Homework 5

Problem 5.1 (Learning the Power Law):

Design, train, and optimize a generic neural network (NN) that can learn and predict the power-function Links to an external site., $f(x) = a \ x^{\lambda}$, for a given power parameter ($\lambda\in \mathbb{R}$), you can assume for simplicity that the constant multiplier $a=1$. Assess the accuracy of the NN prediction of the power function. Your program should start by taking an input from the user specifying the desired power parameter ($\lambda$), assume $\frac{1}{2}\le\lambda\le3$. Then, the program should generate training and testing data (X and Y), and finally, evaluate the performance of the trained NN model. [Hint: Recall the example with the square-root function Links to an external site..]

Problem 5.2 (Autism Clustering):

Use the Autism Dataset , which includes neuroanatomical and functional characterizations of children and young adult brains. Autism is a brain development disorder affecting personal interactions, communications, learning, and behavior. Major clinically relevant questions include: How to discriminate (cluster), predict, or model pediatric normal (controls) and pathological (autism) brains (Y=researchGroup) using some of the other covariates (X) in the dataset?

• Load and prepare the data
• Report (short!) data summaries and show some preliminary visualizations
• Train a k-Means clustering model on the data (exclude certain variables, e.g., researchGroup) , select k
• Evaluate the model performance using bar and silhouette plots and summarize the results
• Tune and plot parameters with k-means++
• Rerun the model with the optimal parameters and interpret the clustering results
• Apply Hierarchical Clustering and compare the corresponding silhouette plots
• Fit a Gaussian mixture model, select the optimal model, report BIC, and display density and classification plots
• Compare the result of the above methods
• From the list of all (~2100) variables, try to identify the most salient (important) features that are associated with autism.