Prediction of Graduate Admissions and Parameter Estimation for Parametric Oscillators

Exercise 1
In this exercise, we consider the file Graduate_admission that you will find on D2L. This dataset is cre?ated for prediction of Graduate Admissions. The dataset comes from Mohan S Acharya, Asfia Armaan, Aneeta S Antony : A Comparison of Regression Models for Prediction of Graduate Admissions, IEEE International Conference on Computational Intelligence in Data Science 2019. It contains several parameters which are considered important during the application for Masters Programs. The parameters included are : GRE Scores ( out of 340 ), TOEFL Scores ( out of 120 ), University Rating ( out of 5 ),
Statement of Purpose and Letter of Recommendation Strength ( out of 5 ), Undergraduate GPA ( out of 10 ), Research Experience ( either 0 or 1 ). The output corresponds to the Chance of Admit (ranging from 0 to 1 ). We want to design an algorithm that can predict the chance of admit of a new student using the different mentioned parameters.
(a) What are the attributes? What is the output? How many samples do we have? What kind of machine learning problem do we want to solve? I what follows this set of data will be divided into a training set and a testing set.
(b) In this question, we only consider the attributes GRE Scores and TOEFL Scores . Using a linear regression, try to approximate the relation between these two parameters and the output. Don’t forget to use cross-validation. Once you have found the optimal value for your regressor, compute the global least squares error (on the whole dataset).
(c) For the rest of this exercise, we now subdivide the output Chance of Admit in five classes: Class 1 (if the change of admit is between 0 and 0.2), Class 2 (if the change of admit is between 0.2 and
0.4), Class 3 (if the change of admit is between 0.4 and 0.6), Class 4 (if the change of admit is between 0.6 and 0.8), Class 5 (if the change of admit is between 0.8 and 1). Justify that we now have a classification problem.
(d) Train a Nearest Neighbors algorithm, using only the attributes GRE Scores and TOEFL Scores.What is the best number of Neighbors? Compute the corresponding accuracy.
(e) Train a classification tree using all the attributes. Prune it if you can. Compute the corresponding accuracy. If you want to use only discrete attribute (which is not mandatory) you can subdivide the continuous values of your attributes into different intervals. For instance, for the attribute Undergraduate GPA, you can consider that the attribute is zero if 0 GPA < 2, one if 2 GPA (f)(BONUS) Test your (optimized) three algorithms with the new set of data which is called Graduate_admission_verification. Compute the skill or accuracy in each case.
Exercise 2
A parametric oscillator is a driven harmonic oscillator in which the drive energy is provided by varying the parameters of the oscillator, such as the damping or restoring force. A familiar example of parametric oscillation is "pumping" on a playground swing. Other examples include electrical systems (Series RLC circuits for instance). More precisely, for such systems, the state x satisfies the following ordinary differential equation:x¨+ 2ζx?+ ω2x = u(t),where ζ is a damping coefficient, ω is the characteristic pulsation of the system and u is an input signal.In what follows, we will consider u as a sinusoidal excitationu(t) = Acos(ω1t).We want to estimate the constant parameters ζ and ω2. We assume that A = 1 and ω1 = 1. You will find on D2L, one file that contains the (noisy) values of t,x(t),x?(t) and x¨(t) (file Data_oscillator).While reading the rows of the csv file (for loop), you should store the corresponding data in a matrix (that has to be defined before). Be careful that the first line should not be considered as it contains only the labels of the columns.
(a)How can you estimate ζ and ω? What would be the problem of a Least Squares algorithm? Using a (recursive) least squares approach, find the values of these two parameters.
(b)We now assume that only the vectors containing the values of t and x(t) are available. Try to estimate ζ and ωin this case. Compare with the previous result.
(c)We now assume that u(t) = cos(ωt) (and thus the input is not known anymore). Try to estimate ζand ω in this case. What is the problem?