Question 4: Consider a data set with the response vector Y = (y₁,..., yn) and the datamatrix-1X12(EB)1Inl=We model the relationship between X and Y using the linear regression model: y =00+0₁₁+0₂x₁2 + €₁, i 1,...,n, where ~ N(0,02). Let the parameter vector bedenoted as (00, 01,02). We wish to minimize the sum of squared residuals: SSE =Σ(yi (00+0₁₁+0₂2))². Let the fitted value be denoted as ĝi = 0 + 0₁x₁1 + 0₂x₁2,and let the fitted value vector be denoted as Ŷ.=2a) Show that SSE = Σ-1(yi - ŷi)².b) Show that SSE = (Y – Ŷ)¹(Y – Ý).c) Show that Ŷ = XO.d) Simplify the derivative equationasse20e) Find the solution of which solves the equation in part d.: 0.

Given a data set with the response vector, and the data matrix .

Answered: Question 4: Consider a data set with…

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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Example 15.11) The following table shows the marks obtained in two tests by 10 students: Marks in Ist Test (X) 9. 8. 8 7 6. 10 4 7 Marks in 2nd Test (Y) 8 7 10 8 10 9. 8 6. (a) Find the least square regression line of Y on X. / (b) Test the hypothesis that marks in the two tests are linearly related.
A set of experimental runs were made to determine a way of predicting a parameter Z in terms of A and B. The results are represented in TABLE 1. 1) Using simple linear regression models, find the missing values in TABLE 1. 2) Using TWO (2) different methods, determine which independent variable is the most linearly related to the dependent variable? Show your calculations and discuss your results. 3) Using a multiple linear regression model, find the missing values in TABLE 1. Compare your results with those obtained in question 1) and discuss your findings. 4) Analyze and discuss the fitness of the multiple linear regression model for the data. TABLE 1. Z 22.1 12 16.5 17.9 7.2 11.8 9.2 19 22.4 12.5 24.4 11.3 14.6 18 20.9 18.9 A 37.8 45.9 a₁ 10.8 48.9 32.8 B 69.2 69.3 58.5 58.4 75 23.5 65.9 46 52.9 b₁ 55.8 18.3 19.1 53.4 22.9 22.9 a2 32.9 47.7 36.6 39.6 20.5 23.9 27.7 16.7 27.1 **The End*
Consider the data, X, 1 4 Y, 4 8 6 10 12 The estimated regression equation for these data is ý- 2.60 + 1.80x. (a) Compute SSE, SST, and SSR using equations SSE - I(r, - ), sST - E(y, - v), and SSR = E(, - . SSE 7.6 SST =40 SSR = 324 (b) Compute the coefficient of determination ?. Comment on the goodness of fit. (For purposes of this exercise, consider a proportion large if it is at least 0.55.) O The least squares line did not provide a good fit as a large proportion of the variability in y has been explained by the least squares line. O The least squares line did not provide a good fit as a small proportion of the variability in y has been explained by the least squares line. O The least squares line provided a good fit as a small proportion of the variability in y has been explained by the least squares line The least squares line provided a good fit as a large proportion of the variability in y has been explained by the least squares ine. (c) Compute the sample correlation…
Solids (grams) obtained from a material as y, with respect to drying time (Hours) as x. Ten experiments were carried out to obtain the following observations: Table on the picture (a) Create a scatter diagram for the data. (b) Estimated regression model according to the data conditions. (c) Calculate Model Accuracy (R²) and Relationship Between Variables (r²)
Derive the least squares estimates of a and ß for the centred form of the simple linear regression model given by Yi = a + B(x; – I) + €; i= 1,2,..,n. Check that the estimates do give a minimum in the same way as we saw for the standard form of the simple linear regression model.
The electricity generation problem: Most of S.As power stations are coal- fired. Assume a random sample of 10 power stations were selected and their coal usage and electricity generated 2017 was obtained. Coal used in 2017 (in million tonnes) 15 6 10 18 9 7 14 11 5 8 Electricity generated (million kilowatt hours) 35 18 24 32 24 20 32 29 14 22 Find the straight-line regression function to estimate electricity generated (y) from coal used (x) by the method of least squares. O A. 9=5.763+ 2.815x O B. =9.367+1.51&r OC.f=9.367+1.518y O D.f=5.763+2.815y
Consider the following linear regression model: Yi = B1 + B2x2i + B3x3i + ei of = o²r What is the weight to be used for generalized (or weighted) least-squares estimation? Select one: 1 1 Ob. 1 Oc. 1 od.
Q2: For the raw data shown in Table below: 1) Find out which of the x₁ or x₂ variables is better correlated to y. 2) Find out the linear regression lines: y₁ = f(x₁) and y2 = f(x₂) depending on the results of 1) above. X1 1 3 4 11 14 X2 1 2 3 12 15 y 1 2 4 9 65 87 99 4 5 сл 7 12 8
(b.) Consider the fictitious set of data shown below, where the line through the data is the fitted simple linear regression line. Sketch a residual plot (It doesn't need to be perfect) on the right side of this graph. What type of transformation is needed to get a proper SLR model? Write the general form of this new SLR model. §
2
Compute the sum-of-squares error (SSE) by hand for the given set of data and linear model. (7, 7), (8, 8), (9, 10); SSE = y = x - 1

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