[MUSIC] I will begin describing the structure of a particular class of models now, the polynomial trend models. These are models that are useful to describe linear trends or polynomial trends in your time series. So if you have a data set, where you have an increasing trend, or a decreasing trend, you would use one of those components in your model. So in general, I'm going to begin with the first order polynomial model, which we have already described. It's the one that has yt is a single parameter, I'm going to call it just theta_t + nu_t. And then a random walk evolution for that single parameters, so that's the mean level of the series. And then we assume that it changes As a random walk, so this is the first order polynomial model. In this model if I want to write it down in short form I would have a quadruple that looks like this. So the F here that goes F transposed times the parameter vector in this case we have a scalar vector, scalar parameter. It's going to be 1 my G that goes next to the state of t -1 is going to also be 1. And then I have vt and Wt here. So this fully defines my model if I think about the forecast function of this model using the representation we had before. Again, we're going to have something of the form F transposed G to the power of h and then the expected value of that theta_t given Dt. F is 1, G is 1, therefore I'm going to end up having just expected value of theta_t given Dt. Which depending on the data that you have is you're just going to have something that is a value that depends on t and it doesn't depend on h. What this model is telling you is that the forecast function, how you expect to see future values of the series h steps ahead is something that looks like the level that you estimated at time t. So that's the forecast function, you is a first order is a zero order polynomial is a constant on h and it's called the first order polynomial model. In the case of a second order polynomial We are going to now think about about a model in which we want to capture things that are not a constant over time but may have an increasing or decreasing linear trend. In this case we're going to need two components in your parameter vector in the state vector. So we have again something that looks like in my observation equation. I'm going to have, I'm going to call it say theta{t,1} Normal vt, and then I'm going to have say theta_{t,1} is going to be of the form to theta_{t-1,1} and there is another component here. The other component enters this equation plus let's call this And then I have finally also I need an evolution for the second component of the process which is going to be again having a random walk type of behavior. So there is different ways in which you can interpret this two parameters but essentially one of them is related to the baseline level of the series the other one is related to the rate of change of the of the series. So if you think about the dlm representation again, these two components, I can collect into the vector wt. and then assume that this wt Is normal. Now this is a bivariate normal. So what would be my F and my G in this model? So again my theta vector has two components My G, so my F is going to be a two dimensional vectors. So I can write down F transposed as the only component that appears at this level is the first component of the vector. I'm going to have 1 and then a zero for F transposed. And then my G here if you think about writing down theta t times G say the t -1 +wt. Then you have that you're G is going to have this form. So for the first component, I have past values of both components. That's why I have a 1 and 1 here for the second component I only have the past value of the second component. So there is a zero and a 1. So this tells me what is the structure of this second order polynomial. If I think about how to obtain the forecast function for this second order polynomial is going to be very similar to what we did before. So you can write it down as F transposed G to the power of h, expected value of theta t given Dt. Now the expected value is going to be vector also with two components because theta_t is a two dimensional vector. The structure here if you look at what G is G to the power of h going to be a matrix, that is going to look like 1, h, 0 1. When you multiply that matrix time this times this F what you're going to end up having is something that looks like 1 h times this expected value of theta t given Dt. So I can think of two components here, so this gives you a constant on h, this part is not going to depend on h. So I can write this down as k t 11 component multiplied by 1 and then I have another constant, multiplied by h. So you can see what happens now is that your forecast function has the form of a linear polynomial. So it's just a linear function on the number of steps ahead. The slope and the intercept related to that linear function are going to depend on the expected value of, theta_t given the all the information I have up to time t. But essentially is a way to model linear trends. So this is what happens with the second order polynomial model. As we included linear trends and constant values in the forecast function, we may want to also incorporate other kinds of trends, polynomial trends in the model. So you may want to have a quadratic form, the forecast function or a cubic forecast function as a function of h. So the all those can be incorporated using the general p-order polynomial model, so I will just describe the form of this model. And in this type of model, the forecast function is going to have order p -1. So your parameter vector is going to have dimension p. So you're going to have theta_t theta t1 to tp. Your F matrix is going to be constant if I write it as a row vector. F transpose is going to be a p dimensional vector with the one in the first entry and zeros everywhere else. My G matrix is going to have this form and there is different parameterizations of this model and I will talk a little bit about this. But one way to parameterize the model is something that looks like this. So you have ones in the diagonal of the matrix, the matrix is going to be a p by p has to be the dimension of the p compatible with the dimension of the state vector. And then you have zeros's below the diagonal above that set of ones that are also ones above the diagonal. So this matrix G is what we call a Jordan block of dimension p of 1. So here 1 is the number that appears in the diagonal. And then I have a p Ip matrix, I have ones in the upper diagonal part. So this is the form of the model, so once again I have the F the G, and the wt. I have my model. The forecast function in this case again can be written as F transposed G to the power of h. And when you simplify times expected value of theta_t, given Dt. Once you simplify those functions you get something that is a polynomial of order p-1 in h. So I just can write this down as kt constant. Plus kt1 h + kt p- 1, h to the p -1, so that's my forecast function. There is an alternative parameterization of this model that has the same F and the same algebraic form of the forecast function, the same form of the forecast function. But instead of having this G matrix, it has a matrix that has ones in the diagonal and ones everywhere above the diagonal. So it's an upper triangular matrix with ones in the diagonal and above the diagonal. That's a different parameterization of the same model is going to have the same general form of the forecast function is a different parameterization. So again, you can consider the way you think about these models is you think what kind of forecast function I want to have for my future? What is the type of predictions that I expect to have in my model? And if they look like a linear trend, I use a second order polynomial. If it looks like a quadratic trend in the forecast then I would use 3rd order polynomial model representation.
