Findthe function of the form y = c + ₁ sin(πx) + c₂ cos(TX)that best fits the data in the following table in a weighted least squares sense; that is, minimizeΣ₁₁ W; (y₁ − c - c₁ sin(πx;) — c₂ cos(πx;))²,where wW3 = 1 and W₂ = W4 = 2.=a)Set up the least squares problem to be solved in the form Ax≈ b.xy010.5 21 01.5 -2

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Answered: Find the function of the form y = + ₁…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Use the Karush-Kuhn-Tucker method for the below optimisation problem and discuss any conclusions regarding local/global optima. min f(x1, x₂) = ₁x₂ + 4x₁+x²23 s.t. x1 + 2x₂ > 20 I1, I2 > 0.
Although we used gradient descent to find the minimizer of ƒ in part (a), we can also use calculus to solve the problem directly. Show that the minimizer of f is â = (A" A + 71)-'A"b, where I is the 4 × 4 identity matrix. What is the exact numerical value of âî for the problem in (a) and how does it compare to your result from gradient descent? What to submnit: your working out.
Only for b and c show the calculation please
Find the linear least squares polynomial approximation to f(x) on the indicated interval if a. f(x) = x² + 3x + 2, [0, 1]; b. f(x)= x³, [0,2];
The goal of this problem is to fit a trigonometric function of the form f(t) = co + c1 sin(t) to the data points (0, –9.5), (5,–11.5), (7, –9.5), (*,-1.5), using least squares. (a) The problem is equivalent to finding the least squares solution to the system Xc= y where X = У — and c = (c1, c2]" (b) Find the coefficients of the best fit by finding the least squares solution to the system in part (a) Co C1
3. Suppose that E(X) = E(X2) = E(X2) = µ, Var(X) = 7, Var(X2) = 13, and Var(X3) = 20. %3D %3D %3D Calculate the mean square error of the point estimate A to u: a =X +X2 +X3.
2. We would like to fit a linear regression estimate to the dataset {(x®,y@),(x), y),., (x(N), g/N)} with x e RM by minimizing the ordinary least square (OLS) objective function: N M -Συ, .(i) J(w): j=1 Specifically, we solve for each coefficient wk (1< k < M) by deriving an expression of Wk from the critical point J(w) the dataset (x(1), y(1)), (x(2), y(2)), . 0. What is the expression for each wk in terms of … , (x(^), y(N) and w1, , wk-1, Wk+1; *** , WM? .. .. Select one: E, (y() –D,-1,j+k W;x;") i=D1 Wk = =1 O Wk
Prepare (by hand) the final (simplified) equation system necessary to obtain the least squares method fitting for the following function type and data. f(x)= ae¹ + b sin(x) x = [0, π/2, π] y = [1, 3, 5] Paste / Upload a picture of your solution below.
Q3. Use Lagrange interpolating polynomials of order а) 1 b) 2 c) 3 to determine y at x=1.32 for the data given below. 1.25 0.2231 Xị 1.20 1.30 1.35 1.40 1.45 1.50 yi 0.1823 0.2624 0.3001 0.3365 0.3716 0.4055
Calculate and simplify the Wronskian of the following two functions: y₁ (t) = 3t+3, y2(t) = 1² + 1 Based on the value of the Wronskian, are y₁ and y2 linearly dependent or linearly independent?
The table gives the readings from a laboratory experiment 2 3. 3. 15 b+x A curve of the form y= to the above data by a+x method of discrete least squares approximation
Which one of the following formulas represent the Generalized IV estimator (or two-stage least squares estimator)? (W'X)-1 W'y (X'X)-1 X'y (X'Ω-1X)-1 X'Ω-1y (X'PwX)-1 X'Pwy All of the above. None of the above.

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