1.1 Binary file grant.doc has changed

2.1 --- a/grant.html Tue Apr 14 02:50:49 2009 -0700 2.2 +++ b/grant.html Tue Apr 14 02:53:00 2009 -0700 2.3 @@ -300,13 +300,11 @@ 2.4 This finds pairs of genes which are most informative (at least at these discretization thresholds) relative to the question, 2.5 &#8220;Is this surface pixel a member of the target area?&#8221;. 2.6 2.7 - 2.8 - 2.9 -Figure 1: Upper left: wwc1. Upper right: mtif2. Lower left: wwc1 + mtif2 (each pixel&#8217;s value on the lower left is the sum 2.10 -of the corresponding pixels in the upper row). Within each picture, the vertical axis roughly corresponds to anterior at the 2.11 -top and posterior at the bottom, and the horizontal axis roughly corresponds to medial at the left and lateral at the right. 2.12 -The red outline is the boundary of region MO. Pixels are colored approximately according to the density of expressing cells 2.13 -underneath each pixel, with red meaning a lot of expression and blue meaning little. 2.14 + 2.15 + 2.16 +Figure 1: The top row shows the three genes which (individually) best predict area AUD, according to logistic regression. 2.17 +The bottom row shows the three genes which (individually) best match area AUD, according to gradient similarity. From 2.18 +left to right and top to bottom, the genes are Ssr1, Efcbp1, Aph1a, Ptk7, Aph1a again, and Lepr 2.19 todo: fig 2.20 Gradient similarity We noticed that the previous two scoring methods, which are pointwise, often found genes whose 2.21 pattern of expression did not look similar in shape to the target region. Fort his reason we designed a non-pointwise local 2.22 @@ -318,48 +316,51 @@ 2.23 of looking for a discrete border, we look for large gradients. Gradient similarity is a symmetric function over two images 2.24 (i.e. two scalar fields). It is is high to the extent that matching pixels which have large values and large gradients also have 2.25 gradients which are oriented in a similar direction. The formula is: 2.26 -&#x2211; 2.27 - 2.28 -pixel<img src="cmsy7-32.png" alt="&#x2208;" />pixels cos(abs(&#x2220;&#x2207;1 - &#x2220;&#x2207;2)) &#x22C5;|&#x2207;1|+|&#x2207;2| 2.29 - 2 &#x22C5; pixel_value1+pixel_value2 2.30 - 2 2.31 + &#x2211; 2.32 + pixel<img src="cmsy7-32.png" alt="&#x2208;" />pixels cos(abs(&#x2220;&#x2207;1 -&#x2220;&#x2207;2)) &#x22C5;|&#x2207;1| + |&#x2207;2| 2.33 + 2 &#x22C5; pixel_value1 + pixel_value2 2.34 + 2 2.35 where &#x2207;1 and &#x2207;2 are the gradient vectors of the two images at the current pixel; &#x2220;&#x2207;i is the angle of the gradient of 2.36 -image i at the current pixel; |&#x2207;1| is the magnitude of the gradient of image i at the current pixel; and pixelvaluei is the 2.37 +image i at the current pixel; |&#x2207;i| is the magnitude of the gradient of image i at the current pixel; and pixel_valuei is the 2.38 value of the current pixel in image i. 2.39 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, 2.40 then both images will have corresponding pixels with large gradients (because this is a border) which are oriented in a 2.41 similar direction (because the borders are similar). 2.42 +Geometric and pointwise scoring methods provide complementary information 2.43 +To show that gradient similarity can provide useful information that cannot be detected via pointwise analyses, consider 2.44 +Fig. . The top row of Fig. displays the 3 genes which most match area AUD, according to a pointwise method9. The 2.45 +bottom row displays the 3 genes which most match AUD according to a method which considers local geometry10 The 2.46 +pointwise method in the top row identifies genes which express more strongly in AUD than outside of it; its weakness is 2.47 +that this includes many areas which don&#8217;t have a salient border matching the areal border. The geometric method identifies 2.48 +genes whose salient expression border seems to partially line up with the border of AUD; its weakness is that this includes 2.49 +genes which don&#8217;t express over the entire area. Genes which have high rankings using both pointwise and border criteria, 2.50 +such as Aph1a in the example, may be particularly good markers. None of these genes are, individually, a perfect marker 2.51 +for AUD; we deliberately chose a &#8220;difficult&#8221; area in order to better contrast pointwise with geometric methods. 2.52 Using combinations of multiple genes is necessary and sufficient to delineate some cortical areas 2.53 Here we give an example of a cortical area which is not marked by any single gene, but which can be identified combi- 2.54 -natorially. according to logistic regression, gene wwc19 is the best fit single gene for predicting whether or not a pixel on 2.55 +natorially. according to logistic regression, gene wwc111 is the best fit single gene for predicting whether or not a pixel on 2.56 +_________________________________________ 2.57 + 9For each gene, a logistic regression in which the response variable was whether or not a surface pixel was within area AUD, and the predictor 2.58 +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 2.59 +they predict area AUD. 2.60 + 10For each gene the gradient similarity (see section ??) between (a) a map of the expression of each gene on the cortical surface and (b) the 2.61 +shape of area AUD, was calculated, and this was used to rank the genes. 2.62 + 11&#8220;WW, C2 and coiled-coil domain containing 1&#8221;; EntrezGene ID 211652 2.63 + 2.64 + 2.65 + 2.66 +Figure 2: Upper left: wwc1. Upper right: mtif2. Lower left: wwc1 + mtif2 (each pixel&#8217;s value on the lower left is the sum 2.67 +of the corresponding pixels in the upper row). Within each picture, the vertical axis roughly corresponds to anterior at the 2.68 +top and posterior at the bottom, and the horizontal axis roughly corresponds to medial at the left and lateral at the right. 2.69 +The red outline is the boundary of region MO. Pixels are colored approximately according to the density of expressing cells 2.70 +underneath each pixel, with red meaning a lot of expression and blue meaning little. 2.71 the cortical surface belongs to the motor area (area MO). The upper-left picture in Figure shows wwc1&#8217;s spatial expression 2.72 pattern over the cortex. The lower-right boundary of MO is represented reasonably well by this gene, however the gene 2.73 overshoots the upper-left boundary. This flattened 2-D representation does not show it, but the area corresponding to the 2.74 overshoot is the medial surface of the cortex. MO is only found on the lateral surface (todo). 2.75 -Gene mtif210 is shown in figure the upper-right of Fig. . Mtif2 captures MO&#8217;s upper-left boundary, but not its lower-right 2.76 +Gene mtif212 is shown in figure the upper-right of Fig. . Mtif2 captures MO&#8217;s upper-left boundary, but not its lower-right 2.77 boundary. Mtif2 does not express very much on the medial surface. By adding together the values at each pixel in these 2.78 two figures, we get the lower-left of Figure . This combination captures area MO much better than any single gene. 2.79 -Geometric and pointwise scoring methods provide complementary information 2.80 -To show that local geometry can provide useful information that cannot be detected via pointwise analyses, consider Fig. 2.81 -. The top row of Fig. displays the 3 genes which most match area AUD, according to a pointwise method11. The bottom 2.82 -_________________________________________ 2.83 - 9&#8220;WW, C2 and coiled-coil domain containing 1&#8221;; EntrezGene ID 211652 2.84 - 10&#8220;mitochondrial translational initiation factor 2&#8221;; EntrezGene ID 76784 2.85 - 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 2.86 -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 2.87 - 2.88 - 2.89 - 2.90 -Figure 2: The top row shows the three genes which (individually) best predict area AUD, according to logistic regression. 2.91 -The bottom row shows the three genes which (individually) best match area AUD, according to gradient similarity. From 2.92 -left to right and top to bottom, the genes are Ssr1, Efcbp1, Aph1a, Ptk7, Aph1a again, and Lepr 2.93 -row displays the 3 genes which most match AUD according to a method which considers local geometry12 The pointwise 2.94 -method in the top row identifies genes which express more strongly in AUD than outside of it; its weakness is that this 2.95 -includes many areas which don&#8217;t have a salient border matching the areal border. The geometric method identifies genes 2.96 -whose salient expression border seems to partially line up with the border of AUD; its weakness is that this includes genes 2.97 -which don&#8217;t express over the entire area. Genes which have high rankings using both pointwise and border criteria, such as 2.98 -Aph1a in the example, may be particularly good markers. None of these genes are, individually, a perfect marker for AUD; 2.99 -we deliberately chose a &#8220;difficult&#8221; area in order to better contrast pointwise with geometric methods. 2.100 Areas which can be identified by single genes 2.101 todo 2.102 Areas can sometimes be marked by underexpression 2.103 @@ -379,13 +380,11 @@ 2.104 Raw dimensionality reduction results 2.105 todo 2.106 (might want to incld nnMF since mentioned above) 2.107 +_________________________________________ 2.108 + 12&#8220;mitochondrial translational initiation factor 2&#8221;; EntrezGene ID 76784 2.109 + 135-fold cross-validation. 2.110 Dimensionality reduction plus K-means or spectral clustering 2.111 Many areas are captured by clusters of genes 2.112 -_________________________________________ 2.113 -they predict area AUD. 2.114 - 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 2.115 -shape of area AUD, was calculated, and this was used to rank the genes. 2.116 - 135-fold cross-validation. 2.117 todo 2.118 todo 2.119 Research plan

3.1 Binary file grant.odt has changed

4.1 Binary file grant.pdf has changed

5.1 --- a/grant.txt Tue Apr 14 02:50:49 2009 -0700 5.2 +++ b/grant.txt Tue Apr 14 02:53:00 2009 -0700 5.3 @@ -242,17 +242,17 @@ 5.4 5.5 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: 5.6 5.7 +\begin{align*} 5.8 \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} 5.9 - 5.10 -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 $i$. 5.11 +\end{align*} 5.12 + 5.13 +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 $i$. 5.14 5.15 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). 5.16 5.17 \vspace{0.3cm}**Geometric and pointwise scoring methods provide complementary information** 5.18 5.19 - 5.20 - 5.21 -To show that local geometry 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 (see section \ref{gradientSim}) 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. 5.22 +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 (see section \ref{gradientSim}) 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. 5.23 5.24 5.25 \begin{figure}\label{AUDgeometry}