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689-613: ECEN SP TP PROB GRAPHICAL MODELS, Spring`09 Final Project Final Project: Identification & Analysis of Gene Regulatory Network Due dates: Part-1 : Implementation of Pearl's inference algorithm for Bayesian Networks. Submission of progress-report (Apr. 28, 2009) Part-2: Identification & analysis of a gene regulatory network Network construction and prediction challenge (May 4, 2009) Submission of final report, in-class presentation (May 5, 2009) PART-1. Implementing the Inference Algorithm Consider a Bayesian Network over X = {X1 , X2 , . . . , XN } with a graph structure G. We assume that G is a singly-connected graph (i.e., a polytree), where the maximum in/out-degree of every node is at most 2. All the random variables Xk X are binary, hence Xk {0, 1}. For this BN, we can use Pearl's messagepassing algorithm for computing the conditional probability P (Xk = xk |E = e), where E X is a set of evidence variables, and e is the set of instantiated values. (1-A) Implement the inference algorithm for computing P (Xk = xk |E = e). (1-B) Consider the BN shown in Fig. 1, where all the random variables take binary values. The conditional probabilities in this BN are specified as follows: P (A1) = 0.7 P (B1) = 0.4 P (C1|A0, B0) = 0.1, P (C1|A1, B0) = 0.3, P (C1|A0, B1) = 0.5, P (C1|A1, B1) = 0.9 P (D1|C0) = 0.8, P (D1|C1) = 0.3 P (E1|C0) = 0.2, P (E1|C1) = 0.6 P (F 1|D0) = 0.1, P (F 1|D1) = 0.7 P (G1|D0) = 0.9, P (G1|D1) = 0.4 Using your algorithm, compute the following probabilities: (i) P (A1|B0), P (A1|D0), P (A1|D0, B0), P (A1|D0, G1) A B C D E F G Figure 1: A simple Bayesian network. (ii) P (B1|A1), P (B1|C1), P (B1|A1, C1), P (B1|C1, F 0) (iii) P (C1), P (C1|A1), P (C1|A1, B0), P (C1|D0), P (C1|D0, F 0) (iv) P (D1), P (D1|E0), P (D1|C0, E0), P (D1|B1, G0), P (D1|B1, G0, F 1) (v) P (E1), P (E1|C1), P (E1|F 0), P (E1|C1, F 0), P (E1|A1, B1) (vi) P (F 1), P (F 1|A1), P (F 1|A1, C0), P (F 1|A1, C0, E0), P (F 1|B1, G0) (vii) P (G1), P (G1|C0), P (G1|C0, D0), P (G1|E0), P (G1|A0, B1) (viii) P (A1, D1|F 0, B1), P (C0, E1|F 1, G0), P (F 0, B1|G0, E1), P (G1, B0|F 1, A0) (1-C) After completing parts (1-A) and (1-B), submit a progress-report that contains the following: Brief description of your source code. Source code of your program. (The code should be well-commented & easy to read) The probabilities in part (1-B) computed using your algorithm. Send your report to the instructor via email. Name your file as "YourName progress report.pdf (or .doc)"Also print out your report and submit it in the class. PART-2. Identification & Analysis of Gene Regulatory Network Consider a gene regulatory network that contains 20 genes. We assume that each gene is either up-regulated ("on") or down-regulated ("off"), hence we represent the genes using binary random variables Xk {0, 1}. Xk = 1 denotes that the kth gene is turned "on", whereas Xk = 0 denotes that the kth gene is "off". We assume that the gene regulatory network of interest can be effectively represented by a Bayesian network over X = {X1 , X2 , . . . , X20 }. However, we do not know the BN structure G nor the underlying conditional probability distributions (CPDs) in the BN. Therefore we have to learn the structure and estimate the CPDs from the available training data. The training data consist of a large number of "samples", where each sample can be viewed as a realization of the 20-dimensional random vector (X1 , X2 , . . . , X20 ) according to the unknown BN. (2-A) Predict the structure G of the underlying Bayesian network based on the training data "train250.txt". This file contains a 250 20 matrix M for 250 samples. The (i, j) element M (i, j) corresponds to the value of the jth gene (i.e., Xj ) in the ith sample. The graph G satisfies the following properties: G is a singly-connected graph. height of the tree 7 maximum in-degree 2 maximum out-degree 2 You can use various approaches (e.g., correlation, mutual information, coefficient of determination (COD), multivariate synergy, finding a minimal I-map, etc.) to find the structure of the gene regulatory network. Try to search for relevant papers to get more ideas be and creative! (2-B) Once the structure G of the Bayesian network has been found, estimate the CPDs in the network. Use pseudocounts to allow small probabilities for events not observed in the training data. (2-C) Use the constructed Bayesian network to perform the following experiments: 1. Download the test data set "test25a.txt". This fine contains a 25 20 matrix T for 25 samples. Consider the jth row, and let (xj , xj , . . . , xj ) = (T (j, 1), T (j, 2), . . . , T (j, 20)). 2 20 1 Let e = {xj , xj , xj } and compute P (Xk = x|e) for x {0, 1}, 11 k 20. 1 2 3 Predict xj = arg maxx P (Xk = x|e), and compare xj with the actual value xj . ^k ^k k Repeat this for all rows and compute the average prediction accuracy. Let pj = maxx P (Xk = x|e) and compute the expected prediction accuracy. Compare it with the k actual prediction accuracy. 2. Repeat the above experiment for the data set "test25b.txt". For the jth row, let e = {xj , xj , . . . , xj } and predict xj and compute pj (11 k 20). ^k 1 2 5 k Compute the average prediction accuracy and compare it with the estimated prediction accuracy. 3. Repeat the above experiment for the data set "test25c.txt". For the jth row, let e = {xj , xj , . . . , xj } and predict xj and compute pj (11 k 20).. ^k 1 2 7 k Compute the average prediction accuracy and compare it with the estimated prediction accuracy. 4. Repeat the above experiment for the data set "test25d.txt". For the jth row, let e = {xj , xj , . . . , xj } and predict xj and compute pj (11 k 20).. ^k 1 2 9 k Compute the average prediction accuracy and compare it with the estimated prediction accuracy. (2-D) After completing parts (2-A)(2-C), summarize your simulation results. The report should contain the following: 1. Include your original progress report. 2. Describe the method that was used to construct the Bayesian network. 3. Draw the predicted network structure, and show the estimated CPDs. 4. Summarize the experimental results: For each data set, summarize xk (predicted values), xk (actual values), and pk in a table. ^j j j For each data set, summarize the average prediction accuracy and the expected prediction accuracy for each gene Xk (k = 11, . . . , 20). Compute the actual overall prediction accuracy as well as the expected overall prediction accuracy for all the genes X11 , . . . , X20 . 5. Discuss the simulation results. Send your report to the instructor via email. Name your file as "YourName final report.pdf (or .doc)" Also print out your final report and submit it in the class. (2-E: Prediction Challenge) Use the constructed Bayesian network to perform the following experiment: 1. Download the dataset "challenge.txt". This file contains a 100 20 matrix C with 25 samples with 17 missing values (rows 1 25), 25 samples with 15 missing values (rows 26 50), 25 samples with 13 missing values (rows 51 75), and 25 samples with 11 missing values (rows 76 100). 2. By using all the known values as the evidence "e", (individually) predict all the missing values. 3. Complete the matrix C. Send your completed matrix C to the instructor vial email. The matrix should be stored in a text file, where the elements in the same row are separated by tabs. Name your file as "YourName challenge.txt" (2-F **optional**) Repeat similar experiments for predicting the pairs (xk , xk+1 ) (for k = 11, . . . , 19) and triplets (xk , xk+1 , xk+2 ) (for k = 11, . . . , 18). Compute the actual (individual & overall) prediction accuracy and compare it with the expected prediction accuracy. Include the results in your report. Grading Your work will be graded as follows: Items (1-A) Algorithm Implementation Evaluation Criteria completeness readability well-commented correctness technical soundness creativity accuracy technical soundness prediction accuracy completeness overall presentation in-depth discussion prediction accuracy technical soundness prediction accuracy Score 10 5 5 10 15 5 10 10 5 5 5 5 10 2.5 2.5 Subtotal 20 10 30 15 (1-B) Probability Computation (2-A/B) BN Construction (2-C) Inference (2-D) Documentation 15 10 5 (2-E) Challenge (2-F) Inference **optional**

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