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2.1 --- a/grant.html Sun Apr 19 15:23:53 2009 -0700 2.2 +++ b/grant.html Sun Apr 19 16:20:50 2009 -0700 2.3 @@ -318,6 +318,9 @@ 2.4 many layer-area intersection clusters, further work is needed to make sense of these). The reason that Gene Finder cannot find marker genes for 2.5 most cortical areas is that in Gene Finder, although the user chooses a seed voxel, Gene Finder chooses the ROI for which genes will be found, 2.6 and it creates that ROI by (pairwise voxel correlation) clustering around the seed. 2.7 + 2.8 + 2.9 + Figure 1: Gene Pitx2 is selectively underexpressed in area SS (somatosensory). 2.10 Preliminary work 2.11 Format conversion between SEV, MATLAB, NIFTI 2.12 We have created software to (politely) download all of the SEV files from the Allen Institute website. We have also created 2.13 @@ -348,13 +351,23 @@ 2.14 In the Research Plan, we describe how we will automatically locate the layer depths. For validation, we have manually 2.15 demarcated the depth of the outer boundary of cortical layer 5 throughout the cortex. 2.16 Feature selection and scoring methods 2.17 +Underexpression of a gene can serve as a marker Underexpression of a gene can sometimes serve as a marker. See, 2.18 +for example, Figure 1. 2.19 Correlation Recall that the instances are surface pixels, and consider the problem of attempting to classify each instance 2.20 as either a member of a particular anatomical area, or not. The target area can be represented as a boolean mask over the 2.21 surface pixels. 2.22 One class of feature selection scoring method are those which calculate some sort of “match” between each gene image 2.23 and the target image. Those genes which match the best are good candidates for features. 2.24 One of the simplest methods in this class is to use correlation as the match score. We calculated the correlation between 2.25 -each gene and each cortical area. The top row of Figure 1 shows the three genes most correlated with area SS. 2.26 +each gene and each cortical area. The top row of Figure 2 shows the three genes most correlated with area SS. 2.27 + 2.28 + 2.29 + 2.30 +Figure 2: Top row: Genes Nfic, A930001M12Rik, C130038G02Rik are the most correlated with area SS (somatosensory 2.31 +cortex). Bottom row: Genes C130038G02Rik, Cacna1i, Car10 are those with the best fit using logistic regression. Within 2.32 +each picture, the vertical axis roughly corresponds to anterior at the top and posterior at the bottom, and the horizontal 2.33 +axis roughly corresponds to medial at the left and lateral at the right. The red outline is the boundary of region MO. Pixels 2.34 +are colored according to correlation, with red meaning high correlation and blue meaning low. 2.35 Conditional entropy An information-theoretic scoring method is to find features such that, if the features (gene 2.36 expression levels) are known, uncertainty about the target (the regional identity) is reduced. Entropy measures uncertainty, 2.37 so what we want is to find features such that the conditional distribution of the target has minimal entropy. The distribution 2.38 @@ -368,18 +381,10 @@ 2.39 expression boolean masks, is minimized. 2.40 This finds pairs of genes which are most informative (at least at these discretization thresholds) relative to the question, 2.41 “Is this surface pixel a member of the target area?”. Its advantage over linear methods such as logistic regression is that it 2.42 - 2.43 - 2.44 - 2.45 -Figure 1: Top row: Genes Nfic, A930001M12Rik, C130038G02Rik are the most correlated with area SS (somatosensory 2.46 -cortex). Bottom row: Genes C130038G02Rik, Cacna1i, Car10 are those with the best fit using logistic regression. Within 2.47 -each picture, the vertical axis roughly corresponds to anterior at the top and posterior at the bottom, and the horizontal 2.48 -axis roughly corresponds to medial at the left and lateral at the right. The red outline is the boundary of region MO. Pixels 2.49 -are colored according to correlation, with red meaning high correlation and blue meaning low. 2.50 takes account of arbitrarily nonlinear relationships; for example, if the XOR of two variables predicts the target, conditional 2.51 entropy would notice, whereas linear methods would not. 2.52 Gradient similarity We noticed that the previous two scoring methods, which are pointwise, often found genes whose 2.53 -pattern of expression did not look similar in shape to the target region. Fort his reason we designed a non-pointwise local 2.54 +pattern of expression did not look similar in shape to the target region. For this reason we designed a non-pointwise local 2.55 scoring method to detect when a gene had a pattern of expression which looked like it had a boundary whose shape is similar 2.56 to the shape of the target region. We call this scoring method “gradient similarity”. 2.57 One might say that gradient similarity attempts to measure how much the border of the area of gene expression and 2.58 @@ -398,9 +403,16 @@ 2.59 The intuition is that we want to see if the borders of the pattern in the two images are similar; if the borders are similar, 2.60 then both images will have corresponding pixels with large gradients (because this is a border) which are oriented in a 2.61 similar direction (because the borders are similar). 2.62 +Most of the genes in Figure 4 were identified via gradient similarity. 2.63 Gradient similarity provides information complementary to correlation 2.64 + 2.65 + 2.66 + 2.67 +Figure 3: The top row shows the three genes which (individually) best predict area AUD, according to logistic regression. 2.68 +The bottom row shows the three genes which (individually) best match area AUD, according to gradient similarity. From 2.69 +left to right and top to bottom, the genes are Ssr1, Efcbp1, Aph1a, Ptk7, Aph1a again, and Lepr 2.70 To show that gradient similarity can provide useful information that cannot be detected via pointwise analyses, consider 2.71 -Fig. 2. The top row of Fig. 2 displays the 3 genes which most match area AUD, according to a pointwise method17. The 2.72 +Fig. 3. The top row of Fig. 3 displays the 3 genes which most match area AUD, according to a pointwise method17. The 2.73 bottom row displays the 3 genes which most match AUD according to a method which considers local geometry18 The 2.74 pointwise method in the top row identifies genes which express more strongly in AUD than outside of it; its weakness is 2.75 that this includes many areas which don’t have a salient border matching the areal border. The geometric method identifies 2.76 @@ -408,7 +420,27 @@ 2.77 genes which don’t express over the entire area. Genes which have high rankings using both pointwise and border criteria, 2.78 such as Aph1a in the example, may be particularly good markers. None of these genes are, individually, a perfect marker 2.79 for AUD; we deliberately chose a “difficult” area in order to better contrast pointwise with geometric methods. 2.80 +Areas which can be identified by single genes Using gradient similarity, we have already found single genes which 2.81 +roughly identify some areas and groupings of areas. For each of these areas, an example of a gene which roughly identifies 2.82 +it is shown in Figure 4. We have not yet cross-verified these genes in other atlases. 2.83 +In addition, there are a number of areas which are almost identified by single genes: COAa+NLOT (anterior part of 2.84 +cortical amygdalar area, nucleus of the lateral olfactory tract), ENT (entorhinal), ACAv (ventral anterior cingulate), VIS 2.85 +(visual), AUD (auditory). 2.86 Combinations of multiple genes are useful and necessary for some areas 2.87 +In Figure 5, we give an example of a cortical area which is not marked by any single gene, but which can be identified 2.88 +combinatorially. 2.89 +Feature selection integrated with prediction As noted earlier, in general, any predictive method can be used for 2.90 +feature selection by running it inside a stepwise wrapper. Also, some predictive methods integrate soft constraints on number 2.91 +of features used. Examples of both of these will be seen in the section “Locating areas with gene expression”. 2.92 +Locating areas with gene expression 2.93 +Forward stepwise logistic regression As a pilot run, for five cortical areas (SS, AUD, RSP, VIS, and MO), we performed 2.94 +forward stepwise logistic regression to find single genes, pairs of genes, and triplets of genes which predict areal identify. 2.95 +This is an example of feature selection integrated with prediction using a stepwise wrapper. Some of the single genes found 2.96 +were shown in various figures throughout this document, and Figure 5 shows a combination of genes which was found. 2.97 +We felt that, for single genes, gradient similarity did a better job than logistic regression at capturing our subjective 2.98 +impression of a “good gene”. 2.99 +SVM on all genes at once 2.100 +In order to see how well one can do when looking at all genes at once, we ran a support vector machine to classify cortical 2.101 _________________________________________ 2.102 17For each gene, a logistic regression in which the response variable was whether or not a surface pixel was within area AUD, and the predictor 2.103 variable was the value of the expression of the gene underneath that pixel. The resulting scores were used to rank the genes in terms of how well 2.104 @@ -416,68 +448,42 @@ 2.105 18For each gene the gradient similarity between (a) a map of the expression of each gene on the cortical surface and (b) the shape of area AUD, 2.106 was calculated, and this was used to rank the genes. 2.107 2.108 - 2.109 - 2.110 -Figure 2: The top row shows the three genes which (individually) best predict area AUD, according to logistic regression. 2.111 -The bottom row shows the three genes which (individually) best match area AUD, according to gradient similarity. From 2.112 -left to right and top to bottom, the genes are Ssr1, Efcbp1, Aph1a, Ptk7, Aph1a again, and Lepr 2.113 -In Figure 3, we give an example of a cortical area which is not marked by any single gene, but which can be identified 2.114 -combinatorially. 2.115 -Underexpression of a gene can serve as a marker Underexpression of a gene can sometimes serve as a marker. 2.116 -See, for example, Figure 4. 2.117 -Feature selection integrated with prediction As noted earlier, in general, any predictive method can be used for 2.118 -feature selection by running it inside a stepwise wrapper. Also, some predictive methods integrate soft constraints on number 2.119 -of features used. Examples of both of these will be seen in the section “Locating areas with gene expression”. 2.120 -Locating areas with gene expression 2.121 -Forward stepwise logistic regression As a pilot run, for five cortical areas (SS, AUD, RSP, VIS, and MO), we performed 2.122 -forward stepwise logistic regression to find single genes, pairs of genes, and triplets of genes which predict areal identify. 2.123 -Some of the single genes found were shown in previous figures, and Figure 3 shows a combination of genes which was found. 2.124 -SVM on all genes at once 2.125 -In order to see how well one can do when looking at all genes at once, we ran a support vector machine to classify cortical 2.126 -surface pixels based on their gene expression profiles. We achieved classification accuracy of about 81%19. As noted above, 2.127 -however, a classifier that looks at all the genes at once isn’t as practically useful as a classifier that uses only a few genes. 2.128 -Decision trees 2.129 -todo 2.130 -Areas which can be identified by single genes 2.131 -Using all of the methods we have tried to far, we have already found single genes which roughly identify some areas and 2.132 -groupings of areas. For each of these areas, an example of a gene which roughly identifies it is shown in Figure 5. We have 2.133 -not yet cross-verified these genes in other atlases. 2.134 -In addition, there are a number of areas which are almost identified by single genes: COAa+NLOT (anterior part of 2.135 -cortical amygdalar area, nucleus of the lateral olfactory tract), ENT (entorhinal), ACAv (ventral anterior cingulate), VIS 2.136 -(visual), AUD (auditory). 2.137 -Data-driven redrawing of the cortical map 2.138 -Raw dimensionality reduction results 2.139 -todo 2.140 -(might want to incld nnMF since mentioned above) 2.141 -Dimensionality reduction plus K-means or spectral clustering 2.142 -_________________________________________ 2.143 - 195-fold cross-validation. 2.144 - 2.145 - 2.146 - 2.147 -Figure 3: Upper left: wwc1. Upper right: mtif2. Lower left: wwc1 + mtif2 (each pixel’s value on the lower left is the 2.148 -sum of the corresponding pixels in the upper row). Acccording to logistic regression, gene wwc1 is the best fit single gene 2.149 -for predicting whether or not a pixel on the cortical surface belongs to the motor area (area MO). The upper-left picture in 2.150 -Figure 3 shows wwc1’s spatial expression pattern over the cortex. The lower-right boundary of MO is represented reasonably 2.151 -well by this gene, however the gene overshoots the upper-left boundary. This flattened 2-D representation does not show 2.152 -it, but the area corresponding to the overshoot is the medial surface of the cortex. MO is only found on the lateral surface. 2.153 -Gene mtif2 is shown in the upper-right. Mtif2 captures MO’s upper-left boundary, but not its lower-right boundary. Mtif2 2.154 -does not express very much on the medial surface. By adding together the values at each pixel in these two figures, we get 2.155 -the lower-left image. This combination captures area MO much better than any single gene. 2.156 - 2.157 - Figure 4: Gene Pitx2 is selectively underexpressed in area SS (somatosensory). 2.158 2.159 2.160 -Figure 5: From left to right and top to bottom, single genes which roughly identify areas SS (somatosensory primary + 2.161 +Figure 4: From left to right and top to bottom, single genes which roughly identify areas SS (somatosensory primary + 2.162 supplemental), SSs (supplemental somatosensory), PIR (piriform), FRP (frontal pole), RSP (retrosplenial), COApm (Corti- 2.163 cal amygdalar, posterior part, medial zone). Grouping some areas together, we have also found genes to identify the groups 2.164 ACA+PL+ILA+DP+ORB+MO (anterior cingulate, prelimbic, infralimbic, dorsal peduncular, orbital, motor), posterior 2.165 and lateral visual (VISpm, VISpl, VISI, VISp; posteromedial, posterolateral, lateral, and primary visual; the posterior and 2.166 lateral visual area is distinguished from its neighbors, but not from the entire rest of the cortex). The genes are Pitx2, 2.167 Aldh1a2, Ppfibp1, Slco1a5, Tshz2, Trhr, Col12a1, Ets1. 2.168 + 2.169 + 2.170 +Figure 5: Upper left: wwc1. Upper right: mtif2. Lower left: wwc1 + mtif2 (each pixel’s value on the lower left is the 2.171 +sum of the corresponding pixels in the upper row). Acccording to logistic regression, gene wwc1 is the best fit single gene 2.172 +for predicting whether or not a pixel on the cortical surface belongs to the motor area (area MO). The upper-left picture in 2.173 +Figure 5 shows wwc1’s spatial expression pattern over the cortex. The lower-right boundary of MO is represented reasonably 2.174 +well by this gene, however the gene overshoots the upper-left boundary. This flattened 2-D representation does not show 2.175 +it, but the area corresponding to the overshoot is the medial surface of the cortex. MO is only found on the lateral surface. 2.176 +Gene mtif2 is shown in the upper-right. Mtif2 captures MO’s upper-left boundary, but not its lower-right boundary. Mtif2 2.177 +does not express very much on the medial surface. By adding together the values at each pixel in these two figures, we get 2.178 +the lower-left image. This combination captures area MO much better than any single gene. 2.179 +surface pixels based on their gene expression profiles. We achieved classification accuracy of about 81%19. As noted above, 2.180 +however, a classifier that looks at all the genes at once isn’t as practically useful as a classifier that uses only a few genes. 2.181 +Data-driven redrawing of the cortical map 2.182 +Raw dimensionality reduction We have applied the following dimensionality reduction algorithms to reduce the di- 2.183 +mensionality of the gene expression profile associated with each voxel: Principal Components Analysis (PCA), Simple 2.184 +PCA (SPCA), Multi-Dimensional Scaling (MDS), Isomap, Landmark Isomap, Laplacian eigenmaps, Local Tangent Space 2.185 +Alignment (LTSA), Hessian locally linear embedding, Diffusion maps, Stochastic Neighbor Embedding (SNE), Stochastic 2.186 +Proximity Embedding (SPE), Fast Maximum Variance Unfolding (FastMVU), Non-negative Matrix Factorization (NNMF). 2.187 +todo 2.188 +(might want to incld nnMF since mentioned above) 2.189 +Dimensionality reduction plus K-means or spectral clustering 2.190 Many areas are captured by clusters of genes 2.191 todo 2.192 todo 2.193 +_________________________________________ 2.194 + 195-fold cross-validation. 2.195 Research plan 2.196 Further work on flatmapping 2.197 In anatomy, the manifold of interest is usually either defined by a combination of two relevant anatomical axes (todo), 2.198 @@ -507,6 +513,11 @@ 2.199 boundary were redrawn slightly, and (b) detect when a difficult area could be combined with adjacent areas to create 2.200 a larger area which can be fit. 2.201 # Linear discriminant analysis 2.202 +Decision trees todo 2.203 +For each cortical area, we used the C4.5 algorithm to find a pruned decision tree and ruleset for that area. We achieved 2.204 +estimated classification accuracy of more than 99.6% on each cortical area (as evaluated on the training data without 2.205 +cross-validation; so actual accuracy is expected to be lower). However, the resulting decision trees each made use of many 2.206 +genes. 2.207 Apply these algorithms to the cortex 2.208 1.Create open source format conversion tools: we will create tools to bulk download the ABA dataset and to convert 2.209 between SEV, NIFTI and MATLAB formats. 2.210 @@ -527,6 +538,7 @@ 2.211 # self-organizing map 2.212 # confirm with EMAGE, GeneAtlas, GENSAT, etc, to fight overfitting 2.213 # compare using clustering scores 2.214 +# multivariate gradient similarity 2.215 Bibliography & References Cited 2.216 [1]Tanya Barrett, Dennis B. Troup, Stephen E. Wilhite, Pierre Ledoux, Dmitry Rudnev, Carlos Evangelista, Irene F. 2.217 Kim, Alexandra Soboleva, Maxim Tomashevsky, and Ron Edgar. NCBI GEO: mining tens of millions of expression

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5.1 --- a/grant.txt Sun Apr 19 15:23:53 2009 -0700 5.2 +++ b/grant.txt Sun Apr 19 16:20:50 2009 -0700 5.3 @@ -260,6 +260,15 @@ 5.4 5.5 === Feature selection and scoring methods === 5.6 5.7 +\vspace{0.3cm}**Underexpression of a gene can serve as a marker** 5.8 +Underexpression of a gene can sometimes serve as a marker. See, for example, Figure \ref{hole}. 5.9 + 5.10 + 5.11 +\begin{figure}\centering 5.12 +\includegraphics[scale=.31]{holeExample_2682_SS_jet.eps} 5.13 +\caption{Gene Pitx2 is selectively underexpressed in area SS (somatosensory).} 5.14 +\label{hole}\end{figure} 5.15 + 5.16 5.17 \vspace{0.3cm}**Correlation** 5.18 Recall that the instances are surface pixels, and consider the problem of attempting to classify each instance as either a member of a particular anatomical area, or not. The target area can be represented as a boolean mask over the surface pixels. 5.19 @@ -294,7 +303,7 @@ 5.20 5.21 5.22 \vspace{0.3cm}**Gradient similarity** 5.23 -We noticed that the previous two scoring methods, which are pointwise, often found genes whose pattern of expression did not look similar in shape to the target region. Fort his reason we designed a non-pointwise local scoring method to detect when a gene had a pattern of expression which looked like it had a boundary whose shape is similar to the shape of the target region. We call this scoring method "gradient similarity". 5.24 +We noticed that the previous two scoring methods, which are pointwise, often found genes whose pattern of expression did not look similar in shape to the target region. For this reason we designed a non-pointwise local scoring method to detect when a gene had a pattern of expression which looked like it had a boundary whose shape is similar to the shape of the target region. We call this scoring method "gradient similarity". 5.25 5.26 One might say that gradient similarity attempts to measure how much the border of the area of gene expression and the border of the target region overlap. However, since gene expression falls off continuously rather than jumping from its maximum value to zero, the spatial pattern of a gene's expression often does not have a discrete border. Therefore, instead of looking for a discrete border, we look for large gradients. Gradient similarity is a symmetric function over two images (i.e. two scalar fields). It is is high to the extent that matching pixels which have large values and large gradients also have gradients which are oriented in a similar direction. The formula is: 5.27 5.28 @@ -306,6 +315,8 @@ 5.29 5.30 The intuition is that we want to see if the borders of the pattern in the two images are similar; if the borders are similar, then both images will have corresponding pixels with large gradients (because this is a border) which are oriented in a similar direction (because the borders are similar). 5.31 5.32 +Most of the genes in Figure \ref{singleSoFar} were identified via gradient similarity. 5.33 + 5.34 \vspace{0.3cm}**Gradient similarity provides information complementary to correlation** 5.35 5.36 To show that gradient similarity can provide useful information that cannot be detected via pointwise analyses, consider Fig. \ref{AUDgeometry}. The top row of Fig. \ref{AUDgeometry} displays the 3 genes which most match area AUD, according to a pointwise method\footnote{For each gene, a logistic regression in which the response variable was whether or not a surface pixel was within area AUD, and the predictor variable was the value of the expression of the gene underneath that pixel. The resulting scores were used to rank the genes in terms of how well they predict area AUD.}. The bottom row displays the 3 genes which most match AUD according to a method which considers local geometry\footnote{For each gene the gradient similarity between (a) a map of the expression of each gene on the cortical surface and (b) the shape of area AUD, was calculated, and this was used to rank the genes.} The pointwise method in the top row identifies genes which express more strongly in AUD than outside of it; its weakness is that this includes many areas which don't have a salient border matching the areal border. The geometric method identifies genes whose salient expression border seems to partially line up with the border of AUD; its weakness is that this includes genes which don't express over the entire area. Genes which have high rankings using both pointwise and border criteria, such as $Aph1a$ in the example, may be particularly good markers. None of these genes are, individually, a perfect marker for AUD; we deliberately chose a "difficult" area in order to better contrast pointwise with geometric methods. 5.37 @@ -323,61 +334,8 @@ 5.38 \label{AUDgeometry}\end{figure} 5.39 5.40 5.41 -\vspace{0.3cm}**Combinations of multiple genes are useful and necessary for some areas** 5.42 - 5.43 -In Figure \ref{MOcombo}, we give an example of a cortical area which is not marked by any single gene, but which can be identified combinatorially. 5.44 - 5.45 -%% wwc1\footnote{"WW, C2 and coiled-coil domain containing 1"; EntrezGene ID 211652} 5.46 -%% mtif2\footnote{"mitochondrial translational initiation factor 2"; EntrezGene ID 76784} 5.47 - 5.48 -%%Acccording to logistic regression, gene wwc1\footnote{"WW, C2 and coiled-coil domain containing 1"; EntrezGene ID 211652} is the best fit single gene for predicting whether or not a pixel on the cortical surface belongs to the motor area (area MO). The upper-left picture in Figure \ref{MOcombo} shows wwc1's spatial expression pattern over the cortex. The lower-right boundary of MO is represented reasonably well by this gene, however the gene overshoots the upper-left boundary. This flattened 2-D representation does not show it, but the area corresponding to the overshoot is the medial surface of the cortex. MO is only found on the lateral surface. 5.49 - 5.50 -%%Gene mtif2\footnote{"mitochondrial translational initiation factor 2"; EntrezGene ID 76784} is shown in figure the upper-right of Fig. \ref{MOcombo}. Mtif2 captures MO's upper-left boundary, but not its lower-right boundary. Mtif2 does not express very much on the medial surface. By adding together the values at each pixel in these two figures, we get the lower-left of Figure \ref{MOcombo}. This combination captures area MO much better than any single gene. 5.51 - 5.52 -\begin{figure}\centering 5.53 -\includegraphics[scale=.36]{MO_vs_Wwc1_jet.eps} 5.54 -\includegraphics[scale=.36]{MO_vs_Mtif2_jet.eps} 5.55 - 5.56 -\includegraphics[scale=.36]{MO_vs_Wwc1_plus_Mtif2_jet.eps} 5.57 -\caption{Upper left: $wwc1$. Upper right: $mtif2$. Lower left: wwc1 + mtif2 (each pixel's value on the lower left is the sum of the corresponding pixels in the upper row). Acccording to logistic regression, gene wwc1 is the best fit single gene for predicting whether or not a pixel on the cortical surface belongs to the motor area (area MO). The upper-left picture in Figure \ref{MOcombo} shows wwc1's spatial expression pattern over the cortex. The lower-right boundary of MO is represented reasonably well by this gene, however the gene overshoots the upper-left boundary. This flattened 2-D representation does not show it, but the area corresponding to the overshoot is the medial surface of the cortex. MO is only found on the lateral surface. Gene mtif2 is shown in the upper-right. Mtif2 captures MO's upper-left boundary, but not its lower-right boundary. Mtif2 does not express very much on the medial surface. By adding together the values at each pixel in these two figures, we get the lower-left image. This combination captures area MO much better than any single gene. } 5.58 -\label{MOcombo}\end{figure} 5.59 - 5.60 - 5.61 - 5.62 - 5.63 - 5.64 - 5.65 -\vspace{0.3cm}**Underexpression of a gene can serve as a marker** 5.66 -Underexpression of a gene can sometimes serve as a marker. See, for example, Figure \ref{hole}. 5.67 - 5.68 - 5.69 -\begin{figure}\centering 5.70 -\includegraphics[scale=.31]{holeExample_2682_SS_jet.eps} 5.71 -\caption{Gene Pitx2 is selectively underexpressed in area SS (somatosensory).} 5.72 -\label{hole}\end{figure} 5.73 - 5.74 - 5.75 -\vspace{0.3cm}**Feature selection integrated with prediction** 5.76 -As noted earlier, in general, any predictive method can be used for feature selection by running it inside a stepwise wrapper. Also, some predictive methods integrate soft constraints on number of features used. Examples of both of these will be seen in the section "Locating areas with gene expression". 5.77 - 5.78 - 5.79 -=== Locating areas with gene expression === 5.80 -\vspace{0.3cm}**Forward stepwise logistic regression** 5.81 -As a pilot run, for five cortical areas (SS, AUD, RSP, VIS, and MO), we performed forward stepwise logistic regression to find single genes, pairs of genes, and triplets of genes which predict areal identify. This is an example of feature selection integrated with prediction using a stepwise wrapper. Some of the single genes found were shown in previous figures, and Figure \ref{MOcombo} shows a combination of genes which was found. 5.82 - 5.83 - 5.84 -\vspace{0.3cm}**SVM on all genes at once** 5.85 - 5.86 -In order to see how well one can do when looking at all genes at once, we ran a support vector machine to classify cortical surface pixels based on their gene expression profiles. We achieved classification accuracy of about 81%\footnote{5-fold cross-validation.}. As noted above, however, a classifier that looks at all the genes at once isn't as practically useful as a classifier that uses only a few genes. 5.87 - 5.88 - 5.89 -\vspace{0.3cm}**Decision trees** 5.90 - 5.91 -todo 5.92 - 5.93 \vspace{0.3cm}**Areas which can be identified by single genes** 5.94 - 5.95 -Using all of the methods we have tried to far, we have already found single genes which roughly identify some areas and groupings of areas. For each of these areas, an example of a gene which roughly identifies it is shown in Figure \ref{singleSoFar}. We have not yet cross-verified these genes in other atlases. 5.96 +Using gradient similarity, we have already found single genes which roughly identify some areas and groupings of areas. For each of these areas, an example of a gene which roughly identifies it is shown in Figure \ref{singleSoFar}. We have not yet cross-verified these genes in other atlases. 5.97 5.98 In addition, there are a number of areas which are almost identified by single genes: COAa+NLOT (anterior part of cortical amygdalar area, nucleus of the lateral olfactory tract), ENT (entorhinal), ACAv (ventral anterior cingulate), VIS (visual), AUD (auditory). 5.99 5.100 @@ -397,9 +355,56 @@ 5.101 5.102 5.103 5.104 +\vspace{0.3cm}**Combinations of multiple genes are useful and necessary for some areas** 5.105 + 5.106 +In Figure \ref{MOcombo}, we give an example of a cortical area which is not marked by any single gene, but which can be identified combinatorially. 5.107 + 5.108 +%% wwc1\footnote{"WW, C2 and coiled-coil domain containing 1"; EntrezGene ID 211652} 5.109 +%% mtif2\footnote{"mitochondrial translational initiation factor 2"; EntrezGene ID 76784} 5.110 + 5.111 +%%Acccording to logistic regression, gene wwc1\footnote{"WW, C2 and coiled-coil domain containing 1"; EntrezGene ID 211652} is the best fit single gene for predicting whether or not a pixel on the cortical surface belongs to the motor area (area MO). The upper-left picture in Figure \ref{MOcombo} shows wwc1's spatial expression pattern over the cortex. The lower-right boundary of MO is represented reasonably well by this gene, however the gene overshoots the upper-left boundary. This flattened 2-D representation does not show it, but the area corresponding to the overshoot is the medial surface of the cortex. MO is only found on the lateral surface. 5.112 + 5.113 +%%Gene mtif2\footnote{"mitochondrial translational initiation factor 2"; EntrezGene ID 76784} is shown in figure the upper-right of Fig. \ref{MOcombo}. Mtif2 captures MO's upper-left boundary, but not its lower-right boundary. Mtif2 does not express very much on the medial surface. By adding together the values at each pixel in these two figures, we get the lower-left of Figure \ref{MOcombo}. This combination captures area MO much better than any single gene. 5.114 + 5.115 +\begin{figure}\centering 5.116 +\includegraphics[scale=.36]{MO_vs_Wwc1_jet.eps} 5.117 +\includegraphics[scale=.36]{MO_vs_Mtif2_jet.eps} 5.118 + 5.119 +\includegraphics[scale=.36]{MO_vs_Wwc1_plus_Mtif2_jet.eps} 5.120 +\caption{Upper left: $wwc1$. Upper right: $mtif2$. Lower left: wwc1 + mtif2 (each pixel's value on the lower left is the sum of the corresponding pixels in the upper row). Acccording to logistic regression, gene wwc1 is the best fit single gene for predicting whether or not a pixel on the cortical surface belongs to the motor area (area MO). The upper-left picture in Figure \ref{MOcombo} shows wwc1's spatial expression pattern over the cortex. The lower-right boundary of MO is represented reasonably well by this gene, however the gene overshoots the upper-left boundary. This flattened 2-D representation does not show it, but the area corresponding to the overshoot is the medial surface of the cortex. MO is only found on the lateral surface. Gene mtif2 is shown in the upper-right. Mtif2 captures MO's upper-left boundary, but not its lower-right boundary. Mtif2 does not express very much on the medial surface. By adding together the values at each pixel in these two figures, we get the lower-left image. This combination captures area MO much better than any single gene. } 5.121 +\label{MOcombo}\end{figure} 5.122 + 5.123 + 5.124 + 5.125 + 5.126 + 5.127 + 5.128 + 5.129 + 5.130 +\vspace{0.3cm}**Feature selection integrated with prediction** 5.131 +As noted earlier, in general, any predictive method can be used for feature selection by running it inside a stepwise wrapper. Also, some predictive methods integrate soft constraints on number of features used. Examples of both of these will be seen in the section "Multivariate Predictive methods". 5.132 + 5.133 + 5.134 +=== Multivariate Predictive methods === 5.135 +\vspace{0.3cm}**Forward stepwise logistic regression** 5.136 +As a pilot run, for five cortical areas (SS, AUD, RSP, VIS, and MO), we performed forward stepwise logistic regression to find single genes, pairs of genes, and triplets of genes which predict areal identify. This is an example of feature selection integrated with prediction using a stepwise wrapper. Some of the single genes found were shown in various figures throughout this document, and Figure \ref{MOcombo} shows a combination of genes which was found. 5.137 + 5.138 +We felt that, for single genes, gradient similarity did a better job than logistic regression at capturing our subjective impression of a "good gene". 5.139 + 5.140 + 5.141 +\vspace{0.3cm}**SVM on all genes at once** 5.142 + 5.143 +In order to see how well one can do when looking at all genes at once, we ran a support vector machine to classify cortical surface pixels based on their gene expression profiles. We achieved classification accuracy of about 81%\footnote{5-fold cross-validation.}. As noted above, however, a classifier that looks at all the genes at once isn't as practically useful as a classifier that uses only a few genes. 5.144 + 5.145 + 5.146 + 5.147 + 5.148 + 5.149 === Data-driven redrawing of the cortical map === 5.150 5.151 -\vspace{0.3cm}**Raw dimensionality reduction results** 5.152 +\vspace{0.3cm}**Raw dimensionality reduction** 5.153 +We have applied the following dimensionality reduction algorithms to reduce the dimensionality of the gene expression profile associated with each voxel: Principal Components Analysis (PCA), Simple PCA (SPCA), Multi-Dimensional Scaling (MDS), Isomap, Landmark Isomap, Laplacian eigenmaps, Local Tangent Space Alignment (LTSA), Hessian locally linear embedding, Diffusion maps, Stochastic Neighbor Embedding (SNE), Stochastic Proximity Embedding (SPE), Fast Maximum Variance Unfolding (FastMVU), Non-negative Matrix Factorization (NNMF). 5.154 + 5.155 5.156 todo 5.157 5.158 @@ -459,6 +464,12 @@ 5.159 # Linear discriminant analysis 5.160 5.161 5.162 +\vspace{0.3cm}**Decision trees** 5.163 +todo 5.164 + 5.165 +For each cortical area, we used the C4.5 algorithm to find a pruned decision tree and ruleset for that area. We achieved estimated classification accuracy of more than 99.6% on each cortical area (as evaluated on the __training data__ without cross-validation; so actual accuracy is expected to be lower). However, the resulting decision trees each made use of many genes. 5.166 + 5.167 + 5.168 \vspace{0.3cm}**Apply these algorithms to the cortex** 5.169 5.170 # Create open source format conversion tools: we will create tools to bulk download the ABA dataset and to convert between SEV, NIFTI and MATLAB formats. 5.171 @@ -489,6 +500,8 @@ 5.172 5.173 # compare using clustering scores 5.174 5.175 +# multivariate gradient similarity 5.176 + 5.177 5.178 \newpage 5.179