Look, you can see there was a falling problem. Assume that Xn is a discrete time process, which is weakly stationary. And that we denote the spectral density of this process by G of X. Remember I ask the following question? How to approximate this process with a moving average process, which depends on previous two events. That is, if I define Yn as A one Xn minus one plus A two Xn minus two. The question is, how to find constance A one, A two such that the process Y over N is the best approximation of the process Xn in the mean squared sense. That is mathematical expectation of Xn minus Yn squared is minimum. This is a very good question because the properties of moving average processes or processes in this form are very well known. Therefore, you can approximate any process with moving the average process. Then you can have a lot of information about this process. And the question is how to solve this problem. Let me first mention that according to the theorem, which I presented in the last subsection, the process Zeta N equal to Xn minus Yn, that is it is equal to Xn minus A one Xn minus one minus A two Xn and minus two. So this transformation, I mean transformation from X to Zeta is in the scope of previous theorem. That is, if X is a weakly stationary process, that Zeta is also a weakly stationary process. So, it is a simple outcome from previous theorem. So, Zeta is also weakly stationary. Moreover, as we know from previous theorem, so the aspect of density of the process Zeta is related to the spectral density of the process X. Here, the following qualities just the spectral density of X multiplied by the Fourier transform of the function rho X squared. Function rho in our situation is equal to indicators that X is equal to zero minus A one multiplied with indicator X two equal to one minus A two multiplied with indicator, ZX is equal to two. Of course, it's Fourier transformed can be easily calculated. So, Fourier transform of rho is just one minus A one multiplied by exponent to the power IX minus A two multiplied with exponent in the power two IX. Well, all of the things are already known, but it is not clear how we can use this theory for solving this particular problem. And now, let me mention that actually what we are asked in this situation is just whether we can find a constant A one and A two such that the variance of the process Zeta N is minimal. Let us consider more precise the variance of process Zeta. We know that Zeta is a weakly stationary process. Therefore, it's covariance function is equal to the autocovariance function, at the point N minus N. That this is a point zero. And you know that the autocovariance function is exactly the Fourier transform of the aspect of density. Therefore, it is equal to the integral of R exponent, here I should write IUX, that is I zero X, multiply it by the spectral density of process Zeta at the point X DX. This exponent is, of course, equal to one. And as for the function GZ of X, we can exactly use this representation. Let me just substitute this formula into this integral. I will get the fuller expression. This is just intergral over R GX of X multiplied. And here, I will take absolute value of one minus A one exponent IX minus A two exponent two IX squared DX. This model of squared can be easily represented by the following property, actually for any of complex number Zeta, models of Zeta squared is equal to Zeta multiplied by complex conjugate of Zeta. And therefore here, we can consider exactly this formula here and guess is equal to one minus A one exponent IX minus A two exponent two IX. And multiply it by complex conjugate, one minus A one exponent minus IX minus A two exponent minus two IX. If you now open the brackets here, you will see that integral which we consider, I mean this integral, they do represent the variance offset and as a quadratic trinomial with respect to A one and A two. So, it is looking rises into the following formula with sum IG from one to two, sum constants BIJ, AIAJ, plus the linear parts, sum for A by from one to two, sum constants CI, AI and plus D. And you can definitely minimize this quadratic polynomial with respect to A one and A two because you just have to take part of derivatives of this function with respect to A one and T twos and you will get a system of linear equations, two equations and two unknown variables. And this task is like a school task for minimizing quadratic polynomial. So, we conclude that the original task, task of finding the constants such as Yn is the best approximation of the process Xn, can be reformulated as a very simple optimization task of the quadratic polynomial. And all of this stuff was made mainly due to the notion of spectral density.
