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CS211 -Programming Assignment II - Solved
This assignment is designed to give you more experience programming in C and using the Unix environment. Your task will be to write one program that implements a simple machine-learning algorithm. This will require file I/O, dynamic memory allocation, and correctly implementing an moderately complex algorithm.
Machine learning (ML) techniques are increasingly used to provide services, such as face recognition in photographs, spelling correction, automated translation, and predicting what YouTube videos you might want to watch next. Implementing a full ML algorithm is beyond the scope of this course, so you will implement a “one shot” learning algorithm that uses historical data to predict house prices based on particular attributes.
For example, a house might have x1 bedrooms, x2 bathrooms, x3 square footage, and be built in year x4. If we had appropriate weights, we could estimate the price of the house y with the formula
y = w0 + w1x1 + w2x2 + w3x3 + w4x4. (1)
The goal of one-shot learning is to find values for the weights wi using a large provided set of training data. Once those weights have been found, they can be used to estimate prices for additional houses.
For example, if the training data includes n houses and has k attributes, this data can be represented as an n × (k + 1) matrix X, of the form
,
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 × 1 matrix Y , of the form
y0 y
1
... ,
yn−1
where each row gives the price of a house.
Finally, the weights will be a (k + 1) × 1 matrix W, of the form
w0 w
1
... ,
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 Y 0 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 X were a square matrix, we could find W by rewriting the equation as W = X−[1]Y , but in general X will not be a square matrix. Thus, we will find its pseudo-inverse by calculating
W = (XTX)−1XTY, (3)
where XT is the transpose of X. XTX 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.
1 Algorithm
Given matrices X and Y , your program will compute (XTX)−1XTY in order to learn W. This will require (1) multiplying, (2) transposing, and (3) inverting matrices. Programming Assignment I already involved matrix multiplication; you may adapt your implementation for this assignment.
Transposing an m × n matrix produces an n × m matrix. Each row of the X becomes a column of XT.
To find the inverse of XTX, you will use a simplified form of Gauss-Jordan elimination.
1.1 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.
The third row operation, which swaps two rows, will not be needed for this assignment. Again, the training data used to grade this assignment will not require swapping rows.
Algorithm 1 is the implementation of Gauss-Jordan your program must implement. Note that the only row operations used are multiplying (or dividing) a row by a number and adding (or subtracting) a multiple of one row to another. 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.
Algorithm 1 Simplified Gauss-Jordan elimination procedure invert(M : n × n matrix)
N ← n × n identity matrix for p ← 0,1,··· ,n − 1 do
f ← Mp,p divide Mp by f divide Np by f for i ← p + 1,··· ,n − 1 do f ← Mi,p subtract Mp × f from Mi subtract Np × f from Ni
end for
end for for p ← n − 1,··· ,0 do
for i ← p − 1,··· ,0 do
f ← Mi,p subtract Mp × f from Mi subtract Np × f from Ni
end for
end for return N
end procedure
To illustrate Gauss-Jordan, we will walk through the process of inverting the matrix
.
As a notational convenience, we will create an augmented matrix A = M|I by adjoining the identity matrix I to M.
2
6
1
4
8
6
1
0
0
0
1
0
It is not necessary for your program to create A; instead, you can represent the two sides of A as two separate matrices.
The goal of Gauss-Jordan is to turn the left half of A into an identity matrix by applying row operations. At each step, we will identify a particular row as the pivot row. The element that lies on the diagonal (that is, element Ap,p) is the pivot element.
The first step is to turn the M 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 41 0 0
0 4 4−1 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 2 41 0 0
0 1 1−14 14 0 .
0 −1 2−1 0 1
To zero out A2,1, we add A1 to A2.
1 2 41 0 0
0 1 1−14 14 0 .
0 0 3−54 14 1
Now the pivot row is A2. To make A2,2 = 1, we divide row A2 by 3.
1 2 41 0 0
0 1 1−14 14 0 .
0 0 1−125 121 13
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 A2 from A1, yielding
1 2 4
0 1 0
0 0 1
Then we subtract 4 × A2 from A0, yielding
1
1
6
−125
0
1
6 1
12
.
1 2 0
0 1 0
0 0 1
8
3
1
6
−125
−31
1
6 1
12
.
Now the elements above A2,2 are 0.
Next, we subtract 2 × A1 from A0, yielding
1
0
0
Now the element above A1,1 is 0.
0
1
0
0
0
1
7
3
1
6
−125
−32
1
6 1
12
.
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.
You are strongly encouraged to write this algorithm in more detailed pseudocode before you begin implementing it in C. Try using it to invert a small square matrix and make sure you understand the operations you must perform at each step.
In particular, ask yourself (1) given a matrix A, how can I multiply (or divide) Ai by a constant c, and (2) given a matrix A, how can I add (or subtract) c × Ai to (or from) Aj?
You MUST use the algorithm as described. 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.
2 Program
You will write a program estimate that uses a training data set to learn weights for a set of house attributes, and then applies those weights to a set of input data to calculate prices for those houses. estimate takes two arguments, which are the paths to files containing the training data and input data.
Training data format The first line will be the word “train”. The second line will contain an integer k, giving the number of attributes. The third line will contain an integer n, giving the number of houses. The next n lines will contain k + 1 floating-point numbers, separated by spaces. Each line gives data for a house. The first k numbers give the values x1 ···xk for that house, and the last number gives its price y.
For example, a file train.txt might contain:
train 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
This file contains data for 7 houses, with 4 attributes and a price for each house. The corresponding matrix X will be 7 × 5 and Y will be 7 × 1. (Recall that column 0 of X is all ones.)
Input data format The first line will be the word “data”. The second line will be an integer k, giving the number of attributes. The third line will be an ineteger m, giving the number of houses. The next m lines will contain k floating-point numbers, separated by spaces. Each line gives data for a house, not including its price.
For example, a file data.txt might contain:
data 4
2
3.000000 2.500000 3560.000000 1965.000000
2.000000 1.000000 1160.000000 1942.000000
This contains data for 2 houses, with 4 attributes for each house. The corresponding matrix X will be 2 × 5.
Output format Your program should output the prices computed for each house in the input data using the weights derived from the training data. Each house price will be printed on a line, rounded to the nearest integer.
To print a floating-point number rounded to the nearest integer, use the formatting code %.0f, as in: printf("%.0f\n", price);
Usage Assuming the files train.txt and data.txt exist in the same directory as estimate:
$ ./estimate train.txt data.txt
737861
203060
Implementation notes The description of Gauss-Jordan elimination given in section 1.1 uses an augmented matrix with twice as many columns as the input matrix X. This is an illustrative tool, and not meant as an implementation requirement. It is simpler to use two matrices that begin as X and the identity matrix I and apply identical row operations to both, until the first matrix becomes I and the second is X−1.
It is recommended to write separate functions to compute the inverse of a matrix, the transpose of a matrix, and the product of two matrices. You may find it simpler to avoid allocating memory within these functions; instead, pass them the input matrix or matrices and a pre-allocated matrix that will be used for the output.
Having separate functions will simplify your development, as you can test your implementations of each separately.
You MUST use double to represent the attributes, weights, and prices. Using float may result in incorrect results due to rounding. To read double values from the training and input data files, you can use fscanf with the format code %lf.
If estimate successfully completes, it MUST return exit code 0.
You MAY assume that the training and input data files are correctly formatted. You MAY assume that the first argument is a training data file and that the second argument is an input data file. However, checking that the training data file begins with “train” and that the input data file begins with “data” may be helpful if you accidentally give the wrong arguments to estimate while you are testing it. To read a string containing up to 5 non-space characters, you can use the fscanf format code %5s.
estimate SHOULD check that the training and input data files specify the same value for k.
If the training or input files do not exist, are not readable, are incorrectly formatted, or specify different values of k, estimate MAY print “error” and return exit code 1.
