1.1 --- a/grant.txt Mon Apr 20 15:08:40 2009 -0700 1.2 +++ b/grant.txt Mon Apr 20 16:23:22 2009 -0700 1.3 @@ -1,6 +1,9 @@ 1.4 \documentclass{nih-blank} 1.5 %%\piname{Stevens, Charles F.} 1.6 1.7 +%%\usepackage{floatflt} 1.8 +\usepackage{wrapfig} 1.9 + 1.10 == Specific aims == 1.11 1.12 Massive new datasets obtained with techniques such as in situ hybridization (ISH), immunohistochemistry, in situ transgenic reporter, microarray voxelation, and others, allow the expression levels of many genes at many locations to be compared. Our goal is to develop automated methods to relate spatial variation in gene expression to anatomy. We want to find marker genes for specific anatomical regions, and also to draw new anatomical maps based on gene expression patterns. We have three specific aims:\\ 1.13 @@ -231,6 +234,13 @@ 1.14 1.15 1.16 === Flatmap of cortex === 1.17 +\begin{wrapfigure}{L}{0.2\textwidth}\centering 1.18 +\includegraphics[scale=.31]{holeExample_2682_SS_jet.eps} 1.19 +\caption{Gene Pitx2 is selectively underexpressed in area SS (somatosensory).} 1.20 +\label{hole}\end{wrapfigure} 1.21 + 1.22 + 1.23 + 1.24 We downloaded the ABA data and applied a mask to select only those voxels which belong to cerebral cortex. We divided the cortex into hemispheres. 1.25 1.26 Using Caret\cite{van_essen_integrated_2001}, we created a mesh representation of the surface of the selected voxels. For each gene, for each node of the mesh, we calculated an average of the gene expression of the voxels "underneath" that mesh node. We then flattened the cortex, creating a two-dimensional mesh. 1.27 @@ -244,6 +254,21 @@ 1.28 * A 2-D matrix whose entries represent the regional label associated with each surface pixel 1.29 * For each gene, a 2-D matrix whose entries represent the average expression level underneath each surface pixel 1.30 1.31 +\begin{wrapfigure}{L}{0.4\textwidth}\centering 1.32 +%%\includegraphics[scale=.31]{singlegene_SS_corr_top_1_2365_jet.eps}\includegraphics[scale=.31]{singlegene_SS_corr_top_2_242_jet.eps}\includegraphics[scale=.31]{singlegene_SS_corr_top_3_654_jet.eps} 1.33 +%%\\ 1.34 +%%\includegraphics[scale=.31]{singlegene_SS_lr_top_1_654_jet.eps}\includegraphics[scale=.31]{singlegene_SS_lr_top_2_685_jet.eps}\includegraphics[scale=.31]{singlegene_SS_lr_top_3_724_jet.eps} 1.35 +%%\caption{Top row: Genes Nfic, A930001M12Rik, C130038G02Rik are the most correlated with area SS (somatosensory cortex). Bottom row: Genes C130038G02Rik, Cacna1i, Car10 are those with the best fit using logistic regression. Within each picture, the vertical axis roughly corresponds to anterior at the top and posterior at the bottom, and the horizontal axis roughly corresponds to medial at the left and lateral at the right. The red outline is the boundary of region MO. Pixels are colored according to correlation, with red meaning high correlation and blue meaning low.} 1.36 + 1.37 +\includegraphics[scale=.31]{singlegene_SS_corr_top_1_2365_jet.eps}\includegraphics[scale=.31]{singlegene_SS_corr_top_2_242_jet.eps} 1.38 +\\ 1.39 +\includegraphics[scale=.31]{singlegene_SS_lr_top_1_654_jet.eps}\includegraphics[scale=.31]{singlegene_SS_lr_top_2_685_jet.eps} 1.40 + 1.41 +\caption{Top row: Genes Nfic and A930001M12Rik are the most correlated with area SS (somatosensory cortex). Bottom row: Genes C130038G02Rik and Cacna1i are those with the best fit using logistic regression. Within each picture, the vertical axis roughly corresponds to anterior at the top and posterior at the bottom, and the horizontal axis roughly corresponds to medial at the left and lateral at the right. The red outline is the boundary of region MO. Pixels are colored according to correlation, with red meaning high correlation and blue meaning low.} 1.42 +\label{SScorrLr}\end{wrapfigure} 1.43 + 1.44 + 1.45 + 1.46 We created a normalized version of the gene expression data by subtracting each gene's mean expression level (over all surface pixels) and dividing each gene by its standard deviation. 1.47 1.48 The features and the target area are both functions on the surface pixels. They can be referred to as scalar fields over the space of surface pixels; alternately, they can be thought of as images which can be displayed on the flatmapped surface. 1.49 @@ -260,15 +285,12 @@ 1.50 1.51 === Feature selection and scoring methods === 1.52 1.53 + 1.54 + 1.55 \vspace{0.3cm}**Underexpression of a gene can serve as a marker** 1.56 Underexpression of a gene can sometimes serve as a marker. See, for example, Figure \ref{hole}. 1.57 1.58 1.59 -\begin{figure}\centering 1.60 -\includegraphics[scale=.31]{holeExample_2682_SS_jet.eps} 1.61 -\caption{Gene Pitx2 is selectively underexpressed in area SS (somatosensory).} 1.62 -\label{hole}\end{figure} 1.63 - 1.64 1.65 \vspace{0.3cm}**Correlation** 1.66 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. 1.67 @@ -277,20 +299,17 @@ 1.68 1.69 One of the simplest methods in this class is to use correlation as the match score. We calculated the correlation between each gene and each cortical area. The top row of Figure \ref{SScorrLr} shows the three genes most correlated with area SS. 1.70 1.71 -\begin{figure}\centering 1.72 -\includegraphics[scale=.31]{singlegene_SS_corr_top_1_2365_jet.eps} 1.73 -\includegraphics[scale=.31]{singlegene_SS_corr_top_2_242_jet.eps} 1.74 -\includegraphics[scale=.31]{singlegene_SS_corr_top_3_654_jet.eps} 1.75 + 1.76 +\begin{wrapfigure}{L}{0.4\textwidth}\centering 1.77 +%%\includegraphics[scale=.31]{singlegene_AUD_lr_top_1_3386_jet.eps}\includegraphics[scale=.31]{singlegene_AUD_lr_top_2_1258_jet.eps}\includegraphics[scale=.31]{singlegene_AUD_lr_top_3_420_jet.eps} 1.78 +%% 1.79 +%%\includegraphics[scale=.31]{singlegene_AUD_gr_top_1_2856_jet.eps}\includegraphics[scale=.31]{singlegene_AUD_gr_top_2_420_jet.eps}\includegraphics[scale=.31]{singlegene_AUD_gr_top_3_2072_jet.eps} 1.80 +%%\caption{The top row shows the three genes which (individually) best predict area AUD, according to logistic regression. The bottom row shows the three genes which (individually) best match area AUD, according to gradient similarity. From left to right and top to bottom, the genes are $Ssr1$, $Efcbp1$, $Aph1a$, $Ptk7$, $Aph1a$ again, and $Lepr$} 1.81 +\includegraphics[scale=.31]{singlegene_AUD_lr_top_1_3386_jet.eps}\includegraphics[scale=.31]{singlegene_AUD_lr_top_2_1258_jet.eps} 1.82 \\ 1.83 -\includegraphics[scale=.31]{singlegene_SS_lr_top_1_654_jet.eps} 1.84 -\includegraphics[scale=.31]{singlegene_SS_lr_top_2_685_jet.eps} 1.85 -\includegraphics[scale=.31]{singlegene_SS_lr_top_3_724_jet.eps} 1.86 - 1.87 - 1.88 -\caption{Top row: Genes Nfic, A930001M12Rik, C130038G02Rik are the most correlated with area SS (somatosensory cortex). Bottom row: Genes C130038G02Rik, Cacna1i, Car10 are those with the best fit using logistic regression. Within each picture, the vertical axis roughly corresponds to anterior at the top and posterior at the bottom, and the horizontal axis roughly corresponds to medial at the left and lateral at the right. The red outline is the boundary of region MO. Pixels are colored according to correlation, with red meaning high correlation and blue meaning low.} 1.89 -\label{SScorrLr}\end{figure} 1.90 - 1.91 - 1.92 +\includegraphics[scale=.31]{singlegene_AUD_gr_top_1_2856_jet.eps}\includegraphics[scale=.31]{singlegene_AUD_gr_top_2_420_jet.eps} 1.93 +\caption{The top row shows the two genes which (individually) best predict area AUD, according to logistic regression. The bottom row shows the two genes which (individually) best match area AUD, according to gradient similarity. From left to right and top to bottom, the genes are $Ssr1$, $Efcbp1$, $Ptk7$, and $Aph1a$.} 1.94 +\label{AUDgeometry}\end{wrapfigure} 1.95 1.96 \vspace{0.3cm}**Conditional entropy** 1.97 An information-theoretic scoring method is to find features such that, if the features (gene expression levels) are known, uncertainty about the target (the regional identity) is reduced. Entropy measures uncertainty, so what we want is to find features such that the conditional distribution of the target has minimal entropy. The distribution to which we are referring is the probability distribution over the population of surface pixels. 1.98 @@ -302,6 +321,14 @@ 1.99 This finds pairs of genes which are most informative (at least at these discretization thresholds) relative to the question, "Is this surface pixel a member of the target area?". Its advantage over linear methods such as logistic regression is that it takes account of arbitrarily nonlinear relationships; for example, if the XOR of two variables predicts the target, conditional entropy would notice, whereas linear methods would not. 1.100 1.101 1.102 +\begin{wrapfigure}{L}{0.4\textwidth}\centering 1.103 +\includegraphics[scale=.31]{MO_vs_Wwc1_jet.eps}\includegraphics[scale=.31]{MO_vs_Mtif2_jet.eps} 1.104 + 1.105 +\includegraphics[scale=.31]{MO_vs_Wwc1_plus_Mtif2_jet.eps} 1.106 +\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).} 1.107 +\label{MOcombo}\end{wrapfigure} 1.108 + 1.109 + 1.110 \vspace{0.3cm}**Gradient similarity** 1.111 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". 1.112 1.113 @@ -322,17 +349,15 @@ 1.114 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. 1.115 1.116 1.117 -\begin{figure}\centering 1.118 -\includegraphics[scale=.31]{singlegene_AUD_lr_top_1_3386_jet.eps} 1.119 -\includegraphics[scale=.31]{singlegene_AUD_lr_top_2_1258_jet.eps} 1.120 -\includegraphics[scale=.31]{singlegene_AUD_lr_top_3_420_jet.eps} 1.121 - 1.122 -\includegraphics[scale=.31]{singlegene_AUD_gr_top_1_2856_jet.eps} 1.123 -\includegraphics[scale=.31]{singlegene_AUD_gr_top_2_420_jet.eps} 1.124 -\includegraphics[scale=.31]{singlegene_AUD_gr_top_3_2072_jet.eps} 1.125 -\caption{The top row shows the three genes which (individually) best predict area AUD, according to logistic regression. The bottom row shows the three genes which (individually) best match area AUD, according to gradient similarity. From left to right and top to bottom, the genes are $Ssr1$, $Efcbp1$, $Aph1a$, $Ptk7$, $Aph1a$ again, and $Lepr$} 1.126 -\label{AUDgeometry}\end{figure} 1.127 - 1.128 + 1.129 +\begin{wrapfigure}{L}{0.4\textwidth}\centering 1.130 +\includegraphics[scale=.31]{singlegene_example_2682_Pitx2_SS_jet.eps}\includegraphics[scale=.31]{singlegene_example_371_Aldh1a2_SSs_jet.eps} 1.131 +\includegraphics[scale=.31]{singlegene_example_2759_Ppfibp1_PIR_jet.eps}\includegraphics[scale=.31]{singlegene_example_3310_Slco1a5_FRP_jet.eps} 1.132 +\includegraphics[scale=.31]{singlegene_example_3709_Tshz2_RSP_jet.eps}\includegraphics[scale=.31]{singlegene_example_3674_Trhr_COApm_jet.eps} 1.133 +\includegraphics[scale=.31]{singlegene_example_925_Col12a1_ACA+PL+ILA+DP+ORB+MO_jet.eps}\includegraphics[scale=.31]{singlegene_example_1334_Ets1_post_lat_vis_jet.eps} 1.134 + 1.135 +\caption{From left to right and top to bottom, single genes which roughly identify areas SS (somatosensory primary + supplemental), SSs (supplemental somatosensory), PIR (piriform), FRP (frontal pole), RSP (retrosplenial), COApm (Cortical amygdalar, posterior part, medial zone). Grouping some areas together, we have also found genes to identify the groups ACA+PL+ILA+DP+ORB+MO (anterior cingulate, prelimbic, infralimbic, dorsal peduncular, orbital, motor), posterior and lateral visual (VISpm, VISpl, VISI, VISp; posteromedial, posterolateral, lateral, and primary visual; the posterior and lateral visual area is distinguished from its neighbors, but not from the entire rest of the cortex). The genes are $Pitx2$, $Aldh1a2$, $Ppfibp1$, $Slco1a5$, $Tshz2$, $Trhr$, $Col12a1$, $Ets1$.} 1.136 +\label{singleSoFar}\end{wrapfigure} 1.137 1.138 \vspace{0.3cm}**Areas which can be identified by single genes** 1.139 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. 1.140 @@ -341,24 +366,13 @@ 1.141 1.142 These results validate our expectation that the ABA dataset can be exploited to find marker genes for many cortical areas, while also validating the relevancy of our new scoring method, gradient similarity. 1.143 1.144 -\begin{figure}\centering 1.145 -\includegraphics[scale=.31]{singlegene_example_2682_Pitx2_SS_jet.eps} 1.146 -\includegraphics[scale=.31]{singlegene_example_371_Aldh1a2_SSs_jet.eps} 1.147 -\includegraphics[scale=.31]{singlegene_example_2759_Ppfibp1_PIR_jet.eps} 1.148 -\includegraphics[scale=.31]{singlegene_example_3310_Slco1a5_FRP_jet.eps} 1.149 -\includegraphics[scale=.31]{singlegene_example_3709_Tshz2_RSP_jet.eps} 1.150 -\includegraphics[scale=.31]{singlegene_example_3674_Trhr_COApm_jet.eps} 1.151 -\includegraphics[scale=.31]{singlegene_example_925_Col12a1_ACA+PL+ILA+DP+ORB+MO_jet.eps} 1.152 -\includegraphics[scale=.31]{singlegene_example_1334_Ets1_post_lat_vis_jet.eps} 1.153 - 1.154 -\caption{From left to right and top to bottom, single genes which roughly identify areas SS (somatosensory primary + supplemental), SSs (supplemental somatosensory), PIR (piriform), FRP (frontal pole), RSP (retrosplenial), COApm (Cortical amygdalar, posterior part, medial zone). Grouping some areas together, we have also found genes to identify the groups ACA+PL+ILA+DP+ORB+MO (anterior cingulate, prelimbic, infralimbic, dorsal peduncular, orbital, motor), posterior and lateral visual (VISpm, VISpl, VISI, VISp; posteromedial, posterolateral, lateral, and primary visual; the posterior and lateral visual area is distinguished from its neighbors, but not from the entire rest of the cortex). The genes are $Pitx2$, $Aldh1a2$, $Ppfibp1$, $Slco1a5$, $Tshz2$, $Trhr$, $Col12a1$, $Ets1$.} 1.155 -\label{singleSoFar}\end{figure} 1.156 - 1.157 1.158 1.159 \vspace{0.3cm}**Combinations of multiple genes are useful and necessary for some areas** 1.160 1.161 -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. This shows that our proposal to develop a method to find combinations of marker genes is both possible and necessary. 1.162 +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. 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. 1.163 + 1.164 +This shows that our proposal to develop a method to find combinations of marker genes is both possible and necessary. 1.165 1.166 %% wwc1\footnote{"WW, C2 and coiled-coil domain containing 1"; EntrezGene ID 211652} 1.167 %% mtif2\footnote{"mitochondrial translational initiation factor 2"; EntrezGene ID 76784} 1.168 @@ -367,13 +381,6 @@ 1.169 1.170 %%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. 1.171 1.172 -\begin{figure}\centering 1.173 -\includegraphics[scale=.36]{MO_vs_Wwc1_jet.eps} 1.174 -\includegraphics[scale=.36]{MO_vs_Mtif2_jet.eps} 1.175 - 1.176 -\includegraphics[scale=.36]{MO_vs_Wwc1_plus_Mtif2_jet.eps} 1.177 -\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. } 1.178 -\label{MOcombo}\end{figure} 1.179 1.180 1.181 1.182 @@ -382,40 +389,37 @@ 1.183 1.184 1.185 === Multivariate Predictive methods === 1.186 + 1.187 \vspace{0.3cm}**Forward stepwise logistic regression** 1.188 Logistic regression is a popular method for predictive modeling of categorial data. 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. 1.189 1.190 We felt that, for single genes, gradient similarity did a better job than logistic regression at capturing our subjective impression of a "good gene". 1.191 1.192 - 1.193 -\vspace{0.3cm}**SVM on all genes at once** 1.194 - 1.195 -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.}. This shows that the genes included in the ABA dataset are sufficient to define much of cortical anatomy. 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. 1.196 - 1.197 - 1.198 - 1.199 - 1.200 - 1.201 -=== Data-driven redrawing of the cortical map === 1.202 - 1.203 -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). Space constraints prevent us from showing many of the results, but as a sample, PCA, NNMF, and landmark Isomap are shown in the second, third, and fourth rows of Figure \ref{dimReduc}. 1.204 - 1.205 -After applying the dimensionality reduction, we ran clustering algorithms on the reduced data. To date we have tried k-means and spectral clustering. The results of k-means after PCA, NNMF, and landmark Isomap are shown in the last row of Figure \ref{dimReduc}. To compare, the first row of Figure \ref{dimReduc} shows some of the major subdivisions of cortex. These results clearly show that different dimensionality reduction techniques capture different aspects of the data and lead to different clusterings, indicating the utility of our proposal to produce a detailed comparion of these techniques as applied to the domain of genomic anatomy. 1.206 - 1.207 -\begin{figure}\centering 1.208 -\includegraphics[scale=.31]{paint_merge3_major.eps} 1.209 -\\ 1.210 +\begin{wrapfigure}{L}{0.6\textwidth}\centering 1.211 \includegraphics[scale=1]{merge3_norm_hv_PCA_ndims50_prototypes_collage_sm_border.eps} 1.212 \includegraphics[scale=1]{nnmf_ndims7_collage_border.eps} 1.213 \includegraphics[scale=1]{merge3_norm_hv_k150_LandmarkIsomap_ndims7_prototypes_collage_sm_border.eps} 1.214 \\ 1.215 -\includegraphics[scale=.31]{merge3_norm_hv_PCA_ndims50_kmeans_7clust.eps} 1.216 -\includegraphics[scale=.31]{norm_hv_NNMF_3_norm_kmeans_4clust.eps} 1.217 -\includegraphics[scale=.31]{merge3_norm_hv_k150_LandmarkIsomap_ndims7_kmeans_7clust.eps} 1.218 -\caption{Top row: 19 of the major subdivisions of the cortex. Second row: the first 6 reduced dimensions, using PCA. Third row: the first 6 reduced dimensions, using NNMF. Fourth row: the first six reduced dimensions, using landmark Isomap. Bottom row: examples of kmeans clustering applied to reduced datasets to find 7 clusters. Left: PCA. Middle: NNMF. Right: Landmark Isomap. Additional details: In the third and fourth rows, 7 dimensions were found, but only 6 displayed. In the last row: for PCA, 50 dimensions were used; for NNMF, 6 dimensions were used; for landmark Isomap, 7 dimensions were used.} 1.219 -\label{dimReduc}\end{figure} 1.220 - 1.221 -todo: nnmf 7 1.222 +\includegraphics[scale=.24]{paint_merge3_major.eps}\includegraphics[scale=.22]{merge3_norm_hv_PCA_ndims50_kmeans_7clust.eps}\includegraphics[scale=.24]{norm_hv_NNMF_6_norm_kmeans_7clust.eps}\includegraphics[scale=.22]{merge3_norm_hv_k150_LandmarkIsomap_ndims7_kmeans_7clust.eps} 1.223 +\caption{First row: the first 6 reduced dimensions, using PCA. Second row: the first 6 reduced dimensions, using NNMF. Third row: the first six reduced dimensions, using landmark Isomap. Bottom row: examples of kmeans clustering applied to reduced datasets to find 7 clusters. Left: 19 of the major subdivisions of the cortex. Second from left: PCA. Third from left: NNMF. Right: Landmark Isomap. Additional details: In the third and fourth rows, 7 dimensions were found, but only 6 displayed. In the last row: for PCA, 50 dimensions were used; for NNMF, 6 dimensions were used; for landmark Isomap, 7 dimensions were used.} 1.224 +\label{dimReduc}\end{wrapfigure} 1.225 + 1.226 + 1.227 +\vspace{0.3cm}**SVM on all genes at once** 1.228 + 1.229 +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.}. This shows that the genes included in the ABA dataset are sufficient to define much of cortical anatomy. 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. 1.230 + 1.231 + 1.232 + 1.233 + 1.234 + 1.235 +=== Data-driven redrawing of the cortical map === 1.236 + 1.237 +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). Space constraints prevent us from showing many of the results, but as a sample, PCA, NNMF, and landmark Isomap are shown in the first, second, and third rows of Figure \ref{dimReduc}. 1.238 + 1.239 +After applying the dimensionality reduction, we ran clustering algorithms on the reduced data. To date we have tried k-means and spectral clustering. The results of k-means after PCA, NNMF, and landmark Isomap are shown in the last row of Figure \ref{dimReduc}. To compare, the leftmost picture on the bottom row of Figure \ref{dimReduc} shows some of the major subdivisions of cortex. These results clearly show that different dimensionality reduction techniques capture different aspects of the data and lead to different clusterings, indicating the utility of our proposal to produce a detailed comparion of these techniques as applied to the domain of genomic anatomy. 1.240 + 1.241 + 1.242 1.243 \vspace{0.3cm}**Many areas are captured by clusters of genes** 1.244