So we often use multivariate decomposition methods to study functional connectivity. 8 00:00:22,450 --> 00:00:23,880 The reason is that they provide a 9 00:00:23,880 --> 00:00:26,500 decomposition of the data into separate components. 10 00:00:26,500 --> 00:00:30,380 And this can be used to define coherent brain networks and also 11 00:00:30,380 --> 00:00:32,300 provide information about how these different 12 00:00:32,300 --> 00:00:34,130 brain regions interact with one another. 13 00:00:35,700 --> 00:00:39,804 Now, the two most common decomposition methods that are used in fMRI data 14 00:00:39,804 --> 00:00:42,656 analysis are principal components analysis, or 15 00:00:42,656 --> 00:00:46,380 PCA, and independent components analysis, or ICA. 16 00:00:46,380 --> 00:00:47,480 There are other methods that are also 17 00:00:47,480 --> 00:00:49,950 used, such as partially squares method, but 18 00:00:49,950 --> 00:00:53,080 these are the two most commonly used, and we'll focus on them in this module. 19 00:00:55,710 --> 00:00:58,020 Throughout, we're going to organize the data in a slightly different 20 00:00:58,020 --> 00:01:00,800 manner than we did when we were doing GLM analysis. 21 00:01:00,800 --> 00:01:04,050 Here we're going to organize the fMRI data over all voxels 22 00:01:04,050 --> 00:01:07,710 in the brain, as an M by N matrix X. 23 00:01:07,710 --> 00:01:09,730 So here, the row dimension is the number of time 24 00:01:09,730 --> 00:01:12,550 points, and the column dimension represents the number of voxels. 25 00:01:13,580 --> 00:01:16,580 So pictorially, we can represent X as follows: 26 00:01:16,580 --> 00:01:21,030 where we have a time by voxels and matrix. 27 00:01:21,030 --> 00:01:25,740 So, to differentiate this from the GLM, we would, would, in the GLM analysis we would 28 00:01:25,740 --> 00:01:30,650 just look at one column of X at a time when we were, building these GLM models. 29 00:01:30,650 --> 00:01:32,980 Here we're looking at all the voxels at the same time. 30 00:01:35,580 --> 00:01:40,020 Now principal components analysis is a multivariate procedure which is concerned 31 00:01:40,020 --> 00:01:42,450 with explaining the variance-covariance structure 32 00:01:42,450 --> 00:01:44,550 of a high dimensional random vector. 33 00:01:44,550 --> 00:01:47,240 So we have a high dimensional random vector, and we want to find the 34 00:01:47,240 --> 00:01:49,720 direction that ex, in that vector that 35 00:01:49,720 --> 00:01:53,180 explains most of the variance, covariance structure. 36 00:01:53,180 --> 00:01:57,780 In PCA a set of correlated variables are transformed, or rotated, to a set of 37 00:01:57,780 --> 00:02:00,620 uncorrelated variables that are ordered by the amount 38 00:02:00,620 --> 00:02:02,670 of variability in the data that they explain. 39 00:02:04,140 --> 00:02:08,090 And so in fMRI principal components analysis involves finding 40 00:02:08,090 --> 00:02:11,550 what they call spatial modes, or eigenimages, in the data. 41 00:02:13,480 --> 00:02:15,505 These patterns, these are patterns that account for 42 00:02:15,505 --> 00:02:18,240 most of the variance-covariance structure in the data. 43 00:02:18,240 --> 00:02:20,830 So basically, the first principal component 44 00:02:20,830 --> 00:02:23,150 corresponds to the first eigenimage, which is 45 00:02:23,150 --> 00:02:27,330 the pattern that account for most of the variance-covariance structure in the data. 46 00:02:27,330 --> 00:02:31,290 The second eigenimage corresponds to that, that explains the second 47 00:02:31,290 --> 00:02:34,720 most variance-covariance conditional, that it's 48 00:02:34,720 --> 00:02:37,250 uncorrelated with the first eigenimage, etcetera. 49 00:02:40,040 --> 00:02:42,860 Now, the nice thing about these eigenimages is that they're ranked 50 00:02:42,860 --> 00:02:46,310 in the, in order of the amount of variation that they explain. 51 00:02:46,310 --> 00:02:47,930 So that tells us a little bit about 52 00:02:47,930 --> 00:02:50,510 which eigenimages are the most important and what 53 00:02:53,360 --> 00:02:53,560 not. 54 00:02:53,560 --> 00:02:55,450 The nice thing is that these eigenimages can 55 00:02:55,450 --> 00:02:59,680 be obtained using singular value decomposition, or SVD, which 56 00:02:59,680 --> 00:03:03,240 decomposes the data into two sets of orthogonal vectors 57 00:03:03,240 --> 00:03:06,390 that ultimate correspond to patterns in space and time. 58 00:03:07,460 --> 00:03:10,240 Principal component analysis is very commonly 59 00:03:10,240 --> 00:03:11,750 used in statistics, but singular value 60 00:03:11,750 --> 00:03:13,470 decomposition is commonly used in linear 61 00:03:13,470 --> 00:03:15,850 algebra and, and lots of other disciplines. 62 00:03:15,850 --> 00:03:20,120 And there's a nice duality between PCA and, and SVD, which allows 63 00:03:20,120 --> 00:03:22,244 us to basically perform SVD analysis 64 00:03:22,244 --> 00:03:24,350 to get these principal components or eigenimages. 65 00:03:27,680 --> 00:03:32,630 In general, singular value decomposition is an operation that takes a matrix 66 00:03:32,630 --> 00:03:38,400 X such as the one that we have, and decomposes that into three new matrices. 67 00:03:38,400 --> 00:03:41,740 One called U, one called S and one called V. 68 00:03:43,050 --> 00:03:45,210 U and V have certain properties. 69 00:03:45,210 --> 00:03:47,380 One is that V transpose V is equal to the 70 00:03:47,380 --> 00:03:50,850 identity, and U transpose U is equal to the identity. 71 00:03:50,850 --> 00:03:53,370 So they both are normal. 72 00:03:53,370 --> 00:03:59,180 Also, S is a diagonal matrix whose elements are called singular values, so 73 00:03:59,180 --> 00:04:03,550 S is 0 everywhere except in the diagonal where it takes these singular values. 74 00:04:04,910 --> 00:04:09,810 And these singular values are often ranked from largest to smallest. 75 00:04:09,810 --> 00:04:14,870 So in the, in the first column, first row, we have the largest singular value, the 76 00:04:14,870 --> 00:04:16,920 second column second row, we have the next 77 00:04:16,920 --> 00:04:20,049 largest, etc, etc, as we go down the diagonal. 78 00:04:22,320 --> 00:04:27,010 So basically, you're performing a single value decomposition of the data x. 79 00:04:27,010 --> 00:04:31,270 We can decompose x into three matrices, u, s, and v. 80 00:04:32,350 --> 00:04:37,040 And so we can write this as x equal to u, s, v, transpose, 81 00:04:37,040 --> 00:04:42,420 or alternatively, we can break it down because we have this diagonal format on s 82 00:04:42,420 --> 00:04:49,720 as the first singular value of s1 times the first, column of u, which we're 83 00:04:49,720 --> 00:04:54,739 going to call u1 times the first column of v, which we're going to call v1 transpose. 84 00:04:56,930 --> 00:04:57,760 Okay? 85 00:04:57,760 --> 00:05:03,490 And so basically, each of these gives us some information about X. 86 00:05:03,490 --> 00:05:08,670 Interestingly, the columns of v, the v1 up to vN, 87 00:05:08,670 --> 00:05:13,580 correspond to these eigenimages that we're saying explain most of the 88 00:05:13,580 --> 00:05:17,810 variance, covariance matrix in order here, and the, the columns 89 00:05:17,810 --> 00:05:21,680 of u correspond to the time courses corresponding to those Eigenimages. 90 00:05:24,130 --> 00:05:30,240 So basically what the last equation shows is that x can be decomposed into a number 91 00:05:30,240 --> 00:05:35,640 of different matrices, which all contain a bit of information about the data. 92 00:05:36,830 --> 00:05:40,150 So here we see that in a different format and so, and this 93 00:05:40,150 --> 00:05:45,990 real data is equal to series of matrices as follows in these green blocks. 94 00:05:45,990 --> 00:05:48,590 And so, each of these matrices consists of three components. 95 00:05:48,590 --> 00:05:51,860 One is s1, which is a singular value. 96 00:05:51,860 --> 00:05:55,120 The other is u1, which is a time course. 97 00:05:55,120 --> 00:05:59,400 And then finally, we have V1 transplus which is the eigenimage. 98 00:05:59,400 --> 00:06:02,397 The second matrix consists of s2, which is 99 00:06:02,397 --> 00:06:05,799 scalar and a singular value, times u2, which is 100 00:06:05,799 --> 00:06:08,958 a time course, times v to transpose, which is 101 00:06:08,958 --> 00:06:13,108 the second eigenimage, et cetera, et cetera, et cetera. 102 00:06:13,108 --> 00:06:16,496 So what we can do is if we take v1, and we, now it's a 103 00:06:16,496 --> 00:06:18,806 vector, but if we reshape it into an 104 00:06:18,806 --> 00:06:22,443 image format, and basically, we get the eigenimage. 105 00:06:22,443 --> 00:06:26,485 And so this is the, the, the pattern of activation that 106 00:06:26,485 --> 00:06:31,662 explains most of the variance covariance structure in the data set X. 107 00:06:31,662 --> 00:06:36,360 Now we see that U1 corresponds to the time course that we see, in this 108 00:06:36,360 --> 00:06:41,686 particular eigenimage, and we see it looks like a sort of drift of some sort. 109 00:06:41,686 --> 00:06:44,474 Similarly, we can reshape V2 to get the 110 00:06:44,474 --> 00:06:49,160 second eigenimage and that shows a, a different pattern. 111 00:06:49,160 --> 00:06:52,850 And that pattern cor, has a time course that corresponds to U2. 112 00:06:54,010 --> 00:06:57,120 So it's pretty neat here that we've gotten, we've taken this data, 113 00:06:57,120 --> 00:07:01,710 X, and now, we've decomposed it into, you know, a series of 114 00:07:01,710 --> 00:07:06,100 spatial patterns and corresponding time courses, which we can use to interpret 115 00:07:06,100 --> 00:07:09,200 and try to figure out what's going on in this data set. 116 00:07:09,200 --> 00:07:11,960 And we did this without even knowing what the status that was all about. 117 00:07:11,960 --> 00:07:14,590 So this is a very data driven way of analyzing the data. 118 00:07:15,990 --> 00:07:20,470 Here's some example, and this is taken from web page of the late Keith Worsley. 119 00:07:20,470 --> 00:07:22,510 Here we,we see for this data set 120 00:07:22,510 --> 00:07:27,650 the corresponding first four, eigenimages as now. 121 00:07:27,650 --> 00:07:30,640 We'll see, it's really more of a three dimensional entity that way. 122 00:07:30,640 --> 00:07:32,204 So we have a bunch of slices here. 123 00:07:32,204 --> 00:07:36,413 And then we have the time course corresponding to that, and we can also, 124 00:07:36,413 --> 00:07:38,897 using the singular values, we can compute 125 00:07:38,897 --> 00:07:42,172 the percent of variance explained by each eigenimage. 126 00:07:42,172 --> 00:07:46,685 So the first eigenimage explains about 15% of the variation. 127 00:07:46,685 --> 00:07:50,267 The second, 8% of the variation, etcetera, etcetera. 128 00:07:50,267 --> 00:07:53,030 >> So they're ranked in order of, you know, relative importance. 129 00:07:54,510 --> 00:07:57,050 So, this is a nice way of just decomposing 130 00:07:57,050 --> 00:08:00,000 an unknown data set into a bunch of spacial and 131 00:08:00,000 --> 00:08:02,260 temporal components, which we later can analyze and try 132 00:08:02,260 --> 00:08:04,190 to figure out what's going on in the data set. 133 00:08:04,190 --> 00:08:08,696 So, that's basically what we try to do in these multi-variant decomposition methods. 134 00:08:09,820 --> 00:08:12,010 The second type of method that we often use 135 00:08:12,010 --> 00:08:16,850 in this class are, is independent component analysis, or ICA. 136 00:08:16,850 --> 00:08:20,930 So, ICA is a family of techniques used to extract independent signals from 137 00:08:20,930 --> 00:08:25,690 some source signal, and so it's often used in some blind source separation problems. 138 00:08:25,690 --> 00:08:28,110 So, ICA provides a method to blindly 139 00:08:28,110 --> 00:08:30,890 separate the data into spatially independent components. 140 00:08:32,640 --> 00:08:35,680 So the key assumption here is that the data set consists 141 00:08:35,680 --> 00:08:40,890 of p spatial independent components, which are linearly mixed and spatially fixed. 142 00:08:40,890 --> 00:08:44,160 So this differentiates a little bit from PCA, where we said 143 00:08:44,160 --> 00:08:48,300 that the spatial components had to be at uncorrelated with each other. 144 00:08:48,300 --> 00:08:51,329 Here they need to be independent which is a slightly stronger condition. 145 00:08:53,400 --> 00:08:58,450 So in general, the ICA model decomposes X into two different matrices, A and S. 146 00:08:59,940 --> 00:09:05,890 Here, A is often referred to as the mixing matrix and S as the source matrix. 147 00:09:05,890 --> 00:09:09,790 And the goal here is, you want to find these sources. 148 00:09:09,790 --> 00:09:13,650 So our goal is to find an un-mixing matrix W such that 149 00:09:13,650 --> 00:09:17,600 Y is equal to W X provides a good approximation to S. 150 00:09:18,670 --> 00:09:20,740 So basically what we want to do is, our 151 00:09:20,740 --> 00:09:24,760 goal is to find these independent sources S, but 152 00:09:24,760 --> 00:09:29,590 it's kind of a tricky problem because here we have X, which is equal to A times S. 153 00:09:30,910 --> 00:09:33,050 Now if A were known, if the mixing 154 00:09:33,050 --> 00:09:35,680 matrix is known, this problem is really straightforward. 155 00:09:35,680 --> 00:09:39,330 We would just the, you know, the inverse of x of some sort, or, or do a 156 00:09:39,330 --> 00:09:41,570 least wear solution or, or, or something like 157 00:09:41,570 --> 00:09:43,700 that, depending on the rank of the mixing matrix. 158 00:09:44,830 --> 00:09:47,150 However, in ICA we try to solve 159 00:09:47,150 --> 00:09:50,480 this problem without knowing the mixing parameters. 160 00:09:50,480 --> 00:09:55,570 So we're basically assuming that both A ans S are unknown. 161 00:09:55,570 --> 00:09:58,170 So this is a very difficult problem, and we 162 00:09:58,170 --> 00:10:01,270 can't solve this without making additional assumptions, so some 163 00:10:01,270 --> 00:10:04,420 constraints on the problem, and in ICA, the key 164 00:10:04,420 --> 00:10:09,160 assumptions are that the, the mixing of sources is linear. 165 00:10:09,160 --> 00:10:14,710 Second is that the components S I, so, so basically the, the rows of S are 166 00:10:14,710 --> 00:10:17,630 going to be statistically independent of one another, and 167 00:10:19,170 --> 00:10:22,000 then finally, the components S I are non-Gaussian. 168 00:10:22,000 --> 00:10:25,070 So this can be relaxed to only one component is 169 00:10:25,070 --> 00:10:28,250 allowed to be Gaussian, but all other have to be non-Gaussian. 170 00:10:28,250 --> 00:10:31,700 So using those assumptions we're actually able solve 171 00:10:31,700 --> 00:10:33,149 this problem in a, in a nice manner. 172 00:10:34,780 --> 00:10:37,680 So how do we perform ICA for fMRI? 173 00:10:37,680 --> 00:10:42,940 Well lets assume that the fMRI data can be modeled by identifying sets of voxels, 174 00:10:42,940 --> 00:10:45,820 whose activity vary both together over time and 175 00:10:45,820 --> 00:10:47,770 are different from the activity in other sets. 176 00:10:49,270 --> 00:10:52,430 So what we do here is we typically try to decompose the data set 177 00:10:52,430 --> 00:10:54,870 into a set of spatially independent component 178 00:10:54,870 --> 00:10:57,590 maps with a set of corresponding time courses. 179 00:10:58,740 --> 00:11:00,600 So here we have the cartoon again. 180 00:11:00,600 --> 00:11:04,440 Here we have our data set X, which is time by voxels, 181 00:11:04,440 --> 00:11:06,800 and so what we want to do is we want to se, we 182 00:11:06,800 --> 00:11:11,040 want to split this into two matrices, A and S, where A 183 00:11:11,040 --> 00:11:14,560 is what we call the mixing matrix, and S is the source matrix. 184 00:11:14,560 --> 00:11:16,170 And what's going to happen here is much like 185 00:11:16,170 --> 00:11:19,710 PCA, the columns of A are going to correspond 186 00:11:19,710 --> 00:11:23,010 to time courses, while the rows of S 187 00:11:23,010 --> 00:11:26,590 are going to correspond to spatially independent components. 188 00:11:26,590 --> 00:11:29,130 And so what we do is we use an ICA 189 00:11:29,130 --> 00:11:33,710 algorithm to find both A and S using simply the data. 190 00:11:33,710 --> 00:11:36,670 So, X is, is the data that we, we observe. 191 00:11:36,670 --> 00:11:40,390 We make assumptions about S, you know, the, and, and, 192 00:11:40,390 --> 00:11:43,510 and then we use this to define both A and S. 193 00:11:45,250 --> 00:11:47,720 Here's an example of the results. 194 00:11:47,720 --> 00:11:54,500 And, so here is one row of S that's kind of strowing out to give us a spatial map. 195 00:11:54,500 --> 00:11:56,950 And here's the corresponding time course. 196 00:11:56,950 --> 00:11:58,940 And this is obtained through the MATLAB toolbot gift. 197 00:11:58,940 --> 00:12:04,150 And as we'll see here there seems to be some sort of kind of periodic signal here, 198 00:12:04,150 --> 00:12:05,770 if you look at the time course, and 199 00:12:05,770 --> 00:12:08,280 there's activation in the visual and the motor tasks. 200 00:12:08,280 --> 00:12:12,500 So this was a visual-motor task where they repeatedly got, you know, 201 00:12:12,500 --> 00:12:15,830 did some finger tapping, and got the, a light flashed in their eyes. 202 00:12:17,340 --> 00:12:21,580 Here's another independent component which is, in this case looks 203 00:12:21,580 --> 00:12:24,610 like its around the, the boundaries of, of the skulls. 204 00:12:24,610 --> 00:12:27,710 So this might be movement related or something like 205 00:12:27,710 --> 00:12:30,240 that, and so its probably some sort of nuisance signal. 206 00:12:30,240 --> 00:12:33,100 So this is something that we can pick up using ICA analysis. 207 00:12:35,440 --> 00:12:40,200 So, unlike PCA which assumes orthomorality constraint, ICA is 208 00:12:40,200 --> 00:12:43,920 that assumes statistical independence among the collection of spatial patterns. 209 00:12:44,930 --> 00:12:48,860 So, independence is a stronger statistical requirement than orthonormality. 210 00:12:50,060 --> 00:12:54,200 However, in ICA the spatially independent components are not ranked 211 00:12:54,200 --> 00:12:57,230 in order of importance as they were when performing the PCA. 212 00:12:57,230 --> 00:13:00,830 So in the PCA analysis, the first eigenimage is 213 00:13:00,830 --> 00:13:03,910 the one that explains most of the variance programs matrix. 214 00:13:03,910 --> 00:13:07,670 In ICA, the components are varied permutation, so we have 215 00:13:07,670 --> 00:13:10,550 no real, they're not ranked by any real rhyme or reason. 216 00:13:10,550 --> 00:13:12,810 So we have to sift through or find some 217 00:13:12,810 --> 00:13:15,679 alternative way to classify them in order of importance. 218 00:13:17,630 --> 00:13:20,320 So in general these multivariable decomposition methods 219 00:13:20,320 --> 00:13:22,500 which is PCA and ICA are extremely 220 00:13:22,500 --> 00:13:24,000 useful when we don't have so much 221 00:13:24,000 --> 00:13:28,170 information about what's going on in the experiment. 222 00:13:28,170 --> 00:13:32,200 For example, when we're doing task for fMRI or resting state fMRI. 223 00:13:32,200 --> 00:13:35,510 They can also be very useful when, you know, unexpected things are happening in 224 00:13:35,510 --> 00:13:40,750 the signal or, or, you, that we can can't model using in a GLM analysis. 225 00:13:40,750 --> 00:13:42,800 So this allows us to kind of look through 226 00:13:42,800 --> 00:13:44,910 the data and try to figure out what's going on. 227 00:13:44,910 --> 00:13:48,660 And so they're actually very, very useful to use in that, and they're also 228 00:13:48,660 --> 00:13:52,200 very good for finding, you know, artifacts in the data and things like that. 229 00:13:55,530 --> 00:13:57,310 Okay, so that's the end of this module. 230 00:13:57,310 --> 00:13:59,320 It was a very brief introduction to 231 00:13:59,320 --> 00:14:01,830 multivariate decomposition methods such as PCA and 232 00:14:01,830 --> 00:14:04,262 ICA, and they're very useful ways for 233 00:14:04,262 --> 00:14:08,690 fin, detecting functional connectivity, and things like that. 234 00:14:08,690 --> 00:14:12,280 In the next module we'll start talking a little bit about effective connectivity. 235 00:14:12,280 --> 00:14:13,750 Okay, I'll see you then. 236 00:14:13,750 --> 00:14:13,850 Bye.