Description
1 Introduction
There is a significant hype and excitement around artificial intelligence (AI)
and machine learning (ML). This assignment is designed to give you a glimpse
of AI/ML. You’ll write a C program to implement a simple “one-shot” machine learning algorithm to predict house prices based on historical data.
2 Problem Statement
There are several factors needs to be considered to determine the price of
a house. 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 + w1x1 + w2x2 + w3x3 + w4x4 (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 wi
from lots of training data.
For example, if the training data includes n houses and has k attributes, this
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data can be represented as an n × (k + 1) matrix X, of the form
1 x0,1 x0,2 … x0,k
1 x1,1 x1,2 … x1,k
. . . . .
. . . . .
. . . . .
1 xn-1,1 xn-1,2 … xn-1,k
where each row corresponds to a house and each column corresponds to an
attribute. Note that the first column contains 1 for all rows: this corresponds
to the weight w0. Similarly, house prices can be represented as an n x 1 matrix
Y , of the form
y0
y1
.
.
.
yn-1
where each row gives the price of a house. Finally, the weights will be a (k
+ 1) x 1 matrix W, of the form
w0
w1
.
.
.
wk+1
where each row gives the weight of an attribute. We can relate the matrices
X, Y , and W with this equation:
XW = Y (2)
Our goal will be to estimate the prices Y0
for some houses with attributes
X0
. This is easily done if we know the weights W, but we do not. Instead,
we can observe the attributes X for houses with known prices Y . We will
use a strategy known as one-shot learning to deduce W, given X and Y. If
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X were a square matrix, we could find W by rewriting the equation as W =
X−1Y , but in general X will not be a square matrix. Thus, we will find its
pseudo-inverse, and calculate
W = (X
T X)
−1X
T Y
where XT
is the transpose of X and XT X is a square matrix, and can be
inverted.1 Once W has been found, it can be used with a new set of house
attributes X0
to estimate prices for those houses by computing X0W = Y
0
.
3 Algorithm
You’ll need to compute (XT X)
−1XT Y in order to learn W for the given
matrices X and Y. This process requires the following matrix operations:
• Multiplication
• Transpose (Transposing an m x n matrix produces an n x m matrix
where each row of the X becomes a column of XT
)
• Inversion (To find the inverse of XT X, you will use a simplified form
of Gauss-Jordan elimination)
3.1 Gauss-Jordan Elimination for Finding Inverses
Gauss-Jordan is a method for solving systems of equations by manipulating
matrices with row operations. This method can also be used to find the
inverse of a matrix. You will implement two of the three row operations in
your program.
• The first multiplies all elements of a particular row by some number
• The second adds the contents of one row to another, element-wise.
More generally, the second operation adds a multiple of the elements
of one row to another so that element xi,k will become xi,k + axj,k
1This is not true in general, but for this assignment you may assume that XT X is
invertable.
3
Figure 1: Pseudocode for Gauss-Jordan Elimination: given a matrix M, we
will use Mi to refer to row i of M and Mi,j to refer to the number in row i,
column j. For this assignment, we will start counting rows and columns from
0.
• The third row operation, which swaps two rows, will not be needed for
this assignment (the training data used to grade this assignment will
not require swapping rows)
An Example
We begin with the matrix we wish to invert:
M =
1 2 4
1 6 7
1 3 2
We create an augmented matrix A = M—I by adjoining the identity matrix
I to M.
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A =
1 2 4 1 0 0
1 6 7 0 1 0
1 3 2 0 0 1
We will then apply row operations to A in order to turn its left half into the
identity matrix. We will write Ai to refer to row i of A. At each step of the
algorithm, we will identify a particular row as the pivot row. The element
in the pivot row that lies on the diagonal (that is, element Ap,p) is the pivot
element.
The first step is to turn the matrix into an upper triangular matrix, where
all elements on the diagonal are 1 and elements below the diagonal are 0.
The pivot row will start at A0 and advance to A2. At each step, we will first
multiply the pivot row by a constant so that the pivot element will become 1.
Next, we will subtract the pivot row from the rows below it, so that the elements below the pivot element become 0. Starting with row A0, we see that
A0,0 is already 1. To make the elements below A0,0 become 0, we subtract
A0 from A1 and A2, yielding
1 2 4 1 0 0
0 4 3 −1 1 0
0 1 −2 −1 0 1
Next, for pivot A1 we see that A1,1 = 4. We divide A1 by 4 (that is, multiply
A4 by 1/4).
1 2 4 1 0 0
0 1 3/4 −1/4 1/4 0
0 1 −2 −1 0 1
Then, we subtract A2 from A3, yielding
1 2 4 1 0 0
0 1 3/4 −1/4 1/4 0
0 0 −11/4 −3/4 −1/4 1
Next, we divide A2 by -11/4, yielding
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1 2 4 1 0 0
0 1 3/4 −1/4 1/4 0
0 0 1 3/11 1/11 −4/11
A is now an upper triangular matrix. To turn the left side of A into an
identity matrix, we will reverse the process and turn the elements above the
diagonal into 0. The pivot row will start at A2 and advance in reverse to A0.
For each step, we will subtract the pivot row from the rows above it so that
the elements above the pivot element become 0. We begin with A2. First,
we subtract 3/4A2 from A1, yielding
1 2 4 1 0 0
0 1 0 −5/11 2/11 3/11
0 0 1 3/11 1/11 −4/11
Then we subtract 4A2 from A0, yielding
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
Now the elements above A2,2 are 0. Next, we subtract 2A1 from A0, yielding
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
Now the element above A1,1 is 0. After that step, the left half of A is the
identity matrix and the right half of A contains M−1
. That is, A = I|M−1
.
The algorithm is complete and the inverse of M has been found.
Tips
• Write this algorithm in pseudocode before you begin implementing it
in C
• Try using it to invert a small square matrix and understand the operations performing at each step
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Note that the augmented matrix is a notational convenience. In your implementation, you may prefer to use two matrices A and B that are initially
equal to M and I. Any time you apply a row operation to A, apply the same
one to B. Once you have A = I, you will also have B = M−1
. It is also
acceptable to create and manipulate an augmented matrix and to extract
M−1
from it later.
Caveat
You MUST use the algorithm demonstrated above. Performing different row
operations, or the same row operations in a different order, may change the
result of your program due to rounding. This may cause your program to
produce results different from the reference result.
4 Program Specification
You can run your program ml as follows:
./ml train-data-file-name test-data-file-name
where train-data-file-name is the name of the training data file with attributes and price of the house. You can assume that the training data file
will exist and that it is well structured. The test-data-file-name is the
name of the test data file with attributes of the house. You have to predict
the price of the house for each entry in the test data file.
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:
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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
• The 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 which is 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:
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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:
p r i n t f ( ”%0.0 l f \n” , val u e ) ;
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:
./ml 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
• 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 -fsanitize=address flags and therefore you are not
allowed to modify the Makefile you are provided with the autograder
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5 Submission
You are provided a tar file named hw2-autograder.tar. Once you untar ($
tar xvf hw2-autograder.tar) it, you’ll have a directory hw2-autograder as
follows:
hw2-autograder
hw2
Makefile
ml.c
testcases
autograder.py
All you need to modify the ml.c file under hw2 directory. Once you are done
with your ml.c file, you can simply tar the hw2 directory ($ tar cvf hw2.tar
hw2) and submit it.
5.1 Autograder
There are two ways to run the autograder to test your program.
First Mode is to test the program when you are writing code with a hw2
folder:
$ python autograder.py
It will run the test cases from the testcases directory and print your scores.
Second Mode is to test your final submission (i.e, hw2.tar)
$ python autograder.py hw2.tar
This command will test your hw2.tar file and print the scores. If you tar the
same directory you tested in the First Mode then both scores should be the
same.
References/Help
You may find these following materials useful:
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• The Pseudo Inverse
• Inverse of a Matrix
https://www.mathsisfun.com/algebra/matrix-inverse.html
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