CHAPTER 06.04: NONLINEAR REGRESSION: Exponential Data: Transformed Data: Part 2 of 2
So let's go ahead and draw a table for that, and be able to see that how can we get all these summations together. So we have ti here, we have gamma-i here, these are the data points which are already given to us, so I'm going to repeat those, 0, 1, 3, 5, 7, 9, so these are all which are given to us, these data values which I'm writing, 1, 0.891, 0.708, 0.562, 0.447, and 0.355, so these values were already given to me as gamma versus t data, but as I know that in order to be able to use my linear regression formulas, I need to calculate my zi, which is nothing but the log of the gamma values. So that's basically taking the natural log of these numbers, that will give me the zi values, and those turn out to be 0, -0.1154, -0.3453, -0.5763, -1.0356, and minus . . . that's about it . . . no, one, two, three, four, I think I skipped a number here, it's -0.8052, and -1.0356. So those are the transformed zi values now which I need, and then of course I need ti times zi, I need that, and I also need ti squared, because these are all needed for the summation. So ti zi will be simply the multiplication of the t values which I have and the z values which I have, so for example, the first one will be 0, the second one will be 1 times -0.1154, next time will 3 times this number here, which turns out to be -1.0359, then I get -2.8813, -5.6364, -9.3207, so those are the multiplication of the t and z values. And then the last column is simply the square of the t values, which I need for the summation, so that will be 0, 1, 9, 25, 49, and 81. So what I need for the formula, for the linear regression formula, are the summations of tis, which is this, and that turns out to be 25. Now, I don't need the summation of gamma-is, because I'm using the transformed data, I need the summation of the zi values, which turns out to be -2.8778. Summation of the ti zi values, all of these values added together gives you -18.9897. And the summation of all the ti values here gives me 165. So I have all the summations which I need to be able to use the formula for calculating a0 and a1 of the transformed data so that I can use the linear regression formula. So what that means is that a1, I'm going to use the a1 formula. a1 is n, which is, how many data points do I have? One, two, three, four, five, six, six data points, summation of the zi ti values, that is this, -18.9897, and then minus the summation of the zi values, which is right here, that's -2.8778, and summation of the ti values, which is 25, which is right there, divided by n, which is, again, 6, times the summation of the ti squared value, which is 165, minus the summation of ti values, which is 25, and square of that, and that number here turns out to be equal to -0.1150, that's the value of a1 which I get for the z is equal to a0, plus a1 t linear regression line. And similarly I'll find a0, a0 is nothing but z-bar, minus a1 times t-bar. Our z-bar values are simply the . . . simply the average of the z values which I have, -2.8878 divided by 6, that's the summation of zi, we have six values, minus a1, which I just found as -0.1150, times the average value of the t values, which is 25 divided by 6, 25 is the summation of the t values, and there are six t values. And this a0 here turns out to be -0.0002615, that's what I get for a0. So basically what I have is off the transformed data, which I . . for which I used linear regression formulas, z is equal to a0, plus a1 t, I get a0 is equal to -0.0002615, and I get a1 equal to -0.1150. But these are the constants of the linear regression model of z versus t. What I need to find out is what are the constants of the exponential model, because z is equal to a e to the power b t. We already established that a is nothing but e to the power a0. So that's e to the power a0, which is this number here, -0.0002615, and that turns out to be 0.9997. And then, if you at b, b is nothing but a1. b is nothing but a1, and a1 turns out to be nothing but whatever this number a1 is, -0.1150. So what you have been able to do is by transforming the data, you have been able to find the constants of the exponential model, a and b, this is the exponential model which you have, where the relationship between relative intensity and time is given. So based on the values of a and b which I have found out, I have been able to find the constants of the regression model. So the regression model is gamma is equal to 0.9997 e to the power -0.11 . . . -0.1150 t. And what we were interested in the beginning was, hey, what is the value of the relative intensity at 24? So all I have to do is just put in the value of 24 for my time, and I will be able to find out what the value of the relative intensity is, and it turns out to be 0.0632. So what that means is that only 6 percent of the isotope which was injected in this person 24 hours ago is left. And this is . . . is very mild, because this is almost the same kind of radiation which we get in everyday life from the sun and other radioactive materials, so we should not be worried about that after 20 hours . . . 24 hours that he has still 6 percent of the isotope in his body. Now, what I would like you to do as homework is as follows, because we know that the relative intensity at time, t equal to 0 is 1, because we are measuring the relative intensity corresponding to time, t equal to 0. What I would like you to do is, since we know that we used gamma is equal to a e to the power b t as our regression model, let's simply use gamma is equal to e to the power b t, because we know that at time, t equal to 1, a is 1. So if you're going to forces the . . . force the relative intensity to be 1 at time, t equal to 0, then the regression model which you're supposed to use is this. There's only one constant of the regression model, which is b. So I'd like you to take the same data, and regress it to gamma is equal to e to the power b t. And that's the end of this segment.