Solution:
t |
y |
t^2 |
ty |
|
1 |
-1 |
1 |
-1 |
|
1 |
0 |
1 |
0 |
|
2 |
1 |
4 |
2 |
|
2 |
2 |
4 |
4 |
|
sum |
6 |
2 |
10 |
5 |
Here n=5 ,
6
2
let
(1)
(2)
Multiplying by x in eqn. (1) and taking sum both side
(3)
Solving eqn. (2) , (3)
y=-0.7143+ 0.9286t
LINEAR ALGEBRA 2. (6 marks) Find the best line y=c+dt to fit y=1, 1, 2, 2 at times t=-1, 0, 1, 2. (Use the least squares approximation.)
Find the best line y = c+dt to fit y = 1, 1, 2, 2 at times t=-1, 0, 1, 2. (Use the least squares approximation.) 9
1. (4 marks) Show that the set of vectors {7: = (1,1,1,1), 7x = (1,0,−1,0), 7: = (0,1,0, -1)} is orthogonal. Use those vectors in the set to get an orthonormal set {to, to, s}. 2. (6 marks) Find the best line y =c+dt to fit y=1, 1, 2, 2 at times t = -1, 0, 1, 2. (Use the least squares approximation.)
3. Find the best straight-line fit (least squares) to the measurements t -2, b 4 at at t =-1 b 1 at t0 b 0 at t = 2. Then find the projection p of 3 b 0 onto the column space of A = 1 10 1 2
q We would like to fit a line y = cx + d to the following data х у -1 -3 0 -1 0 2 3 using the method of least squares. (a) Write down the (overdetermined) linear system this problem gives rise to in the form Ax = b, where x = (..) (b) Find the best-fit line by computing the least-squares solution of the system Ax = b.
(a) Sketch the line that appears to be the best fit for the given points. (b) Find the least squares regression line. y(x)= (c) Calculate the sum of squared error. 11. [-13.22 Points] DETAILS LARLINALG8 2.6.017. Consider the following. 5 (1,5) 4 (2, 4) 3 2 (2, 2) (3, 1) - Х - 1 2 3 4 -14 (a) Sketch the line that appears to be the best fit for the given points.
FITTING A STRAIGHT LINE 1-6 Fit a straight line to the given points (x, y) by least squares. Using MATLAB 5. Average speed. Estimate the average speed vav of a car traveling according to s v - t [km] (s = distance traveled, [hr]time) from (t, s)= (9,140), (10,220) (11, 310), (12, 410). FITTING A STRAIGHT LINE 1-6 Fit a straight line to the given points (x, y) by least squares. Using MATLAB 5. Average speed. Estimate the average speed...
6. Start with the three points (a) Find the least squares line through the points. (b) Find the best curve of the form. I, C + D2r through the points. (c) Sketch the points, the least squares line, and the curve you found in part graph. Which gives a better fit, the line or the curve? (b) on the same 6. Start with the three points (a) Find the least squares line through the points. (b) Find the best curve...
2. For the data plotted below, draw (visually) a best-fit line. Then write down an equation for the best-fit line you have drawn. Finally, extrapolate (i.e. pred value when the independent variable is at 3.5. It is not necessary to calculate the least-squares best-fit line! lict) the dependent variable 3.5 2.5 1.5 0.5 0 0.5 1 1.522.5 3 3.5 3.5 T 2.5 1.5 0.5 1.5 2.5
Find the line y = a + bx which best fits the data points (x, y): (0, 1), (1, 1), (1, 2) in the least squares sense. must use matrix