In the proof consider x_i = t_i for i=0,1,...,n.
Given knots a = to 〈 ti < . .. < tn-b, and data points (ti, yi), for i = 0, . . . , n, prove the clamped cubic s...
5. Given knots a-to <t1 < < tn-b, and data points (ti, yi), for i - 0,... ,n, prove the clamped cubic spline S satisfying S,(a)-a, S,(b) satisfies for all g E C2[a,b] interpolating the data points and satisfying g'(a)-α, g'(b)-β 5. Given knots a-to
5. Given knots a-to <ti<..< tn b, and data points (tiy), for i-0,..., n, prove the clamped cubic spline S satisfying S,(a) , S,(b)-β satisfies C2[a,b] interpolating the data points and satisfying g'(a)-α, g'(b)-β. for all g 5. Given knots a-to
Question is highlighted, thank you! 2. On each interval I ]. i-0,..n 1, with length h, the cubic spline is given by Write down the (4n) conditions that determine the nat ural cubic spline and the clamped cubic spline. Recall. on each interval 1.-Fez,+1],に0.- n-1. with length h, the cubic spline is given by The equations which define a cubic spline (using the textbook's notation and that used in dass), that is the coefficients satisfy 41- 3h, and the c,...
Problem 2. Given the data points (xi. yi), with xi 2 02 4 yil 5 1 1.25 find the following interpolating polynomials, and use MATLAB to graph both the interpolating polynomials and the data points: a) The piecewise linear Lagrange interpolating polynomialx) b) The piecewise quadratic Lagrange interpolating polynomial q(x) c) Newton's divided difference interpolation pa(x) of degree s 4 Problem 2. Given the data points (xi. yi), with xi 2 02 4 yil 5 1 1.25 find the following...
In the simple linear regression with zero-constant item for (xi , yi) where i = 1, 2, · · · , n, Yi = βxi + i where {i} n i=1 are i.i.d. N(0, σ2 ). (a) Derive the normal equation that the LS estimator, βˆ, satisfies. (b) Show that the LS estimator of β is given by βˆ = Pn i=1 P xiYi n i=1 x 2 i . (c) Show that E(βˆ) = β, V ar(βˆ) = σ...
6. Let p;(xi = 0,... , n}, with degp;(x) = i, be a set of orthogonal polynomials with respect to the inner product f f(x)g(x) dx. Given a < b, let q(x) be the line mapping a to -1 and b to 1. Prove {p;(q(x))|i = 0,... , n} is a set of orthogonal polynomials with respect to the inner product f(x)g(x) dz, satisfying deg p;(q(x))= i - 6. Let p;(xi = 0,... , n}, with degp;(x) = i, be...
1) Consider n data points with 3 covariates and observations {xil, Гіг, xī,3, yi); i-1,.,n, and you fit the following model, y Bo+B+B32+Br+e that is yi-An + ßiXiut Ali,2 + Asri,3 + Ei where є,'s are independent normal distribution with mean zero and variance ơ2 For a observed covariate vector-(1, ri, ^2, r3) (with the intercept and three regressor variables) and observed yg at that point a) write the expression for estimated variance for the fit zs at z. (Let...
linear stat modeling & regression 1) Consider n data points with 3 covariates and observations {xn, ^i2, xi3,yid; i,,n, and you fit the following model, y Bi+Br2+Br+e that is yi A) +Ari,1 +Ari,2 +Buri,3 + єї where є,'s are independent normal distribution with mean zero and variance ơ2 . H the vectors of (Y1, . . . ,Yn). Assume the covariates are centered: Σίχί,,-0, k = 1,2,3. ere, n = 50, Let L are Assume, X'X is a diagonal matrix...
Question 4 (4 marks) (a) State TWO advantages of using B-spline curve (b) Given a set of data points Po- (0,0,0), P(2,-1,-1), P-(1,2.2) on the curve Q. A B-spline curve calculated by: P(,)-Σ N,,0)I, where N,,0-11 fort, sl < 0 otherwise" 1) Nikiのwith 3 control points and knot vector defined 1-1.) N'서の+ N,,(t)- Itt by [0 0 0 1 1 刂is used to represent the curve. (2 marks) What type of B-spline curve is this? (i) (2 marks) (i) What...
4) Consider n data points with 2 covariates and observation {xi,i, Vi,2, yi); i -1,... ,n, where yi 's are indicator variable for the experiment that is if a particular medicine is effective on some individual. Here, xi1 and ri.2 are age and blood pressure of i th individual, respectively. Our assumption is that the log odds ratio follows a linear model. That is p-P(i-1) and 10i b) What should be a good estimator for ?,A, e) Suppose. On, A,n...