IDS575_PS2_Q1&2

Q1 Linear Regression Basic 22 Points Q1.1 5 Points Linear Regression is a supervised machine learning model/algorithm for predicting a continuous output variable. Q1.2 5 Points Which of the following offsets does the linear regression uses for its least square line fitting? Feel free to assume that the horizontal axis is an independent variable and the vertical axis is a dependent variable? True  False  Vertical offset  Perpendicular offset  Both, depending on the situation  None of above 

Q1.3 7 Points Which of the following evaluation metrics can be properly used to evaluate a model that predicts a continuous output variable? (select all correct) Q1.4 5 Points Given the pictures of vertical and perpendicular offsets, choose the right one for residual. Q2 Multivariate Calculus Basic 30 Points Q2.1 10 Points Absolute Error  ∣ y − ∣ y ^ Squared Error  ( y − ) y ^ 2 Cubic Error ( y − ) y ^ 3 0-1 Error 1{ y = } y ^ Measured by vertical offsets. Higher is better.  Measured by vertical offsets. Lower is better.  Measured by perpendicular offsets. Higher is better.  Measured by perpendicular offsets. Lower is better.  None of the above. 

Q3 Linear Regression 48 Points In class, we derived linear regression and various learning algorithms based on gradient descent. In addition to the least square objective, we also learned its probabilistic perspective where each observation is assumed to have a Gaussian noise. (i.e., Noise of each example is an independent and identically distributed sample from a normal distribution) In this problem, you are supposed to deal with the following regression model that includes one feature that relates linearly/quadratically and another linear feature . where Q3.1 5 Points The above equation says that linear regresion assumes is also a random variable due the amount of uncertainty given by the noise term . What distribution would the random output variable follow? Tiny change in by (with the fixed ) will change by .  x 1 ϵ x 2 h s Tiny change in by (with the fixed ) will change by .  x 2 ϵ x 1 h s Tiny change in by (with the fixed ) will change by .  x 1 ϵ x 2 h ϵs Tiny change in by (with the fixed ) will change by .  x 2 ϵ x 1 h ϵs None of the above  x 1 x 2 y = θ + 0 θ x + 1 1 θ x + 2 2 θ x + 3 1 2 ϵ ϵ ∼ N (0, σ ) 2 y ϵ y uniform distribution  poisson distribution  normal distribution  Bernoulli distribution  multinomial distribution 