cg
changeset 40:cb2ac88dd526
.
author | bshanks@bshanks.dyndns.org |
---|---|
date | Tue Apr 14 02:50:49 2009 -0700 (16 years ago) |
parents | 9365a696c0b8 |
children | 34e681823d3a |
files | grant.doc grant.html grant.odt grant.pdf grant.txt |
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1.1 Binary file grant.doc has changed
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2.2 +++ b/grant.html Tue Apr 14 02:50:49 2009 -0700
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2.4 ∙For each gene, a 2-D matrix whose entries represent the average expression level underneath each surface pixel
2.5 We created a normalized version of the gene expression data by subtracting each gene’s mean expression level (over all
2.6 surface pixels) and dividing each gene by its standard deviation.
2.7 +The features and the target area are both functions on the surface pixels. They can be referred to as scalar fields over
2.8 +the space of surface pixels; alternately, they can be thought of as images which can be displayed on the flatmapped surface.
2.9 To move beyond a single average expression level for each surface pixel, we plan to create a separate matrix for each
2.10 cortical layer to represent the average expression level within that layer. Cortical layers are found at different depths in
2.11 different parts of the cortex. In preparation for extracting the layer-specific datasets, we have extended Caret with routines
2.12 @@ -279,10 +281,8 @@
2.13 Correlation Recall that the instances are surface pixels, and consider the problem of attempting to classify each instance
2.14 as either a member of a particular anatomical area, or not. The target area can be represented as a binary mask over the
2.15 surface pixels.
2.16 -The features and the target area are both functions on the surface pixels; alternately, they can be thought of as images
2.17 -which can be displayed on the flatmapped surface. One class of feature selection scoring method are those which calculate
2.18 -some sort of “match” between each gene image and the target image. Those genes which match the best are good candidates
2.19 -for features.
2.20 +One class of feature selection scoring method are those which calculate some sort of “match” between each gene image
2.21 +and the target image. Those genes which match the best are good candidates for features.
2.22 One of the simplest methods in this class is to use correlation as the match score. We calculated the correlation between
2.23 each gene and each cortical area.
2.24 todo: fig
2.25 @@ -292,8 +292,8 @@
2.26 to which we are referring is the probability distribution over the population of surface pixels.
2.27 The simplest way to use information theory is on discrete data, so we discretized our gene expression data by creating,
2.28 for each gene, five thresholded binary masks of the gene data. For each gene, we created a binary mask of its expression
2.29 -levels over pixels using each of these thresholds: the mean of that gene, the mean minus one standard deviation, the mean
2.30 -minus two standard deviations, the mean plus one standard deviation, the mean plus two standard deviations.
2.31 +levels using each of these thresholds: the mean of that gene, the mean minus one standard deviation, the mean minus two
2.32 +standard deviations, the mean plus one standard deviation, the mean plus two standard deviations.
2.33 Now, for each region, we created and ran a forward stepwise procedure which attempted to find pairs of gene expression
2.34 binary masks such that the conditional entropy of the target area’s binary mask, conditioned upon the pair of gene expression
2.35 binary masks, is minimized.
2.36 @@ -311,9 +311,24 @@
2.37 Gradient similarity We noticed that the previous two scoring methods, which are pointwise, often found genes whose
2.38 pattern of expression did not look similar in shape to the target region. Fort his reason we designed a non-pointwise local
2.39 scoring method to detect when a gene had a pattern of expression which looked like it had a boundary whose shape is similar
2.40 -to the shape of the target region.
2.41 -had shape of the pattern of expression did not seem to match the shape of the target area.
2.42 -todo
2.43 +to the shape of the target region. We call this scoring method “gradient similarity”.
2.44 +One might say that gradient similarity attempts to measure how much the border of the area of gene expression and
2.45 +the border of the target region overlap. However, since gene expression falls off continuously rather than jumping from its
2.46 +maximum value to zero, the spatial pattern of a gene’s expression often does not have a discrete border. Therefore, instead
2.47 +of looking for a discrete border, we look for large gradients. Gradient similarity is a symmetric function over two images
2.48 +(i.e. two scalar fields). It is is high to the extent that matching pixels which have large values and large gradients also have
2.49 +gradients which are oriented in a similar direction. The formula is:
2.50 +∑
2.51 +
2.52 +pixel<img src="cmsy7-32.png" alt="∈" />pixels cos(abs(∠∇1 - ∠∇2)) ⋅|∇1|+|∇2|
2.53 + 2 ⋅ pixel_value1+pixel_value2
2.54 + 2
2.55 +where ∇1 and ∇2 are the gradient vectors of the two images at the current pixel; ∠∇i is the angle of the gradient of
2.56 +image i at the current pixel; |∇1| is the magnitude of the gradient of image i at the current pixel; and pixelvaluei is the
2.57 +value of the current pixel in image i.
2.58 +The intuition is that we want to see if the borders of the pattern in the two images are similar; if the borders are similar,
2.59 +then both images will have corresponding pixels with large gradients (because this is a border) which are oriented in a
2.60 +similar direction (because the borders are similar).
2.61 Using combinations of multiple genes is necessary and sufficient to delineate some cortical areas
2.62 Here we give an example of a cortical area which is not marked by any single gene, but which can be identified combi-
2.63 natorially. according to logistic regression, gene wwc19 is the best fit single gene for predicting whether or not a pixel on
2.64 @@ -327,6 +342,17 @@
2.65 Geometric and pointwise scoring methods provide complementary information
2.66 To show that local geometry can provide useful information that cannot be detected via pointwise analyses, consider Fig.
2.67 . The top row of Fig. displays the 3 genes which most match area AUD, according to a pointwise method11. The bottom
2.68 +_________________________________________
2.69 + 9“WW, C2 and coiled-coil domain containing 1”; EntrezGene ID 211652
2.70 + 10“mitochondrial translational initiation factor 2”; EntrezGene ID 76784
2.71 + 11For each gene, a logistic regression in which the response variable was whether or not a surface pixel was within area AUD, and the predictor
2.72 +variable was the value of the expression of the gene underneath that pixel. The resulting scores were used to rank the genes in terms of how well
2.73 +
2.74 +
2.75 +
2.76 +Figure 2: The top row shows the three genes which (individually) best predict area AUD, according to logistic regression.
2.77 +The bottom row shows the three genes which (individually) best match area AUD, according to gradient similarity. From
2.78 +left to right and top to bottom, the genes are Ssr1, Efcbp1, Aph1a, Ptk7, Aph1a again, and Lepr
2.79 row displays the 3 genes which most match AUD according to a method which considers local geometry12 The pointwise
2.80 method in the top row identifies genes which express more strongly in AUD than outside of it; its weakness is that this
2.81 includes many areas which don’t have a salient border matching the areal border. The geometric method identifies genes
2.82 @@ -335,20 +361,6 @@
2.83 Aph1a in the example, may be particularly good markers. None of these genes are, individually, a perfect marker for AUD;
2.84 we deliberately chose a “difficult” area in order to better contrast pointwise with geometric methods.
2.85 Areas which can be identified by single genes
2.86 -_________________________________________
2.87 - 9“WW, C2 and coiled-coil domain containing 1”; EntrezGene ID 211652
2.88 - 10“mitochondrial translational initiation factor 2”; EntrezGene ID 76784
2.89 - 11For each gene, a logistic regression in which the response variable was whether or not a surface pixel was within area AUD, and the predictor
2.90 -variable was the value of the expression of the gene underneath that pixel. The resulting scores were used to rank the genes in terms of how well
2.91 -they predict area AUD.
2.92 - 12For each gene the gradient similarity (see section ??) between (a) a map of the expression of each gene on the cortical surface and (b) the
2.93 -shape of area AUD, was calculated, and this was used to rank the genes.
2.94 -
2.95 -
2.96 -
2.97 -Figure 2: The top row shows the three genes which (individually) best predict area AUD, according to logistic regression.
2.98 -The bottom row shows the three genes which (individually) best match area AUD, according to gradient similarity. From
2.99 -left to right and top to bottom, the genes are Ssr1, Efcbp1, Aph1a, Ptk7, Aph1a again, and Lepr
2.100 todo
2.101 Areas can sometimes be marked by underexpression
2.102 todo
2.103 @@ -369,10 +381,13 @@
2.104 (might want to incld nnMF since mentioned above)
2.105 Dimensionality reduction plus K-means or spectral clustering
2.106 Many areas are captured by clusters of genes
2.107 -todo
2.108 -todo
2.109 _________________________________________
2.110 - 135-fold cross-validation.
2.111 +they predict area AUD.
2.112 + 12For each gene the gradient similarity (see section ??) between (a) a map of the expression of each gene on the cortical surface and (b) the
2.113 +shape of area AUD, was calculated, and this was used to rank the genes.
2.114 + 135-fold cross-validation.
2.115 +todo
2.116 +todo
2.117 Research plan
2.118 todo amongst other things:
2.119 Develop algorithms that find genetic markers for anatomical regions
3.1 Binary file grant.odt has changed
4.1 Binary file grant.pdf has changed
5.1 --- a/grant.txt Tue Apr 14 02:31:37 2009 -0700
5.2 +++ b/grant.txt Tue Apr 14 02:50:49 2009 -0700
5.3 @@ -202,6 +202,8 @@
5.4
5.5 We created a normalized version of the gene expression data by subtracting each gene's mean expression level (over all surface pixels) and dividing each gene by its standard deviation.
5.6
5.7 +The features and the target area are both functions on the surface pixels. They can be referred to as scalar fields over the space of surface pixels; alternately, they can be thought of as images which can be displayed on the flatmapped surface.
5.8 +
5.9 To move beyond a single average expression level for each surface pixel, we plan to create a separate matrix for each cortical layer to represent the average expression level within that layer. Cortical layers are found at different depths in different parts of the cortex. In preparation for extracting the layer-specific datasets, we have extended Caret with routines that allow the depth of the ROI for volume-to-surface projection to vary.
5.10
5.11 In the Research Plan, we describe how we will automatically locate the layer depths. For validation, we have manually demarcated the depth of the outer boundary of cortical layer 5 throughout the cortex.
5.12 @@ -218,7 +220,7 @@
5.13 \vspace{0.3cm}**Correlation**
5.14 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 binary mask over the surface pixels.
5.15
5.16 -The features and the target area are both functions on the surface pixels; alternately, they can be thought of as images which can be displayed on the flatmapped surface. One class of feature selection scoring method are those 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.
5.17 +One class of feature selection scoring method are those 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.
5.18
5.19 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.
5.20
5.21 @@ -227,7 +229,7 @@
5.22 \vspace{0.3cm}**Conditional entropy**
5.23 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.
5.24
5.25 -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 binary masks of the gene data. For each gene, we created a binary mask of its expression levels over pixels 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.
5.26 +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 binary masks of the gene data. For each gene, we created a binary 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.
5.27
5.28 Now, for each region, we created and ran a forward stepwise procedure which attempted to find pairs of gene expression binary masks such that the conditional entropy of the target area's binary mask, conditioned upon the pair of gene expression binary masks, is minimized.
5.29
5.30 @@ -236,32 +238,15 @@
5.31 todo: fig
5.32
5.33 \vspace{0.3cm}**Gradient similarity**
5.34 -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.
5.35 -
5.36 -
5.37 -
5.38 -had shape of the pattern of expression did not seem to match the shape of the target area.
5.39 -
5.40 -todo
5.41 -
5.42 -
5.43 -
5.44 -
5.45 -\vspace{0.3cm}**Using combinations of multiple genes is necessary and sufficient to delineate some cortical areas**
5.46 -
5.47 -Here 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\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 (todo).
5.48 -
5.49 -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.
5.50 -
5.51 -\begin{figure}\label{MOcombo}
5.52 -\includegraphics[scale=.36]{MO_vs_Wwc1_jet.eps}
5.53 -\includegraphics[scale=.36]{MO_vs_Mtif2_jet.eps}
5.54 -
5.55 -\includegraphics[scale=.36]{MO_vs_Wwc1_plus_Mtif2_jet.eps}
5.56 -\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). Within each picture, the vertical axis roughly corresponds to anterior at the top and posterior at the bottom, and the horizontal axis roughly corresponds to medial at the left and lateral at the right. The red outline is the boundary of region MO. Pixels are colored approximately according to the density of expressing cells underneath each pixel, with red meaning a lot of expression and blue meaning little.}
5.57 -\end{figure}
5.58 -
5.59 -
5.60 +We noticed that the previous two scoring methods, which are pointwise, often found genes whose pattern of expression did not look similar in shape to the target region. Fort his reason we designed a non-pointwise local scoring method to detect when a gene had a pattern of expression which looked like it had a boundary whose shape is similar to the shape of the target region. We call this scoring method "gradient similarity".
5.61 +
5.62 +One might say that gradient similarity attempts to measure how much the border of the area of gene expression and the border of the target region overlap. However, since gene expression falls off continuously rather than jumping from its maximum value to zero, the spatial pattern of a gene's expression often does not have a discrete border. Therefore, instead of looking for a discrete border, we look for large gradients. Gradient similarity is a symmetric function over two images (i.e. two scalar fields). It is is high to the extent that matching pixels which have large values and large gradients also have gradients which are oriented in a similar direction. The formula is:
5.63 +
5.64 +\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}
5.65 +
5.66 +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; $\vert \nabla_1 \vert$ 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 $i$.
5.67 +
5.68 +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).
5.69
5.70 \vspace{0.3cm}**Geometric and pointwise scoring methods provide complementary information**
5.71
5.72 @@ -282,6 +267,24 @@
5.73 \end{figure}
5.74
5.75
5.76 +\vspace{0.3cm}**Using combinations of multiple genes is necessary and sufficient to delineate some cortical areas**
5.77 +
5.78 +Here 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\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 (todo).
5.79 +
5.80 +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.
5.81 +
5.82 +\begin{figure}\label{MOcombo}
5.83 +\includegraphics[scale=.36]{MO_vs_Wwc1_jet.eps}
5.84 +\includegraphics[scale=.36]{MO_vs_Mtif2_jet.eps}
5.85 +
5.86 +\includegraphics[scale=.36]{MO_vs_Wwc1_plus_Mtif2_jet.eps}
5.87 +\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). Within each picture, the vertical axis roughly corresponds to anterior at the top and posterior at the bottom, and the horizontal axis roughly corresponds to medial at the left and lateral at the right. The red outline is the boundary of region MO. Pixels are colored approximately according to the density of expressing cells underneath each pixel, with red meaning a lot of expression and blue meaning little.}
5.88 +\end{figure}
5.89 +
5.90 +
5.91 +
5.92 +
5.93 +
5.94 \vspace{0.3cm}**Areas which can be identified by single genes**
5.95
5.96 todo