CHAPTER 06.04: NONLINEAR REGRESSION: Power Model: Transformed Data: Example: Part 1 of 2
In this segment, we're going to take an example of a power model of regression. So let's go ahead and see that how do we use a power model for regressing y versus x data, and we are going to take the example with . . . by transforming the data, and we'll see what that means in a little bit. So let's look at the example itself. The example is of a fire hose, so if you have a fire hose, the relationship between the amount of water which is coming out of the fire hose is related to the pressure at which it is being disposed. So the relationship between the flow rate, so let's suppose F is the flow rate, in gallons per minute, let's suppose, let's suppose somebody tells me that the flow rate is in gallons per minute, is equal to a times whatever the pressure is and raised to the power b. So this is the power model, so a and b are the constants of the . . . the constants of the power model, and let's suppose p is given in psi, let's suppose, those are the units. So if somebody gives us flow rate versus pressure data, what we want to be able to find out is what are the constants of the model, a and b. So this is the data which is given to us. So we have pressure here, we have flow rate here, this is given in psi, flow rate is in gallons per minute, and for 10, 16, 25, 40, and 60, we are given data for five different pressures, and the flow rate corresponding to them are 94, 118, 147, 180, and 230. Now, what we're going to do is we're going to transform this data so that we can use our linear regression formulas to find out what the constants of the model, a and b, are. Again, keep in mind that we are not linearizing the model itself. What we're going to do is we're going to transform this data so that we can use the linear regression formulas to find the constants of some linear model, and then be able to find out what the values of a and b are. What does that mean? That means as follows, if you have F is equal to a p raised to the power b, I'm going to take the log of both sides, get log of F is equal to log of a, plus b log of p. And now what I'm going to do is I'm going to take log of F to be some z value, let's suppose, I'm going to call log of a to be a0, b to be a1, and log of p to be w, let's suppose. So if I take these substitutions, what I will get is z is equal to a0, plus a1 w. So what you are finding out is that there's a linear relationship between the z and the w variable, where z is nothing but the log of the F values which I have, which is the flow rate values which I have, and w are the log of the pressure values which I have. So what that means is that, since z versus w is a linear relationship, if I am able to find a0 and a1, then I can find out a from this relationship, where log of a is a0, and a1 is nothing but b itself. So log of a is equal to a0, which implies then that a is equal to e to the power a0, that's how I'll be able to find out what the constant of the power model is, and then of course b is nothing but a1 itself, to any transformation there. So what that means is that I have to find z versus w data, so that I can find my a0 and a1 by using the linear regression formulas, which are like this, a1 will be n times summation, wi zi, minus summation, wi, summation, zi, divided by n times summation, wi squared, minus summation, wi, whole squared. All these summations which you are seeing here in this formula for a1, they are all going from 1 to n, where n is the number of data points. a0 will be nothing but your z-bar, minus a1 w-bar, where z-bar is the average value of the z values, and w is the average of the w values. So what that means is that not only do I have to first do the transformation of the data, because I've got to find my w values and zi values. and where am I going to get these wi and zi values? I'm going to get the wi and zi values from here, because the zi values will be taking the log of each of the flow rate data points which are given to me, and wis will be the log of the pressure values which I have there, and then I have to find these summations, I have to find the summation of this, summation of wis, summation of zis, and summation of wi squares. So all those things have to be found to be able to find out a0 and a1, and then I can backtrack to find a and b. So let's go ahead and draw a table of these data values which I need . . . which I have, and then which I need, so that I can find out what the constants of the power model are.