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changeset 92:b4b79f107b2a
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author | bshanks@bshanks.dyndns.org |
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date | Tue Apr 21 14:28:12 2009 -0700 (16 years ago) |
parents | 7c5d98f0cd5a |
children | 9f36acf8d9a8 |
files | grant.html grant.odt grant.pdf grant.txt |
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1.1 --- a/grant.html Tue Apr 21 06:11:15 2009 -0700
1.2 +++ b/grant.html Tue Apr 21 14:28:12 2009 -0700
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1.4 efficiently find marker genes for anatomical regions, or to use gene expression to discover new anatomical patterning. As
1.5 described above, marker genes have a variety of uses in the development of drugs and experimental manipulations, and in
1.6 the anatomical characterization of tissue samples. The discovery of new ways to carve up anatomical structures into regions
1.7 -will widely impact all areas of biology.
1.8 +may lead to the discovery of new anatomical subregions in various structures, which will widely impact all areas of biology.
1.9 +Although our particular application involves the 3D spatial distribution of gene expression, we anticipate that the
1.10 +methods developed in aims (1) and (2) will not be limited to gene expression data, but rather will generalize to any sort of
1.11 +high-dimensional data over points located in a low-dimensional space.
1.12 The approach: Preliminary Studies
1.13 +Format conversion between SEV, MATLAB, NIFTI
1.14 +We have created software to (politely) download all of the SEV files16 from the Allen Institute website. We have also created
1.15 +software to convert between the SEV, MATLAB, and NIFTI file formats, as well as some of Caret’s file formats.
1.16 +Flatmap of cortex
1.17 +We downloaded the ABA data and applied a mask to select only those voxels which belong to cerebral cortex. We divided
1.18 +the cortex into hemispheres.
1.19 +Using Caret[7], we created a mesh representation of the surface of the selected voxels. For each gene, for each node of
1.20 +the mesh, we calculated an average of the gene expression of the voxels “underneath” that mesh node. We then flattened
1.21 +the cortex, creating a two-dimensional mesh.
1.22 +We sampled the nodes of the irregular, flat mesh in order to create a regular grid of pixel values. We converted this grid
1.23 +into a MATLAB matrix.
1.24 +We manually traced the boundaries of each of 49 cortical areas from the ABA coronal reference atlas slides. We then
1.25 +converted these manual traces into Caret-format regional boundary data on the mesh surface. We projected the regions
1.26 +onto the 2-d mesh, and then onto the grid, and then we converted the region data into MATLAB format.
1.27 +At this point, the data are in the form of a number of 2-D matrices, all in registration, with the matrix entries representing
1.28 +a grid of points (pixels) over the cortical surface:
1.29 +_
1.30 + 16SEV is a sparse format for spatial data. It is the format in which the ABA data is made available.
1.31 +
1.32
1.33
1.34 Figure 1: Top row: Genes Nfic and
1.35 @@ -337,35 +359,9 @@
1.36 The red outline is the boundary of region
1.37 SS. Pixels are colored according to correla-
1.38 tion, with red meaning high correlation and
1.39 -blue meaning low. Format conversion between SEV, MATLAB, NIFTI
1.40 - We have created software to (politely) download all of the SEV files16 from
1.41 - the Allen Institute website. We have also created software to convert between
1.42 - the SEV, MATLAB, and NIFTI file formats, as well as some of Caret’s file
1.43 - formats.
1.44 - Flatmap of cortex
1.45 - We downloaded the ABA data and applied a mask to select only those voxels
1.46 - which belong to cerebral cortex. We divided the cortex into hemispheres.
1.47 - Using Caret[7], we created a mesh representation of the surface of the se-
1.48 - lected voxels. For each gene, for each node of the mesh, we calculated an
1.49 - average of the gene expression of the voxels “underneath” that mesh node. We
1.50 - then flattened the cortex, creating a two-dimensional mesh.
1.51 - We sampled the nodes of the irregular, flat mesh in order to create a regular
1.52 - grid of pixel values. We converted this grid into a MATLAB matrix.
1.53 - We manually traced the boundaries of each of 49 cortical areas from the
1.54 - ABA coronal reference atlas slides. We then converted these manual traces
1.55 - into Caret-format regional boundary data on the mesh surface. We projected
1.56 - the regions onto the 2-d mesh, and then onto the grid, and then we converted
1.57 - the region data into MATLAB format.
1.58 - At this point, the data are in the form of a number of 2-D matrices, all in
1.59 - registration, with the matrix entries representing a grid of points (pixels) over
1.60 - the cortical surface:
1.61 - ∙ A 2-D matrix whose entries represent the regional label associated with each
1.62 - surface pixel
1.63 - ∙ For each gene, a 2-D matrix whose entries represent the average expression
1.64 - level underneath each surface pixel
1.65 -_________________________________________
1.66 - 16SEV is a sparse format for spatial data. It is the format in which the ABA data is made available.
1.67 -
1.68 +blue meaning low.
1.69 +∙A 2-D matrix whose entries represent the regional label associated with each surface pixel
1.70 +∙For each gene, a 2-D matrix whose entries represent the average expression level underneath each surface pixel
1.71
1.72 Figure 2: Gene Pitx2
1.73 is selectively underex-
1.74 @@ -380,8 +376,8 @@
1.75 Cortical layers are found at different depths in different parts of the cortex. In preparation for
1.76 extracting the layer-specific datasets, we have extended Caret with routines that allow the depth
1.77 of the ROI for volume-to-surface projection to vary.
1.78 -In the Research Plan, we describe how we will automatically locate the layer depths. For validation, we have manually
1.79 -demarcated the depth of the outer boundary of cortical layer 5 throughout the cortex.
1.80 + In the Research Plan, we describe how we will automatically locate the layer depths. For
1.81 +validation, we have manually demarcated the depth of the outer boundary of cortical layer 5 throughout the cortex.
1.82 Feature selection and scoring methods
1.83 Underexpression of a gene can serve as a marker Underexpression of a gene can sometimes serve as a marker. See,
1.84 for example, Figure 2.
1.85 @@ -392,6 +388,7 @@
1.86 image and the target image. Those genes which match the best are good candidates for features.
1.87 One of the simplest methods in this class is to use correlation as the match score. We calculated the correlation between
1.88 each gene and each cortical area. The top row of Figure 1 shows the three genes most correlated with area SS.
1.89 +
1.90
1.91
1.92 Figure 3: The top row shows the two genes
1.93 @@ -423,78 +420,53 @@
1.94 sion is that it takes account of arbitrarily nonlinear relationships; for example,
1.95 if the XOR of two variables predicts the target, conditional entropy would
1.96 notice, whereas linear methods would not.
1.97 - Gradient similarity We noticed that the previous two scoring methods,
1.98 -which are pointwise, often found genes whose pattern of expression did not look similar in shape to the target region. For
1.99 -this reason we designed a non-pointwise local scoring method to detect when a gene had a pattern of expression which
1.100 -looked like it had a boundary whose shape is similar to the shape of the target region. We call this scoring method “gradient
1.101 -similarity”.
1.102 -One might say that gradient similarity attempts to measure how much the border of the area of gene expression and
1.103 -the border of the target region overlap. However, since gene expression falls off continuously rather than jumping from its
1.104 -maximum value to zero, the spatial pattern of a gene’s expression often does not have a discrete border. Therefore, instead
1.105 -of looking for a discrete border, we look for large gradients. Gradient similarity is a symmetric function over two images
1.106 -(i.e. two scalar fields). It is is high to the extent that matching pixels which have large values and large gradients also have
1.107 -gradients which are oriented in a similar direction. The formula is:
1.108 - ∑
1.109 - pixel<img src="cmsy7-32.png" alt="∈" />pixels cos(abs(∠∇1 -∠∇2)) ⋅|∇1| + |∇2|
1.110 - 2 ⋅ pixel_value1 + pixel_value2
1.111 - 2
1.112
1.113
1.114 Figure 4: Upper left: wwc1. Upper right:
1.115 mtif2. Lower left: wwc1 + mtif2 (each
1.116 pixel’s value on the lower left is the sum of
1.117 -the corresponding pixels in the upper row). where ∇1 and ∇2 are the gradient vectors of the two images at the current
1.118 - pixel; ∠∇i is the angle of the gradient of image i at the current pixel; |∇i| is
1.119 - the magnitude of the gradient of image i at the current pixel; and pixel_valuei
1.120 - is the value of the current pixel in image i.
1.121 - The intuition is that we want to see if the borders of the pattern in the
1.122 - two images are similar; if the borders are similar, then both images will have
1.123 - corresponding pixels with large gradients (because this is a border) which are
1.124 - oriented in a similar direction (because the borders are similar).
1.125 - Most of the genes in Figure 5 were identified via gradient similarity.
1.126 - Gradient similarity provides information complementary to cor-
1.127 - relation
1.128 - To show that gradient similarity can provide useful information that cannot
1.129 - be detected via pointwise analyses, consider Fig. 3. The top row of Fig. 3
1.130 - displays the 3 genes which most match area AUD, according to a pointwise
1.131 - method17. The bottom row displays the 3 genes which most match AUD ac-
1.132 - cording to a method which considers local geometry18 The pointwise method
1.133 - in the top row identifies genes which express more strongly in AUD than out-
1.134 - side of it; its weakness is that this includes many areas which don’t have a
1.135 - salient border matching the areal border. The geometric method identifies
1.136 -genes whose salient expression border seems to partially line up with the border of AUD; its weakness is that this includes
1.137 -genes which don’t express over the entire area. Genes which have high rankings using both pointwise and border criteria,
1.138 -such as Aph1a in the example, may be particularly good markers. None of these genes are, individually, a perfect marker
1.139 -for AUD; we deliberately chose a “difficult” area in order to better contrast pointwise with geometric methods.
1.140 -Areas which can be identified by single genes Using gradient similarity, we have already found single genes which
1.141 -roughly identify some areas and groupings of areas. For each of these areas, an example of a gene which roughly identifies
1.142 -it is shown in Figure 5. We have not yet cross-verified these genes in other atlases.
1.143 -In addition, there are a number of areas which are almost identified by single genes: COAa+NLOT (anterior part of
1.144 -cortical amygdalar area, nucleus of the lateral olfactory tract), ENT (entorhinal), ACAv (ventral anterior cingulate), VIS
1.145 -(visual), AUD (auditory).
1.146 -These results validate our expectation that the ABA dataset can be exploited to find marker genes for many cortical
1.147 -areas, while also validating the relevancy of our new scoring method, gradient similarity.
1.148 -Combinations of multiple genes are useful and necessary for some areas
1.149 -In Figure 4, we give an example of a cortical area which is not marked by any single gene, but which can be identified
1.150 -combinatorially. Acccording to logistic regression, gene wwc1 is the best fit single gene for predicting whether or not a
1.151 -pixel on the cortical surface belongs to the motor area (area MO). The upper-left picture in Figure 4 shows wwc1’s spatial
1.152 -expression pattern over the cortex. The lower-right boundary of MO is represented reasonably well by this gene, but the
1.153 -gene overshoots the upper-left boundary. This flattened 2-D representation does not show it, but the area corresponding
1.154 -to the overshoot is the medial surface of the cortex. MO is only found on the dorsal surface. Gene mtif2 is shown in the
1.155 -upper-right. Mtif2 captures MO’s upper-left boundary, but not its lower-right boundary. Mtif2 does not express very much
1.156 -on the medial surface. By adding together the values at each pixel in these two figures, we get the lower-left image. This
1.157 -combination captures area MO much better than any single gene.
1.158 -This shows that our proposal to develop a method to find combinations of marker genes is both possible and necessary.
1.159 -Feature selection integrated with prediction As noted earlier, in general, any predictive method can be used for
1.160 -feature selection by running it inside a stepwise wrapper. Also, some predictive methods integrate soft constraints on number
1.161 -of features used. Examples of both of these will be seen in the section “Multivariate Predictive methods”.
1.162 +the corresponding pixels in the upper row). Gradient similarity We noticed that the previous two scoring methods,
1.163 + which are pointwise, often found genes whose pattern of expression did not
1.164 + look similar in shape to the target region. For this reason we designed a
1.165 + non-pointwise local scoring method to detect when a gene had a pattern of
1.166 + expression which looked like it had a boundary whose shape is similar to the
1.167 + shape of the target region. We call this scoring method “gradient similarity”.
1.168 + One might say that gradient similarity attempts to measure how much the
1.169 + border of the area of gene expression and the border of the target region over-
1.170 + lap. However, since gene expression falls off continuously rather than jumping
1.171 + from its maximum value to zero, the spatial pattern of a gene’s expression often
1.172 + does not have a discrete border. Therefore, instead of looking for a discrete
1.173 + border, we look for large gradients. Gradient similarity is a symmetric function
1.174 + over two images (i.e. two scalar fields). It is is high to the extent that matching
1.175 + pixels which have large values and large gradients also have gradients which
1.176 + are oriented in a similar direction. The formula is:
1.177 + ∑
1.178 + pixel<img src="cmsy7-32.png" alt="∈" />pixels cos(abs(∠∇1 -∠∇2)) ⋅|∇1| + |∇2|
1.179 + 2 ⋅ pixel_value1 + pixel_value2
1.180 + 2
1.181 + where ∇1 and ∇2 are the gradient vectors of the two images at the current
1.182 +pixel; ∠∇i is the angle of the gradient of image i at the current pixel; |∇i| is the magnitude of the gradient of image i at
1.183 +the current pixel; and pixel_valuei is the value of the current pixel in image i.
1.184 +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,
1.185 +then both images will have corresponding pixels with large gradients (because this is a border) which are oriented in a
1.186 +similar direction (because the borders are similar).
1.187 +Most of the genes in Figure 5 were identified via gradient similarity.
1.188 +Gradient similarity provides information complementary to correlation
1.189 +To show that gradient similarity can provide useful information that cannot be detected via pointwise analyses, consider
1.190 +Fig. 3. The top row of Fig. 3 displays the 3 genes which most match area AUD, according to a pointwise method17. The
1.191 +bottom row displays the 3 genes which most match AUD according to a method which considers local geometry18 The
1.192 +pointwise method in the top row identifies genes which express more strongly in AUD than outside of it; its weakness is
1.193 +that this includes many areas which don’t have a salient border matching the areal border. The geometric method identifies
1.194 _________________________________________
1.195 17For each gene, a logistic regression in which the response variable was whether or not a surface pixel was within area AUD, and the predictor
1.196 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
1.197 they predict area AUD.
1.198 18For each gene the gradient similarity between (a) a map of the expression of each gene on the cortical surface and (b) the shape of area AUD,
1.199 was calculated, and this was used to rank the genes.
1.200 -
1.201 +genes whose salient expression border seems to partially line up with the border of AUD; its weakness is that this includes
1.202 +genes which don’t express over the entire area. Genes which have high rankings using both pointwise and border criteria,
1.203 +such as Aph1a in the example, may be particularly good markers. None of these genes are, individually, a perfect marker
1.204 +for AUD; we deliberately chose a “difficult” area in order to better contrast pointwise with geometric methods.
1.205
1.206
1.207
1.208 @@ -518,7 +490,59 @@
1.209 bors, but not from the entire rest of the
1.210 cortex). The genes are Pitx2, Aldh1a2,
1.211 Ppfibp1, Slco1a5, Tshz2, Trhr, Col12a1,
1.212 -Ets1. Multivariate Predictive methods
1.213 +Ets1. Areas which can be identified by single genes Using gradient simi-
1.214 + larity, we have already found single genes which roughly identify some areas
1.215 + and groupings of areas. For each of these areas, an example of a gene which
1.216 + roughly identifies it is shown in Figure 5. We have not yet cross-verified these
1.217 + genes in other atlases.
1.218 + In addition, there are a number of areas which are almost identified by single
1.219 + genes: COAa+NLOT (anterior part of cortical amygdalar area, nucleus of the
1.220 + lateral olfactory tract), ENT (entorhinal), ACAv (ventral anterior cingulate),
1.221 + VIS (visual), AUD (auditory).
1.222 + These results validate our expectation that the ABA dataset can be ex-
1.223 + ploited to find marker genes for many cortical areas, while also validating the
1.224 + relevancy of our new scoring method, gradient similarity.
1.225 + Combinations of multiple genes are useful and necessary for some
1.226 + areas
1.227 + In Figure 4, we give an example of a cortical area which is not marked by
1.228 + any single gene, but which can be identified combinatorially. Acccording to
1.229 + logistic regression, gene wwc1 is the best fit single gene for predicting whether
1.230 + or not a pixel on the cortical surface belongs to the motor area (area MO).
1.231 + The upper-left picture in Figure 4 shows wwc1’s spatial expression pattern over
1.232 + the cortex. The lower-right boundary of MO is represented reasonably well by
1.233 + this gene, but the gene overshoots the upper-left boundary. This flattened 2-D
1.234 + representation does not show it, but the area corresponding to the overshoot is
1.235 + the medial surface of the cortex. MO is only found on the dorsal surface. Gene
1.236 + mtif2 is shown in the upper-right. Mtif2 captures MO’s upper-left boundary,
1.237 + but not its lower-right boundary. Mtif2 does not express very much on the
1.238 + medial surface. By adding together the values at each pixel in these two figures,
1.239 + we get the lower-left image. This combination captures area MO much better
1.240 + than any single gene.
1.241 + This shows that our proposal to develop a method to find combinations of
1.242 + marker genes is both possible and necessary.
1.243 + Feature selection integrated with prediction As noted earlier, in gen-
1.244 + eral, any predictive method can be used for feature selection by running it
1.245 + inside a stepwise wrapper. Also, some predictive methods integrate soft con-
1.246 + straints on number of features used. Examples of both of these will be seen in
1.247 + the section “Multivariate Predictive methods”.
1.248 + Multivariate Predictive methods
1.249 + Forward stepwise logistic regression Logistic regression is a popular
1.250 + method for predictive modeling of categorial data. As a pilot run, for five
1.251 + cortical areas (SS, AUD, RSP, VIS, and MO), we performed forward stepwise
1.252 + logistic regression to find single genes, pairs of genes, and triplets of genes
1.253 + which predict areal identify. This is an example of feature selection integrated
1.254 + with prediction using a stepwise wrapper. Some of the single genes found
1.255 + were shown in various figures throughout this document, and Figure 4 shows
1.256 + a combination of genes which was found.
1.257 + We felt that, for single genes, gradient similarity did a better job than
1.258 + logistic regression at capturing our subjective impression of a “good gene”.
1.259 +SVM on all genes at once
1.260 +In order to see how well one can do when looking at all genes at once, we ran a support vector machine to classify cortical
1.261 +surface pixels based on their gene expression profiles. We achieved classification accuracy of about 81%19. This shows that
1.262 +_________________________________________
1.263 + 195-fold cross-validation.
1.264 +the genes included in the ABA dataset are sufficient to define much of cortical anatomy. However, as noted above, a classifier
1.265 +that looks at all the genes at once isn’t as practically useful as a classifier that uses only a few genes.
1.266
1.267
1.268
1.269 @@ -531,91 +555,97 @@
1.270 from left: NNMF. Right: Landmark Isomap. Additional details: In the
1.271 third and fourth rows, 7 dimensions were found, but only 6 displayed. In
1.272 the last row: for PCA, 50 dimensions were used; for NNMF, 6 dimensions
1.273 -were used; for landmark Isomap, 7 dimensions were used. Forward stepwise logistic regression Lo-
1.274 - gistic regression is a popular method for pre-
1.275 - dictive modeling of categorial data. As a pi-
1.276 - lot run, for five cortical areas (SS, AUD, RSP,
1.277 - VIS, and MO), we performed forward stepwise
1.278 - logistic regression to find single genes, pairs of
1.279 - genes, and triplets of genes which predict areal
1.280 - identify. This is an example of feature selec-
1.281 - tion integrated with prediction using a stepwise
1.282 - wrapper. Some of the single genes found were
1.283 - shown in various figures throughout this doc-
1.284 - ument, and Figure 4 shows a combination of
1.285 - genes which was found.
1.286 - We felt that, for single genes, gradient simi-
1.287 - larity did a better job than logistic regression at
1.288 - capturing our subjective impression of a “good
1.289 - gene”.
1.290 - SVM on all genes at once
1.291 - In order to see how well one can do when
1.292 - looking at all genes at once, we ran a support
1.293 - vector machine to classify cortical surface pix-
1.294 - els based on their gene expression profiles. We
1.295 - achieved classification accuracy of about 81%19.
1.296 - This shows that the genes included in the ABA
1.297 - dataset are sufficient to define much of cortical
1.298 - anatomy. However, as noted above, a classifier
1.299 - that looks at all the genes at once isn’t as prac-
1.300 - tically useful as a classifier that uses only a few
1.301 - genes.
1.302 - Data-driven redrawing of the cor-
1.303 +were used; for landmark Isomap, 7 dimensions were used. Data-driven redrawing of the cor-
1.304 tical map
1.305 + We have applied the following dimensionality
1.306 + reduction algorithms to reduce the dimension-
1.307 + ality of the gene expression profile associated
1.308 + with each voxel: Principal Components Anal-
1.309 + ysis (PCA), Simple PCA (SPCA), Multi-Dimensional
1.310 + Scaling (MDS), Isomap, Landmark Isomap, Lapla-
1.311 + cian eigenmaps, Local Tangent Space Alignment
1.312 + (LTSA), Hessian locally linear embedding, Dif-
1.313 + fusion maps, Stochastic Neighbor Embedding
1.314 + (SNE), Stochastic Proximity Embedding (SPE),
1.315 + Fast Maximum Variance Unfolding (FastMVU),
1.316 + Non-negative Matrix Factorization (NNMF). Space
1.317 + constraints prevent us from showing many of
1.318 + the results, but as a sample, PCA, NNMF, and
1.319 + landmark Isomap are shown in the first, second,
1.320 + and third rows of Figure 6.
1.321 + After applying the dimensionality reduction,
1.322 + we ran clustering algorithms on the reduced data.
1.323 + To date we have tried k-means and spectral
1.324 + clustering. The results of k-means after PCA,
1.325 + NNMF, and landmark Isomap are shown in the
1.326 + last row of Figure 6. To compare, the left-
1.327 + most picture on the bottom row of Figure 6
1.328 + shows some of the major subdivisions of cor-
1.329 + tex. These results clearly show that different di-
1.330 + mensionality reduction techniques capture dif-
1.331 + ferent aspects of the data and lead to differ-
1.332 + ent clusterings, indicating the utility of our pro-
1.333 + posal to produce a detailed comparion of these
1.334 + techniques as applied to the domain of genomic
1.335 + anatomy.
1.336
1.337 Figure 7: Prototypes corresponding to sample gene clusters,
1.338 clustered by gradient similarity. Region boundaries for the
1.339 -region that most matches each prototype are overlayed. We have applied the following dimensionality reduction al-
1.340 - gorithms to reduce the dimensionality of the gene expression
1.341 - profile associated with each voxel: Principal Components
1.342 - Analysis (PCA), Simple PCA (SPCA), Multi-Dimensional
1.343 - Scaling (MDS), Isomap, Landmark Isomap, Laplacian eigen-
1.344 - maps, Local Tangent Space Alignment (LTSA), Hessian lo-
1.345 - cally linear embedding, Diffusion maps, Stochastic Neigh-
1.346 - bor Embedding (SNE), Stochastic Proximity Embedding
1.347 - (SPE), Fast Maximum Variance Unfolding (FastMVU),
1.348 - Non-negative Matrix Factorization (NNMF). Space con-
1.349 - straints prevent us from showing many of the results, but as
1.350 - a sample, PCA, NNMF, and landmark Isomap are shown in
1.351 - the first, second, and third rows of Figure 6.
1.352 - After applying the dimensionality reduction, we ran clus-
1.353 - tering algorithms on the reduced data. To date we have tried
1.354 -k-means and spectral clustering. The results of k-means after PCA, NNMF, and landmark Isomap are shown in the last
1.355 -row of Figure 6. To compare, the leftmost picture on the bottom row of Figure 6 shows some of the major subdivisions of
1.356 -cortex. These results clearly show that different dimensionality reduction techniques capture different aspects of the data
1.357 -and lead to different clusterings, indicating the utility of our proposal to produce a detailed comparion of these techniques
1.358 -as applied to the domain of genomic anatomy.
1.359 -Many areas are captured by clusters of genes We also clustered the genes using gradient similarity to see if the
1.360 -_________________________________________
1.361 - 195-fold cross-validation.
1.362 -spatial regions defined by any clusters matched known anatomical regions. Figure 7 shows, for ten sample gene clusters, each
1.363 -cluster’s average expression pattern, compared to a known anatomical boundary. This suggests that it is worth attempting
1.364 -to cluster genes, and then to use the results to cluster voxels.
1.365 -The approach: what we plan to do
1.366 -Flatmap and segment cortical layers
1.367 -There are multiple ways to flatten 3-D data into 2-D. We will compare mappings from manifolds to planes which attempt
1.368 -to preserve size (such as the one used by Caret[7]) with mappings which preserve angle (conformal maps). Our method will
1.369 -include a statistical test that warns the user if the assumption of 2-D structure seems to be wrong.
1.370 +region that most matches each prototype are overlayed. Many areas are captured by clusters of genes We
1.371 + also clustered the genes using gradient similarity to see if
1.372 + the spatial regions defined by any clusters matched known
1.373 + anatomical regions. Figure 7 shows, for ten sample gene
1.374 + clusters, each cluster’s average expression pattern, compared
1.375 + to a known anatomical boundary. This suggests that it is
1.376 + worth attempting to cluster genes, and then to use the re-
1.377 + sults to cluster voxels.
1.378 + The approach: what we plan to do
1.379 + Flatmap cortex and segment cortical layers
1.380 + There are multiple ways to flatten 3-D data into 2-D. We
1.381 + will compare mappings from manifolds to planes which at-
1.382 + tempt to preserve size (such as the one used by Caret[7])
1.383 + with mappings which preserve angle (conformal maps). Our
1.384 + method will include a statistical test that warns the user if
1.385 +the assumption of 2-D structure seems to be wrong.
1.386 We have not yet made use of radial profiles. While the radial profiles may be used “raw”, for laminar structures like the
1.387 cortex another strategy is to group together voxels in the same cortical layer; each surface pixel would then be associated
1.388 with one expression level per gene per layer. We will develop a segmentation algorithm to automatically identify the layer
1.389 boundaries.
1.390 Develop algorithms that find genetic markers for anatomical regions
1.391 -1.Develop scoring measures for evaluating how good individual genes are at marking areas: we will compare pointwise,
1.392 -geometric, and information-theoretic measures.
1.393 -2.Develop a procedure to find single marker genes for anatomical regions: for each cortical area, by using or combining
1.394 -the scoring measures developed, we will rank the genes by their ability to delineate each area.
1.395 -3.Extend the procedure to handle difficult areas by using combinatorial coding: for areas that cannot be identified by any
1.396 -single gene, identify them with a handful of genes. We will consider both (a) algorithms that incrementally/greedily
1.397 -combine single gene markers into sets, such as forward stepwise regression and decision trees, and also (b) supervised
1.398 -learning techniques which use soft constraints to minimize the number of features, such as sparse support vector
1.399 -machines.
1.400 -4.Extend the procedure to handle difficult areas by combining or redrawing the boundaries: An area may be difficult
1.401 -to identify because the boundaries are misdrawn, or because it does not “really” exist as a single area, at least on the
1.402 -genetic level. We will develop extensions to our procedure which (a) detect when a difficult area could be fit if its
1.403 -boundary were redrawn slightly, and (b) detect when a difficult area could be combined with adjacent areas to create
1.404 -a larger area which can be fit.
1.405 -# Linear discriminant analysis
1.406 +We will develop scoring methods for evaluating how good individual genes are at marking areas. We will compare pointwise,
1.407 +geometric, and information-theoretic measures. We already developed one entirely new scoring method (gradient similarity),
1.408 +but we may develop more. Scoring measures that we will explore will include the L1 norm, correlation, expression energy
1.409 +ratio, conditional entropy, gradient similarity, Jaccard similarity, Dice similarity, Hough transform, and statistical tests such
1.410 +as Student’s t-test, and the Mann-Whitney U test (a non-parametric test). In addition, any predictive procedure induces a
1.411 +scoring measure on genes by taking the prediction error when using that gene to predict the target.
1.412 +Using some combination of these measures, we will develop a procedure to find single marker genes for anatomical regions:
1.413 +for each cortical area, we will rank the genes by their ability to delineate each area.
1.414 +Some cortical areas have no single marker genes but can be identified by combinatorial coding. This requires multivariate
1.415 +scoring measures and feature selection procedures. Many of the measures, such as expression energy, gradient similarity,
1.416 +Jaccard, Dice, Hough, Student’s t, and Mann-Whitney U are univariate. We will extend these scoring measures for use
1.417 +in multivariate feature selection, that is, for scoring how well combinations of genes, rather than individual genes, can
1.418 +distinguish a target area. There are existing multivariate forms of some of the univariate scoring measures, for example,
1.419 +Hotelling’s T-square is a multivariate analog of Student’s t.
1.420 +We will develop a feature selection procedure for choosing the best small set of marker genes for a given anatomical
1.421 +area. In addition to using the scoring measures that we develop, we will also explore (a) feature selection using a stepwise
1.422 +wrapper over “vanilla” predictive methods such as logistic regression, (b) predictive methods such as decision trees which
1.423 +incrementally/greedily combine single gene markers into sets, and (c) predictive methods which use soft constraints to
1.424 +minimize number of features used, such as sparse support vector machines.
1.425 +todo
1.426 +Some of these methods, such as the Hough transform, are designed to be resistant to registration error and error in the
1.427 +anatomical map.
1.428 +We will also consider extensions to scoring measures that may improve their robustness to registration error and to
1.429 +error in the anatomical map; for example, a wrapper that runs a scoring method on small displacements and distortions
1.430 +of the data adds robustness to registration error at the expense of computation time. It is possible that some areas in the
1.431 +anatomical map do not correspond to natural domains of gene expression.
1.432 +# Extend the procedure to handle difficult areas by combining or redrawing the boundaries: An area may be difficult to
1.433 +identify because the boundaries are misdrawn, or because it does not “really” exist as a single area, at least on the genetic
1.434 +level. We will develop extensions to our procedure which (a) detect when a difficult area could be fit if its boundary were
1.435 +redrawn slightly, and (b) detect when a difficult area could be combined with adjacent areas to create a larger area which
1.436 +can be fit.
1.437 +A future publication on the method that we develop in Aim 1 will review the scoring measures and quantitatively compare
1.438 +their performance in order to provide a foundation for future research of methods of marker gene finding. We will measure
1.439 +the robustness of the scoring measures as well as their absolute performance on our dataset.
1.440 Decision trees todo
1.441 20.
1.442 # confirm with EMAGE, GeneAtlas, GENSAT, etc, to fight overfitting, two hemis
1.443 @@ -627,21 +657,23 @@
1.444 4.Explore clustering algorithms applied to genes: including gene shaving, TODO
1.445 5.Develop an algorithm to use dimensionality reduction and/or hierarchial clustering to create anatomical maps
1.446 6.Run this algorithm on the cortex: present a hierarchial, genoarchitectonic map of the cortex
1.447 +_________________________________________
1.448 + 20Already, for each cortical area, we have used the C4.5 algorithm to find a decision tree for that area. We achieved good classification accuracy
1.449 +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
1.450 +trees that use fewer genes
1.451 # Linear discriminant analysis
1.452 # jbt, coclustering
1.453 # self-organizing map
1.454 +# Linear discriminant analysis
1.455 # compare using clustering scores
1.456 -__________
1.457 - 20Already, for each cortical area, we have used the C4.5 algorithm to find a decision tree for that area. We achieved good classification accuracy
1.458 -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
1.459 -trees that use fewer genes
1.460 # multivariate gradient similarity
1.461 # deep belief nets
1.462 Apply these algorithms to the cortex
1.463 Using the methods developed in Aim 1, we will present, for each cortical area, a short list of markers to identify that
1.464 area; and we will also present lists of “panels” of genes that can be used to delineate many areas at once. Using the methods
1.465 developed in Aim 2, we will present one or more hierarchial cortical maps. We will identify and explain how the statistical
1.466 -structure in the gene expression data led to any unexpected or interesting features of these maps.
1.467 +structure in the gene expression data led to any unexpected or interesting features of these maps, and we will provide
1.468 +biological hypotheses to interpret any new cortical areas, or groupings of areas, which are discovered.
1.469 Timeline and milestones
1.470 Finding marker genes
1.471 ∙September-November 2009: Develop an automated mechanism for segmenting the cortical voxels into layers
2.1 Binary file grant.odt has changed
3.1 Binary file grant.pdf has changed
4.1 --- a/grant.txt Tue Apr 21 06:11:15 2009 -0700
4.2 +++ b/grant.txt Tue Apr 21 14:28:12 2009 -0700
4.3 @@ -231,7 +231,27 @@
4.4
4.5 The method developed in aim (2) will provide a genoarchitectonic viewpoint that will contribute to the creation of a better map. The development of present-day cortical maps was driven by the application of histological stains. If a different set of stains had been available which identified a different set of features, then today's cortical maps may have come out differently. It is likely that there are many repeated, salient spatial patterns in the gene expression which have not yet been captured by any stain. Therefore, cortical anatomy needs to incorporate what we can learn from looking at the patterns of gene expression.
4.6
4.7 -While we do not here propose to analyze human gene expression data, it is conceivable that the methods we propose to develop could be used to suggest modifications to the human cortical map as well. In fact, the methods we will develop will be applicable to other datasets beyond the brain. We will provide an open-source toolbox to allow other researchers to easily use our methods. With these methods, researchers with gene expression for any area of the body will be able to efficiently find marker genes for anatomical regions, or to use gene expression to discover new anatomical patterning. As described above, marker genes have a variety of uses in the development of drugs and experimental manipulations, and in the anatomical characterization of tissue samples. The discovery of new ways to carve up anatomical structures into regions will widely impact all areas of biology.
4.8 +While we do not here propose to analyze human gene expression data, it
4.9 +is conceivable that the methods we propose to develop could be used to
4.10 +suggest modifications to the human cortical map as well. In fact, the
4.11 +methods we will develop will be applicable to other datasets beyond
4.12 +the brain. We will provide an open-source toolbox to allow other
4.13 +researchers to easily use our methods. With these methods, researchers
4.14 +with gene expression for any area of the body will be able to
4.15 +efficiently find marker genes for anatomical regions, or to use gene
4.16 +expression to discover new anatomical patterning. As described above,
4.17 +marker genes have a variety of uses in the development of drugs and
4.18 +experimental manipulations, and in the anatomical characterization of
4.19 +tissue samples. The discovery of new ways to carve up anatomical
4.20 +structures into regions may lead to the discovery of
4.21 +new anatomical subregions in various structures, which will widely
4.22 +impact all areas of biology.
4.23 +
4.24 +Although our particular application involves the 3D spatial
4.25 +distribution of gene expression, we anticipate that the methods
4.26 +developed in aims (1) and (2) will not be limited to gene expression
4.27 +data, but rather will generalize to any sort of
4.28 +high-dimensional data over points located in a low-dimensional space.
4.29
4.30
4.31
4.32 @@ -431,19 +451,19 @@
4.33
4.34 === Data-driven redrawing of the cortical map ===
4.35
4.36 +
4.37 +
4.38 +
4.39 +We have applied the following dimensionality reduction algorithms to reduce the dimensionality of the gene expression profile associated with each voxel: Principal Components Analysis (PCA), Simple PCA (SPCA), Multi-Dimensional Scaling (MDS), Isomap, Landmark Isomap, Laplacian eigenmaps, Local Tangent Space Alignment (LTSA), Hessian locally linear embedding, Diffusion maps, Stochastic Neighbor Embedding (SNE), Stochastic Proximity Embedding (SPE), Fast Maximum Variance Unfolding (FastMVU), Non-negative Matrix Factorization (NNMF). Space constraints prevent us from showing many of the results, but as a sample, PCA, NNMF, and landmark Isomap are shown in the first, second, and third rows of Figure \ref{dimReduc}.
4.40 +
4.41 +After applying the dimensionality reduction, we ran clustering algorithms on the reduced data. To date we have tried k-means and spectral clustering. The results of k-means after PCA, NNMF, and landmark Isomap are shown in the last row of Figure \ref{dimReduc}. To compare, the leftmost picture on the bottom row of Figure \ref{dimReduc} shows some of the major subdivisions of cortex. These results clearly show that different dimensionality reduction techniques capture different aspects of the data and lead to different clusterings, indicating the utility of our proposal to produce a detailed comparion of these techniques as applied to the domain of genomic anatomy.
4.42 +
4.43 +
4.44 \begin{wrapfigure}{L}{0.5\textwidth}\centering
4.45 \includegraphics[scale=.2]{cosine_similarity1_rearrange_colorize.eps}
4.46 \caption{Prototypes corresponding to sample gene clusters, clustered by gradient similarity. Region boundaries for the region that most matches each prototype are overlayed.}
4.47 \label{geneClusters}\end{wrapfigure}
4.48
4.49 -
4.50 -
4.51 -We have applied the following dimensionality reduction algorithms to reduce the dimensionality of the gene expression profile associated with each voxel: Principal Components Analysis (PCA), Simple PCA (SPCA), Multi-Dimensional Scaling (MDS), Isomap, Landmark Isomap, Laplacian eigenmaps, Local Tangent Space Alignment (LTSA), Hessian locally linear embedding, Diffusion maps, Stochastic Neighbor Embedding (SNE), Stochastic Proximity Embedding (SPE), Fast Maximum Variance Unfolding (FastMVU), Non-negative Matrix Factorization (NNMF). Space constraints prevent us from showing many of the results, but as a sample, PCA, NNMF, and landmark Isomap are shown in the first, second, and third rows of Figure \ref{dimReduc}.
4.52 -
4.53 -After applying the dimensionality reduction, we ran clustering algorithms on the reduced data. To date we have tried k-means and spectral clustering. The results of k-means after PCA, NNMF, and landmark Isomap are shown in the last row of Figure \ref{dimReduc}. To compare, the leftmost picture on the bottom row of Figure \ref{dimReduc} shows some of the major subdivisions of cortex. These results clearly show that different dimensionality reduction techniques capture different aspects of the data and lead to different clusterings, indicating the utility of our proposal to produce a detailed comparion of these techniques as applied to the domain of genomic anatomy.
4.54 -
4.55 -
4.56 -
4.57 \vspace{0.3cm}**Many areas are captured by clusters of genes**
4.58 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 voxels.
4.59
4.60 @@ -453,7 +473,9 @@
4.61 == The approach: what we plan to do ==
4.62
4.63
4.64 -\vspace{0.3cm}**Flatmap and segment cortical layers**
4.65 +%%\vspace{0.3cm}**Flatmap cortex and segment cortical layers**
4.66 +
4.67 +=== Flatmap cortex and segment cortical layers ===
4.68
4.69 %%In anatomy, the manifold of interest is usually either defined by a combination of two relevant anatomical axes (todo), or by the surface of the structure (as is the case with the cortex). In the former case, the manifold of interest is a plane, but in the latter case it is curved. If the manifold is curved, there are various methods for mapping the manifold into a plane.
4.70
4.71 @@ -466,20 +488,35 @@
4.72
4.73 We have not yet made use of radial profiles. While the radial profiles may be used "raw", for laminar structures like the cortex another strategy is to group together voxels in the same cortical layer; each surface pixel would then be associated with one expression level per gene per layer. We will develop a segmentation algorithm to automatically identify the layer boundaries.
4.74
4.75 -\vspace{0.3cm}**Develop algorithms that find genetic markers for anatomical regions**
4.76 -
4.77 -
4.78 -
4.79 -
4.80 -
4.81 -
4.82 -
4.83 -# Develop scoring measures for evaluating how good individual genes are at marking areas: we will compare pointwise, geometric, and information-theoretic measures.
4.84 -# Develop a procedure to find single marker genes for anatomical regions: for each cortical area, by using or combining the scoring measures developed, we will rank the genes by their ability to delineate each area.
4.85 -# Extend the procedure to handle difficult areas by using combinatorial coding: for areas that cannot be identified by any single gene, identify them with a handful of genes. We will consider both (a) algorithms that incrementally/greedily combine single gene markers into sets, such as forward stepwise regression and decision trees, and also (b) supervised learning techniques which use soft constraints to minimize the number of features, such as sparse support vector machines.
4.86 +%%\vspace{0.3cm}**Develop algorithms that find genetic markers for anatomical regions**
4.87 +%%\vspace{0.3cm}****
4.88 +
4.89 +
4.90 +=== Develop algorithms that find genetic markers for anatomical regions ===
4.91 +
4.92 +%%\vspace{0.3cm}**Scoring measures and feature selection**
4.93 +
4.94 +%%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).
4.95 +
4.96 +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 predictive procedure induces a scoring measure on genes by taking the prediction error when using that gene to predict the target.
4.97 +
4.98 +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.
4.99 +
4.100 +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.
4.101 +
4.102 +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" predictive methods such as logistic regression, (b) predictive methods such as decision trees which incrementally/greedily combine single gene markers into sets, and (c) predictive methods which use soft constraints to minimize number of features used, such as sparse support vector machines.
4.103 +
4.104 +todo
4.105 +
4.106 +Some of these methods, such as the Hough transform, are designed to be resistant to registration error and error in the anatomical map.
4.107 +
4.108 +We will also consider extensions to scoring measures that may improve their robustness to registration error and to error in the anatomical map; 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. It is possible that some areas in the anatomical map do not correspond to natural domains of gene expression.
4.109 +
4.110 # Extend the procedure to handle difficult areas by combining or redrawing the boundaries: An area may be difficult to identify because the boundaries are misdrawn, or because it does not "really" exist as a single area, at least on the genetic level. We will develop extensions to our procedure which (a) detect when a difficult area could be fit if its boundary were redrawn slightly, and (b) detect when a difficult area could be combined with adjacent areas to create a larger area which can be fit.
4.111
4.112 -# Linear discriminant analysis
4.113 +
4.114 +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.
4.115 +
4.116
4.117
4.118 \vspace{0.3cm}**Decision trees**
4.119 @@ -493,6 +530,7 @@
4.120
4.121
4.122
4.123 +
4.124 \vspace{0.3cm}**Develop algorithms to suggest a division of a structure into anatomical parts**
4.125
4.126 # Explore dimensionality reduction algorithms applied to pixels: including TODO
4.127 @@ -508,6 +546,7 @@
4.128
4.129 # self-organizing map
4.130
4.131 +# Linear discriminant analysis
4.132
4.133
4.134 # compare using clustering scores
4.135 @@ -520,7 +559,7 @@
4.136
4.137 \vspace{0.3cm}**Apply these algorithms to the cortex**
4.138
4.139 -Using the methods developed in Aim 1, we will present, for each cortical area, a short list of markers to identify that area; and we will also present lists of "panels" of genes that can be used to delineate many areas at once. Using the methods developed in Aim 2, we will present one or more hierarchial cortical maps. We will identify and explain how the statistical structure in the gene expression data led to any unexpected or interesting features of these maps.
4.140 +Using the methods developed in Aim 1, we will present, for each cortical area, a short list of markers to identify that area; and we will also present lists of "panels" of genes that can be used to delineate many areas at once. Using the methods developed in Aim 2, we will present one or more hierarchial cortical maps. We will identify and explain how the statistical structure in the gene expression data led to any unexpected or interesting features of these maps, and we will provide biological hypotheses to interpret any new cortical areas, or groupings of areas, which are discovered.
4.141
4.142
4.143 %%# note: slice artifact