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Suppose Y = N(XB.Z) where X âˆˆ R^(pÃ—8), Z âˆˆ R^(nÃ—n), Y âˆˆ R^(nÃ—1). The Least Squares Problem is to find B such that

B = arg min ||Xe – Y||^2

The solution is given by the least-squares estimator, B_ls = (X^T X)^(-1) X^T Y. We will derive B_ls with standard calculus. Use the definition of ||.||^2 summation, i.e., ||x||^2 = Î£(x_i)^2, x âˆˆ R^n, to rewrite ||Xe – Y||^2. Sum of squared terms. (6 pts) Take the partial derivative with respect to B, i.e., âˆ‚/âˆ‚B and set it equal to zero to create a system of equations. (6 pts) Show that the system of equations in (b) is equivalent to

X^T X B = X^T Y

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03:10

Find the least square estimator Î²s for m < n (i.e. there are more variables than equations, since m is the number of data pairs recorded and n is the number of parameters) under the minimum norm consideration by stating the minimization problem as follows:minimize ||Î²||â‚ subject to AÎ² = y (with Î² âˆˆ â„â¿)

05:23

Consider the boundary value ordinary differential equation (BVODE) u” + 2u + x^2 = 0, with u(0) = u(1) = 0. By means of a cubic polynomial, determine the least squares solution that approximates the solution to the given BVODE.

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