The subject matter for this assignment is financial data, specifically global prices of a selection of basic goods.

Coding for this requires multiple uses of linear regression, which is the solution of an over- determined linear equation. You may use the MATLAB builtin function linsolve, or the“back- slash”operator /, or any builtin function. You may not use functions for cross-validation, or any other functions from a MATLAB toolbox, because you are expected to code these by yourself.

The technical problem in linear regression, for this assignment, is to estimate a weight vector w⃗ for a design matrix A that is “tall thin”, and a data vector ⃗c. The vector w⃗ is an approximate solution to the regression problem

A⃗w ≈ ⃗c

Please read the details and instructions carefully before you begin to work on the problem. The second question in this assignment is modestly difficult because it is intended to be a practical introduction to a method of evaluating algorithms for linear regression.  There must be a single results section and a single discussion section on your report. The results section of the report must contain two tables and up to one figure; more or fewer, of either tables or figures, may produce deductions from your grade on this assignment.

Statement of Academic Integrity

This assignment is copyrighted by the instructor, so unauthorized dissemination of this assign- ment may be a violation of copyright law and may constitute a departure from academic integrity.

Sharing of all or part of a solution to this assignment, whether as code or as a report, will be interpreted as a departure from academic integrity. This includes sharing of the assignment after the due date and after completion of this course.

Learning Outcomes

On completion, a successful student will be able to:

• Import data from a Comma Separated Values (CSV) file into MATLAB

• Develop and implement a method for managing missing data

• Standardize data for use in linear regression

• Compare different regressions in a consistent manner

• Implement a 5-fold cross-validation of linear regression

• Evaluate the results of a cross-validation

Preliminary: The Data

This assignment uses data that the instructor gathered from the U.S. Federal Reserve Bank of St.  Louis Economic Data (FRED) website.  The data are world prices of a select subset of commonly traded basic commodities.  Dates for the prices were recorded quarterly, or four per year.

The data are in the file goods .csv that is in the ZIP file for this assignment. The data can be loaded using the csvread function that is provided in MATLAB. You will need to“skip”the first row and the first column when reading the data for numerical processing. For your report, you will need to understand the names of the variables.

Preliminary: The Code

The starter code is structured as three functions. The base function, which is the function that will be called when the graders invoke your file name, is a2.

You can, if you wish, change the name of this function to match the name of your submission file. MATLAB will execute the code correctly even if the function name and the file name differ.

The base function will then invoke two functions in succession.

The base function will invoke the code for the first question. This first code will return two variables: the RMS errors of linear regression and the index of the lowest RMS error. The base function will then invoke the code for the second question. This invocation uses the index from the first question and computes the 5-fold cross-validation for the corresponding variable.

The base function will return four values. The TAs will examine these values and use them as

part of the grade for your assignment.

DO NOT MODIFY THE BASE FUNCTION. MODIFY ONLY THESE FUNCTIONS: a2q1       a2q2      mykfold   (optional)

Question 1: Variable of Best Regression                               8% of Final Grade

For this question you will need to modify the function a2q1 in the starter code.

The technical problem for this question is to find the variable in the data that is best explained by the other variables. You are expected to do this by selecting the variable that, when chosen as the ⃗c vector, has the lowest RMS error of fit when the remaining variables are gathered into a data matrix A. In this document, the data are assumed to have been converted into a data matrix Amat that has the commodities as variables and the prices as observations.

You should use every available observation of the data to search for the regression variable. You should describe what, if any, data standardization you used.  You should also describe and justify whether or not you used an intercept term in your linear regression (this is done by adding a column of 1s to Amat).

The starter code for this assignment will return a row of small numbers; your completed code should return the RMS error for the regression of each variable in terms of all of the other variables.

For example, if there were three variables in the data set, the function a2 00000000 would return a matrix such as

[0 . 1  0 .2  0 .3]

The starter code will also return the index of the variable with the smallest RMS error; for the above example, this would be

1

You may report RMS errors either in the units of the relevant commodity or in standardized units, depending on your choice of whether or not to use data standardization.

In your report, the values of the RMS errors must be presented in Table 1.  The caption of the table must state the index of the variable that is best explained by the other variables and the corresponding name of the commodity. You can, optionally, plot the dependent variable and the regression. An example of the optional figure for this assignment is Figure 1.

Linear Regression of Best-Modeled Chemical (not to scale)

1

0.8

0.6

0.4

0.2

0

0                   2000                4000                 6000                8000               10000

Time (arbitrary units)

Figure 1: Example of an optional plot of data and a linear regression to these as dependent data. For this plot, the commodity prices were scaled to a maximum of 1.

1.1: Linear Regression

Each column of Amat will need to be considered as a dependent variable, in this document referred to as cvec. The other columns of Amat will need to be considered as the independent variables, in this document referred to as Xmat.  You may wish to standardize the data.  If you choose to use an intercept term, you will need to augment Xmat with a column that is the  vector.

The RMS error of regression of each cvec must be computed and returned as the correspond- ing entry of the variable rmsvars of the function a2q1. The index of the smallest RMS error must be computed and returned as the variable lowndx of the function a2q1.

1.2: Methods

Your methods must include a narrative description of your computation. For example, in find- ing the best data variable, do not simply say something like“I used a for loop”. You should give a reader the logic that underlies your code, not a line-by-line description of its implementation. In addition to your algorithmic description, your methods section should include 1)your decision of whether or not standardize the data, 2)your decision of whether or not to use an intercept term, and 3)an explanation of your decision for 1) and 2).

1.3: Discussing The Results

Your discussion should compare and contrast your numerical results and any observation you have on, for example, the merits of using the second-best commodity as the dependent variable. You should describe any other effects related to the choices you made in your implementation.

For this assignment, you are encouraged to provide a modest amount of creativity – it is a good idea to try to engage the readers of your report. For example, are the results what you expected them to be?

Question 2: Cross Validation of Regression                         7% of Final Grade

The problem for this question is to determine the reliability of choosing one commodity to act as a proxy for the other commodities in the given data. This reliability will be found by performing a 5-fold cross-validation of the given data.

Important: RMS errors must be reported in the units of the commodity that is the dependent variable.  If you standardized the data for linear regression, the standardization may need to be inverted, or recomputed with an intercept term, to find the correct units.

You can select the five folds of data by a method of your choice. In the methods section of your report, you must state your choice and verbally defend the choice.  The instructor’s notes recommend a random selection, which requires a form of indirect indexing that is more complex than the indexing that was used in previous assignments.

For each fold, you must first use 4/5 of the data to“train”your regression. This training is the computation of the weight vector of a linear regression.

For each fold, you must then use 1/5 of the data to“test”your regression.  This testing is the application of the weight vector from the training phase; you must use the same choices of standardization and an intercept term that you used in the training phase.

For each fold, the RMS error of training must be computed and returned as the corresponding entry of the variable rmstrain of the function a2q2. For each fold, the RMS error of testing must be computed and returned as the corresponding entry of the variable rmstest of the function a2q2.

In your report, the values of the RMS errors of the folds must be presented in Table 2. The caption of the table must state the index of the variable that is best explained by the other variables and the corresponding name of the commodity. You can, optionally, summarize the RMS errors of testing and the RMS errors of testing as two overall values.

2.1: Loading The Data

The data must be loaded exactly as they were loaded for Question 1 of this assignment. You must use the column specified as the value of the input variable as the dependent vector ⃗c, and the remaining data as the data matrix A.

2.2: Fold Selection

There are many ways to select folds, including algorithmic selection and random selection. You must describe and verbally defend your choice in the methods section of your report. Some choices require more elaborate coding than others and some choices introduce biases into the results, so this choice has consequences.

The starter code includes the MATLAB function mykfold, which is the instructor’s starter code for randomly selecting the folds. Because you do not need to use this code, this code is not called from the other starter code.

0.0.1    2.3: Discussing The Results

Your discussion should compare the RMS errors between the training and testing data in each fold as well as between folds. The mean/standard deviation of the training RMS and testing RMS may be useful in guiding this discussion. It is important to think what choices you made in the implementation could have impacted these results.

3: Grading Guide

We will test your code by invoking the function that you uploaded. Your grade will be reduced if: you plot more or fewer than the specified number of figures; your code outputs anything other than the specified values; or you otherwise deviate in your implementation from these specifica- tions.

The TA’s have been instructed to use this guide when they mark your assignment. Your grade will be based on the numerical results and on the report. The distribution of points for the assign- ment grade are:

9/60 points: all and only the numerical values that are produced by the code and that are presented in the results

15/60 points: quality of the code in the modified“starter”functions, and any other changes in the submission file that was used to generate values and plots for the report

36/60 points: quality of the report, especially including the figures and descriptions; clarity may be assessed, in part, by the written introduction, verbal defense of choices, and the discussion of results. Make sure to read this PDF carefully to understand what specific points to touch on in the methods section (described in Section 1.2 and 2.2) and discussion section (described in Section 1.3 and 2.3) of your report.