Derive the least squares normal equations for the model :
Using the principle of least squares, we have to determine the constants c1 and c2 so that
is minimum
Equating to zero, the partial derivates of E with respect to c1 and c2 as
The normal equations are
Apply least squares fitting to derive the normal equations and solve for the coefficients by hand (using Cramer's rule) for the model y a1x + a2x2. Use the data below to evaluate the values of the coefficients. Also solve the normal equations in MATLAB (using backslash) and verify your hand calculations. Lastly, plot the data N) 25 70 380 SS0 points and the model in MATLAB and submit plot with handwork. (m/s)102 F. (N) 25 70 380550 610 1220 830...
Find a least-squares solution of Ax b by (a) constructing the normal equations for x and (b) solving for x. 1-3 -1 3 a. Construct the normal equations for x without solving x-(Simplify your answers.)
2. Derive the sum of squares decomposition for the two-factor model with interaction.
Fitting a Line to Data The method of least squares is a standard approach to the approximate solution of overdeter- mined systems, i.e., sets of equations in which there are more equations than unknowns. The term "least squares" means that the overall solution minimizes the sum of the squares of the errors made in the results of every single equation. In this worksheet you will derive the general for- mula for the slope and y-intercept of a least squares line....
3. Derive the expression for b1 from the normal equations:
Example 1: Least Squares Fit to a Data Set by a Linear Function. Compute the coefficients of the best linear least-squares fit to the following data. x2.4 3.6 3.64 4.7 5.3 y| 33.8 34.7 35.5 36.0 37.5 38.1 Plot both the linear function and the data points on the same axis system Solution We can solve the problem with the following MATLAB commands x[2.4;3.6; 3.6;4.1;4.7;5.3]; y-L33.8;34.7;35.5;36.0;37.5;38.1 X [ones ( size (x)),x); % build the matrix X for linear model %...
Given the following data, use least-squares regression to derive a trend equation: Period 1 2 3 4 5 6 Demand 6 8 5 8 7 13 The least-squares regression equation that shows the best relationship between demand and period is (round your responses to two decimal places): y = ? + ?x where y = demand and x = period
1. For the general multivariate regression model, the least squares estimator is given by Show that for the slope estimator in the simple (bivariate) regression case, this is equivalent to ja! įs] 2. In the general multivariate regression model, the variance of the least squares estimator, Va( is σ2(XX)". Show that for the simple regression case, this is equivalent to a. Var(B- b. Var(B)o i, Σ (Xi-X) 2 C. What is the covariance between β° and β,?
Q.8 In a regression model, the assumptions of the method of least squares include: [I] Relationship between x and y is linear [II] the values of X are fixed (non-random) [III] the error terms must be correlated with each other [IV] X is independent of Y [V] the error term is normal and is identically and independently distributed about the mean of zero [VI] the error term is normal but non random a. I, II, V b. II, III, VI...
Consider a factorial design model with 2 levels of Factor A, 3 levels of Factor B, and 2 observations at each combination of factor levels. Write this model in matrix notation as Y = X8+ €. Compute the matrices X'X and X'Y and use them to derive the least-squares estimators of all appropriate parameters from the normal equations XXB = X'Y Note: Do not write the matrices for a general factorial design model. Consider the particular factorial design described here.