1.1 --- a/grant.txt Sun Apr 19 15:23:53 2009 -0700 1.2 +++ b/grant.txt Sun Apr 19 16:24:10 2009 -0700 1.3 @@ -260,6 +260,15 @@ 1.4 1.5 === Feature selection and scoring methods === 1.6 1.7 +\vspace{0.3cm}**Underexpression of a gene can serve as a marker** 1.8 +Underexpression of a gene can sometimes serve as a marker. See, for example, Figure \ref{hole}. 1.9 + 1.10 + 1.11 +\begin{figure}\centering 1.12 +\includegraphics[scale=.31]{holeExample_2682_SS_jet.eps} 1.13 +\caption{Gene Pitx2 is selectively underexpressed in area SS (somatosensory).} 1.14 +\label{hole}\end{figure} 1.15 + 1.16 1.17 \vspace{0.3cm}**Correlation** 1.18 Recall that the instances are surface pixels, and consider the problem of attempting to classify each instance as either a member of a particular anatomical area, or not. The target area can be represented as a boolean mask over the surface pixels. 1.19 @@ -294,7 +303,7 @@ 1.20 1.21 1.22 \vspace{0.3cm}**Gradient similarity** 1.23 -We noticed that the previous two scoring methods, which are pointwise, often found genes whose pattern of expression did not look similar in shape to the target region. Fort his reason we designed a non-pointwise local scoring method to detect when a gene had a pattern of expression which looked like it had a boundary whose shape is similar to the shape of the target region. We call this scoring method "gradient similarity". 1.24 +We noticed that the previous two scoring methods, which are pointwise, often found genes whose pattern of expression did not look similar in shape to the target region. For this reason we designed a non-pointwise local scoring method to detect when a gene had a pattern of expression which looked like it had a boundary whose shape is similar to the shape of the target region. We call this scoring method "gradient similarity". 1.25 1.26 One might say that gradient similarity attempts to measure how much the border of the area of gene expression and the border of the target region overlap. However, since gene expression falls off continuously rather than jumping from its maximum value to zero, the spatial pattern of a gene's expression often does not have a discrete border. Therefore, instead of looking for a discrete border, we look for large gradients. Gradient similarity is a symmetric function over two images (i.e. two scalar fields). It is is high to the extent that matching pixels which have large values and large gradients also have gradients which are oriented in a similar direction. The formula is: 1.27 1.28 @@ -306,6 +315,8 @@ 1.29 1.30 The intuition is that we want to see if the borders of the pattern in the two images are similar; if the borders are similar, then both images will have corresponding pixels with large gradients (because this is a border) which are oriented in a similar direction (because the borders are similar). 1.31 1.32 +Most of the genes in Figure \ref{singleSoFar} were identified via gradient similarity. 1.33 + 1.34 \vspace{0.3cm}**Gradient similarity provides information complementary to correlation** 1.35 1.36 To show that gradient similarity can provide useful information that cannot be detected via pointwise analyses, consider Fig. \ref{AUDgeometry}. The top row of Fig. \ref{AUDgeometry} displays the 3 genes which most match area AUD, according to a pointwise method\footnote{For each gene, a logistic regression in which the response variable was whether or not a surface pixel was within area AUD, and the predictor variable was the value of the expression of the gene underneath that pixel. The resulting scores were used to rank the genes in terms of how well they predict area AUD.}. The bottom row displays the 3 genes which most match AUD according to a method which considers local geometry\footnote{For each gene the gradient similarity between (a) a map of the expression of each gene on the cortical surface and (b) the shape of area AUD, was calculated, and this was used to rank the genes.} The pointwise method in the top row identifies genes which express more strongly in AUD than outside of it; its weakness is that this includes many areas which don't have a salient border matching the areal border. The geometric method identifies genes whose salient expression border seems to partially line up with the border of AUD; its weakness is that this includes genes which don't express over the entire area. Genes which have high rankings using both pointwise and border criteria, such as $Aph1a$ in the example, may be particularly good markers. None of these genes are, individually, a perfect marker for AUD; we deliberately chose a "difficult" area in order to better contrast pointwise with geometric methods. 1.37 @@ -323,61 +334,8 @@ 1.38 \label{AUDgeometry}\end{figure} 1.39 1.40 1.41 -\vspace{0.3cm}**Combinations of multiple genes are useful and necessary for some areas** 1.42 - 1.43 -In Figure \ref{MOcombo}, we give an example of a cortical area which is not marked by any single gene, but which can be identified combinatorially. 1.44 - 1.45 -%% wwc1\footnote{"WW, C2 and coiled-coil domain containing 1"; EntrezGene ID 211652} 1.46 -%% mtif2\footnote{"mitochondrial translational initiation factor 2"; EntrezGene ID 76784} 1.47 - 1.48 -%%Acccording to logistic regression, gene wwc1\footnote{"WW, C2 and coiled-coil domain containing 1"; EntrezGene ID 211652} is the best fit single gene for predicting whether or not a pixel on the cortical surface belongs to the motor area (area MO). The upper-left picture in Figure \ref{MOcombo} shows wwc1's spatial expression pattern over the cortex. The lower-right boundary of MO is represented reasonably well by this gene, however the gene overshoots the upper-left boundary. This flattened 2-D representation does not show it, but the area corresponding to the overshoot is the medial surface of the cortex. MO is only found on the lateral surface. 1.49 - 1.50 -%%Gene mtif2\footnote{"mitochondrial translational initiation factor 2"; EntrezGene ID 76784} is shown in figure the upper-right of Fig. \ref{MOcombo}. Mtif2 captures MO's upper-left boundary, but not its lower-right boundary. Mtif2 does not express very much on the medial surface. By adding together the values at each pixel in these two figures, we get the lower-left of Figure \ref{MOcombo}. This combination captures area MO much better than any single gene. 1.51 - 1.52 -\begin{figure}\centering 1.53 -\includegraphics[scale=.36]{MO_vs_Wwc1_jet.eps} 1.54 -\includegraphics[scale=.36]{MO_vs_Mtif2_jet.eps} 1.55 - 1.56 -\includegraphics[scale=.36]{MO_vs_Wwc1_plus_Mtif2_jet.eps} 1.57 -\caption{Upper left: $wwc1$. Upper right: $mtif2$. Lower left: wwc1 + mtif2 (each pixel's value on the lower left is the sum of the corresponding pixels in the upper row). Acccording to logistic regression, gene wwc1 is the best fit single gene for predicting whether or not a pixel on the cortical surface belongs to the motor area (area MO). The upper-left picture in Figure \ref{MOcombo} shows wwc1's spatial expression pattern over the cortex. The lower-right boundary of MO is represented reasonably well by this gene, however the gene overshoots the upper-left boundary. This flattened 2-D representation does not show it, but the area corresponding to the overshoot is the medial surface of the cortex. MO is only found on the lateral surface. Gene mtif2 is shown in the upper-right. Mtif2 captures MO's upper-left boundary, but not its lower-right boundary. Mtif2 does not express very much on the medial surface. By adding together the values at each pixel in these two figures, we get the lower-left image. This combination captures area MO much better than any single gene. } 1.58 -\label{MOcombo}\end{figure} 1.59 - 1.60 - 1.61 - 1.62 - 1.63 - 1.64 - 1.65 -\vspace{0.3cm}**Underexpression of a gene can serve as a marker** 1.66 -Underexpression of a gene can sometimes serve as a marker. See, for example, Figure \ref{hole}. 1.67 - 1.68 - 1.69 -\begin{figure}\centering 1.70 -\includegraphics[scale=.31]{holeExample_2682_SS_jet.eps} 1.71 -\caption{Gene Pitx2 is selectively underexpressed in area SS (somatosensory).} 1.72 -\label{hole}\end{figure} 1.73 - 1.74 - 1.75 -\vspace{0.3cm}**Feature selection integrated with prediction** 1.76 -As noted earlier, in general, any predictive method can be used for feature selection by running it inside a stepwise wrapper. Also, some predictive methods integrate soft constraints on number of features used. Examples of both of these will be seen in the section "Locating areas with gene expression". 1.77 - 1.78 - 1.79 -=== Locating areas with gene expression === 1.80 -\vspace{0.3cm}**Forward stepwise logistic regression** 1.81 -As a pilot run, for five cortical areas (SS, AUD, RSP, VIS, and MO), we performed forward stepwise logistic regression to find single genes, pairs of genes, and triplets of genes which predict areal identify. This is an example of feature selection integrated with prediction using a stepwise wrapper. Some of the single genes found were shown in previous figures, and Figure \ref{MOcombo} shows a combination of genes which was found. 1.82 - 1.83 - 1.84 -\vspace{0.3cm}**SVM on all genes at once** 1.85 - 1.86 -In order to see how well one can do when looking at all genes at once, we ran a support vector machine to classify cortical surface pixels based on their gene expression profiles. We achieved classification accuracy of about 81%\footnote{5-fold cross-validation.}. As noted above, however, a classifier that looks at all the genes at once isn't as practically useful as a classifier that uses only a few genes. 1.87 - 1.88 - 1.89 -\vspace{0.3cm}**Decision trees** 1.90 - 1.91 -todo 1.92 - 1.93 \vspace{0.3cm}**Areas which can be identified by single genes** 1.94 - 1.95 -Using all of the methods we have tried to far, we have already found single genes which roughly identify some areas and groupings of areas. For each of these areas, an example of a gene which roughly identifies it is shown in Figure \ref{singleSoFar}. We have not yet cross-verified these genes in other atlases. 1.96 +Using gradient similarity, we have already found single genes which roughly identify some areas and groupings of areas. For each of these areas, an example of a gene which roughly identifies it is shown in Figure \ref{singleSoFar}. We have not yet cross-verified these genes in other atlases. 1.97 1.98 In addition, there are a number of areas which are almost identified by single genes: COAa+NLOT (anterior part of cortical amygdalar area, nucleus of the lateral olfactory tract), ENT (entorhinal), ACAv (ventral anterior cingulate), VIS (visual), AUD (auditory). 1.99 1.100 @@ -397,9 +355,56 @@ 1.101 1.102 1.103 1.104 +\vspace{0.3cm}**Combinations of multiple genes are useful and necessary for some areas** 1.105 + 1.106 +In Figure \ref{MOcombo}, we give an example of a cortical area which is not marked by any single gene, but which can be identified combinatorially. 1.107 + 1.108 +%% wwc1\footnote{"WW, C2 and coiled-coil domain containing 1"; EntrezGene ID 211652} 1.109 +%% mtif2\footnote{"mitochondrial translational initiation factor 2"; EntrezGene ID 76784} 1.110 + 1.111 +%%Acccording to logistic regression, gene wwc1\footnote{"WW, C2 and coiled-coil domain containing 1"; EntrezGene ID 211652} is the best fit single gene for predicting whether or not a pixel on the cortical surface belongs to the motor area (area MO). The upper-left picture in Figure \ref{MOcombo} shows wwc1's spatial expression pattern over the cortex. The lower-right boundary of MO is represented reasonably well by this gene, however the gene overshoots the upper-left boundary. This flattened 2-D representation does not show it, but the area corresponding to the overshoot is the medial surface of the cortex. MO is only found on the lateral surface. 1.112 + 1.113 +%%Gene mtif2\footnote{"mitochondrial translational initiation factor 2"; EntrezGene ID 76784} is shown in figure the upper-right of Fig. \ref{MOcombo}. Mtif2 captures MO's upper-left boundary, but not its lower-right boundary. Mtif2 does not express very much on the medial surface. By adding together the values at each pixel in these two figures, we get the lower-left of Figure \ref{MOcombo}. This combination captures area MO much better than any single gene. 1.114 + 1.115 +\begin{figure}\centering 1.116 +\includegraphics[scale=.36]{MO_vs_Wwc1_jet.eps} 1.117 +\includegraphics[scale=.36]{MO_vs_Mtif2_jet.eps} 1.118 + 1.119 +\includegraphics[scale=.36]{MO_vs_Wwc1_plus_Mtif2_jet.eps} 1.120 +\caption{Upper left: $wwc1$. Upper right: $mtif2$. Lower left: wwc1 + mtif2 (each pixel's value on the lower left is the sum of the corresponding pixels in the upper row). Acccording to logistic regression, gene wwc1 is the best fit single gene for predicting whether or not a pixel on the cortical surface belongs to the motor area (area MO). The upper-left picture in Figure \ref{MOcombo} shows wwc1's spatial expression pattern over the cortex. The lower-right boundary of MO is represented reasonably well by this gene, however the gene overshoots the upper-left boundary. This flattened 2-D representation does not show it, but the area corresponding to the overshoot is the medial surface of the cortex. MO is only found on the lateral surface. Gene mtif2 is shown in the upper-right. Mtif2 captures MO's upper-left boundary, but not its lower-right boundary. Mtif2 does not express very much on the medial surface. By adding together the values at each pixel in these two figures, we get the lower-left image. This combination captures area MO much better than any single gene. } 1.121 +\label{MOcombo}\end{figure} 1.122 + 1.123 + 1.124 + 1.125 + 1.126 + 1.127 + 1.128 + 1.129 + 1.130 +\vspace{0.3cm}**Feature selection integrated with prediction** 1.131 +As noted earlier, in general, any predictive method can be used for feature selection by running it inside a stepwise wrapper. Also, some predictive methods integrate soft constraints on number of features used. Examples of both of these will be seen in the section "Multivariate Predictive methods". 1.132 + 1.133 + 1.134 +=== Multivariate Predictive methods === 1.135 +\vspace{0.3cm}**Forward stepwise logistic regression** 1.136 +As a pilot run, for five cortical areas (SS, AUD, RSP, VIS, and MO), we performed forward stepwise logistic regression to find single genes, pairs of genes, and triplets of genes which predict areal identify. This is an example of feature selection integrated with prediction using a stepwise wrapper. Some of the single genes found were shown in various figures throughout this document, and Figure \ref{MOcombo} shows a combination of genes which was found. 1.137 + 1.138 +We felt that, for single genes, gradient similarity did a better job than logistic regression at capturing our subjective impression of a "good gene". 1.139 + 1.140 + 1.141 +\vspace{0.3cm}**SVM on all genes at once** 1.142 + 1.143 +In order to see how well one can do when looking at all genes at once, we ran a support vector machine to classify cortical surface pixels based on their gene expression profiles. We achieved classification accuracy of about 81%\footnote{5-fold cross-validation.}. As noted above, however, a classifier that looks at all the genes at once isn't as practically useful as a classifier that uses only a few genes. 1.144 + 1.145 + 1.146 + 1.147 + 1.148 + 1.149 === Data-driven redrawing of the cortical map === 1.150 1.151 -\vspace{0.3cm}**Raw dimensionality reduction results** 1.152 +\vspace{0.3cm}**Raw dimensionality reduction** 1.153 +We have applied the following dimensionality reduction algorithms to reduce the dimensionality of the gene expression profile associated with each voxel: Principal Components Analysis (PCA), Simple PCA (SPCA), Multi-Dimensional Scaling (MDS), Isomap, Landmark Isomap, Laplacian eigenmaps, Local Tangent Space Alignment (LTSA), Hessian locally linear embedding, Diffusion maps, Stochastic Neighbor Embedding (SNE), Stochastic Proximity Embedding (SPE), Fast Maximum Variance Unfolding (FastMVU), Non-negative Matrix Factorization (NNMF). 1.154 + 1.155 1.156 todo 1.157 1.158 @@ -459,6 +464,12 @@ 1.159 # Linear discriminant analysis 1.160 1.161 1.162 +\vspace{0.3cm}**Decision trees** 1.163 +todo 1.164 + 1.165 +For each cortical area, we used the C4.5 algorithm to find a pruned decision tree and ruleset for that area. We achieved estimated classification accuracy of more than 99.6% on each cortical area (as evaluated on the __training data__ without cross-validation; so actual accuracy is expected to be lower). However, the resulting decision trees each made use of many genes. 1.166 + 1.167 + 1.168 \vspace{0.3cm}**Apply these algorithms to the cortex** 1.169 1.170 # Create open source format conversion tools: we will create tools to bulk download the ABA dataset and to convert between SEV, NIFTI and MATLAB formats. 1.171 @@ -489,6 +500,8 @@ 1.172 1.173 # compare using clustering scores 1.174 1.175 +# multivariate gradient similarity 1.176 + 1.177 1.178 \newpage 1.179