5

Problem 5 (3+3 pts) Consider multiple regression modelYi = Bo + B1xli + + Bprpi + €i for i = 1, n . Let Y (Y;:Yn)T , € = (€1,- en)T , 8 (Bo, B1, B...

Question

Problem 5 (3+3 pts) Consider multiple regression modelYi = Bo + B1xli + + Bprpi + €i for i = 1, n . Let Y (Y;:Yn)T , € = (€1,- en)T , 8 (Bo, B1, Bp)T and X [1,X1,.. Xp] where (1, 1)T and Xj (1j1. Tjn)T for j 1, ~p. Then one can rewrite the model asY =XB +e.Let 8 is the ordinary least squares estimator of BShow that, for any product in R"RP+l <XB,Y >=< XB,XB >_ Here a,6 >= aTb is the usual inner2. Show that the vector XB has the least angle with the vector Y among the v

Problem 5 (3+3 pts) Consider multiple regression model Yi = Bo + B1xli + + Bprpi + €i for i = 1, n . Let Y (Y;:Yn)T , € = (€1,- en)T , 8 (Bo, B1, Bp)T and X [1,X1,.. Xp] where (1, 1)T and Xj (1j1. Tjn)T for j 1, ~p. Then one can rewrite the model as Y =XB +e. Let 8 is the ordinary least squares estimator of B Show that, for any product in R" RP+l <XB,Y >=< XB,XB >_ Here a,6 >= aTb is the usual inner 2. Show that the vector XB has the least angle with the vector Y among the vectors in the column space of X. (Hint_ a,6 >= llallllbll cos 0 where 0 is the angle between vector a and b.)



Answers

A certain experiment produces the data $(1,1.8),(2,2.7),$ $(3,3.4),(4,3.8),(5,3.9) .$ Describe the model that produces a least-squares fit of these points by a function of the form
$y=\beta_{1} x+\beta_{2} x^{2}$
Such a function might arise, for example, as the revenue from the sale of $x$ units of a product, when the amount offered for sale affects the price to be set for the product.
a. Give the design matrix, the observation vector, and the unknown parameter vector.
b. [M] Find the associated least-squares curve for the data.

Okay. So I multiply my first equation by -15 x six. And so that gives me a -15 b. Which will then cancel out with the 15 B from the second equation those cancel out. And so I'm going to have 55 times six plus 15 square over six times a prepare. That's a minus equals minus 59 Us 48.8 which is -10.2. So I end up getting equals minus 10.2 times six, divided by 55 times 6 -15 square. And this gives me -0.58 approx. So from this second sulphur B I can use this equation here all the way down here, So it's gonna be 23.6 -1580, divided by six. Okay. And that will give me approximately 5.39. So those are the values of A. And B.

We can write a genetic regression line problem in this way where we have on the first column the wise off the points. Then we have the big Matics X, which is the mother lode of coefficients, where we have in the first column all once and in the second columns the axis of the points that is my exact multiplies the Matics Beetle, which contains the parameters that we want to find a better one and bitter, too. And then we have the air or the residuals now to solve the problem. Let me just write a the metrics one over end. Well, all ones in a row. And we upset. Right now that is one over end. The first column of acts. Well, transport's because we put it as a raw and we're going to use this observation later. So now let's multiply from the left by a because remember that why bar and X bar are the average is off the points and well, when we multiply from the left by A were basically obtaining the means so well different from the left. The previous equation we have a wise equal to X beat up, plus a Absalon. An hour's observed on the left hand side. We know that it's just why bar So the average of the wise and on the right inside will a X is going to be the vector. Well, the first entries one because the average is awful ones. And then the second column is the average of the axis. So just x bar. And then that multiplies the metrics beetle, which is better one bitter too, and then plus a Absalom. And so basically, the point is, we want to show that a absolute zero. And if we do that, the proof is over. So we know that the residuals saw Absalon is or for gonna to the columns off X or other way of writing that is that a genetic column of X. We put it, transports and we multiply by Absalon. We should get zero. But we observed that the first column Oh X is a so a upside down. He's one over end. The first column of Ex Put as a row so transposed the multiplies Absalon and therefore that zero. Therefore from the previous page, we see that why bar is equal to be one plus beetle to x bar, which is we wanted to show

So for Question 46 I'm given. This plane is supposed to be fitted to the following points. So I have the points 000 011 111 and one zero negative one and one for the values of a B and C that minimize this whole some. So I am waiting to minimize it. So it's gonna be a malfunction of eth. So I'm gonna write out my F variables A, B and C That's going to be that whole some a X K Must be why Kay plus C minus Z k. So, first I want to get rid of that sum. I am given my point, So there's gonna be like my at the ones cake want cake was two cables three vehicles for And so when I plug those in to hear, you should get C Square plus B plus C minus Once were plus a plus B plus C possum or minus 13 minus one squared, and the last one is a plus C plus one squared. So I have my function here. I don't have any constraints. This is not gonna be where we use equation one or two And so this is just going to be us taking the partial derivative with respect to each variable here, setting an equal to zero and finding my values of A, B and C Those were going to be critical points. So this is just what we did in the last section. And whatever critical point has a low value is going to be arm in points. So taking the partial derivative of each of the he's on first take partial derivative with respect, eh? So that will go zero that would go to zero. Here I use my chain rule. So I end up with do times a plus B plus C minus one and then two times a policy, this one. And so we simplify that out. You should get for a plus two b plus foresee And I'm saying that equal to zero take the partial derivatives like to be. I now get these two have a being and somebody to change Well knows. So I get two times B plus C minus one plus two times a plus B plus C minus one. And when I simplify that out, I get to a plus four b plus four C minus for is equal to zero. And then I take partial derivative with respect to see so I get to see, plus two times people see plus one nice one plus two times the 3rd 1 And that last one, which comes out to for a that's for B Plus eight C minus two is equal to zero. So I'm end up with this system of equations. This is gonna give me my critical points, and if you go ahead and solve the system of equations, you think substitution and what not You should get values that a is equal to negative. 1/2 B is equal to three halves and C is equal. It's a negative 1/4.

Okay, so we're trying to find the values of a B and C that minimize this caution here. The square of Xscape was Burk plus e minus. Zk do que starts at one and go support. We have four points the points for a x y and z on kz Gonna want to be in for over here, given the equation is easy with the expressed view I policy. So I could go on this question. We just need to know about it. Oh, for each level of Kate. So let's start with K Big oil. He had 000 Um, so we'll have in order to minimize this, we could see that was easy. You zero was equal to that a kind of zero. So if you just crossed it out plus B Tom zero plus c So obviously for this occurs there seems be equal to zero and then to minimize it Totally read, Need A and B is equal also equal to zero. So a and was equal to be issued must be equal to see must be equal to zero freaky cause what for? Kate was too. We have zero loin boil. So in this case will have. Let's see a X, which x zero So a kid just be there as well be Tom's wife. So just be plus C must be who'd minus disease or view Negative oil so we'll have or under the side of Z is equal to B plus C. So we want to minimize this. We could just say that disease one we could just have stayed. Hat B is equal to zero and C zero toy or these equal to one C is equal to zero. So for this case, we have two options. We have prince sinks A equals B is equal to zero comma. It's easy for a while or is equal to see which is equal to zero BZ go to one. Either these cases will minimize Are folks in? So it was a long Takeko stream When we have one boy who have okay, class and be Percy violist busy or is easy, who don't want who have their Y is equal to a course to be Percy. So for this to work out, for instance, the three again three instances, uh, we couldn't try. I would have a few instances. Actually, we could have a is equal to C zero toy and then be equally negative one or any or any one of those on combinations. Because if we look here at a formula, reconsider a Chris is plus plus see will be too. But then it's attract. Launch will be never be one again. And then you subtract one from Z visa the cases to minimize so we could have any any number combinations like this year or you could have a is equal to B is equal to zero. And it's easy, Good toy that one will have, you know, still minimize it. Where and inside Everything told was up to zero. Okay, and then let's move on. The cake was for where we have 10 negative one. Oh, we'll have a plus. See move press one within the currency and then we'll have, um negative one is gonna be equal to a because, See, So in this case for April, see equal negative oil we could have Hey, is he going to one season? Go to negative oil? If we do that out on labels, awful ese would want seasonal into zero. Do it with that. In that case, everybody quotes a negative one c equals zero. And that would minimize the punching if we also had B equals zero. Because if you look in the function, we'd have to have a is equal negative. Olympus equals zero plus one. So that would be zero and the whole thing be minimized. We could also have, For instance, a is equal to negative two. See Equalling. Why be equaling zero? Because if Dan, if you put it in here, we get negative too. It'll be 92 plus two, actually, which would be zero again. Thing is to minimize. So either those could work out. And so these will be some of our minimal values for are for this for this function.


Similar Solved Questions

5 answers
A029kg puck is initially stationary on anice surface wth negligible friction At time t = 0,4 horizontal force beginsto mave the puck The (orce Is given by F (13.8 - 3.991' with / ncwtons and in seconds, and itacts until its magnitude Is rero (a) What Is the marnitudc of thc impulsc on the puck from the force betwecnt-0.476 and t # 1.84,? (b) What /s the change In momentum ofthc puck bctwcen [ Oand the instant at which FNumbcrUnitsNumberUqlts
A029kg puck is initially stationary on anice surface wth negligible friction At time t = 0,4 horizontal force beginsto mave the puck The (orce Is given by F (13.8 - 3.991' with / ncwtons and in seconds, and itacts until its magnitude Is rero (a) What Is the marnitudc of thc impulsc on the puck ...
5 answers
Question 44 ptsIn the following rational function, define the horizontal and vertical asymptote 322 _ ~2x-1 y = 9x2 _9Vertical asymptoteSelect ]Horizontal asymptote[Select ]Refer: Finding Horizontal and Vertical asymptote of Rational Functions
Question 4 4 pts In the following rational function, define the horizontal and vertical asymptote 322 _ ~2x-1 y = 9x2 _9 Vertical asymptote Select ] Horizontal asymptote [Select ] Refer: Finding Horizontal and Vertical asymptote of Rational Functions...
5 answers
1t 29 tne Euclidcon Abojodot ond the discrete tapolog} on R . Befin f.(R,v)_ (R,t) by {()= * Detcrmine Luhekher Guiddow Uodo Guiddow closed mafping3qconkinuos
1t 29 tne Euclidcon Abojodot ond the discrete tapolog} on R . Befin f.(R,v)_ (R,t) by {()= * Detcrmine Luhekher Guiddow Uodo Guiddow closed mafping 3q conkinuos...
5 answers
Convert the rectangular equation to an equation in (a) cylindrical coordinates and (b) spherical coordinates. $4left(x^{2}+y^{2}ight)=z^{2}$
Convert the rectangular equation to an equation in (a) cylindrical coordinates and (b) spherical coordinates. $4left(x^{2}+y^{2} ight)=z^{2}$...
5 answers
Solve the following:2x+3y= 5 3x-5y= -2 (1,1) (1,2) (2,1)(2,.2)
Solve the following: 2x+3y= 5 3x-5y= -2 (1,1) (1,2) (2,1) (2,.2)...
1 answers
Two $2.00 \mathrm{cm} \times 2.00 \mathrm{cm}$ plates that form a parallel-plate capacitor are charged to $\pm 0.708 \mathrm{nC}$. What are the electric field strength inside and the potential difference across the capacitor if the spacing between the plates is (a) $1.00 \mathrm{mm}$ and (b) $2.00 \mathrm{mm} ?$
Two $2.00 \mathrm{cm} \times 2.00 \mathrm{cm}$ plates that form a parallel-plate capacitor are charged to $\pm 0.708 \mathrm{nC}$. What are the electric field strength inside and the potential difference across the capacitor if the spacing between the plates is (a) $1.00 \mathrm{mm}$ and (b) $2.00 \...
5 answers
3 3 3 7 3 1 a2e 9 8 J ; J I { 2 8 1 1 { iV 1 1 0 3 3 2 1 | 2 0 3 { L L 1 1
3 3 3 7 3 1 a2e 9 8 J ; J I { 2 8 1 1 { iV 1 1 0 3 3 2 1 | 2 0 3 { L L 1 1...
1 answers
Peaks and valleys The following functions have exactly one isolated peak or one isolated depression (one local maximum or minimum). Use a graphing utility to approximate the coordinates of the peak or depression. $$h(x, y)=1-e^{-\left(x^{2}+y^{2}-2 x\right)}$$
Peaks and valleys The following functions have exactly one isolated peak or one isolated depression (one local maximum or minimum). Use a graphing utility to approximate the coordinates of the peak or depression. $$h(x, y)=1-e^{-\left(x^{2}+y^{2}-2 x\right)}$$...
5 answers
Simplify each rational expression. Find all numbers that must be excluded from the domain of the simplified rational expression.$$ rac{y^{2}-4 y-5}{y^{2}+5 y+4}$$
Simplify each rational expression. Find all numbers that must be excluded from the domain of the simplified rational expression. $$ \frac{y^{2}-4 y-5}{y^{2}+5 y+4} $$...
5 answers
0.30-35 Match type of immunodeficiency with characteristics30.RAG deficiencyX-linked SCIDADA deficiencyAutosomal SCIDPNP deficiencyX-linked hyper IgM syndromeSignaling IL-ZRY chain deficiencyX-linked agammaglobulinemiaDeficiency in dependent and Me activationDiGeorge syndromeRecuMutation in the tyrosine kinase gene of B cell
0.30-35 Match type of immunodeficiency with characteristics 30. RAG deficiency X-linked SCID ADA deficiency Autosomal SCID PNP deficiency X-linked hyper IgM syndrome Signaling IL-ZRY chain deficiency X-linked agammaglobulinemia Deficiency in dependent and Me activation DiGeorge syndrome Recu Mutatio...
5 answers
12 [-/9 Points]DETAILSSPRECALC6 11.1.052A graphing calculator recommended _(a) Find equations for the family of parabolas with vertex at the origin, focus on the positive Y-axis and with focal diameters 1, 2, 4 and 8Focal diameterEquationDraw the graphsGraph LayersAftor you add an object to the graph you can use Graph Layers to view and edit its properties.
12 [-/9 Points] DETAILS SPRECALC6 11.1.052 A graphing calculator recommended _ (a) Find equations for the family of parabolas with vertex at the origin, focus on the positive Y-axis and with focal diameters 1, 2, 4 and 8 Focal diameter Equation Draw the graphs Graph Layers Aftor you add an object to...
5 answers
Question Simplify the following assuming = > 0,y > 0,and 2 > 0 3381"y{2882"yProvide your answer below:
Question Simplify the following assuming = > 0,y > 0,and 2 > 0 3381"y {2882"y Provide your answer below:...
5 answers
02652 {0ro 6 05 Fut deu Inyou ducrdon MDaF inte IFo SrSt Ireoaiaht Yo4 KJT Ueiad ni et Yuuf Mur IFE Quetlion Jontuln Dac Alaaetom udalaem'e l merbuejamin 0 74njo enini Lan Crie0 0Uetant
02652 {0ro 6 05 Fut deu Inyou ducrdon MDaF inte IFo SrSt Ireoaiaht Yo4 KJT Ueiad ni et Yuuf Mur IFE Quetlion Jontuln Dac Alaaetom udalaem 'e l merbuejamin 0 74njo enini Lan Crie 0 0 Uetant...
5 answers
Resolve the vector given in the indicated figure into its X component and y componentA=50.9 0=130.08Ax= Ay=L (Round to the nearest tenth as needed: )
Resolve the vector given in the indicated figure into its X component and y component A=50.9 0=130.08 Ax= Ay=L (Round to the nearest tenth as needed: )...
5 answers
4 Let zelR. Given 2 V/and 24 _ are (ESpectively 2 WXa + ZVn = wX tz +o X And Y 4heo Lonvergent Pcove {t,
4 Let zelR. Given 2 V/ and 24 _ are (ESpectively 2 WXa + ZVn = wX tz +o X And Y 4heo Lonvergent Pcove {t,...
5 answers
20. Let Xi;_ Xn be iid. real random variables with Xi Pois(A), A > 0. What is the limit a ina? X2 +_+X2 n-0Point]@ = 1 @ = 1. @ = [email protected] = ATA @ = 0.
20. Let Xi;_ Xn be iid. real random variables with Xi Pois(A), A > 0. What is the limit a in a? X2 +_+X2 n-0 Point] @ = 1 @ = 1. @ = A @ = ATA @ = 0....

-- 0.023202--