CHAPTER 06.04: NONLINEAR REGRESSION: Power Model: Derivation: Part 1 of 2
In this segment, we will talk about power model regression, and we're going to derive the formula for the power model. So again, the problem statement is given x1, y1, x2, y2, so you're given n data points, let's suppose, and what you want to do is you want to best fit y is equal to a x to the power b, and the reason why it's called power model, because x is raised to the power b, and b is the power of x. Some of the examples of power model regression are like, for example, if you have a fire hose, so if you have a fire hose, the amount of water which is . . . the rate at which the water is coming out, which is the flow rate, let's suppose, the flow rate will be a times the pressure raised to the power b. So you're going to get flow rate F is equal to a p raised to the power b. so that'll be the model which you will use for the flow rate which is coming out of a pressure hose, so more the pressure, the higher the flow rate, but you do want to find out what the constants of the model are so you can relate the flow rate to the pressure. So let's go ahead and see that how we will go ahead and derive how we will find out the values of a and b in there. So we're going to start again from the sum of the square of the residuals. Sum of the square of the residuals is i is equal to 1 to n, yi minus a xi raised to the power b. So the residual is the difference between the observed value, so this is observed . . . this is observed, and this is predicted. So that's the observed value and that's the predicted value, you're taking the difference between the two, and then you're going to square each one of those differences, which is called the residual, and add them up, and what you now want to do is minimize with respect to a and b. So how do we go about minimizing with respect to a and b will depend on simply taking the derivatives of this quantity. So I'm going to . . . so your Sr is summation, let me write it down again here, it is yi minus a xi raised to the power b, whole squared. So that is the sum of the square of the residuals which you are getting, and what you want to do is you want to minimize with respect to a and b, so you're going to take the summation, sum of square of the residuals with respect to a, put that equal to 0, and that's going to give you one equation, and take del Sr by del b, and put that equal to 0, and that's going to give you the second equation. So what that means is that by taking the derivative with respect to a and b, put that equal to 0, you are going to get a local maximum or a minimum, so I'm not going into the proof that why, when you find out the values of a and b that you will get the local minimum which you are looking for to minimize the value of Sr, you do, if you're getting multiple solutions, then you have to figure out whether you are getting actually the absolute minimum for the value of Sr. So once you solve those two equations, two unknowns, so you're going to set up one equation here, one equation here, you have two unknowns, a and b, you'll be able to find out what a and b is. In the next segment, I will show you what kind of expressions do we get for a and b, when we try to simplify it. And this is the end of this partial segment. |