CHAPTER 06.04: NONLINEAR REGRESSION: Power Model: Transformed Data: Example: Part 2 of 2
So what I'm going to do is I'm going to repeat the data which is given to me, so this is pis and Fis are given to me, and those are given to me as 10, 16, 25, 40, and 60, and the corresponding flow rate values given will be 94 gallons per minute, 118, 147, 180, and 230. Now, what I do need is I need the values of w. So wi is nothing but the log of the pi values, that's what I need. So then these log of pi values are as follows, I get 2.3026, that's the log of 10, natural log of 10, then 2.7726, 3.2189, 3.6889, and 40943. So what I have been able to do is I've been able to transform the data of the pis to wis by simply taking the natural log of each of the pressure values. And then similarly, zi will be nothing but the log of the Fi values. So this is basically taking each individual flow rate numbers, and taking the natural log of those numbers to be able to generate the z values, which turn out to be 4.5433, 4.7707, 4.9904, 5.1930, 5.4381. So that's what you get for zi values, and then I need the other . . . I need other values for the summation formulas which are existing in my a1 and a0 formulas. So I need wi zi . . . I need wi zi, then I also need wi squared. So those are the other numbers which I need basically for the formula itself. And these values here, the wi zi is 10.4613, which is simply the multiplication of these two numbers gives me that, multiplication of these two numbers, for example, gives me 13.2271, then I get 16.0636, 19.1562, 22.2654. And also I need the wi squared values, which are simply the square of these numbers here, because that's what wi squared stands for, and that's 5.3019, 7.6872, 10.3612, 13.6078, and 16.7637. So let's go ahead and see what summations do I need. So I don't need the summations of pi and Fi values, I don't need these summations, because they are not existing in the formula. So I need this summation, which is the summation of all the w values, which turns out to be 16.077, all the summation of the zi values, which turns out to be 24.935, summation of all the wi zi values, which turns out to be 81.174, and summation of all the w squared values, which turns out to be 53.722. So now we have all the summations available to us to be able to apply them in the formula for the linear model which we have between z and w. So a1 is n, n is, how many data points do I have? 5 times the summation of wi zi values, which is this, 81.174, minus the summation of all the w values, which is this, which is 16.077, and the summation of zi values, which is this, which is 24.935, divided by n, which is 5, times the summation of all the w squared values, which is 253.722, minus the summation of wi squared values, which is this, 16.077 squared, and this a1 turns out to be equal to 0.4910. Now, if I want to find a0, a0 is nothing but the average values of the z, minus a1 the average values of the ws. So the z-bar values will be the summation of the zis, which is 24.9335, divided by five data points which I have, minus a1, which I just found out, which is 0.4910, and w-bar is simply the . . . taking this summation of ws and divided by 5, because we have five data points, 16.017 . . . 077, divided by 5, and the a0 value turns out to be equal to 3.4083. So these are the constants of the model of z versus w which we are getting here, and we need to find out what the constants of the actual power model are. go ahead and do that. So we have z is equal to a0, plus a1 w. a0 we just found out to be 3.4083, a1 we just found to be 0.4910 w. So that is the linear relationship between z and w, but that's not what we are interested in. We are interested in finding a and b of our power model, but a is given as e to the power a0, and a0 is nothing but 3.4083, so that number turns out to be 30.213, and b is nothing but a1 itself, so we don't have to do any kind of mathematical transformation with that, so you get 0.4910. So what you are basically finding out is that the relationship between for . . .between the flow rate and the pressure is nothing but a p raised to the power b, and a is nothing but 30.213, p raised to the power b, which is 0.4910, that's what you get as the relationship between flow rate and pressure. And if somebody asked you, hey, can you find out the flow rate for a pressure of 22.5 psi, all you have to do is to plug in 22.5 to 30.213 22.5 raised to the power 0.4910, and the value turns out to be 139.4 gallons per minute. So that's how you are using the regression model to be able to find out the transformed data to find out the constants of the regression model, and then be able to figure out, or to be able to predict the values of the flow rate at given pressure. As part of your homework, what I would like you to do is, let's suppose somebody said, hey, F is equal to a p raised to the power 0.5. So somebody fixed the exponent of pressure, let's suppose. F is equal to a p raised to the power 0.5, in fact, you'll find out it's very close to 0.5. So let's suppose the flow rate is a p raised to the power 0.5. Can you find out the constants of . . . the constant of this regression model, which is just a now? So you have only one constant of the regression model, which is a, and somebody is telling you this exponent is 0.5, is fixed. Can you find out what the constant of the regression model, a, is, and that's your homework. And that's the end of this segment. |