In this segment, we're going to take an example of the exponential model of regression. So
we have an exponential model which we want to
be able to use for regressing to a particular data, and what we're going to do is we're
going to use transformed data. We're going to use transformed data
to so do, and we'll talk about, a little bit later, about what the transformed data means.
In the previous segment, I already talked the theory about using the transformed data
to develop the exponential model. So let's talk about the
example. The example is about an isotope of technetium, which is used for gallbladder
scanning, and an isotope, technetium 99m isotope is
used to be able to do a CT scan of a gallbladder.
Now, as we know that this is a radioactive material, so we want to be able
to figure out that how long does it take for a person to
lose this isotope from the body. The half-life of technetium 99m is about 6
hours, so it doesn't take too long for one to go back to the normal levels of radiation
which we get from the sun, for example. But let's go ahead and see, figure out for ourselves
that if we take the intensity of the isotope versus time data, will we be able to figure
out that how much of the isotope is still in the body? So what I'm going to
do is I'm going to give you time, which is in hours, versus relative intensity
data. So that's relative intensity. What I mean
by relative intensity is that this is the intensity of
the isotope when related to what is happening at 0 hours.
So what that means is that at 0, the relative intensity would
be, of course, 1, because you are dividing by the intensity
at time, t equal to 0. At other points, the data is given to us,
at 1, it is 0.891, at 3, it is 0.708,
at 5, it is 0.562, at 7, it is 0.447,
at 9, it is 0.355. So you're already seeing that down around
6 hours or so, you're losing about half of it,
so the half-life is, of technetium 99m actually is 6 hours. But let's
go ahead and figure that what happens, what is the relative intensity at 24 hours. Does
it go down very low so that the patient is not worried
about having radioactive material in their bodies? So what we're going to do
is we are going to use the exponential model to be able to figure out
what the relative intensity of the technetium isotope is at 24 hours.
So we're going to use the exponential model to do the regression. So,
but we're going to use the transformed data for
this. So what that means is that if I have gamma is equal to a e to the power
b t, I'm going to transform the data corresponding to this model
so that I can use the linear regression formulas. So what does that mean? It's that I'm going
to take the log of both sides, for example, in this case, that's
how I'll be able to do the transformation of the data
so that I can use the linear regression model. So I get log of gamma is equal to
log of a, plus b t, and what I'm going to do is I'm going to take this to be
my z, I'm going to assume this to be my a0, this to be my a1,
that's what I'm going to take, do those substitutions. So if I do those
substitutions, I'll get z is equal to a0, plus a1 times t,
that's what I'm getting. Now this is a linear . . . this is a
linear relationship between z and t, as you can see that it's a straight line, or a first-order
polynomial, relationship between z and t. Now, what that
means is that if I am able to find z versus t
data, then I'll be able to find a0 and a1. So if I am able to find a0 and a1, then I
can find my a and b, which b is same as a1, while
log of a is equal to a0, so a will be equal to e to
the power a0. So that means that, let's suppose, for example,
once I find a0, log of a is a0, so a will be e to the power
a0, so that's how I'll be able to find my a, and b is nothing but
the value of a1, so that's how I will be able to find the constants of the exponential model.
Keep in again mind that the linear relationship is between z and t.
So in no way should you consider that we are linearizing the model. We're not linearizing
the model. All we are doing is we're transforming the
data itself so that there's a linear relationship between
z and t, not between gamma and t, not between the relative intensity and time, it's
an exponential model. We are developing, trying to see if there's a linear relationship between
z and t, and from that we can backtrack the constants of the exponential model, which
are a and b. So as you can see that z is equal
to the log of gamma, which means that what we'll have to do is we'll have to take
all the individual gamma values, and calculate our zi
values to be able to do that. Now, how will I find a0 and a1 of this?
a1 will be equal to n times summation, zi ti, minus summation zi, summation,
ti, divided by n times summation, ti squared, minus summation,
ti, whole squared. So that's the . . . coming from the linear regression
formulas, that's what I'm going to use. All these summations are going from 1 to n, so
that's why I have not put the limits of the summation
there, I should have, but all the limits of the summation are 1 to n, where
n is the number of data points which I have. And a0 is nothing
but z-bar, minus a1 t-bar, where z-bar is the average value of the z
values, and t-bar is the average value of the t values. So that's how I will be able
to find a1 and a0 from the z versus t . . . z versus t data, and then I'll be able to
find a and b, which are the constants of the exponential model from the values of a0
and a1 which I find from the linear regression formulas here. So I'm going to
draw a table so that I can calculate these zi value, which are the values which
I need from the original data. So I'm going to calculate the
zi values, and then also I'm going to develop a table so that I can show the summations
which I need. I need the summation of zi ti, summation of zi, summation of ti,
and summation of ti squared, that's what I need.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.