Reference no: EM133640096

Scientific Computing

Instruction:

You may use MATLAB or Python for writing your programs. You are allowed to use the library function for calculating the matrix inverse, but are not allowed to use any library functions for calculating the pseudo or Moore-Penrose inverse.

Upload all your programs to the homework dropbox (do not put this in a word document or a pdf file. I'll should be able to run your programs for evaluation)

Learning Objective: To understand that there may exist a unique solution for a square matrix system of equations (using matrix inverse) and there exists a unique least squares solution to a rectangular matrix system of equations when there are more number of training data than the number of unknown model parameters to be estimated.

Homework Problem:

Question 1. Write a program to determine a unique polynomial function y(x) that passes through all the given data {(x₁, y₁), . . . , (x_n, y_n)}, for an arbitrary choice of n. Confirm that your estimates of the model parameters agree with the data by plotting the model predicted values of y_i for various values of x_i.

Data Generation : To evaluate your algorithms (programs), generate test data

{(x_i, y_i)}_i=1ⁿ that has a cubic relationship as follows:

(a) Let the underlying relationship between variables x_i (independent) and y_i (dependent) be cubic. i.e. y_i = β₀ + β₁x_i + β₂x₂ + β₃x₃.

(b) Choose any non-zero values for the model parameters β₀, β₁, β₂ and β₃.

(c) Generate exactly 4 unique data points from the cubic model: {(x_i, yi)}⁴_i=1, where y_i = β₀ + β₁x_i + β₂x₂ + β₃x₃. For example, choosei x_i = {-2, 0, 2, 4} ; β₀ = 3; β₁ = -1; β₂ = -3 and β₃ = 1 and estimate the model predicted values yi. i.e. these data points lie exactly on the cubic model. (ancillary question: how should one choose {x_i}⁴_i=1 for simulating the data governed by the model?)

Model validation

(a) Plot the data {(x_i, y_i)}⁴ _i=1 used for estimating the model parameters {β_i, i = 0, . . . , 3}

(b) Once the model parameters are estimated, estimate yˆ_i for {x_i = -2, -1.9, -1.8, . . . , -0.1, 0, 0.1, . . . , 3.8, 3.9, 4.0}

(c) In the same figure, plot the data {(x_i, y^_i)} predicted by the model in the previous step.

(d) Write your observations from the plot as well as by comparing your parameter estimates with the true model parameters.

Question 2: Write a program to determine a polynomial function y(x) that passes through all the given data {(x₁, y₁), . . . , (x_n, y_n)} in a least squared error sense as discussed in class (for an arbitrary choice of n). Confirm that your estimates of the model parameters agree with the data by plotting the model predicted values of y_i for various values of x_i

Data Generation : To evaluate your algorithms (programs), generate test data {(x_i, y_i)}ⁿ_i=1 that has a cubic relationship as follows:

a) Same as in the first problem.
b) Same as in the first problem.
c) Same as in the first problem.
d) Add random variations to each of the data points as follows:

y_i = β₀ + β₁x_i + β₂x₂ + β₃x₃ + ∈_i, where the random variable ∈_i follows a normal distribution with 0 mean and σ standard deviation. i.e. ∈_i ∼ Normal(0, σ²). You can choose any variance σ, e.g. σ = 2. Hint: In Matlab, you can use eps = normrnd(mu, sigma, n) to draw n random ∈_i from a normal distribution with mean mu = 0 and standard deviation sigma.

Model validation:

(a) Plot the data {(x_i, y_i)}⁴_i=1 used for estimating the model parameters {β_i, i = 0, . . . , 3}

(b) Once the model parameters are estimated, estimate yˆ_i for {x_i = -2, -1.9, -1.8, . . . , -0.1, 0, 0.1, . . . , 3.8, 3.9, 4.0}

(c) In the same figure, plot the data {(x_i, yˆ_i)} predicted by the model in the previous step.

(d) Write your observations from the plot as well as by comparing your parameter estimates with the true model parameters.