Linear Algebra Question: Least Squares
Linear Algebra Question: Least Squares {(0, b), (1, b2),, , . (k, bk)) İs a set of k points in R2. Show that in the horizontal line 5. Suppose P of best fit f(x) A, A is the average of the numbers b1...
{(0, b), (1, b2),, , . (k, bk)) İs a set of k points in R2. Show that in the horizontal line 5. Suppose P of best fit f(x) A, A is the average of the numbers b1,. b {(0, b), (1, b2),, , . (k, bk)) İs a set of k points in R2. Show that in the horizontal line 5. Suppose P of best fit f(x) A, A is the average of the numbers b1,. b
Projections and Least Squares 3. Consider the points P (0,0), (1,8),(2,8),(3,20)) in R2, For each of the given function types f(x) below, . Find values for A, B, C that give the least squares fit to the set of points P . Graph your solution along with P (feel free to graph all functions on the same graph). . Compute sum of squares error ((O) -0)2((1) 8)2 (f(2) -8)2+ (f(3) - 20)2 for the least squares fit you found (a)...
3. Now suppose that (a,b), (a2, b2),..., (aq, be) are l distinct points on R2. Let X be the set formed by these l points. Prove that there are l vector fields F1, F2,..., Fe, each defined on R2X (the set R2 without the points in X), with the following properties: (i) curl F; = 0 on RP X for all i = 1, ..., l. (ii) (“linearly independent”) If C1,C2, ..., Ce are real numbers such that the vector...
2.4 We have defined the simple linear regression model to be y =B1 + B2x+e. Suppose however that we knew, for a fact, that ßı = 0. (a) What does the linear regression model look like, algebraically, if ßı = 0? (b) What does the linear regression model look like, graphically, if ßı = 0? (c) If Bi=0 the least squares "sum of squares" function becomes S(R2) = Gyi - B2x;)?. Using the data, x 1 2 3 4 5...
5. (2 points) When a least-squares linear regression equation is constructed based upon a data set, and a line is constructed from this equation, which (Gif any) of the following is a. The point (F,) must be on the regression line. b. The point (0,b) must be on the regression line. c. The point (0,b) must be on the regression line. d. None of the above statements are false. All of the above statements are true. ons for ss is...
9) Suppose you are given n points: (x,y)(, y). And we wish to fit a cirele to the data. A general circle, as we all know, is Cr-y+-k. So the question becomes: What are h, k, and r so that the circle becomes the best least squares fit? Show that this problem becomes Th .e. What is a, B and what is M? B, When fitting the cirele to the data points (0,2), (1,2),3,-),(0,-D,6,0) what are the normal equations? GIVE...
Suppose we fit the points XK 0 | Yk 2 /6 -0.5 /4 -1 / 3 -1.2 31/4 3 2/3 2 5/6 1.5 by the curve f(x) = a + b cos x+csin 2x using least squares approach. Find the system of three linear equations that can be used to solve for a, b, c. (You may present your answer with 3 decimal digits and you are not required to solve the system).
3. Let L be the linear transformation on R2 that reflects each point P across the line y kx (k>0) a) (2 marks) Show that v and v2 - 1 are eigenvectors of L. b) (1 mark) What is the eigenvalue corresponding to each eigenvector? (Hint: No need to calculate the characteristic polynomial or solve a matrix equation. Geometric reasoning should suffice to solve this problem. Drawing a diagram is recommended!) 3. Let L be the linear transformation on R2...
2. [40 points) Consider the following pairs of observations: X 5 3 2 6 6 0 1 1 7 5 у 3 3 1 4 1 (a) (15 points] Use the method of least squares to fit a straight line to the seven data points in the table. (b) [5 pointsSpecify the null and alternative hypotheses you would use to test whether the data provide sufficient evidence to indicate that x contributes infor- mation for the (linear) prediction of y....