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changeset 104:d6ecbc494f0b

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author bshanks@bshanks.dyndns.org
date Wed Apr 22 07:35:14 2009 -0700 (16 years ago)
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2.1 --- a/grant.html Wed Apr 22 07:26:09 2009 -0700 2.2 +++ b/grant.html Wed Apr 22 07:35:14 2009 -0700 2.3 @@ -37,22 +37,17 @@ 2.4 conclusion of each section, we will summarize why our strategy is different from what has been done before. At 2.5 the end of this section, we will describe the potential impact. 2.6 Aim 1: Given a map of regions, find genes that mark the regions 2.7 - 2.8 -Figure 1: Gene Pitx2 2.9 -is selectively underex- 2.10 -pressed in area SS. Machine learning terminology: classifiers The task of looking for marker genes for 2.11 - known anatomical regions means that one is looking for a set of genes such that, if 2.12 - the expression level of those genes is known, then the locations of the regions can be 2.13 - inferred. 2.14 - If we define the regions so that they cover the entire anatomical structure to be 2.15 - subdivided, we may say that we are using gene expression in each voxel to assign 2.16 - that voxel to the proper area. We call this a classification task, because each voxel 2.17 - is being assigned to a class (namely, its region). An understanding of the relationship 2.18 - between the combination of their expression levels and the locations of the regions may 2.19 - be expressed as a function. The input to this function is a voxel, along with the gene 2.20 - expression levels within that voxel; the output is the regional identity of the target voxel, 2.21 -that is, the region to which the target voxel belongs. We call this function a classifier. In general, the input to a 2.22 -classifier is called an instance, and the output is called a label (or a class label). 2.23 +Machine learning terminology: classifiers The task of looking for marker genes for known anatomical regions 2.24 +means that one is looking for a set of genes such that, if the expression level of those genes is known, then the 2.25 +locations of the regions can be inferred. 2.26 + If we define the regions so that they cover the entire anatomical structure to be subdivided, we may say that 2.27 +we are using gene expression in each voxel to assign that voxel to the proper area. We call this a classification 2.28 +task, because each voxel is being assigned to a class (namely, its region). An understanding of the relationship 2.29 +between the combination of their expression levels and the locations of the regions may be expressed as a 2.30 +function. The input to this function is a voxel, along with the gene expression levels within that voxel; the output is 2.31 +the regional identity of the target voxel, that is, the region to which the target voxel belongs. We call this function 2.32 +a classifier. In general, the input to a classifier is called an instance, and the output is called a label (or a class 2.33 +label). 2.34 The object of aim 1 is not to produce a single classifier, but rather to develop an automated method for 2.35 determining a classifier for any known anatomical structure. Therefore, we seek a procedure by which a gene 2.36 expression dataset may be analyzed in concert with an anatomical atlas in order to produce a classifier. The 2.37 @@ -76,60 +71,33 @@ 2.38 calculate a voxel’s sub-score, then we say it is a local scoring method. If only information from the voxel itself is 2.39 used to calculate a voxel’s sub-score, then we say it is a pointwise scoring method. 2.40 Both gene expression data and anatomical atlases have errors, due to a variety of factors. Individual subjects 2.41 - 1Strictly speaking, the features are gene expression levels, but we’ll call them genes. 2.42 have idiosyncratic anatomy. Subjects may be improperly registered to the atlas. The method used to measure 2.43 gene expression may be noisy. The atlas may have errors. It is even possible that some areas in the anatomical 2.44 + 1Strictly speaking, the features are gene expression levels, but we’ll call them genes. 2.45 atlas are “wrong” in that they do not have the same shape as the natural domains of gene expression to which 2.46 they correspond. These sources of error can affect the displacement and the shape of both the gene expression 2.47 data and the anatomical target areas. Therefore, it is important to use feature selection methods which are 2.48 robust to these kinds of errors. 2.49 Our strategy for Aim 1 2.50 - 2.51 - 2.52 -Figure 2: Top row: Genes Nfic 2.53 -and A930001M12Rik are the most 2.54 -correlated with area SS (somatosen- 2.55 -sory cortex). Bottom row: Genes 2.56 -C130038G02Rik and Cacna1i are 2.57 -those with the best fit using logistic 2.58 -regression. Within each picture, the 2.59 -vertical axis roughly corresponds to 2.60 -anterior at the top and posterior at the 2.61 -bottom, and the horizontal axis roughly 2.62 -corresponds to medial at the left and 2.63 -lateral at the right. The red outline is 2.64 -the boundary of region SS. Pixels are 2.65 -colored according to correlation, with 2.66 -red meaning high correlation and blue 2.67 -meaning low. Key questions when choosing a learning method are: What are the 2.68 - instances? What are the features? How are the features chosen? 2.69 - Here are four principles that outline our answers to these questions. 2.70 - Principle 1: Combinatorial gene expression 2.71 - It is too much to hope that every anatomical region of interest will be 2.72 - identified by a single gene. For example, in the cortex, there are some 2.73 - areas which are not clearly delineated by any gene included in the 2.74 - Allen Brain Atlas (ABA) dataset. However, at least some of these areas 2.75 - can be delineated by looking at combinations of genes (an example 2.76 - of an area for which multiple genes are necessary and sufficient is 2.77 - provided in Preliminary Studies, Figure 4). Therefore, each instance 2.78 - should contain multiple features (genes). 2.79 - Principle 2: Only look at combinations of small numbers of 2.80 - genes 2.81 - When the classifier classifies a voxel, it is only allowed to look at 2.82 - the expression of the genes which have been selected as features. 2.83 - The more data that are available to a classifier, the better that it can do. 2.84 - For example, perhaps there are weak correlations over many genes 2.85 - that add up to a strong signal. So, why not include every gene as a 2.86 - feature? The reason is that we wish to employ the classifier in situations 2.87 - in which it is not feasible to gather data about every gene. For example, 2.88 - if we want to use the expression of marker genes as a trigger for some 2.89 - regionally-targeted intervention, then our intervention must contain a 2.90 - molecular mechanism to check the expression level of each marker 2.91 - gene before it triggers. It is currently infeasible to design a molecular 2.92 - trigger that checks the level of more than a handful of genes. Similarly, 2.93 - if the goal is to develop a procedure to do ISH on tissue samples in 2.94 - order to label their anatomy, then it is infeasible to label more than a 2.95 - few genes. Therefore, we must select only a few genes as features. 2.96 +Key questions when choosing a learning method are: What are the instances? What are the features? How are 2.97 +the features chosen? Here are four principles that outline our answers to these questions. 2.98 +Principle 1: Combinatorial gene expression 2.99 +It is too much to hope that every anatomical region of interest will be identified by a single gene. For example, 2.100 +in the cortex, there are some areas which are not clearly delineated by any gene included in the Allen Brain Atlas 2.101 +(ABA) dataset. However, at least some of these areas can be delineated by looking at combinations of genes 2.102 +(an example of an area for which multiple genes are necessary and sufficient is provided in Preliminary Studies, 2.103 +Figure 4). Therefore, each instance should contain multiple features (genes). 2.104 +Principle 2: Only look at combinations of small numbers of genes 2.105 +When the classifier classifies a voxel, it is only allowed to look at the expression of the genes which have 2.106 +been selected as features. The more data that are available to a classifier, the better that it can do. For example, 2.107 +perhaps there are weak correlations over many genes that add up to a strong signal. So, why not include every 2.108 +gene as a feature? The reason is that we wish to employ the classifier in situations in which it is not feasible to 2.109 +gather data about every gene. For example, if we want to use the expression of marker genes as a trigger for 2.110 +some regionally-targeted intervention, then our intervention must contain a molecular mechanism to check the 2.111 +expression level of each marker gene before it triggers. It is currently infeasible to design a molecular trigger that 2.112 +checks the level of more than a handful of genes. Similarly, if the goal is to develop a procedure to do ISH on 2.113 +tissue samples in order to label their anatomy, then it is infeasible to label more than a few genes. Therefore, we 2.114 +must select only a few genes as features. 2.115 The requirement to find combinations of only a small number of genes limits us from straightforwardly ap- 2.116 plying many of the most simple techniques from the field of supervised machine learning. In the parlance of 2.117 machine learning, our task combines feature selection with supervised learning. 2.118 @@ -146,49 +114,35 @@ 2.119 humans to visualize and work with 2-D data. Therefore, when possible, the instances should represent pixels, 2.120 not voxels. 2.121 Related work 2.122 - 2.123 - 2.124 -Figure 3: The top row shows the two 2.125 -genes which (individually) best predict 2.126 -area AUD, according to logistic regres- 2.127 -sion. The bottom row shows the two 2.128 -genes which (individually) best match 2.129 -area AUD, according to gradient sim- 2.130 -ilarity. From left to right and top to 2.131 -bottom, the genes are Ssr1, Efcbp1, 2.132 -Ptk7, and Aph1a. There is a substantial body of work on the analysis of gene expres- 2.133 - sion data, most of this concerns gene expression data which are not 2.134 - fundamentally spatial2. 2.135 - As noted above, there has been much work on both supervised 2.136 - learning and there are many available algorithms for each. However, 2.137 - the algorithms require the scientist to provide a framework for repre- 2.138 - senting the problem domain, and the way that this framework is set 2.139 - up has a large impact on performance. Creating a good framework 2.140 - can require creatively reconceptualizing the problem domain, and is 2.141 - not merely a mechanical “fine-tuning” of numerical parameters. For 2.142 - example, we believe that domain-specific scoring measures (such as 2.143 - gradient similarity, which is discussed in Preliminary Studies) may be 2.144 - necessary in order to achieve the best results in this application. 2.145 - We now turn to efforts to find marker genes using spatial gene ex- 2.146 - pression data using automated methods. 2.147 - GeneAtlas[5] and EMAGE [26] allow the user to construct a search 2.148 - query by demarcating regions and then specifying either the strength of 2.149 - expression or the name of another gene or dataset whose expression 2.150 - pattern is to be matched. Neither GeneAtlas nor EMAGE allow one to 2.151 - search for combinations of genes that define a region in concert but not 2.152 - separately. 2.153 - [15] describes AGEA, ”Anatomic Gene Expression Atlas”. AGEA 2.154 -has three components. Gene Finder: The user selects a seed voxel and the system (1) chooses a cluster which 2.155 -includes the seed voxel, (2) yields a list of genes which are overexpressed in that cluster. Correlation: The 2.156 -user selects a seed voxel and the system then shows the user how much correlation there is between the gene 2.157 -expression profile of the seed voxel and every other voxel. Clusters: will be described later. [6] looks at the mean 2.158 -expression level of genes within anatomical regions, and applies a Student’s t-test with Bonferroni correction to 2.159 -determine whether the mean expression level of a gene is significantly higher in the target region. [15] and [6] 2.160 -differ from our Aim 1 in at least three ways. First, [15] and [6] find only single genes, whereas we will also look 2.161 -for combinations of genes. Second, [15] and [6] can only use overexpression as a marker, whereas we will also 2.162 -search for underexpression. Third, [15] and [6] use scores based on pointwise expression levels, whereas we 2.163 -will also use geometric scores such as gradient similarity (described in Preliminary Studies). Figures 4, 1, and 3 2.164 -in the Preliminary Studies section contain evidence that each of our three choices is the right one. 2.165 +There is a substantial body of work on the analysis of gene expression data, most of this concerns gene expres- 2.166 +sion data which are not fundamentally spatial2. 2.167 +As noted above, there has been much work on both supervised learning and there are many available 2.168 +algorithms for each. However, the algorithms require the scientist to provide a framework for representing the 2.169 +problem domain, and the way that this framework is set up has a large impact on performance. Creating a 2.170 +good framework can require creatively reconceptualizing the problem domain, and is not merely a mechanical 2.171 +“fine-tuning” of numerical parameters. For example, we believe that domain-specific scoring measures (such 2.172 +_________________________________________ 2.173 + 2By “fundamentally spatial” we mean that there is information from a large number of spatial locations indexed by spatial coordinates; 2.174 +not just data which have only a few different locations or which is indexed by anatomical label. 2.175 +as gradient similarity, which is discussed in Preliminary Studies) may be necessary in order to achieve the best 2.176 +results in this application. 2.177 +We now turn to efforts to find marker genes using spatial gene expression data using automated methods. 2.178 +GeneAtlas[5] and EMAGE [26] allow the user to construct a search query by demarcating regions and then 2.179 +specifying either the strength of expression or the name of another gene or dataset whose expression pattern 2.180 +is to be matched. Neither GeneAtlas nor EMAGE allow one to search for combinations of genes that define a 2.181 +region in concert but not separately. 2.182 +[15 ] describes AGEA, ”Anatomic Gene Expression Atlas”. AGEA has three components. Gene Finder: The 2.183 +user selects a seed voxel and the system (1) chooses a cluster which includes the seed voxel, (2) yields a list of 2.184 +genes which are overexpressed in that cluster. Correlation: The user selects a seed voxel and the system then 2.185 +shows the user how much correlation there is between the gene expression profile of the seed voxel and every 2.186 +other voxel. Clusters: will be described later. [6] looks at the mean expression level of genes within anatomical 2.187 +regions, and applies a Student’s t-test with Bonferroni correction to determine whether the mean expression 2.188 +level of a gene is significantly higher in the target region. [15] and [6] differ from our Aim 1 in at least three 2.189 +ways. First, [15] and [6] find only single genes, whereas we will also look for combinations of genes. Second, 2.190 +[15 ] and [6] can only use overexpression as a marker, whereas we will also search for underexpression. Third, 2.191 +[15 ] and [6] use scores based on pointwise expression levels, whereas we will also use geometric scores such 2.192 +as gradient similarity (described in Preliminary Studies). Figures 4, 2, and 3 in the Preliminary Studies section 2.193 +contain evidence that each of our three choices is the right one. 2.194 [10 ] describes a technique to find combinations of marker genes to pick out an anatomical region. They use 2.195 an evolutionary algorithm to evolve logical operators which combine boolean (thresholded) images in order to 2.196 match a target image. 2.197 @@ -201,9 +155,6 @@ 2.198 referred to as unsupervised learning in the jargon of machine learning. One thing that you can do with such a 2.199 dataset is to group instances together. A set of similar instances is called a cluster, and the activity of finding 2.200 grouping the data into clusters is called clustering or cluster analysis. 2.201 -_________________________________________ 2.202 - 2By “fundamentally spatial” we mean that there is information from a large number of spatial locations indexed by spatial coordinates; 2.203 -not just data which have only a few different locations or which is indexed by anatomical label. 2.204 The task of deciding how to carve up a structure into anatomical regions can be put into these terms. The 2.205 instances are once again voxels (or pixels) along with their associated gene expression profiles. We make 2.206 the assumption that voxels from the same anatomical region have similar gene expression profiles, at least 2.207 @@ -215,31 +166,18 @@ 2.208 Similarity scores A crucial choice when designing a clustering method is how to measure similarity, across 2.209 either pairs of instances, or clusters, or both. There is much overlap between scoring methods for feature 2.210 selection (discussed above under Aim 1) and scoring methods for similarity. 2.211 - 2.212 - 2.213 -Figure 4: Upper left: wwc1. Upper 2.214 -right: mtif2. Lower left: wwc1 + mtif2 2.215 -(each pixel’s value on the lower left is 2.216 -the sum of the corresponding pixels in 2.217 -the upper row). Spatially contiguous clusters; image segmentation We have 2.218 - shown that aim 2 is a type of clustering task. In fact, it is a special 2.219 - type of clustering task because we have an additional constraint on 2.220 - clusters; voxels grouped together into a cluster must be spatially con- 2.221 - tiguous. In Preliminary Studies, we show that one can get reasonable 2.222 - results without enforcing this constraint; however, we plan to compare 2.223 - these results against other methods which guarantee contiguous clus- 2.224 - ters. 2.225 - Image segmentation is the task of partitioning the pixels in a digital 2.226 - image into clusters, usually contiguous clusters. Aim 2 is similar to an 2.227 - image segmentation task. There are two main differences; in our task, 2.228 - there are thousands of color channels (one for each gene), rather than 2.229 - just three3. A more crucial difference is that there are various cues 2.230 - which are appropriate for detecting sharp object boundaries in a visual 2.231 - scene but which are not appropriate for segmenting abstract spatial 2.232 - data such as gene expression. Although many image segmentation 2.233 - algorithms can be expected to work well for segmenting other sorts of 2.234 - spatially arranged data, some of these algorithms are specialized for 2.235 -visual images. 2.236 +Spatially contiguous clusters; image segmentation We have shown that aim 2 is a type of clustering 2.237 +task. In fact, it is a special type of clustering task because we have an additional constraint on clusters; voxels 2.238 +grouped together into a cluster must be spatially contiguous. In Preliminary Studies, we show that one can get 2.239 +reasonable results without enforcing this constraint; however, we plan to compare these results against other 2.240 +methods which guarantee contiguous clusters. 2.241 +Image segmentation is the task of partitioning the pixels in a digital image into clusters, usually contiguous 2.242 +clusters. Aim 2 is similar to an image segmentation task. There are two main differences; in our task, there are 2.243 +thousands of color channels (one for each gene), rather than just three3. A more crucial difference is that there 2.244 +are various cues which are appropriate for detecting sharp object boundaries in a visual scene but which are not 2.245 +appropriate for segmenting abstract spatial data such as gene expression. Although many image segmentation 2.246 +algorithms can be expected to work well for segmenting other sorts of spatially arranged data, some of these 2.247 +algorithms are specialized for visual images. 2.248 Dimensionality reduction In this section, we discuss reducing the length of the per-pixel gene expression 2.249 feature vector. By “dimension”, we mean the dimension of this vector, not the spatial dimension of the underlying 2.250 data. 2.251 @@ -256,74 +194,58 @@ 2.252 we could have one reduced feature for each gene cluster. 2.253 Gene clusters could also be used to directly yield a clustering on instances. This is because many genes have 2.254 an expression pattern which seems to pick out a single, spatially contiguous region. This suggests the following 2.255 +procedure: cluster together genes which pick out similar regions, and then to use the more popular common 2.256 +regions as the final clusters. In Preliminary Studies, Figure 7, we show that a number of anatomically recognized 2.257 +cortical regions, as well as some “superregions” formed by lumping together a few regions, are associated with 2.258 +gene clusters in this fashion. 2.259 +Related work 2.260 +Some researchers have attempted to parcellate cortex on the basis of non-gene expression data. For example, 2.261 +[18 ], [2 ], [19], and [1] associate spots on the cortex with the radial profile5 of response to some stain ([12] uses 2.262 +MRI), extract features from this profile, and then use similarity between surface pixels to cluster. 2.263 +[23 ] describes an analysis of the anatomy of the hippocampus using the ABA dataset. In addition to manual 2.264 +analysis, two clustering methods were employed, a modified Non-negative Matrix Factorization (NNMF), and 2.265 +a hierarchical recursive bifurcation clustering scheme based on correlation as the similarity score. The paper 2.266 +yielded impressive results, proving the usefulness of computational genomic anatomy. We have run NNMF on 2.267 +the cortical dataset 2.268 +AGEA[15] includes a preset hierarchical clustering of voxels based on a recursive bifurcation algorithm with 2.269 +correlation as the similarity metric. EMAGE[26] allows the user to select a dataset from among a large number 2.270 +of alternatives, or by running a search query, and then to cluster the genes within that dataset. EMAGE clusters 2.271 +via hierarchical complete linkage clustering. 2.272 +[6 ] clusters genes. For each cluster, prototypical spatial expression patterns were created by averaging the 2.273 +genes in the cluster. The prototypes were analyzed manually, without clustering voxels. 2.274 +[10 ] applies their technique for finding combinations of marker genes for the purpose of clustering genes 2.275 +around a “seed gene”. 2.276 +In summary, although these projects obtained clusterings, there has not been much comparison between 2.277 +different algorithms or scoring methods, so it is likely that the best clustering method for this application has not 2.278 +yet been found. The projects using gene expression on cortex did not attempt to make use of the radial profile 2.279 +of gene expression. Also, none of these projects did a separate dimensionality reduction step before clustering 2.280 _________________________________________ 2.281 3There are imaging tasks which use more than three colors, for example multispectral imaging and hyperspectral imaging, which are 2.282 often used to process satellite imagery. 2.283 4First, because the number of features in the reduced dataset is less than in the original dataset, the running time of clustering 2.284 algorithms may be much less. Second, it is thought that some clustering algorithms may give better results on reduced data. 2.285 -procedure: cluster together genes which pick out similar regions, and then to use the more popular common 2.286 -regions as the final clusters. In Preliminary Studies, Figure 7, we show that a number of anatomically recognized 2.287 -cortical regions, as well as some “superregions” formed by lumping together a few regions, are associated with 2.288 -gene clusters in this fashion. 2.289 + 5A radial profile is a profile along a line perpendicular to the cortical surface. 2.290 +pixels, none tried to cluster genes first in order to guide automated clustering of pixels into spatial regions, and 2.291 +none used co-clustering algorithms. 2.292 +Aim 3: apply the methods developed to the cerebral cortex 2.293 2.294 2.295 - 2.296 - 2.297 -Figure 5: From left to right and top 2.298 -to bottom, single genes which roughly 2.299 -identify areas SS (somatosensory pri- 2.300 -mary + supplemental), SSs (supple- 2.301 -mental somatosensory), PIR (piriform), 2.302 -FRP (frontal pole), RSP (retrosple- 2.303 -nial), COApm (Cortical amygdalar, pos- 2.304 -terior part, medial zone). Grouping 2.305 -some areas together, we have also 2.306 -found genes to identify the groups 2.307 -ACA+PL+ILA+DP+ORB+MO (anterior 2.308 -cingulate, prelimbic, infralimbic, dor- 2.309 -sal peduncular, orbital, motor), poste- 2.310 -rior and lateral visual (VISpm, VISpl, 2.311 -VISI, VISp; posteromedial, posterolat- 2.312 -eral, lateral, and primary visual; the 2.313 -posterior and lateral visual area is dis- 2.314 -tinguished from its neighbors, but not 2.315 -from the entire rest of the cortex). The 2.316 -genes are Pitx2, Aldh1a2, Ppfibp1, 2.317 -Slco1a5, Tshz2, Trhr, Col12a1, Ets1. Related work 2.318 - Some researchers have attempted to parcellate cortex on the basis of 2.319 - non-gene expression data. For example, [18], [2], [19], and [1] asso- 2.320 - ciate spots on the cortex with the radial profile5 of response to some 2.321 - stain ([12] uses MRI), extract features from this profile, and then use 2.322 - similarity between surface pixels to cluster. 2.323 - [23] describes an analysis of the anatomy of the hippocampus us- 2.324 - ing the ABA dataset. In addition to manual analysis, two clustering 2.325 - methods were employed, a modified Non-negative Matrix Factoriza- 2.326 - tion (NNMF), and a hierarchical recursive bifurcation clustering scheme 2.327 - based on correlation as the similarity score. The paper yielded impres- 2.328 - sive results, proving the usefulness of computational genomic anatomy. 2.329 - We have run NNMF on the cortical dataset 2.330 - AGEA[15] includes a preset hierarchical clustering of voxels based 2.331 - on a recursive bifurcation algorithm with correlation as the similarity 2.332 - metric. EMAGE[26] allows the user to select a dataset from among a 2.333 - large number of alternatives, or by running a search query, and then to 2.334 - cluster the genes within that dataset. EMAGE clusters via hierarchical 2.335 - complete linkage clustering. 2.336 - [6] clusters genes. For each cluster, prototypical spatial expression 2.337 - patterns were created by averaging the genes in the cluster. The pro- 2.338 - totypes were analyzed manually, without clustering voxels. 2.339 - [10] applies their technique for finding combinations of marker genes 2.340 - for the purpose of clustering genes around a “seed gene”. 2.341 - In summary, although these projects obtained clusterings, there has 2.342 - not been much comparison between different algorithms or scoring 2.343 - methods, so it is likely that the best clustering method for this appli- 2.344 - cation has not yet been found. The projects using gene expression 2.345 - on cortex did not attempt to make use of the radial profile of gene ex- 2.346 - pression. Also, none of these projects did a separate dimensionality 2.347 - reduction step before clustering pixels, none tried to cluster genes first 2.348 - in order to guide automated clustering of pixels into spatial regions, and 2.349 - none used co-clustering algorithms. 2.350 - Aim 3: apply the methods developed to the cerebral cortex 2.351 - Background 2.352 +Figure 1: Top row: Genes Nfic 2.353 +and A930001M12Rik are the most 2.354 +correlated with area SS (somatosen- 2.355 +sory cortex). Bottom row: Genes 2.356 +C130038G02Rik and Cacna1i are 2.357 +those with the best fit using logistic 2.358 +regression. Within each picture, the 2.359 +vertical axis roughly corresponds to 2.360 +anterior at the top and posterior at the 2.361 +bottom, and the horizontal axis roughly 2.362 +corresponds to medial at the left and 2.363 +lateral at the right. The red outline is 2.364 +the boundary of region SS. Pixels are 2.365 +colored according to correlation, with 2.366 +red meaning high correlation and blue 2.367 +meaning low. Background 2.368 The cortex is divided into areas and layers. Because of the cortical 2.369 columnar organization, the parcellation of the cortex into areas can be 2.370 drawn as a 2-D map on the surface of the cortex. In the third dimension, 2.371 @@ -333,58 +255,30 @@ 2.372 six-layered cake6. 2.373 It is known that different cortical areas have distinct roles in both 2.374 normal functioning and in disease processes, yet there are no known 2.375 -_________________________________________ 2.376 - 5A radial profile is a profile along a line perpendicular to the cortical surface. 2.377 - 6Outside of isocortex, the number of layers varies. 2.378 marker genes for most cortical areas. When it is necessary to divide a 2.379 tissue sample into cortical areas, this is a manual process that requires 2.380 a skilled human to combine multiple visual cues and interpret them in 2.381 the context of their approximate location upon the cortical surface. 2.382 - 2.383 - 2.384 - 2.385 - 2.386 -Figure 6: First row: the first 6 reduced dimensions, using PCA. Sec- 2.387 -ond row: the first 6 reduced dimensions, using NNMF. Third row: 2.388 -the first six reduced dimensions, using landmark Isomap. Bottom 2.389 -row: examples of kmeans clustering applied to reduced datasets 2.390 -to find 7 clusters. Left: 19 of the major subdivisions of the cortex. 2.391 -Second from left: PCA. Third from left: NNMF. Right: Landmark 2.392 -Isomap. Additional details: In the third and fourth rows, 7 dimen- 2.393 -sions were found, but only 6 displayed. In the last row: for PCA, 2.394 -50 dimensions were used; for NNMF, 6 dimensions were used; for 2.395 -landmark Isomap, 7 dimensions were used. Even the questions of how many ar- 2.396 - eas should be recognized in cortex, and 2.397 - what their arrangement is, are still not com- 2.398 - pletely settled. A proposed division of the 2.399 - cortex into areas is called a cortical map. 2.400 - In the rodent, the lack of a single agreed- 2.401 - upon map can be seen by contrasting the 2.402 - recent maps given by Swanson[22] on the 2.403 - one hand, and Paxinos and Franklin[17] on 2.404 - the other. While the maps are certainly 2.405 - very similar in their general arrangement, 2.406 - significant differences remain. 2.407 - The Allen Mouse Brain Atlas dataset 2.408 - The Allen Mouse Brain Atlas (ABA) 2.409 - data were produced by doing in-situ hy- 2.410 - bridization on slices of male, 56-day-old 2.411 - C57BL/6J mouse brains. Pictures were 2.412 - taken of the processed slice, and these pic- 2.413 - tures were semi-automatically analyzed to 2.414 - create a digital measurement of gene ex- 2.415 - pression levels at each location in each 2.416 - slice. Per slice, cellular spatial resolution 2.417 - is achieved. Using this method, a single 2.418 - physical slice can only be used to measure 2.419 - one single gene; many different mouse 2.420 - brains were needed in order to measure 2.421 - the expression of many genes. 2.422 - An automated nonlinear alignment pro- 2.423 - cedure located the 2D data from the var- 2.424 -ious slices in a single 3D coordinate system. In the final 3D coordinate system, voxels are cubes with 200 2.425 -microns on a side. There are 67x41x58 = 159,326 voxels in the 3D coordinate system, of which 51,533 are in 2.426 -the brain[15]. 2.427 + Even the questions of how many areas should be recognized in 2.428 + cortex, and what their arrangement is, are still not completely settled. 2.429 + A proposed division of the cortex into areas is called a cortical map. 2.430 + In the rodent, the lack of a single agreed-upon map can be seen by 2.431 + contrasting the recent maps given by Swanson[22] on the one hand, 2.432 + and Paxinos and Franklin[17] on the other. While the maps are cer- 2.433 + tainly very similar in their general arrangement, significant differences 2.434 + remain. 2.435 + The Allen Mouse Brain Atlas dataset 2.436 + The Allen Mouse Brain Atlas (ABA) data were produced by doing in- 2.437 + situ hybridization on slices of male, 56-day-old C57BL/6J mouse brains. 2.438 + Pictures were taken of the processed slice, and these pictures were 2.439 + semi-automatically analyzed to create a digital measurement of gene 2.440 + expression levels at each location in each slice. Per slice, cellular spa- 2.441 + tial resolution is achieved. Using this method, a single physical slice 2.442 +can only be used to measure one single gene; many different mouse brains were needed in order to measure 2.443 +the expression of many genes. 2.444 +An automated nonlinear alignment procedure located the 2D data from the various slices in a single 3D 2.445 +coordinate system. In the final 3D coordinate system, voxels are cubes with 200 microns on a side. There are 2.446 +67x41x58 = 159,326 voxels in the 3D coordinate system, of which 51,533 are in the brain[15]. 2.447 Mus musculus is thought to contain about 22,000 protein-coding genes[28]. The ABA contains data on about 2.448 20,000 genes in sagittal sections, out of which over 4,000 genes are also measured in coronal sections. Our 2.449 dataset is derived from only the coronal subset of the ABA7. 2.450 @@ -394,13 +288,14 @@ 2.451 Related work 2.452 [15 ] describes the application of AGEA to the cortex. The paper describes interesting results on the structure 2.453 of correlations between voxel gene expression profiles within a handful of cortical areas. However, this sort 2.454 +_________________________________________ 2.455 + 6Outside of isocortex, the number of layers varies. 2.456 + 7The sagittal data do not cover the entire cortex, and also have greater registration error[15]. Genes were selected by the Allen 2.457 +Institute for coronal sectioning based on, “classes of known neuroscientific interest... or through post hoc identification of a marked 2.458 +non-ubiquitous expression pattern”[15]. 2.459 of analysis is not related to either of our aims, as it neither finds marker genes, nor does it suggest a cortical 2.460 map based on gene expression data. Neither of the other components of AGEA can be applied to cortical 2.461 areas; AGEA’s Gene Finder cannot be used to find marker genes for the cortical areas; and AGEA’s hierarchical 2.462 -_________________________________________ 2.463 - 7The sagittal data do not cover the entire cortex, and also have greater registration error[15]. Genes were selected by the Allen 2.464 -Institute for coronal sectioning based on, “classes of known neuroscientific interest... or through post hoc identification of a marked 2.465 -non-ubiquitous expression pattern”[15]. 2.466 clustering does not produce clusters corresponding to the cortical areas8. 2.467 In summary, for all three aims, (a) only one of the previous projects explores combinations of marker genes, 2.468 (b) there has been almost no comparison of different algorithms or scoring methods, and (c) there has been no 2.469 @@ -411,25 +306,19 @@ 2.470 different methods. 2.471 Significance 2.472 2.473 -Figure 7: Prototypes corresponding to sample gene 2.474 -clusters, clustered by gradient similarity. Region bound- 2.475 -aries for the region that most matches each prototype 2.476 -are overlaid. The method developed in aim (1) will be applied to 2.477 - each cortical area to find a set of marker genes such 2.478 - that the combinatorial expression pattern of those 2.479 - genes uniquely picks out the target area. Finding 2.480 - marker genes will be useful for drug discovery as well 2.481 - as for experimentation because marker genes can be 2.482 - used to design interventions which selectively target 2.483 - individual cortical areas. 2.484 - The application of the marker gene finding algo- 2.485 - rithm to the cortex will also support the development 2.486 - of new neuroanatomical methods. In addition to find- 2.487 - ing markers for each individual cortical areas, we will 2.488 - find a small panel of genes that can find many of the 2.489 - areal boundaries at once. This panel of marker genes 2.490 -will allow the development of an ISH protocol that will allow experimenters to more easily identify which anatom- 2.491 -ical areas are present in small samples of cortex. 2.492 +Figure 2: Gene Pitx2 2.493 +is selectively underex- 2.494 +pressed in area SS. The method developed in aim (1) will be applied to each cortical area to find a set of 2.495 + marker genes such that the combinatorial expression pattern of those genes uniquely 2.496 + picks out the target area. Finding marker genes will be useful for drug discovery as 2.497 + well as for experimentation because marker genes can be used to design interventions 2.498 + which selectively target individual cortical areas. 2.499 + The application of the marker gene finding algorithm to the cortex will also support 2.500 + the development of new neuroanatomical methods. In addition to finding markers for 2.501 + each individual cortical areas, we will find a small panel of genes that can find many of 2.502 + the areal boundaries at once. This panel of marker genes will allow the development of 2.503 + an ISH protocol that will allow experimenters to more easily identify which anatomical 2.504 + areas are present in small samples of cortex. 2.505 The method developed in aim (2) will provide a genoarchitectonic viewpoint that will contribute to the creation 2.506 of a better map. The development of present-day cortical maps was driven by the application of histological 2.507 stains. If a different set of stains had been available which identified a different set of features, then today’s 2.508 @@ -452,11 +341,11 @@ 2.509 of the voxels “underneath” that mesh node. We then flattened the cortex, creating a two-dimensional mesh. We 2.510 sampled the nodes of the irregular, flat mesh in order to create a regular grid of pixel values. We converted this 2.511 grid into a MATLAB matrix. We manually traced the boundaries of each of 49 cortical areas from the ABA coronal 2.512 +reference atlas slides. We then converted these manual traces into Caret-format regional boundary data on the 2.513 8In both cases, the cause is that pairwise correlations between the gene expression of voxels in different areas but the same layer 2.514 are often stronger than pairwise correlations between the gene expression of voxels in different layers but the same area. Therefore, a 2.515 pairwise voxel correlation clustering algorithm will tend to create clusters representing cortical layers, not areas. 2.516 9SEV is a sparse format for spatial data. It is the format in which the ABA data is made available. 2.517 -reference atlas slides. We then converted these manual traces into Caret-format regional boundary data on the 2.518 mesh surface. We projected the regions onto the 2-d mesh, and then onto the grid, and then we converted the 2.519 region data into MATLAB format. 2.520 At this point, the data are in the form of a number of 2-D matrices, all in registration, with the matrix entries 2.521 @@ -474,21 +363,38 @@ 2.522 Research Plan, we describe how we will automatically locate the layer depths. For validation, we have manually 2.523 demarcated the depth of the outer boundary of cortical layer 5 throughout the cortex. 2.524 Feature selection and scoring methods 2.525 -Underexpression of a gene can serve as a marker Underexpression of a gene can sometimes serve as a 2.526 -marker. See, for example, Figure 1. 2.527 -Correlation Recall that the instances are surface pixels, and consider the problem of attempting to classify 2.528 -each instance as either a member of a particular anatomical area, or not. The target area can be represented 2.529 -as a boolean mask over the surface pixels. 2.530 -We calculated the correlation between each gene and each cortical area. The top row of Figure 2 shows the 2.531 -three genes most correlated with area SS. 2.532 -Conditional entropy 2.533 -For each region, we created and ran a forward stepwise procedure which attempted to find pairs of gene 2.534 -expression boolean masks such that the conditional entropy of the target area’s boolean mask, conditioned 2.535 -upon the pair of gene expression boolean masks, is minimized. 2.536 -This finds pairs of genes which are most informative (at least at these discretization thresholds) relative to the 2.537 -question, “Is this surface pixel a member of the target area?”. Its advantage over linear methods such as logistic 2.538 -regression is that it takes account of arbitrarily nonlinear relationships; for example, if the XOR of two variables 2.539 -predicts the target, conditional entropy would notice, whereas linear methods would not. 2.540 + 2.541 + 2.542 +Figure 3: The top row shows the two 2.543 +genes which (individually) best predict 2.544 +area AUD, according to logistic regres- 2.545 +sion. The bottom row shows the two 2.546 +genes which (individually) best match 2.547 +area AUD, according to gradient sim- 2.548 +ilarity. From left to right and top to 2.549 +bottom, the genes are Ssr1, Efcbp1, 2.550 +Ptk7, and Aph1a. Underexpression of a gene can serve as a marker Underexpression 2.551 + of a gene can sometimes serve as a marker. See, for example, Figure 2.552 + 2. 2.553 + Correlation Recall that the instances are surface pixels, and con- 2.554 + sider the problem of attempting to classify each instance as either a 2.555 + member of a particular anatomical area, or not. The target area can be 2.556 + represented as a boolean mask over the surface pixels. 2.557 + We calculated the correlation between each gene and each cortical 2.558 + area. The top row of Figure 1 shows the three genes most correlated 2.559 + with area SS. 2.560 + Conditional entropy 2.561 + For each region, we created and ran a forward stepwise procedure 2.562 + which attempted to find pairs of gene expression boolean masks such 2.563 + that the conditional entropy of the target area’s boolean mask, condi- 2.564 + tioned upon the pair of gene expression boolean masks, is minimized. 2.565 + This finds pairs of genes which are most informative (at least at 2.566 + these discretization thresholds) relative to the question, “Is this surface 2.567 + pixel a member of the target area?”. Its advantage over linear methods 2.568 + such as logistic regression is that it takes account of arbitrarily nonlin- 2.569 + ear relationships; for example, if the XOR of two variables predicts the 2.570 + target, conditional entropy would notice, whereas linear methods would 2.571 + not. 2.572 Gradient similarity We noticed that the previous two scoring methods, which are pointwise, often found 2.573 genes whose pattern of expression did not look similar in shape to the target region. For this reason we designed 2.574 a non-pointwise scoring method to detect when a gene had a pattern of expression which looked like it had a 2.575 @@ -505,20 +411,32 @@ 2.576 are similar, then both images will have corresponding pixels with large gradients (because this is a border) which 2.577 are oriented in a similar direction (because the borders are similar). 2.578 Gradient similarity provides information complementary to correlation 2.579 -To show that gradient similarity can provide useful information that cannot be detected via pointwise analyses, 2.580 -consider Fig. 3. The pointwise method in the top row identifies genes which express more strongly in AUD than 2.581 -outside of it; its weakness is that this includes many areas which don’t have a salient border matching the areal 2.582 -border. The geometric method identifies genes whose salient expression border seems to partially line up with 2.583 -the border of AUD; its weakness is that this includes genes which don’t express over the entire area. 2.584 -Areas which can be identified by single genes Using gradient similarity, we have already found single 2.585 -genes which roughly identify some areas and groupings of areas. For each of these areas, an example of 2.586 -a gene which roughly identifies it is shown in Figure 5. We have not yet cross-verified these genes in other 2.587 -atlases. 2.588 -In addition, there are a number of areas which are almost identified by single genes: COAa+NLOT (anterior 2.589 -part of cortical amygdalar area, nucleus of the lateral olfactory tract), ENT (entorhinal), ACAv (ventral anterior 2.590 -cingulate), VIS (visual), AUD (auditory). 2.591 -These results validate our expectation that the ABA dataset can be exploited to find marker genes for many 2.592 -cortical areas, while also validating the relevancy of our new scoring method, gradient similarity. 2.593 + 2.594 + 2.595 +Figure 4: Upper left: wwc1. Upper 2.596 +right: mtif2. Lower left: wwc1 + mtif2 2.597 +(each pixel’s value on the lower left is 2.598 +the sum of the corresponding pixels in 2.599 +the upper row). To show that gradient similarity can provide useful information that 2.600 + cannot be detected via pointwise analyses, consider Fig. 3. The 2.601 + pointwise method in the top row identifies genes which express more 2.602 + strongly in AUD than outside of it; its weakness is that this includes 2.603 + many areas which don’t have a salient border matching the areal bor- 2.604 + der. The geometric method identifies genes whose salient expression 2.605 + border seems to partially line up with the border of AUD; its weakness 2.606 + is that this includes genes which don’t express over the entire area. 2.607 + Areas which can be identified by single genes Using gradient 2.608 + similarity, we have already found single genes which roughly identify 2.609 + some areas and groupings of areas. For each of these areas, an ex- 2.610 + ample of a gene which roughly identifies it is shown in Figure 5. We 2.611 + have not yet cross-verified these genes in other atlases. 2.612 + In addition, there are a number of areas which are almost identified 2.613 + by single genes: COAa+NLOT (anterior part of cortical amygdalar area, 2.614 + nucleus of the lateral olfactory tract), ENT (entorhinal), ACAv (ventral 2.615 + anterior cingulate), VIS (visual), AUD (auditory). 2.616 + These results validate our expectation that the ABA dataset can be 2.617 +exploited to find marker genes for many cortical areas, while also validating the relevancy of our new scoring 2.618 +method, gradient similarity. 2.619 Combinations of multiple genes are useful and necessary for some areas 2.620 In Figure 4, we give an example of a cortical area which is not marked by any single gene, but which 2.621 can be identified combinatorially. According to logistic regression, gene wwc1 is the best fit single gene for 2.622 @@ -546,58 +464,128 @@ 2.623 anatomy. However, as noted above, a classifier that looks at all the genes at once isn’t as practically useful as a 2.624 classifier that uses only a few genes. 2.625 Data-driven redrawing of the cortical map 2.626 -We have applied the following dimensionality reduction algorithms to reduce the dimensionality of the gene ex- 2.627 -pression profile associated with each pixel: Principal Components Analysis (PCA), Simple PCA, Multi-Dimensional 2.628 -Scaling, Isomap, Landmark Isomap, Laplacian eigenmaps, Local Tangent Space Alignment, Stochastic Proximity 2.629 -Embedding, Fast Maximum Variance Unfolding, Non-negative Matrix Factorization (NNMF). Space constraints 2.630 -prevent us from showing many of the results, but as a sample, PCA, NNMF, and landmark Isomap are shown in 2.631 -the first, second, and third rows of Figure 6. 2.632 -_ 2.633 + 2.634 + 2.635 + 2.636 + 2.637 +Figure 5: From left to right and top 2.638 +to bottom, single genes which roughly 2.639 +identify areas SS (somatosensory pri- 2.640 +mary + supplemental), SSs (supple- 2.641 +mental somatosensory), PIR (piriform), 2.642 +FRP (frontal pole), RSP (retrosple- 2.643 +nial), COApm (Cortical amygdalar, pos- 2.644 +terior part, medial zone). Grouping 2.645 +some areas together, we have also 2.646 +found genes to identify the groups 2.647 +ACA+PL+ILA+DP+ORB+MO (anterior 2.648 +cingulate, prelimbic, infralimbic, dor- 2.649 +sal peduncular, orbital, motor), poste- 2.650 +rior and lateral visual (VISpm, VISpl, 2.651 +VISI, VISp; posteromedial, posterolat- 2.652 +eral, lateral, and primary visual; the 2.653 +posterior and lateral visual area is dis- 2.654 +tinguished from its neighbors, but not 2.655 +from the entire rest of the cortex). The 2.656 +genes are Pitx2, Aldh1a2, Ppfibp1, 2.657 +Slco1a5, Tshz2, Trhr, Col12a1, Ets1. We have applied the following dimensionality reduction algorithms 2.658 + to reduce the dimensionality of the gene expression profile associ- 2.659 + ated with each pixel: Principal Components Analysis (PCA), Simple 2.660 + PCA, Multi-Dimensional Scaling, Isomap, Landmark Isomap, Laplacian 2.661 + eigenmaps, Local Tangent Space Alignment, Stochastic Proximity Em- 2.662 + bedding, Fast Maximum Variance Unfolding, Non-negative Matrix Fac- 2.663 + torization (NNMF). Space constraints prevent us from showing many of 2.664 + the results, but as a sample, PCA, NNMF, and landmark Isomap are 2.665 + shown in the first, second, and third rows of Figure 6. 2.666 + After applying the dimensionality reduction, we ran clustering algo- 2.667 + rithms on the reduced data. To date we have tried k-means and spec- 2.668 + tral clustering. The results of k-means after PCA, NNMF, and landmark 2.669 + Isomap are shown in the last row of Figure 6. To compare, the leftmost 2.670 + picture on the bottom row of Figure 6 shows some of the major subdivi- 2.671 + sions of cortex. These results clearly show that different dimensionality 2.672 + reduction techniques capture different aspects of the data and lead to 2.673 + different clusterings, indicating the utility of our proposal to produce a 2.674 + detailed comparison of these techniques as applied to the domain of 2.675 + genomic anatomy. 2.676 + Many areas are captured by clusters of genes We also clustered 2.677 + the genes using gradient similarity to see if the spatial regions defined 2.678 + by any clusters matched known anatomical regions. Figure 7 shows, for 2.679 + ten sample gene clusters, each cluster’s average expression pattern, 2.680 + compared to a known anatomical boundary. This suggests that it is 2.681 + worth attempting to cluster genes, and then to use the results to cluster 2.682 + pixels. 2.683 + The approach: what we plan to do 2.684 + Flatmap cortex and segment cortical layers 2.685 + There are multiple ways to flatten 3-D data into 2-D. We will compare 2.686 + mappings from manifolds to planes which attempt to preserve size 2.687 + (such as the one used by Caret[7]) with mappings which preserve an- 2.688 + gle (conformal maps). Our method will include a statistical test that 2.689 + warns the user if the assumption of 2-D structure seems to be wrong. 2.690 + We have not yet made use of radial profiles. While the radial pro- 2.691 + files may be used “raw”, for laminar structures like the cortex another 2.692 + strategy is to group together voxels in the same cortical layer; each sur- 2.693 + face pixel would then be associated with one expression level per gene 2.694 + per layer. We will develop a segmentation algorithm to automatically 2.695 + identify the layer boundaries. 2.696 + Develop algorithms that find genetic markers for anatomical re- 2.697 + gions 2.698 + Scoring measures and feature selection We will develop scoring 2.699 + methods for evaluating how good individual genes are at marking ar- 2.700 + eas. We will compare pointwise, geometric, and information-theoretic 2.701 +_________________________________________ 2.702 105-fold cross-validation. 2.703 -After applying the dimensionality reduction, we ran clustering algorithms on the reduced data. To date we 2.704 -have tried k-means and spectral clustering. The results of k-means after PCA, NNMF, and landmark Isomap are 2.705 -shown in the last row of Figure 6. To compare, the leftmost picture on the bottom row of Figure 6 shows some 2.706 -of the major subdivisions of cortex. These results clearly show that different dimensionality reduction techniques 2.707 -capture different aspects of the data and lead to different clusterings, indicating the utility of our proposal to 2.708 -produce a detailed comparison of these techniques as applied to the domain of genomic anatomy. 2.709 -Many areas are captured by clusters of genes We also clustered the genes using gradient similarity to 2.710 -see if the spatial regions defined by any clusters matched known anatomical regions. Figure 7 shows, for ten 2.711 -sample gene clusters, each cluster’s average expression pattern, compared to a known anatomical boundary. 2.712 -This suggests that it is worth attempting to cluster genes, and then to use the results to cluster pixels. 2.713 -The approach: what we plan to do 2.714 -Flatmap cortex and segment cortical layers 2.715 -There are multiple ways to flatten 3-D data into 2-D. We will compare mappings from manifolds to planes which 2.716 -attempt to preserve size (such as the one used by Caret[7]) with mappings which preserve angle (conformal 2.717 -maps). Our method will include a statistical test that warns the user if the assumption of 2-D structure seems to 2.718 -be wrong. 2.719 -We have not yet made use of radial profiles. While the radial profiles may be used “raw”, for laminar structures 2.720 -like the cortex another strategy is to group together voxels in the same cortical layer; each surface pixel would 2.721 -then be associated with one expression level per gene per layer. We will develop a segmentation algorithm to 2.722 -automatically identify the layer boundaries. 2.723 -Develop algorithms that find genetic markers for anatomical regions 2.724 -Scoring measures and feature selection We will develop scoring methods for evaluating how good individual 2.725 -genes are at marking areas. We will compare pointwise, geometric, and information-theoretic measures. We 2.726 -already developed one entirely new scoring method (gradient similarity), but we may develop more. Scoring 2.727 -measures that we will explore will include the L1 norm, correlation, expression energy ratio, conditional entropy, 2.728 -gradient similarity, Jaccard similarity, Dice similarity, Hough transform, and statistical tests such as Student’s t- 2.729 -test, and the Mann-Whitney U test (a non-parametric test). In addition, any classifier induces a scoring measure 2.730 -on genes by taking the prediction error when using that gene to predict the target. 2.731 -Using some combination of these measures, we will develop a procedure to find single marker genes for 2.732 -anatomical regions: for each cortical area, we will rank the genes by their ability to delineate each area. We 2.733 -will quantitatively compare the list of single genes generated by our method to the lists generated by previous 2.734 -methods which are mentioned in Aim 1 Related Work. 2.735 -Some cortical areas have no single marker genes but can be identified by combinatorial coding. This requires 2.736 -multivariate scoring measures and feature selection procedures. Many of the measures, such as expression 2.737 -energy, gradient similarity, Jaccard, Dice, Hough, Student’s t, and Mann-Whitney U are univariate. We will extend 2.738 -these scoring measures for use in multivariate feature selection, that is, for scoring how well combinations of 2.739 -genes, rather than individual genes, can distinguish a target area. There are existing multivariate forms of some 2.740 -of the univariate scoring measures, for example, Hotelling’s T-square is a multivariate analog of Student’s t. 2.741 -We will develop a feature selection procedure for choosing the best small set of marker genes for a given 2.742 -anatomical area. In addition to using the scoring measures that we develop, we will also explore (a) feature 2.743 -selection using a stepwise wrapper over “vanilla” classifiers such as logistic regression, (b) supervised learning 2.744 -methods such as decision trees which incrementally/greedily combine single gene markers into sets, and (c) 2.745 -supervised learning methods which use soft constraints to minimize number of features used, such as sparse 2.746 -support vector machines (SVMs). 2.747 +measures. We already developed one entirely new scoring method (gradient similarity), but we may develop 2.748 +more. Scoring measures that we will explore will include the L1 norm, correlation, expression energy ratio, con- 2.749 +ditional entropy, gradient similarity, Jaccard similarity, Dice similarity, Hough transform, and statistical tests such 2.750 +as Student’s t-test, and the Mann-Whitney U test (a non-parametric test). In addition, any classifier induces a 2.751 +scoring measure on genes by taking the prediction error when using that gene to predict the target. 2.752 + 2.753 + 2.754 + 2.755 + 2.756 +Figure 6: First row: the first 6 reduced dimensions, using PCA. Sec- 2.757 +ond row: the first 6 reduced dimensions, using NNMF. Third row: 2.758 +the first six reduced dimensions, using landmark Isomap. Bottom 2.759 +row: examples of kmeans clustering applied to reduced datasets 2.760 +to find 7 clusters. Left: 19 of the major subdivisions of the cortex. 2.761 +Second from left: PCA. Third from left: NNMF. Right: Landmark 2.762 +Isomap. Additional details: In the third and fourth rows, 7 dimen- 2.763 +sions were found, but only 6 displayed. In the last row: for PCA, 2.764 +50 dimensions were used; for NNMF, 6 dimensions were used; for 2.765 +landmark Isomap, 7 dimensions were used. Using some combination of these mea- 2.766 + sures, we will develop a procedure to 2.767 + find single marker genes for anatomical 2.768 + regions: for each cortical area, we will 2.769 + rank the genes by their ability to delineate 2.770 + each area. We will quantitatively compare 2.771 + the list of single genes generated by our 2.772 + method to the lists generated by previous 2.773 + methods which are mentioned in Aim 1 Re- 2.774 + lated Work. 2.775 + Some cortical areas have no single 2.776 + marker genes but can be identified by com- 2.777 + binatorial coding. This requires multivari- 2.778 + ate scoring measures and feature selec- 2.779 + tion procedures. Many of the measures, 2.780 + such as expression energy, gradient sim- 2.781 + ilarity, Jaccard, Dice, Hough, Student’s t, 2.782 + and Mann-Whitney U are univariate. We 2.783 + will extend these scoring measures for use 2.784 + in multivariate feature selection, that is, for 2.785 + scoring how well combinations of genes, 2.786 + rather than individual genes, can distin- 2.787 + guish a target area. There are existing 2.788 + multivariate forms of some of the univariate 2.789 + scoring measures, for example, Hotelling’s 2.790 + T-square is a multivariate analog of Stu- 2.791 + dent’s t. 2.792 + We will develop a feature selection pro- 2.793 + cedure for choosing the best small set of 2.794 +marker genes for a given anatomical area. In addition to using the scoring measures that we develop, we will 2.795 +also explore (a) feature selection using a stepwise wrapper over “vanilla” classifiers such as logistic regression, 2.796 +(b) supervised learning methods such as decision trees which incrementally/greedily combine single gene mark- 2.797 +ers into sets, and (c) supervised learning methods which use soft constraints to minimize number of features 2.798 +used, such as sparse support vector machines (SVMs). 2.799 Since errors of displacement and of shape may cause genes and target areas to match less than they should, 2.800 we will consider the robustness of feature selection methods in the presence of error. Some of these methods, 2.801 such as the Hough transform, are designed to be resistant in the presence of error, but many are not. We will 2.802 @@ -609,6 +597,10 @@ 2.803 recognized by anatomists. We will extend our procedure to handle difficult areas by combining areas or redrawing 2.804 their boundaries. We will develop extensions to our procedure which (a) detect when a difficult area could be 2.805 fit if its boundary were redrawn slightly11, and (b) detect when a difficult area could be combined with adjacent 2.806 +_________________________________________ 2.807 + 11Not just any redrawing is acceptable, only those which appear to be justified as a natural spatial domain of gene expression by 2.808 +multiple sources of evidence. Interestingly, the need to detect “natural spatial domains of gene expression” in a data-driven fashion 2.809 +means that the methods of Aim 2 might be useful in achieving Aim 1, as well – particularly discriminative dimensionality reduction. 2.810 areas to create a larger area which can be fit. 2.811 A future publication on the method that we develop in Aim 1 will review the scoring measures and quantita- 2.812 tively compare their performance in order to provide a foundation for future research of methods of marker gene 2.813 @@ -621,13 +613,24 @@ 2.814 naive bayes), kernel density estimation, instance-based learning methods (such as k-nearest neighbor), genetic 2.815 algorithms, and artificial neural networks. 2.816 Develop algorithms to suggest a division of a structure into anatomical parts 2.817 -Dimensionality reduction on gene expression profiles We have already described the application of ten 2.818 -dimensionality reduction algorithms for the purpose of replacing the gene expression profiles, which are vectors 2.819 -of about 4000 gene expression levels, with a smaller number of features. We plan to further explore and interpret 2.820 -these results, as well as to apply other unsupervised learning algorithms, including independent components 2.821 -analysis, self-organizing maps, and generative models such as deep Boltzmann machines. We will explore ways 2.822 -to quantitatively compare the relevance of the different dimensionality reduction methods for identifying cortical 2.823 -areal boundaries. 2.824 + 2.825 +Figure 7: Prototypes corresponding to sample gene 2.826 +clusters, clustered by gradient similarity. Region bound- 2.827 +aries for the region that most matches each prototype 2.828 +are overlaid. Dimensionality reduction on gene expression pro- 2.829 + files We have already described the application of 2.830 + ten dimensionality reduction algorithms for the pur- 2.831 + pose of replacing the gene expression profiles, which 2.832 + are vectors of about 4000 gene expression levels, 2.833 + with a smaller number of features. We plan to fur- 2.834 + ther explore and interpret these results, as well as to 2.835 + apply other unsupervised learning algorithms, includ- 2.836 + ing independent components analysis, self-organizing 2.837 + maps, and generative models such as deep Boltz- 2.838 + mann machines. We will explore ways to quantitatively 2.839 + compare the relevance of the different dimensionality 2.840 + reduction methods for identifying cortical areal bound- 2.841 + aries. 2.842 Dimensionality reduction on pixels Instead of applying dimensionality reduction to the gene expression 2.843 profiles, the same techniques can be applied instead to the pixels. It is possible that the features generated in 2.844 this way by some dimensionality reduction techniques will directly correspond to interesting spatial regions. 2.845 @@ -648,10 +651,7 @@ 2.846 Co-clustering There are some algorithms which simultaneously incorporate clustering on instances and on 2.847 features (in our case, genes and pixels), for example, IRM[11]. These are called co-clustering or biclustering 2.848 _________________________________________ 2.849 - 11Not just any redrawing is acceptable, only those which appear to be justified as a natural spatial domain of gene expression by 2.850 -multiple sources of evidence. Interestingly, the need to detect “natural spatial domains of gene expression” in a data-driven fashion 2.851 -means that the methods of Aim 2 might be useful in achieving Aim 1, as well – particularly discriminative dimensionality reduction. 2.852 - 12Actually, we have already begun to explore decision trees. For each cortical area, we have used the C4.5 algorithm to find a decision 2.853 + 12Actually, we have already begun to explore decision trees. For each cortical area, we have used the C4.5 algorithm to find a decision 2.854 tree for that area. We achieved good classification accuracy on our training set, but the number of genes that appeared in each tree was 2.855 too large. We plan to implement a pruning procedure to generate trees that use fewer genes. 2.856 algorithms.
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5.1 --- a/grant.txt Wed Apr 22 07:26:09 2009 -0700 5.2 +++ b/grant.txt Wed Apr 22 07:35:14 2009 -0700 5.3 @@ -226,6 +226,18 @@ 5.4 5.5 5.6 === Aim 3: apply the methods developed to the cerebral cortex === 5.7 +\begin{wrapfigure}{L}{0.35\textwidth}\centering 5.8 +%%\includegraphics[scale=.27]{singlegene_SS_corr_top_1_2365_jet.eps}\includegraphics[scale=.27]{singlegene_SS_corr_top_2_242_jet.eps}\includegraphics[scale=.27]{singlegene_SS_corr_top_3_654_jet.eps} 5.9 +%%\\ 5.10 +%%\includegraphics[scale=.27]{singlegene_SS_lr_top_1_654_jet.eps}\includegraphics[scale=.27]{singlegene_SS_lr_top_2_685_jet.eps}\includegraphics[scale=.27]{singlegene_SS_lr_top_3_724_jet.eps} 5.11 +%%\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 SS. Pixels are colored according to correlation, with red meaning high correlation and blue meaning low.} 5.12 + 5.13 +\includegraphics[scale=.27]{singlegene_SS_corr_top_1_2365_jet.eps}\includegraphics[scale=.27]{singlegene_SS_corr_top_2_242_jet.eps} 5.14 +\\ 5.15 +\includegraphics[scale=.27]{singlegene_SS_lr_top_1_654_jet.eps}\includegraphics[scale=.27]{singlegene_SS_lr_top_2_685_jet.eps} 5.16 + 5.17 +\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 SS. Pixels are colored according to correlation, with red meaning high correlation and blue meaning low.} 5.18 +\label{SScorrLr}\end{wrapfigure} 5.19 5.20 5.21 \vspace{0.3cm}**Background** 5.22 @@ -256,6 +268,8 @@ 5.23 5.24 === Related work === 5.25 5.26 + 5.27 + 5.28 \cite{ng_anatomic_2009} describes the application of AGEA to the cortex. The paper describes interesting results on the structure of correlations between voxel gene expression profiles within a handful of cortical areas. However, this sort of analysis is not related to either of our aims, as it neither finds marker genes, nor does it suggest a cortical map based on gene expression data. Neither of the other components of AGEA can be applied to cortical areas; AGEA's Gene Finder cannot be used to find marker genes for the cortical areas; and AGEA's hierarchical clustering does not produce clusters corresponding to the cortical areas\footnote{In both cases, the cause is that pairwise correlations between the gene expression of voxels in different areas but the same layer are often stronger than pairwise correlations between the gene expression of voxels in different layers but the same area. Therefore, a pairwise voxel correlation clustering algorithm will tend to create clusters representing cortical layers, not areas.}. 5.29 5.30 %% (there may be clusters which presumably correspond to the intersection of a layer and an area, but since one area will have many layer-area intersection clusters, further work is needed to make sense of these). The reason that Gene Finder cannot the find marker genes for cortical areas is that, although the user chooses a seed voxel, Gene Finder chooses the ROI for which genes will be found, and it creates that ROI by (pairwise voxel correlation) clustering around the seed. 5.31 @@ -268,6 +282,10 @@ 5.32 5.33 5.34 == Significance == 5.35 +\begin{wrapfigure}{L}{0.2\textwidth}\centering 5.36 +\includegraphics[scale=.27]{holeExample_2682_SS_jet.eps} 5.37 +\caption{Gene $Pitx2$ is selectively underexpressed in area SS.} 5.38 +\label{hole}\end{wrapfigure} 5.39 5.40 5.41 5.42 @@ -293,20 +311,6 @@ 5.43 \vspace{0.3cm}\hrule 5.44 5.45 == The approach: Preliminary Studies == 5.46 -\begin{wrapfigure}{L}{0.35\textwidth}\centering 5.47 -%%\includegraphics[scale=.27]{singlegene_SS_corr_top_1_2365_jet.eps}\includegraphics[scale=.27]{singlegene_SS_corr_top_2_242_jet.eps}\includegraphics[scale=.27]{singlegene_SS_corr_top_3_654_jet.eps} 5.48 -%%\\ 5.49 -%%\includegraphics[scale=.27]{singlegene_SS_lr_top_1_654_jet.eps}\includegraphics[scale=.27]{singlegene_SS_lr_top_2_685_jet.eps}\includegraphics[scale=.27]{singlegene_SS_lr_top_3_724_jet.eps} 5.50 -%%\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 SS. Pixels are colored according to correlation, with red meaning high correlation and blue meaning low.} 5.51 - 5.52 -\includegraphics[scale=.27]{singlegene_SS_corr_top_1_2365_jet.eps}\includegraphics[scale=.27]{singlegene_SS_corr_top_2_242_jet.eps} 5.53 -\\ 5.54 -\includegraphics[scale=.27]{singlegene_SS_lr_top_1_654_jet.eps}\includegraphics[scale=.27]{singlegene_SS_lr_top_2_685_jet.eps} 5.55 - 5.56 -\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 SS. Pixels are colored according to correlation, with red meaning high correlation and blue meaning low.} 5.57 -\label{SScorrLr}\end{wrapfigure} 5.58 - 5.59 - 5.60 5.61 === Format conversion between SEV, MATLAB, NIFTI === 5.62 We have created software to (politely) download all of the SEV files\footnote{SEV is a sparse format for spatial data. It is the format in which the ABA data is made available.} from the Allen Institute website. We have also created software to convert between the SEV, MATLAB, and NIFTI file formats, as well as some of Caret's file formats. 5.63 @@ -328,61 +332,6 @@ 5.64 5.65 5.66 === Feature selection and scoring methods === 5.67 - 5.68 -\begin{wrapfigure}{L}{0.2\textwidth}\centering 5.69 -\includegraphics[scale=.27]{holeExample_2682_SS_jet.eps} 5.70 -\caption{Gene $Pitx2$ is selectively underexpressed in area SS.} 5.71 -\label{hole}\end{wrapfigure} 5.72 - 5.73 - 5.74 - 5.75 -\vspace{0.3cm}**Underexpression of a gene can serve as a marker** 5.76 -Underexpression of a gene can sometimes serve as a marker. See, for example, Figure \ref{hole}. 5.77 - 5.78 - 5.79 - 5.80 - 5.81 -\vspace{0.3cm}**Correlation** 5.82 -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. 5.83 - 5.84 -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. 5.85 - 5.86 -%%One class of feature selection scoring methods contains methods which calculate some sort of "match" between each gene image and the target image. Those genes which match the best are good candidates for features. 5.87 - 5.88 -%%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. 5.89 - 5.90 - 5.91 - 5.92 -\vspace{0.3cm}**Conditional entropy** 5.93 -%%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. 5.94 - 5.95 -%%The simplest way to use information theory is on discrete data, so we discretized our gene expression data by creating, for each gene, five thresholded boolean masks of the gene data. For each gene, we created a boolean mask of its expression levels using each of these thresholds: the mean of that gene, the mean minus one standard deviation, the mean minus two standard deviations, the mean plus one standard deviation, the mean plus two standard deviations. 5.96 - 5.97 -%%Now, for each region, we created and ran a forward stepwise procedure which attempted to find pairs of gene expression boolean masks such that the conditional entropy of the target area's boolean mask, conditioned upon the pair of gene expression boolean masks, is minimized. 5.98 - 5.99 -For each region, we created and ran a forward stepwise procedure which attempted to find pairs of gene expression boolean masks such that the conditional entropy of the target area's boolean mask, conditioned upon the pair of gene expression boolean masks, is minimized. 5.100 - 5.101 -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. 5.102 - 5.103 - 5.104 - 5.105 - 5.106 -\vspace{0.3cm}**Gradient similarity** 5.107 -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 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". The formula is: 5.108 - 5.109 -%%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. 5.110 - 5.111 - 5.112 - 5.113 -\begin{align*} 5.114 -\sum_{pixel \in pixels} cos(abs(\angle \nabla_1 - \angle \nabla_2)) \cdot \frac{\vert \nabla_1 \vert + \vert \nabla_2 \vert}{2} \cdot \frac{pixel\_value_1 + pixel\_value_2}{2} 5.115 -\end{align*} 5.116 - 5.117 -where $\nabla_1$ and $\nabla_2$ are the gradient vectors of the two images at the current pixel; $\angle \nabla_i$ is the angle of the gradient of image $i$ at the current pixel; $\vert \nabla_i \vert$ is the magnitude of the gradient of image $i$ at the current pixel; and $pixel\_value_i$ is the value of the current pixel in image $i$. 5.118 - 5.119 -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). 5.120 - 5.121 -\vspace{0.3cm}**Gradient similarity provides information complementary to correlation** 5.122 \begin{wrapfigure}{L}{0.35\textwidth}\centering 5.123 %%\includegraphics[scale=.27]{singlegene_AUD_lr_top_1_3386_jet.eps}\includegraphics[scale=.27]{singlegene_AUD_lr_top_2_1258_jet.eps}\includegraphics[scale=.27]{singlegene_AUD_lr_top_3_420_jet.eps} 5.124 %% 5.125 @@ -396,44 +345,54 @@ 5.126 5.127 5.128 5.129 -To show that gradient similarity can provide useful information that cannot be detected via pointwise analyses, consider Fig. \ref{AUDgeometry}. 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. 5.130 - 5.131 -%%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. 5.132 - 5.133 -%% 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.} 5.134 - 5.135 -%% Genes which have high rankings using both pointwise and border criteria, such as $Aph1a$ in the example, may be particularly good markers. 5.136 - 5.137 -\vspace{0.3cm}**Areas which can be identified by single genes** 5.138 -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. 5.139 - 5.140 -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). 5.141 - 5.142 -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. 5.143 - 5.144 - 5.145 - 5.146 -\vspace{0.3cm}**Combinations of multiple genes are useful and necessary for some areas** 5.147 - 5.148 -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. According 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, but 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 dorsal 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. 5.149 - 5.150 -This shows that our proposal to develop a method to find combinations of marker genes is both possible and necessary. 5.151 - 5.152 -%% wwc1\footnote{"WW, C2 and coiled-coil domain containing 1"; EntrezGene ID 211652} 5.153 -%% mtif2\footnote{"mitochondrial translational initiation factor 2"; EntrezGene ID 76784} 5.154 - 5.155 -%%According 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, but 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. 5.156 - 5.157 -%%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. 5.158 - 5.159 - 5.160 - 5.161 - 5.162 -%%\vspace{0.3cm}**Feature selection integrated with prediction** 5.163 -%%As noted earlier, in general, any classifier can be used for feature selection by running it inside a stepwise wrapper. Also, some learning algorithms integrate soft constraints on number of features used. Examples of both of these will be seen in the section "Multivariate supervised learning". 5.164 - 5.165 - 5.166 -=== Multivariate supervised learning === 5.167 + 5.168 +\vspace{0.3cm}**Underexpression of a gene can serve as a marker** 5.169 +Underexpression of a gene can sometimes serve as a marker. See, for example, Figure \ref{hole}. 5.170 + 5.171 + 5.172 + 5.173 + 5.174 +\vspace{0.3cm}**Correlation** 5.175 +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. 5.176 + 5.177 +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. 5.178 + 5.179 +%%One class of feature selection scoring methods contains methods which calculate some sort of "match" between each gene image and the target image. Those genes which match the best are good candidates for features. 5.180 + 5.181 +%%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. 5.182 + 5.183 + 5.184 + 5.185 +\vspace{0.3cm}**Conditional entropy** 5.186 +%%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. 5.187 + 5.188 +%%The simplest way to use information theory is on discrete data, so we discretized our gene expression data by creating, for each gene, five thresholded boolean masks of the gene data. For each gene, we created a boolean mask of its expression levels using each of these thresholds: the mean of that gene, the mean minus one standard deviation, the mean minus two standard deviations, the mean plus one standard deviation, the mean plus two standard deviations. 5.189 + 5.190 +%%Now, for each region, we created and ran a forward stepwise procedure which attempted to find pairs of gene expression boolean masks such that the conditional entropy of the target area's boolean mask, conditioned upon the pair of gene expression boolean masks, is minimized. 5.191 + 5.192 +For each region, we created and ran a forward stepwise procedure which attempted to find pairs of gene expression boolean masks such that the conditional entropy of the target area's boolean mask, conditioned upon the pair of gene expression boolean masks, is minimized. 5.193 + 5.194 +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. 5.195 + 5.196 + 5.197 + 5.198 + 5.199 +\vspace{0.3cm}**Gradient similarity** 5.200 +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 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". The formula is: 5.201 + 5.202 +%%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. 5.203 + 5.204 + 5.205 + 5.206 +\begin{align*} 5.207 +\sum_{pixel \in pixels} cos(abs(\angle \nabla_1 - \angle \nabla_2)) \cdot \frac{\vert \nabla_1 \vert + \vert \nabla_2 \vert}{2} \cdot \frac{pixel\_value_1 + pixel\_value_2}{2} 5.208 +\end{align*} 5.209 + 5.210 +where $\nabla_1$ and $\nabla_2$ are the gradient vectors of the two images at the current pixel; $\angle \nabla_i$ is the angle of the gradient of image $i$ at the current pixel; $\vert \nabla_i \vert$ is the magnitude of the gradient of image $i$ at the current pixel; and $pixel\_value_i$ is the value of the current pixel in image $i$. 5.211 + 5.212 +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). 5.213 + 5.214 +\vspace{0.3cm}**Gradient similarity provides information complementary to correlation** 5.215 \begin{wrapfigure}{L}{0.35\textwidth}\centering 5.216 \includegraphics[scale=.27]{MO_vs_Wwc1_jet.eps}\includegraphics[scale=.27]{MO_vs_Mtif2_jet.eps} 5.217 5.218 @@ -443,6 +402,47 @@ 5.219 5.220 5.221 5.222 + 5.223 +To show that gradient similarity can provide useful information that cannot be detected via pointwise analyses, consider Fig. \ref{AUDgeometry}. 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. 5.224 + 5.225 +%%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. 5.226 + 5.227 +%% 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.} 5.228 + 5.229 +%% Genes which have high rankings using both pointwise and border criteria, such as $Aph1a$ in the example, may be particularly good markers. 5.230 + 5.231 +\vspace{0.3cm}**Areas which can be identified by single genes** 5.232 +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. 5.233 + 5.234 +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). 5.235 + 5.236 +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. 5.237 + 5.238 + 5.239 + 5.240 +\vspace{0.3cm}**Combinations of multiple genes are useful and necessary for some areas** 5.241 + 5.242 +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. According 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, but 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 dorsal 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. 5.243 + 5.244 +This shows that our proposal to develop a method to find combinations of marker genes is both possible and necessary. 5.245 + 5.246 +%% wwc1\footnote{"WW, C2 and coiled-coil domain containing 1"; EntrezGene ID 211652} 5.247 +%% mtif2\footnote{"mitochondrial translational initiation factor 2"; EntrezGene ID 76784} 5.248 + 5.249 +%%According 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, but 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. 5.250 + 5.251 +%%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. 5.252 + 5.253 + 5.254 + 5.255 + 5.256 +%%\vspace{0.3cm}**Feature selection integrated with prediction** 5.257 +%%As noted earlier, in general, any classifier can be used for feature selection by running it inside a stepwise wrapper. Also, some learning algorithms integrate soft constraints on number of features used. Examples of both of these will be seen in the section "Multivariate supervised learning". 5.258 + 5.259 + 5.260 +=== Multivariate supervised learning === 5.261 + 5.262 + 5.263 \vspace{0.3cm}**Forward stepwise logistic regression** 5.264 Logistic regression is a popular method for predictive modeling of categorical 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. 5.265 5.266 @@ -459,24 +459,6 @@ 5.267 5.268 === Data-driven redrawing of the cortical map === 5.269 5.270 - 5.271 - 5.272 - 5.273 - 5.274 -We have applied the following dimensionality reduction algorithms to reduce the dimensionality of the gene expression profile associated with each pixel: Principal Components Analysis (PCA), Simple PCA, Multi-Dimensional Scaling, Isomap, Landmark Isomap, Laplacian eigenmaps, Local Tangent Space Alignment, Stochastic Proximity Embedding, Fast Maximum Variance Unfolding, 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}. 5.275 - 5.276 - 5.277 - 5.278 -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 comparison of these techniques as applied to the domain of genomic anatomy. 5.279 - 5.280 - 5.281 - 5.282 - 5.283 -\vspace{0.3cm}**Many areas are captured by clusters of genes** 5.284 -We also clustered the genes using gradient similarity to see if the spatial regions defined by any clusters matched known anatomical regions. Figure \ref{geneClusters} shows, for ten sample gene clusters, each cluster's average expression pattern, compared to a known anatomical boundary. This suggests that it is worth attempting to cluster genes, and then to use the results to cluster pixels. 5.285 - 5.286 - 5.287 -== The approach: what we plan to do == 5.288 \begin{wrapfigure}{L}{0.35\textwidth}\centering 5.289 \includegraphics[scale=.27]{singlegene_example_2682_Pitx2_SS_jet.eps}\includegraphics[scale=.27]{singlegene_example_371_Aldh1a2_SSs_jet.eps} 5.290 \includegraphics[scale=.27]{singlegene_example_2759_Ppfibp1_PIR_jet.eps}\includegraphics[scale=.27]{singlegene_example_3310_Slco1a5_FRP_jet.eps} 5.291 @@ -489,6 +471,24 @@ 5.292 5.293 5.294 5.295 + 5.296 + 5.297 + 5.298 +We have applied the following dimensionality reduction algorithms to reduce the dimensionality of the gene expression profile associated with each pixel: Principal Components Analysis (PCA), Simple PCA, Multi-Dimensional Scaling, Isomap, Landmark Isomap, Laplacian eigenmaps, Local Tangent Space Alignment, Stochastic Proximity Embedding, Fast Maximum Variance Unfolding, 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}. 5.299 + 5.300 + 5.301 + 5.302 +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 comparison of these techniques as applied to the domain of genomic anatomy. 5.303 + 5.304 + 5.305 + 5.306 + 5.307 +\vspace{0.3cm}**Many areas are captured by clusters of genes** 5.308 +We also clustered the genes using gradient similarity to see if the spatial regions defined by any clusters matched known anatomical regions. Figure \ref{geneClusters} shows, for ten sample gene clusters, each cluster's average expression pattern, compared to a known anatomical boundary. This suggests that it is worth attempting to cluster genes, and then to use the results to cluster pixels. 5.309 + 5.310 + 5.311 +== The approach: what we plan to do == 5.312 + 5.313 %%\vspace{0.3cm}**Flatmap cortex and segment cortical layers** 5.314 5.315 === Flatmap cortex and segment cortical layers === 5.316 @@ -509,30 +509,6 @@ 5.317 5.318 5.319 === Develop algorithms that find genetic markers for anatomical regions === 5.320 - 5.321 -\vspace{0.3cm}**Scoring measures and feature selection** 5.322 -%%We will develop scoring methods for evaluating how good individual genes are at marking areas. We will compare pointwise, geometric, and information-theoretic measures. We already developed one entirely new scoring method (gradient similarity), but we may develop more. Scoring measures that we will explore will include the L1 norm, correlation, expression energy ratio, conditional entropy, gradient similarity, Jaccard similarity, Dice similarity, Hough transform, and statistical tests such as Hotelling's T-square test (a multivariate generalization of Student's t-test), ANOVA, and a multivariate version of the Mann-Whitney U test (a non-parametric test). 5.323 -We will develop scoring methods for evaluating how good individual genes are at marking areas. We will compare pointwise, geometric, and information-theoretic measures. We already developed one entirely new scoring method (gradient similarity), but we may develop more. Scoring measures that we will explore will include the L1 norm, correlation, expression energy ratio, conditional entropy, gradient similarity, Jaccard similarity, Dice similarity, Hough transform, and statistical tests such as Student's t-test, and the Mann-Whitney U test (a non-parametric test). In addition, any classifier induces a scoring measure on genes by taking the prediction error when using that gene to predict the target. 5.324 - 5.325 -Using some combination of these measures, we will develop a procedure to find single marker genes for anatomical regions: for each cortical area, we will rank the genes by their ability to delineate each area. We will quantitatively compare the list of single genes generated by our method to the lists generated by previous methods which are mentioned in Aim 1 Related Work. 5.326 - 5.327 - 5.328 -Some cortical areas have no single marker genes but can be identified by combinatorial coding. This requires multivariate scoring measures and feature selection procedures. Many of the measures, such as expression energy, gradient similarity, Jaccard, Dice, Hough, Student's t, and Mann-Whitney U are univariate. We will extend these scoring measures for use in multivariate feature selection, that is, for scoring how well combinations of genes, rather than individual genes, can distinguish a target area. There are existing multivariate forms of some of the univariate scoring measures, for example, Hotelling's T-square is a multivariate analog of Student's t. 5.329 - 5.330 -We will develop a feature selection procedure for choosing the best small set of marker genes for a given anatomical area. In addition to using the scoring measures that we develop, we will also explore (a) feature selection using a stepwise wrapper over "vanilla" classifiers such as logistic regression, (b) supervised learning methods such as decision trees which incrementally/greedily combine single gene markers into sets, and (c) supervised learning methods which use soft constraints to minimize number of features used, such as sparse support vector machines (SVMs). 5.331 - 5.332 -Since errors of displacement and of shape may cause genes and target areas to match less than they should, we will consider the robustness of feature selection methods in the presence of error. Some of these methods, such as the Hough transform, are designed to be resistant in the presence of error, but many are not. We will consider extensions to scoring measures that may improve their robustness; for example, a wrapper that runs a scoring method on small displacements and distortions of the data adds robustness to registration error at the expense of computation time. 5.333 - 5.334 -An area may be difficult to identify because the boundaries are misdrawn in the atlas, or because the shape of the natural domain of gene expression corresponding to the area is different from the shape of the area as recognized by anatomists. We will extend our procedure to handle difficult areas by combining areas or redrawing their boundaries. We will develop extensions to our procedure which (a) detect when a difficult area could be fit if its boundary were redrawn slightly\footnote{Not just any redrawing is acceptable, only those which appear to be justified as a natural spatial domain of gene expression by multiple sources of evidence. Interestingly, the need to detect "natural spatial domains of gene expression" in a data-driven fashion means that the methods of Aim 2 might be useful in achieving Aim 1, as well -- particularly discriminative dimensionality reduction.}, and (b) detect when a difficult area could be combined with adjacent areas to create a larger area which can be fit. 5.335 - 5.336 -A future publication on the method that we develop in Aim 1 will review the scoring measures and quantitatively compare their performance in order to provide a foundation for future research of methods of marker gene finding. We will measure the robustness of the scoring measures as well as their absolute performance on our dataset. 5.337 - 5.338 -\vspace{0.3cm}**Classifiers** 5.339 -We will explore and compare different classifiers. As noted above, this activity is not separate from the previous one, because some supervised learning algorithms include feature selection, and any classifier can be combined with a stepwise wrapper for use as a feature selection method. We will explore logistic regression (including spatial models\cite{paciorek_computational_2007}), decision trees\footnote{Actually, we have already begun to explore decision trees. For each cortical area, we have used the C4.5 algorithm to find a decision tree for that area. We achieved good classification accuracy on our training set, but the number of genes that appeared in each tree was too large. We plan to implement a pruning procedure to generate trees that use fewer genes.}, sparse SVMs, generative mixture models (including naive bayes), kernel density estimation, instance-based learning methods (such as k-nearest neighbor), genetic algorithms, and artificial neural networks. 5.340 - 5.341 - 5.342 - 5.343 -=== Develop algorithms to suggest a division of a structure into anatomical parts === 5.344 \begin{wrapfigure}{L}{0.6\textwidth}\centering 5.345 \includegraphics[scale=1]{merge3_norm_hv_PCA_ndims50_prototypes_collage_sm_border.eps} 5.346 \includegraphics[scale=.98]{nnmf_ndims7_collage_border.eps} 5.347 @@ -542,33 +518,58 @@ 5.348 \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.} 5.349 \label{dimReduc}\end{wrapfigure} 5.350 5.351 -\vspace{0.3cm}**Dimensionality reduction on gene expression profiles** 5.352 -We have already described the application of ten dimensionality reduction algorithms for the purpose of replacing the gene expression profiles, which are vectors of about 4000 gene expression levels, with a smaller number of features. We plan to further explore and interpret these results, as well as to apply other unsupervised learning algorithms, including independent components analysis, self-organizing maps, and generative models such as deep Boltzmann machines. We will explore ways to quantitatively compare the relevance of the different dimensionality reduction methods for identifying cortical areal boundaries. 5.353 - 5.354 -\vspace{0.3cm}**Dimensionality reduction on pixels** 5.355 -Instead of applying dimensionality reduction to the gene expression profiles, the same techniques can be applied instead to the pixels. It is possible that the features generated in this way by some dimensionality reduction techniques will directly correspond to interesting spatial regions. 5.356 - 5.357 -%% \footnote{Consider a matrix whose rows represent pixel locations, and whose columns represent genes. An entry in this matrix represents the gene expression level at a given pixel. One can look at this matrix as a collection of pixels, each corresponding to a vector of many gene expression levels; or one can look at it as a collection of genes, each corresponding to a vector giving that gene's expression at each pixel. Similarly, dimensionality reduction can be used to replace a large number of genes with a small number of features, or it can be used to replace a large number of pixels with a small number of features.} 5.358 - 5.359 -\vspace{0.3cm}**Clustering and segmentation on pixels** 5.360 -We will explore clustering and segmentation algorithms in order to segment the pixels into regions. We will explore k-means, spectral clustering, gene shaving\cite{hastie_gene_2000}, recursive division clustering, multivariate generalizations of edge detectors, multivariate generalizations of watershed transformations, region growing, active contours, graph partitioning methods, and recursive agglomerative clustering with various linkage functions. These methods can be combined with dimensionality reduction. 5.361 - 5.362 -\vspace{0.3cm}**Clustering on genes** 5.363 -We have already shown that the procedure of clustering genes according to gradient similarity, and then creating an averaged prototype of each cluster's expression pattern, yields some spatial patterns which match cortical areas. We will further explore the clustering of genes. 5.364 - 5.365 -In addition to using the cluster expression prototypes directly to identify spatial regions, this might be useful as a component of dimensionality reduction. For example, one could imagine clustering similar genes and then replacing their expression levels with a single average expression level, thereby removing some redundancy from the gene expression profiles. One could then perform clustering on pixels (possibly after a second dimensionality reduction step) in order to identify spatial regions. It remains to be seen whether removal of redundancy would help or hurt the ultimate goal of identifying interesting spatial regions. 5.366 - 5.367 -\vspace{0.3cm}**Co-clustering** 5.368 -There are some algorithms which simultaneously incorporate clustering on instances and on features (in our case, genes and pixels), for example, IRM\cite{kemp_learning_2006}. These are called co-clustering or biclustering algorithms. 5.369 - 5.370 -\vspace{0.3cm}**Radial profiles** 5.371 -We wil explore the use of the radial profile of gene expression under each pixel. 5.372 +\vspace{0.3cm}**Scoring measures and feature selection** 5.373 +%%We will develop scoring methods for evaluating how good individual genes are at marking areas. We will compare pointwise, geometric, and information-theoretic measures. We already developed one entirely new scoring method (gradient similarity), but we may develop more. Scoring measures that we will explore will include the L1 norm, correlation, expression energy ratio, conditional entropy, gradient similarity, Jaccard similarity, Dice similarity, Hough transform, and statistical tests such as Hotelling's T-square test (a multivariate generalization of Student's t-test), ANOVA, and a multivariate version of the Mann-Whitney U test (a non-parametric test). 5.374 +We will develop scoring methods for evaluating how good individual genes are at marking areas. We will compare pointwise, geometric, and information-theoretic measures. We already developed one entirely new scoring method (gradient similarity), but we may develop more. Scoring measures that we will explore will include the L1 norm, correlation, expression energy ratio, conditional entropy, gradient similarity, Jaccard similarity, Dice similarity, Hough transform, and statistical tests such as Student's t-test, and the Mann-Whitney U test (a non-parametric test). In addition, any classifier induces a scoring measure on genes by taking the prediction error when using that gene to predict the target. 5.375 + 5.376 +Using some combination of these measures, we will develop a procedure to find single marker genes for anatomical regions: for each cortical area, we will rank the genes by their ability to delineate each area. We will quantitatively compare the list of single genes generated by our method to the lists generated by previous methods which are mentioned in Aim 1 Related Work. 5.377 + 5.378 + 5.379 +Some cortical areas have no single marker genes but can be identified by combinatorial coding. This requires multivariate scoring measures and feature selection procedures. Many of the measures, such as expression energy, gradient similarity, Jaccard, Dice, Hough, Student's t, and Mann-Whitney U are univariate. We will extend these scoring measures for use in multivariate feature selection, that is, for scoring how well combinations of genes, rather than individual genes, can distinguish a target area. There are existing multivariate forms of some of the univariate scoring measures, for example, Hotelling's T-square is a multivariate analog of Student's t. 5.380 + 5.381 +We will develop a feature selection procedure for choosing the best small set of marker genes for a given anatomical area. In addition to using the scoring measures that we develop, we will also explore (a) feature selection using a stepwise wrapper over "vanilla" classifiers such as logistic regression, (b) supervised learning methods such as decision trees which incrementally/greedily combine single gene markers into sets, and (c) supervised learning methods which use soft constraints to minimize number of features used, such as sparse support vector machines (SVMs). 5.382 + 5.383 +Since errors of displacement and of shape may cause genes and target areas to match less than they should, we will consider the robustness of feature selection methods in the presence of error. Some of these methods, such as the Hough transform, are designed to be resistant in the presence of error, but many are not. We will consider extensions to scoring measures that may improve their robustness; for example, a wrapper that runs a scoring method on small displacements and distortions of the data adds robustness to registration error at the expense of computation time. 5.384 + 5.385 +An area may be difficult to identify because the boundaries are misdrawn in the atlas, or because the shape of the natural domain of gene expression corresponding to the area is different from the shape of the area as recognized by anatomists. We will extend our procedure to handle difficult areas by combining areas or redrawing their boundaries. We will develop extensions to our procedure which (a) detect when a difficult area could be fit if its boundary were redrawn slightly\footnote{Not just any redrawing is acceptable, only those which appear to be justified as a natural spatial domain of gene expression by multiple sources of evidence. Interestingly, the need to detect "natural spatial domains of gene expression" in a data-driven fashion means that the methods of Aim 2 might be useful in achieving Aim 1, as well -- particularly discriminative dimensionality reduction.}, and (b) detect when a difficult area could be combined with adjacent areas to create a larger area which can be fit. 5.386 + 5.387 +A future publication on the method that we develop in Aim 1 will review the scoring measures and quantitatively compare their performance in order to provide a foundation for future research of methods of marker gene finding. We will measure the robustness of the scoring measures as well as their absolute performance on our dataset. 5.388 + 5.389 +\vspace{0.3cm}**Classifiers** 5.390 +We will explore and compare different classifiers. As noted above, this activity is not separate from the previous one, because some supervised learning algorithms include feature selection, and any classifier can be combined with a stepwise wrapper for use as a feature selection method. We will explore logistic regression (including spatial models\cite{paciorek_computational_2007}), decision trees\footnote{Actually, we have already begun to explore decision trees. For each cortical area, we have used the C4.5 algorithm to find a decision tree for that area. We achieved good classification accuracy on our training set, but the number of genes that appeared in each tree was too large. We plan to implement a pruning procedure to generate trees that use fewer genes.}, sparse SVMs, generative mixture models (including naive bayes), kernel density estimation, instance-based learning methods (such as k-nearest neighbor), genetic algorithms, and artificial neural networks. 5.391 + 5.392 + 5.393 + 5.394 + 5.395 +=== Develop algorithms to suggest a division of a structure into anatomical parts === 5.396 5.397 \begin{wrapfigure}{L}{0.5\textwidth}\centering 5.398 \includegraphics[scale=.2]{cosine_similarity1_rearrange_colorize.eps} 5.399 \caption{Prototypes corresponding to sample gene clusters, clustered by gradient similarity. Region boundaries for the region that most matches each prototype are overlaid.} 5.400 \label{geneClusters}\end{wrapfigure} 5.401 5.402 +\vspace{0.3cm}**Dimensionality reduction on gene expression profiles** 5.403 +We have already described the application of ten dimensionality reduction algorithms for the purpose of replacing the gene expression profiles, which are vectors of about 4000 gene expression levels, with a smaller number of features. We plan to further explore and interpret these results, as well as to apply other unsupervised learning algorithms, including independent components analysis, self-organizing maps, and generative models such as deep Boltzmann machines. We will explore ways to quantitatively compare the relevance of the different dimensionality reduction methods for identifying cortical areal boundaries. 5.404 + 5.405 +\vspace{0.3cm}**Dimensionality reduction on pixels** 5.406 +Instead of applying dimensionality reduction to the gene expression profiles, the same techniques can be applied instead to the pixels. It is possible that the features generated in this way by some dimensionality reduction techniques will directly correspond to interesting spatial regions. 5.407 + 5.408 +%% \footnote{Consider a matrix whose rows represent pixel locations, and whose columns represent genes. An entry in this matrix represents the gene expression level at a given pixel. One can look at this matrix as a collection of pixels, each corresponding to a vector of many gene expression levels; or one can look at it as a collection of genes, each corresponding to a vector giving that gene's expression at each pixel. Similarly, dimensionality reduction can be used to replace a large number of genes with a small number of features, or it can be used to replace a large number of pixels with a small number of features.} 5.409 + 5.410 +\vspace{0.3cm}**Clustering and segmentation on pixels** 5.411 +We will explore clustering and segmentation algorithms in order to segment the pixels into regions. We will explore k-means, spectral clustering, gene shaving\cite{hastie_gene_2000}, recursive division clustering, multivariate generalizations of edge detectors, multivariate generalizations of watershed transformations, region growing, active contours, graph partitioning methods, and recursive agglomerative clustering with various linkage functions. These methods can be combined with dimensionality reduction. 5.412 + 5.413 +\vspace{0.3cm}**Clustering on genes** 5.414 +We have already shown that the procedure of clustering genes according to gradient similarity, and then creating an averaged prototype of each cluster's expression pattern, yields some spatial patterns which match cortical areas. We will further explore the clustering of genes. 5.415 + 5.416 +In addition to using the cluster expression prototypes directly to identify spatial regions, this might be useful as a component of dimensionality reduction. For example, one could imagine clustering similar genes and then replacing their expression levels with a single average expression level, thereby removing some redundancy from the gene expression profiles. One could then perform clustering on pixels (possibly after a second dimensionality reduction step) in order to identify spatial regions. It remains to be seen whether removal of redundancy would help or hurt the ultimate goal of identifying interesting spatial regions. 5.417 + 5.418 +\vspace{0.3cm}**Co-clustering** 5.419 +There are some algorithms which simultaneously incorporate clustering on instances and on features (in our case, genes and pixels), for example, IRM\cite{kemp_learning_2006}. These are called co-clustering or biclustering algorithms. 5.420 + 5.421 +\vspace{0.3cm}**Radial profiles** 5.422 +We wil explore the use of the radial profile of gene expression under each pixel. 5.423 + 5.424 \vspace{0.3cm}**Compare different methods** 5.425 In order to tell which method is best for genomic anatomy, for each experimental method we will compare the cortical map found by unsupervised learning to a cortical map derived from the Allen Reference Atlas. We will explore various quantitative metrics that purport to measure how similar two clusterings are, such as Jaccard, Rand index, Fowlkes-Mallows, variation of information, Larsen, Van Dongen, and others. 5.426