Description
Problem 1 You have a set of 𝑁 training inputs 𝐱𝑛 ∈ ℝ𝑀, 𝑛 = 1, 2, … , 𝑁, 𝑁 ≫ 𝑀. The target outputs of the training inputs are 𝑡𝑛 ∈ ℝ, 𝑛 = 1, 2, … , 𝑁. Build a linear regression model to predict the target value by 𝐰𝑇𝐱𝑛. Derive the closed-form solution for the weight vector 𝐰 ∈ ℝ𝑀 that minimizes the error function 𝐸(𝐰) =
{𝐰𝑇𝐱𝑛 − 𝑡𝑛 }2.
Problem 2 The Pima Indians diabetes data set (pima-indians-diabetes.xlsx) is a data set used to diagnostically predict whether or not a patient has diabetes, based on certain diagnostic measurements included in the dataset. All patients here are females at least 21 years old of Pima Indian heritage. The dataset consists of M = 8 attributes and one target variable, Outcome (1 represents diabetes, 0 represents no diabetes). The 8 attributes include Pregnancies, Glucose, BloodPressure, BMI, insulin level, age, and so on. There are N=768 data samples.
Randomly select n samples from the “diabetes” class and n samples from the “no diabetes” class, and use them as the training samples. The remaining data samples are the test samples. Build a linear regression model as described in Problem 1 with the training set, and test your model on the test samples to predict whether or not a test patient has diabetes or not. Assume the predicted outcome of a test sample is 𝑡̂, if 𝑡̂ ≥ 0.5 (closer to 1), classify it as “diabetes”; if 𝑡̂ < 0.5 (closer to 0), classify it as “no diabetes”. Run 1000 independent experiments, and calculate the prediction accuracy rate as 𝑡ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑐𝑜𝑟𝑟𝑒𝑐𝑡 𝑝𝑟𝑒𝑑𝑖𝑐𝑡𝑖𝑜𝑛𝑠 %. Let n=40, 80, 120, 160, 200, plot the
