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Project 3: Two-dimensional Arrays and Multilinear Regression, Slides of Computer Science

Biju Patnaik University of Technology Computer Science

Project 3 for computer science 106 and computing in engineering and science, focusing on two-dimensional arrays and multilinear regression. The project involves solving simultaneous linear equations using given data and library programs. Students are required to compute the coefficients (b0 to bk) by solving k + 1 simultaneous linear equations.

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Project 3 – Two-dimensional arrays May 9, 2006

1

Third Programming Project

–

–Two

Two-

-Dimensional Arrays

Dimensional Arrays

Larry Caretto

Computer Science 106

Computing in Engineering

and Science

May 9, 2006

2

Outline

• Quiz three on Thursday for full lab

period

– See sample quiz on home page of course

web site

• Project three

– Multilinear regression

– Equations to be solved

– Data structures to be used

– Library program use in separate code file

3

Files You Can Download

• Project instructions on home page and

projects page

• Other files on projects page only

– Data file for project

– Library program for solving simultaneous

linear equations

– Excel file with answers

• This lecture on home page and

laboratory presentations page

4

Exercise 8 Linear Regression

• Fit linear relation-

ship to measured

data pairs (xi, yi)

• y = a + bx

• Have equations to

determine a and b

• Get goodness-of-

fit measures

Fitted Line

y

x

indicates data points

y

i

x

i

y

ˆ

• Fitted value at

xiis ii bx+a=y

ˆ

5

Project Three Regression

• Have a result that depends on more

than one variable

– Example is emissions from diesel engine

that depends on fuel properties

– emissions = b0+ b1(cetane) +

b2(aromatics) + b3(density)

• Use measured data on emissions,

cetane, aromatics to find b0, b1, b2, b3

6

General Regression

• Use notation so that we can write code

for any number of predictive variables

• Call predictive variables x1, x2, x3, etc.

• Call response variable y

– In previous example, x1= cetane, x2=

aromatics, x3= density, and y = emissions

– For three variables the equation is y = b0+

b1 x1+ b2 x2+ b3 x3

Discover Slides of Computer Science Biju Patnaik University of Technology

Partial preview of the text

Download Project 3: Two-dimensional Arrays and Multilinear Regression and more Slides Computer Science in PDF only on Docsity!

Third Programming Project Third Programming Project

Two–Two--Dimensional ArraysDimensional Arrays

Larry Caretto Computer Science 106

Computing in Engineering

and Science

May 9, 2006

2

Outline

Quiz three on Thursday for full lab

period

See sample quiz on home page of course web site
Project three
Multilinear regression
Equations to be solved
Data structures to be used
Library program use in separate code file

3

Files You Can Download

Project instructions on home page and

projects page

Other files on projects page only
- Data file for project
- Library program for solving simultaneous linear equations
- Excel file with answers
This lecture on home page and

laboratory presentations page

4

Exercise 8 Linear Regression

Fit linear relation-

ship to measured

data pairs (xi, yi)

y = a + bx
Have equations to

determine a and b

Get goodness-of-

fit measures

Fitted Line

y

x indicates data points

y (^) i x (^) i

y ˆ i

Fitted value at xi is yˆi =a+bxi

5

Project Three Regression

Have a result that depends on more

than one variable

Example is emissions from diesel engine that depends on fuel properties
emissions = b 0 + b 1 (cetane) + b 2 (aromatics) + b 3 (density)
Use measured data on emissions,

cetane, aromatics to find b 0 , b 1 , b 2 , b 3

6

General Regression

Use notation so that we can write code

for any number of predictive variables

Call predictive variables x 1 , x 2 , x 3 , etc.
Call response variable y
- In previous example, x 1 = cetane, x 2 = aromatics, x 3 = density, and y = emissions
- For three variables the equation is y = b 0 + b 1 x 1 + b 2 x 2 + b 3 x 3

7

General Equation and Data

In general we can have K predictive

variables, x 1 to x k

General model equation:
How do we represent the data?
- Each data set consists of one value of y and one value for each of the x (^) j variables
- For data set m, we can call the value of y, ym , and we can call the value of x (^) j for data set m xjm

j

K

j

y b ∑ bj x

=

1

0

8

Data Set with K = 3 and N = 8

m y x 1 x 2 x 3 0 2.55 3.00 440 500 1 1.95 3.47 350 400 2 1.89 3.14 440 540 3 2.24 3.46 350 370 4 2.31 3.59 450 480 5 1.74 1.75 200 320 6 1.87 3.03 310 470 7 0.83 3.18 290 400

Each data

set, m, has a

value for y

and each xj

What are

y 4 , x 23 , x 31?

y 0

x 12

x 35

9

Summary of Data

We use K different variables (x 1 to xK) to

predict the value of another variable, y

We have N sets of data
- numbered from m = 0 to m = N-
- each data set has one value of y, called y (^) m , and one value of each xj , called xjm
- all x and y values from file input
- all data used to determine b 0 to b (^) K

10

How do we find b j?

Define x0m = 1 for all m = 0, ... , N-
- Note that there is no x 0 in model
- Setting x0m = 1 used to simplify equations
Values of b 0 , ... , b (^) K found by solving set

of K + 1 simultaneous linear equations

A (^) i0 b 0 + A (^) i1 b 1 + A (^) i2 b 2 +.....+ A (^) iK b (^) K = c (^) i
Compute A (^) ij and c (^) i from input data
Use library program to solve equations

11

Equations to be Used

Compute the Aij

coefficients ∑

−

=

1

0

N

m

Aij ximxjm

−

=

1

0

N

m

Compute the ci ximym

c i coefficients

Use the “library”

routine provided

to solve for b j

A b ci i K

K

j

ij j^0 , ,

0

= = K

=

12

Computing one A[i][j] (A[1][2])

m y x 0 x 1 x 2 x 3 0 2.55 1 3.00 440 500 1 1.95 1 3.47 350 400 2 1.89 1 3.14 440 540 3 2.24 1 3.46 350 370 4 2.31 1 3.59 450 480 5 1.74 1 1.75 200 320 6 1.87 1 3.03 310 470 7 0.83 1 3.18 290 400

A 12 = x 10 x 20 +

x 11 x 21 + x 12 x 22

x 13 x 23 + x 14 x 24

+ x 15 x 25 +

x 16 x 26 + x 16 x 26

+ x 17 x 27 = A 21

−

=

1

0

N

m

Aij ximxjm

19

Arrays for Computing Aij and c i

A (^) ij is 2D array with both dimensions at least the number of variables plus two
c (^) i is 1D array with same dimension as A (^) ij
xim is 2D array with different dimensions
- First is the number of variables plus one
- Second is maximum data points
ym is 1D array; dimension is maximum data points

−

=

1

0

N

m

Aij ximxjm ∑

−

=

1

0

N

m

ci ximym

20

Code for Computing A[i][j]

−

=

1

0

N

m

Aij ximxjm

Use typical code for computing a sum
A[i][j] += x[i][m] * x[j][m]
What is the for loop for this sum?
for ( m = 0; m < N; m++ )
How do we compute all the A[i][j]?
- We need to place the computation of one A[i][j] inside two for loops over i and j

21

Library Function to Get bj

Code variables use typical notation for

system of linear equations shown above

a (^) ij are left-hand side coefficients
b (^) i are right-hand-side coefficients
xj are unknowns
There are N equations to be solved

0 , , 1

1

0

= = −

−

=

∑ a^ x bi i N

N

j

ij j K

22

Library Function Definition

Function prototype (second dimension

required for passing 2D arrays to functions)

bool GaussianElim( double a[][MAX_VAR], double b[], double x[], int N_eqn );

a[][MAX_VAR] is left-hand side array (aij)

b[] is right hand side array (b i)

x[] is unknown array (xj)

N_eqn is the number of equations (N)

23

Using bool GaussianElim

Function returns an error flag in the

function name

True indicates an error
False indicates a valid solution if ( GaussianElim( A, c, b, K+1 ) ) { cout << “No solution for b[j]”; return EXIT_FAILURE; } // remainder of code

24

Compare prototype and call

if ( GaussianElim( A, c, b, K+1 ) ) { cout << “No solution for b[j]”; return EXIT_FAILURE; }

bool GaussianElim( double a[][MAX_VAR], double b[], double x[], int N_eqn ); Left-side

Right-side Unknowns

Number of equations

Prototype

25

Passing the Second Dimension

Use two files: your code and the library

program

Place prototype for library program in your code
Make sure that the second dimension for a[i][j] (or A[i][j]) is the same for three cases - The header in the library program - The prototype in your code - Your declaration of the A[i][j] array

26

Project Three Code Summary

Input data on K, N, ym, and xjm
Set each x0m = 1
Compute A (^) ij and c (^) i
Use library program that takes Aij and c (^) i

as inputs and returns solution for b j

Use b (^) j to compute predicted values:
Compute R^2 and output results

yˆ m

Project 3: Two-dimensional Arrays and Multilinear Regression, Slides of Computer Science

Related documents

Partial preview of the text

Download Project 3: Two-dimensional Arrays and Multilinear Regression and more Slides Computer Science in PDF only on Docsity!

Third Programming Project Third Programming Project

Computing in Engineering

and Science

May 9, 2006

Outline

period

Files You Can Download

projects page

laboratory presentations page

Exercise 8 Linear Regression

ship to measured

data pairs (xi, yi)

determine a and b

fit measures

Project Three Regression

than one variable

cetane, aromatics to find b 0 , b 1 , b 2 , b 3

General Regression

for any number of predictive variables

General Equation and Data

variables, x 1 to x k

y b ∑ bj x

Data Set with K = 3 and N = 8

set, m, has a

value for y

and each xj

y 4 , x 23 , x 31?

y 0

x 12

x 35

Summary of Data

predict the value of another variable, y

How do we find b j?

of K + 1 simultaneous linear equations

Equations to be Used

coefficients ∑

Aij ximxjm

c i coefficients

routine provided

to solve for b j

A b ci i K

ij j^0 , ,

= = K

Computing one A[i][j] (A[1][2])

A 12 = x 10 x 20 +

x 11 x 21 + x 12 x 22

x 13 x 23 + x 14 x 24

+ x 15 x 25 +

x 16 x 26 + x 16 x 26

+ x 17 x 27 = A 21

Arrays for Computing Aij and c i

Aij ximxjm ∑

ci ximym

Code for Computing A[i][j]

Aij ximxjm

Library Function to Get bj

system of linear equations shown above

∑ a^ x bi i N

Library Function Definition

required for passing 2D arrays to functions)

a[][MAX_VAR] is left-hand side array (aij)

b[] is right hand side array (b i)

x[] is unknown array (xj)

N_eqn is the number of equations (N)

Using bool GaussianElim

function name

Compare prototype and call

program

Project Three Code Summary

as inputs and returns solution for b j