Project #2
This will be an exercise which might approximate one form of a machine learning algorithm, in which we fit various functions to data. (A true machine learning system might use neural networks or expert systems to learn patterns, both of which are techniques outside the scope of this class). We are going to use regression analysis, correlation analysis, and function generation to complete this task.
You will find a data file named ‘data.mat’ on your CANVAS page. This file can be read by MATLAB using the ‘LOAD’ command. As in a real data analysis task, figuring how to parse (or divide up) the data into separate fields (or vectors) is part of the task.
This data file should be copied into your main MATLAB user directory (typically /Documents/MATLAB), the same one where you would place your m-files.
The data is a 400 row by 5 column matrix, each row consisting of 4 independent variables (call them X1, X2, X3, and X4, if you wish), and a corresponding Y value which is based on some function of these four X values. There is some random variation built into all the variables. The Y element will be the first one in each row, followed by (X1, X2, X3, X4), in order.
Your task is to try different prototype functions, of your making, and find the one which best fits the data. You may use regression analysis like we used in class (any of the methods) to fit the data Y to any desired combination of the variables (X1, X2, X3, X4). Then you are to find the correlation between the vector of the Y result predicted by your model to the Y vector in the dataset. You will simply then report which model produces the highest correlation (i.e., the best fit).
The most challenging part of this assignment will be to write code to generate these ‘prototype’ functions, which are then to be fit to the dataset. I do not want you to simply guess at functions to be tested – these are all to be generated by an algorithm.
Hints to the solution:
•The solution is a simple linear polynomial fit of Y to (X1, X2, X3, X4). No advanced functions like trigonometric or exponential functions are involved.
•This polynomial will be a sum of at most 5 terms. Each of these five terms is a constant times a product of one or more or the independent variables, each of which could be raised to an integer power between[-1,3] (endpoints inclusive).
•At most one of these 5 additive terms will contain more than two of the independent variables, and that one exception may contain only 3 of them.
Example: a test function might be something like
One would then proceed by making a new matrix of independent variables corresponding to each of these ‘cross’ terms from the given X variables, and then doing a linear regression fit of Y to this new variable matrix. In this case, our new X matrix to fit (but don’t erase the old X values – you’ll need them again for the next trial run) would have each row looking like
Competitive grading: 30% of your final score on this project will be of a competitive nature. This is to intended to help curb sharing of algorithms and results with each other. I will give more points to those who get better results, and less to those with not so good results. I will then apply some sort of curve to all these results. Sharing your work so that others benefit from it will thus reduce your score and as well as all those who had a higher score than the one helped in such a manner.
Report Requirement
In your project report, you need to
Describe the process of getting the answers and explain your graph or table. You have to make it understandable to students with basic knowledge of the pre-requisites and the course materials covered so far. In general, you should type it up in a word document and copy and paste any relevant results from MATLAB.
Append your MATLAB programs at the end of your document or submit them as separate files.
Organize your report according to the questions asked.
Submitting only MATLAB files will result in very few points.
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