Solved–Programming Assignment II –Solution

Description

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 recog-nition 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 w_i 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

0	1	x0;1	x0;2			x0;k		1
B1...		x1...;1	x1...;2		. . .	x1...;k		C	;
B	1	x	x	n 1;2		x	n 1;k	C
B		n 1;1						C
@								A

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

• 1

y₀

• y₁ C

• ._. ^C;

• C @ . A

^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

• 1

w₀

• w₁ C

• ._. ^C;

• C @ . A

^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 ¹Y , but in general X will not be a square matrix. Thus, we will find its pseudo-inverse by calculating

W = (X^T X) ¹X^T Y; (3)

where X^T is the transpose of X. X^T 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.

1 Algorithm

Given matrices X and Y , your program will compute (X^T X) ¹X^T Y 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 X^T .

To find the inverse of X^T X, 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 x_i;k will become x_i;k + ax_j;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.

¹This is not true in general, but for this assignment you may assume that X^T X is invertable.

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 M_i to refer to row i of

• and M_i;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 identity matrix

for p 0; 1; ; n 1 do

• ^Mp;p

divide M_p by f

divide N_p 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	1;; 0 do
for p n
for i	p 1; ; 0 do
f	Mi;p

subtract M_p f from M_i

subtract N_p f from N_i

end for

end for

return N

end procedure

To illustrate Gauss-Jordan, we will walk through the process of inverting the matrix

• 3

1 2 4

M = 4	1	6	8	5	:
	1	1	6

As a notational convenience, we will create an augmented matrix A = MjI by adjoining the identity matrix I to M.

• 3

A = 4	1	2	4	1	0	0	5
	1	6	8	0	1	0
	1	1	6	0	0	1

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 A_p;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 A_0;0 is already 1. To make the elements below A_0;0 become 0, we subtract A₀ from A₁ and A₂, yielding

2						3
1	2	4	1	0	0	5 :
4 0	4	4	1	1	0

• 12101

Next, for pivot A₁ we see that A_1;1 = 4. We divide A₁ by 4 (that is, multiply A₄ by ¹₄ ).

2	0	1	1	4			4			0	3	:
4	1	2	4	1			0			0	5
					1		1
	0	1	2	1			0			1

To zero out A_2;1, we add A₁ to A₂.

• 3

6	1	2	4	1		0			0	7
	0	1	1		1	1			0		:
					4	4
	0	0	3						1
4				4		4				5
					5	1

Now the pivot row is A₂. To make A_2;2 = 1, we divide row A₂ by 3.

2	1	2	4	1					0					0			3
6																	7
	0	1	1	1					1					0				:
				12					12
	0	0	1											3
4				4					4								5
				5					1					1

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

Next, we subtract 2 A1 from A0, yielding
2	0	1	0		6				6			3		3	:
	1	0	0			7					2	2
6						3					3	3		7
						1				1			1
	0	0	1
						12			12			3
4					5				1			1		5

Now the element above A_1;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 = IjM 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) A_i by a constant c, and (2) given a matrix A, how can I add (or subtract) c A_i to (or from) A_j?

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.

• 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 x_k 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. Your code will not be tested with these scenarios.

• Grading

Your submission will be awarded up to 100 points, based on how many test cases your program completes successfully.

The auto-grader provided for students includes half of the test cases that will be used during grading. Thus, it will award up to 50 points.

Make sure that your programs meet the specifications given, even if no test case explicitly checks it. It is advisable to perform additional tests of your own devising.

3.1 Academic integrity

You must submit your own work. You should not copy or even see code for this project written by anyone else, nor should you look at code written for other classes. We will be using state of the art plagiarism detectors. Projects which the detectors deem similar will be reported to the Office of Student Conduct.

Do not post your code on-line or anywhere publically readable. If another student copies your code and submits it, both of you will be reported.

• Submission

Your solution to the assignment will be submitted through Sakai. You will submit a Tar archive file containing the source code and makefiles for your project. Your archive should not include any compiled code or object files.

The remainder of this section describes the directory structure, the requirements for your makefiles, how to create the archive, how to use the provided auto-grader, and how to create your own test files to supplement the auto-grader.

4.1 Directory structure

Your project should be stored in a directory named src, which will contain (1) a makefile, and (2) any source files needed to compile your program. Typically, you will provide a single C file named for the program (estimate.c).

This diagram shows the layout of a typical project:

src

+- Makefile

+- estimate.c

Note that your code and makefile go directly in src, without any subdirectories.

4.2 Makefiles

We will use make to manage compilation. Your src directory will contain a file named Makefile that describes at least two targets. The first target must compile the program. An additional target, clean, must delete any files created when compiling the program (typically just the compiled program).

For reference, this makefile is provided with the auto-grader:

TARGET = estimate

29. = clang

OPT=

CFLAGS = -g -std=c99 -Wall -Wvla -Werror -fsanitize=address \

$(if $(findstring clang,$(CC)),-fsanitize=undefined) $(OPT)

$(TARGET): $(TARGET).c

$(CC) $(CFLAGS) $^ -o $@

clean:

rm -f $(TARGET) *.o *.a *.dylib *.dSYM

Note that the command for compiling estimate uses warnings and the address sanitizer, and includes the undefined behavior sanitizer when the compiler is clang. Your score may be reduced if your makefile does not include these options.

Use of -g is recommended, but not required.

This makefile can be adapted to work with multiple source files. Contact the professor for directions if that interests you.

Note that the makefile format requires that lines be indented using a single tab, not spaces. Be aware that copying and pasting text from this document will “helpfully” convert the indentation to spaces. You will need to replace them with tabs (literal tab characters), or simply type the makefile yourself. You are advised to use make when compiling your program, as this will ensure (1) that your makefile works, and (2) that you are testing your program with the same compiler options that the auto-grader will use.

Working with GDB The default makefile includes -g, which instructs the compiler to include source information for the debugger. Depending on your program, clang may optimize too aggressively and remove variables whose values you would like to examine. In that case, you may want to recompile with gcc. This can be done without modifying your makefile by including CC=gcc in your arguments to make. Similarly, you may use OPT=-O0 (the letter ‘O’ followed by the number zero) to disable optimization.

pa2/src$ make CC=gcc OPT=-O0

gcc -g -std=c99 -Wall -Wvla -Werror -fsanitize=address -O0 estimate.c -o estimate

If you have not changed your source code, you can give make the option -B to force recompilation, or use make clean before recompiling.

Note that clang is still the recommended compiler for normal usage, because it provides the undefined behavior sanitizer and its version of AddressSanitizer has better error messages.

4.3 Creating the archive

We will use tar to create the archive file. To create the archive, first ensure that your src directory contains only the source code and makefiles needed to compile your project. Any compiled programs, object files, or other additional files must be moved or removed.

Next, move to the directory containing src and execute this command:

pa2$ tar -czvf pa2.tar src/Makefile src/estimate.c

tar will create a file pa2.tar that contains your makefile and source code. (If you are using multiple source files, or have re-named your source code, you will need to adjust this command accordingly.) This file can now be submitted through Sakai.

To verify that the archive contains the necessary files, you can print a list of the files contained in the archive with this command:

pa2$ tar -tf pa2.tar

You should also use the auto-grader to confirm that your archive is correctly structured.

pa2$ ./grader.py -a pa2.tar

4.4 Using the auto-grader

We have provided a tool for checking the correctness of your project. The auto-grader will compile your programs and execute them several times with different arguments, comparing the results against the expected results.

Setup The auto-grader is distributed as an archive file pa2_grader.tar. To unpack the archive, move the archive to a directory and use this command:

$ tar -xf pa2_grader.tar

This will create a directory pa2 containing the auto-grader itself, grader.py, a library autograde.py, a directory of test cases data, and directory src containing files Makefile and estimate.c.

Do not modify any of the files provided by the auto-grader, aside from those in src. Doing so may prevent the auto-grader from correctly assessing your program.

You may freely edit estimate.c or Makefile, or delete the directory and create your own in a location of your choosing. However, it is simplest to use the structure provided.

Usage While in the same directory as grader.py and src, use this command:

pa2$ ./grader.py

The auto-grader will compile and execute the program in the directory src, assuming src has the structure described in section 4.1. The compiled program will be in a directory build, which is created by the auto-grader.

During development, you may prefer to use the –stop or -1 option, which produces more program output but stops after the first failed test case.

pa2$ ./grader.py -1

To obtain usage information, use the -h option.

Program output By default, the auto-grader will not print the output from your programs, except for lines that are incorrect. To see all program output for all unsuccessful tests, use the –verbose or -v option:

pa2$ ./grader.py -v

To see program output for all tests, use -vv. To see no program output, use -q.

Checking your archive We recommend that you use the auto-grader to check an archive before submitting. To do this, use the –archive or -a option with the archive file name. For example,

pa2$ ./grader.py -a pa2.tar

This will unpack the archive into a temporary directory, grade the programs, and then delete the temporary directory.

Specifying source directory If your src directory is not located in the same directory as grader.py, you may specify it using the –src or -s option. For example,

pa2$ ./grader.py -s ../path/to/src

10