## Description

This assignment is designed to provide you some experience writing programs with the C programming

language. You will write a C program that implements simple “one-shot” machine learning algorithm

for predicting house prices in your area.

There is significant hype and excitement around artificial intelligence (AI) and machine learning.

CS 211 students will get a glimpse of AI/ML by implementing a simple machine learning algorithm to

predict house prices based on historical data.

For example, the price of the house (y) can depend on certain attributes of the house: number of

bedrooms (x1), total size of the house (x2), number of baths (x3), and the year the house was built (x4).

Then, the price of the house can be computed by the following equation:

y = w0 + w1.x1 + w2.x2 + w3.x3 + w4.x4 (1)

Given a house, we know the attributes of the house (i.e., x1, x2, x3, x4). However, we don’t know

the weights for these attributes: w0, w1, w2, w3 and w4. The goal of the machine learning algorithm in

our context is to learn the weights for the attributes of the house from lots of training data.

Let’s say we have N examples in your training data set that provide the values of the attributes and

the price. Let’s say there are K attributes. We can represent the attributes from all the examples in the

training data as a Nx(K + 1) matrix as follows, which we call X:

[

1, x0,1 , x0,2 , x0,3 , x0,4

1, x1,1 , x1,2 , x1,3 , x1,4

1, x2,1 , x2,2 , x2,3 , x2,4

1, x3,1 , x3,2 , x3,3 , x3,4

..

1, xn,1 , xn,2 , xn,3 , xn,4

]

where n is N − 1. We can represent the prices of the house from the examples in the training data

as a Nx1 matrix, which we call Y .

[

y0

y1

..

yn

]

Similarly, we can represent the weights for each attribute as a (K + 1)x1 matrix, which we call W.

[

w0

w1

..

wk

]

1

The goal of our machine learning algorithm is to learn this matrix from the training data.

Now in the matrix notation, entire learning process can be represented by the following equation,

where X, Y , and W are matrices as described above.

X.W = Y (2)

Using the training data, we can learn the weights using the below equation:

W = (XT

.X)

−1

.XT

.Y (3)

where XT

is the transpose of the matrix X, (XT

.X)

−1

is the inverse of the matrix XT

.X.

Your main task in this part to implement a program to read the training data and

learn the weights for each of the attributes. You have to implement functions to multiply matrices,

transpose matrices, and compute the inverses of the matrix. You will use the learned weights to predict

the house prices for the examples in the test data set.

Want to learn more about One-shot Learning? The theory behind this learning is not important

for the purposes of this class. The algorithm you are implementing is known as linear regression with

least square error as the error measure. The matrix ((XT

.X)

−1

.XT

) is also known as the pseudo-inverse

of matrix X. If you are curious, you can learn more about this algorithm at https://www.youtube.

com/watch?v=FIbVs5GbBlQ&hd=1.

Computing the Inverse using Gauss-Jordan Elimination

To compute the weights above, your program has to compute the inverse of matrix. There are numerous

methods to compute the inverse of a matrix. We want you to implement a specific method for

computing the inverse of a matrix known as Guass-Jordan elimination, which is described

below. If you compute inverse using any other method, you will risk losing all points for this part.

An inverse of a square matrix A is another square matrix B, such that A.B = B.A = I, where I is

the identity matrix.

Gauss-Jordan Elimination for computing inverses

Below, we give a sketch of Gauss-Jordan elimination method. Given a matrix A whose inverse needs to

be computed, you create a new matrix Aaug, which is called the augmented matrix of A, by concatenating

identity matrix with A as shown below.

Let say matrix A, whose inverse you want to compute is shown below:

[

1 2 4

1 6 7

1 3 2

]

The augmented matrix (Aaug) of A is:

[

1 2 4 1 0 0

1 6 7 0 1 0

1 3 2 0 0 1

]

The augmented matrix essentially has the original matrix and the identity matrix. Next, we perform

row operations on the augmented matrix so that the original matrix part of the augmented matrix turns

into an identity matrix.

The valid row operations to compute the inverse (for this assignment) are:

• You can divide the entire row by a constant

• You can subtract a row by another row

2

• You can subtract a row by another row multiplied by a constant

However, you are not allowed to swap the rows. In the traditional Gauss-Jordan elimination method

you are allowed to swap the rows. For simplicity, we do not allow you to swap the rows.

Let’s see this method with the above augmented matrix Aaug.

• Our goal is to transform A part of the augmented matrix into an identity matrix.

Since Aaug[1][0]! = 0, we will subtract the first row from the second row because we want to make

Aaug[1][0] = 0. Hence, we perform the operation R1 = R1 − R0, where R1 and R0 represents the

second and first row of the augmented matrix. Augmented matrix Aaug after R1 = R1 − R0

[

1 2 4 1 0 0

0 4 3 −1 1 0

1 3 2 0 0 1

]

• Now we want to make Aaug[1][1] = 1. Hence, we perform the operation R1 = R1/4. The augmented

matrix Aaug after R1 = R1/4 is:

[

1 2 4 1 0 0

0 1 3

4

−1

4

1

4

0

1 3 2 0 0 1

]

• Next, we want to make Aaug[2][0] = 0. Hence, we perform the operation R2 = R2 − R0. The

augmented matrix Aaug after R2 = R2 − R0 is:

[

1 2 4 1 0 0

0 1 3

4

−1

4

1

4

0

0 1 -2 -1 0 1

]

• Next, we want to make Aaug[2][1] = 0. Hence, we perform the operation R2 = R2 − R1. The

augmented matrix Aaug after R2 = R2 − R1 is:

[

1 2 4 1 0 0

0 1 3

4

−1

4

1

4

0

0 0 −11

4

−3

4

−1

4

1

]

• Now, we want to make Aaug[2, 2] = 1, Hence, we perform the operation R3 = R3 ∗

−4

11 . Then, Aaug

is:

[

1 2 4 1 0 0

0 1 3

4

−1

4

1

4

0

0 0 1 3

11

1

11

−4

11

]

3

• Next, we want to make Aaug[1, 2] = 0, Hence, we perform the operation R1 = R1 −

3

4

∗ R2. Then,

Aaug is:

[

1 2 4 1 0 0

0 1 0 −5

11

2

11

3

11

0 0 1 3

11

1

11

−4

11

]

• Next, we want to make Aaug[0, 2] = 0, Hence, we perform the operation R0 = R0 − 4 ∗ R2. Then,

Aaug is:

[

1 2 0 1

11

−4

11

16

11

0 1 0 −5

11

2

11

3

11

0 0 1 3

11

1

11

−4

11

]

• Next, we want to make Aaug[0, 1] = 0, Hence, we perform the operation R0 = R0 − 2 ∗ R1. Then,

Aaug is:

[

1 0 0 9

11

−8

11

10

11

0 1 0 −5

11

2

11

3

11

0 0 1 3

11

1

11

−4

11

]

• At this time, the A part of the augmented matrix is an identity matrix. Hence, the inverse of A

matrix is:

[

9

11

−8

11

10

11

−5

11

2

11

3

11

3

11

1

11

−4

11

]

Your goal is to write a program to compute the inverse of a matrix to perform one-shot learning.

Input/Output specification

Usage interface

Your program learn will be executed as follows:

./learn

where

the house. You can assume that the training data file will exist and that it is well structured. The

predict the price of the house for each entry in the test data file.

Input specification

The input to the program will be a training data file and a test data file.

4

Structure of the training data file

The first line in the training file will be an integer that provides the number of attributes (K) in the

training set. The second line in the training data file will be an integer (N) providing the number of

training examples in the training data set. The next N lines represent the N training examples. Each

line for the example will be a list of comma-separated double precision floating point values. The first

K double precision values represent the values for the attributes of the house. The last double precision

value in the line represents the price of the house.

An example training data file (train1.txt) is shown below:

4

7

3.000000,1.000000,1180.000000,1955.000000,221900.000000

3.000000,2.250000,2570.000000,1951.000000,538000.000000

2.000000,1.000000,770.000000,1933.000000,180000.000000

4.000000,3.000000,1960.000000,1965.000000,604000.000000

3.000000,2.000000,1680.000000,1987.000000,510000.000000

4.000000,4.500000,5420.000000,2001.000000,1230000.000000

3.000000,2.250000,1715.000000,1995.000000,257500.000000

In the example above, there are 4 attributes and 7 training data examples. Each example has values

for the attributes and last value is the price of the house. To illustrate, consider the training example

below

3.000000,1.000000,1180.000000,1955.000000,221900.000000

The first attribute has value 3.000000, the second attribute has value 1.000000, third attribute has value

1180.000000, and the fourth attribute has value 1955.000000. The price of the house for these set of

attributes is provided as the last value in the line: 221900.000000

Structure of the test data file

The first line in the training file will be an integer (M) that provides the number of test data points in

the file. Each line will have K attributes. The value of K is defined in the training data file. Your goal

is predict the price of house for each line in the test data file. The next M lines represent the M test

points for which you have to predict the price of the house. Each line will be a list of comma-separated

double precision floating point values. There will be K double precision values that represent the values

for the attributes of the house.

An example test data file (test1.txt) is shown below:

2

3.000000,2.500000,3560.000000,1965.000000

2.000000,1.000000,1160.000000,1942.000000

It indicates that you have to predict the price of the house using your training data for 2 houses. The

attributes of each house is listed in the subsequent lines.

Output specification

Your program should print the price of the house for each line in the test data file. Your program should

not produce any additional output. If the price of the house is a fractional value, then your program

should round it to the nearest integer, which you can accomplish with the following printf statement:

printf(“%0.0lf\n”, value);

where value is the price of the house and its type is double in C.

Your program should predict the price of the entry in the test data file by substituting the attributes

and the weights (learned from the training data set) in Equation (1).

A sample output of the execution when you execute your program as shown below,

5

./learn train1.txt test1.txt

should be

737861

203060

Hints and suggestions

• You are allowed to use functions from standard libraries but you cannot use third-party libraries

downloaded from the Internet (or from anywhere else). If you are unsure whether you can use

something, ask us.

• We will compile and test your program on the iLab machines so you should make sure that your

program compiles and runs correctly on these machines. You must compile all C code using the

gcc compiler with the -Wall -Werror -fsanitize=address flags.

• You should test your program with the autograder provided with the assignment.

Submission

You have to e-submit the assignment using Sakai. Your submission should be a tar file named pa2.tar.

To create this file, put everything that you are submitting into a directory named pa2. Then, cd into

the directory containing pa2 (that is, pa2’s parent directory) and run the following command:

tar cvf pa2.tar pa2

To check that you have correctly created the tar file, you should copy it (pa2.tar) into an empty

directory and run the following command:

tar xvf pa2.tar

This should create a directory named pa2 in the (previously) empty directory.

The pa2 directory in your tar file must have:

• Makefile: There should be at least two rules in your Makefile:

1. learn: build your learn executable.

2. clean: prepare for rebuilding from scratch.

• source code: all source code files necessary for building your programs. Your code should contain

at least two files: learn.h and learn.c.

Grading guidelines

The grading will be automatically graded using the autograder.

Automated grading phase

This phase will be based on programmatic checking of your program using the autograder. We will

build a binary using the Makefile and source code that you submit, and then test the binary for correct

functionality and efficiency against a set of inputs.

• We should be able build your program by just running make.

• Your program should follow the format specified above for both both the parts.

6

• Your program should strictly follow the input and output specifications mentioned. Note: This

is perhaps the most important guideline: failing to follow it might result in you losing

all or most of your points for this assignment. Make sure your program’s output

format is exactly as specified. Any deviation will cause the automated grader to mark

your output as “incorrect”. REQUESTS FOR RE-EVALUATIONS OF PROGRAMS

REJECTED DUE TO IMPROPER FORMAT WILL NOT BE ENTERTAINED.

• We will check all solutions pair-wise from all sections of this course to detect cheating

using moss software and related tools. If two submissions are found to be similar, they will

instantly be awarded zero points and reported to office of student conduct. See Rutgers CS’s academic integrity policy at: https://www.cs.rutgers.edu/academic-integrity/introduction.

Autograder

We provide the AutoGrader to test your assignment. AutoGrader is provided as autograder.tar. Executing the following command will create the autograder folder.

$tar xvf autograder.tar

There are two modes available for testing your assignment with the AutoGrader.

First mode

Testing when you are writing code with a pa2 folder

1. Lets say you have a pa2 folder with the directory structure as described in the assignment.

2. Copy the folder to the directory of the autograder

3. Run the autograder with the following command

$python pa2 autograder.py

It will run the test cases and print your scores.

Second mode

This mode is to test your final submission (i.e, pa2.tar)

1. Copy pa2.tar to the autograder directory

2. Run the autograder with pa2.tar as the argument. The command line is:

$python pa2 autograder.py pa2.tar

7