Linear least squares Consider the relation between hours of study and quiz grade for three different students. -- Student 1 did not spend any time and received 5 in the quiz. P = (0, 5) -- Student 2 spent an hour studying and received 5 in the quiz. P2 = (1, 5) -- Student 3 spent two hours studying and received 8 in the quiz. P3 = (2, 8). We want to find a linear relation between hours of study and grade of the quiz y = mx + q where m and q are our unknowns, x represents the hours studied for the quiz and y the quiz grade. (a) Show that there is no line y = mx + q that passes through P , P. and P3 using an inconsistent system of equations. Hint: Substitute the x and y values of P , P2, Pz in the line equation and show that the 3x2 system with unknowns m and q has no solutions. (b) Write the system of equations for part a) in matrix form A = b where x = (m, q)". (c) Compute (A" A)- using the determinant formula in Chapter 3.2 of the Kuttler textbook. (d) Compute x = (AT A)-' A™b and find m = x1 and q = x2. (e) Plot the line obtained by replacing m and q and the points P , P2 and P3. Optional - What do you observe? In what way is this line a good representative of the data. This method is called least-square estimation and can be applied to larger sets of points.
Linear least squares Consider the relation between hours of study and quiz grade for three different students. -- Student 1 did not spend any time and received 5 in the quiz. P = (0, 5) -- Student 2 spent an hour studying and received 5 in the quiz. P2 = (1, 5) -- Student 3 spent two hours studying and received 8 in the quiz. P3 = (2, 8). We want to find a linear relation between hours of study and grade of the quiz y = mx + q where m and q are our unknowns, x represents the hours studied for the quiz and y the quiz grade. (a) Show that there is no line y = mx + q that passes through P , P. and P3 using an inconsistent system of equations. Hint: Substitute the x and y values of P , P2, Pz in the line equation and show that the 3x2 system with unknowns m and q has no solutions. (b) Write the system of equations for part a) in matrix form A = b where x = (m, q)". (c) Compute (A" A)- using the determinant formula in Chapter 3.2 of the Kuttler textbook. (d) Compute x = (AT A)-' A™b and find m = x1 and q = x2. (e) Plot the line obtained by replacing m and q and the points P , P2 and P3. Optional - What do you observe? In what way is this line a good representative of the data. This method is called least-square estimation and can be applied to larger sets of points.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Transcribed Image Text:Linear least squares
Consider the relation between hours of study and quiz grade for three different students.
-- Student 1 did not spend any time and received 5 in the quiz. P = (0, 5)
-- Student 2 spent an hour studying and received 5 in the quiz. P2 = (1, 5)
-- Student 3 spent two hours studying and received 8 in the quiz. P3 = (2, 8).
We want to find a linear relation between hours of study and grade of the quiz y = mx + q where m and q are
our unknowns, x represents the hours studied for the quiz and y the quiz grade.
(a) Show that there is no line y = mx + q that passes through Pi , P2 and P3 using an inconsistent system of
equations.
Hint: Substitute the x and y values of P1, P,, P3 in the line equation and show that the 3x2 system with
unknowns m and q has no solutions.
(b) Write the system of equations for part a) in matrix form Ax = b where x = (m, q)" .
(c) Compute (A' A)¯ using the determinant formula in Chapter 3.2 of the Kuttler textbook.
(d) Compute x = (A" A)~!A" b and find m = x1 and q = x2.
(e) Plot the line obtained by replacing m and q and the points P1 , P2 and P3. Optional - What do you observe?
In what way is this line a good representative of the data. This method is called least-square estimation and can
be applied to larger sets of points.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 4 steps
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
