We can use the superposition principle to build models that have different kinds of components. The main idea is to think about what is the general structure we want for the forecast function and then isolate the different components of the forecast function and think about the classes of dynamic linear models that are represented in each of those components. Each of those components has a class and then we can build the general dynamic linear model with all those pieces together using this principle. I will illustrate how to do that with an example. Let's say that you want to create a model here with a forecast function that has a linear trend component. Let's say we have a linear function as a function of the number of steps ahead that you want to consider. Then suppose you also have a covariate here that you want to include in your model as a regression component. Let's say we have a K_t2 and then we have X_t plus h, this is my covariate. Again, the k's here are just constants, as of constants in terms of h, they are dependent on time here. This is the general structure we want to have for the forecast function. Now you can see that when I look at the forecast function, I can isolate here and separate these two components. I have a component that looks like a linear trend and then I have a component that is a regression component. Each of this can be set in terms of two forecast functions. I'm going to call the forecast function F_1t h, this is just the first piece. Then I have my second piece here. I'm going to call it F_2t, is just this piece here with the regression component. We know how to represent this forecast function in terms of a dynamic linear model. I can write down a model that has an F, G, and some V, and some W that I'm going to just leave here and not specify them explicitly because the important components for the structure of the model are the F and the G. If you'll recall the F in the case of a forecast function with a linear trend like this, is just my E_2 vector, which is a two-dimensional vector. The first entry is one, and the second one is a zero. Then the G in this case is just this upper triangular matrix that has 1, 1 in the first row and 0, 1 in the second one. Remember, in this case we have a two-dimensional state vector where one of the components in the vector is telling me information about the level of the time series, the other component is telling me about the rate of change in that level. This is a representation that corresponds to this forecast function. For this other forecast function, we have a single covariate, it's just a regression and I can represent these in terms of an F_2, G_2, and then some observational variance and some system variance here in the case of a single covariate and this one depends on t. We have F_2t is X_t and my G here is simply going to be one. This is a one-dimensional vector in terms of the state parameter vector. We have a single state vector and it's just going to tell me about the changes, the coefficient that goes with the X_t covariate. Once I have these, I can create my final model and I'm going to just say that my final model is F, G, and then I have some observational variance and some covariance also for the system where the F is going to be an F that has, you just concatenate the two Fs. You're going to get 1, 0 and then you're going to put the next component here. Again, this one is dependent on time because this component is time dependent and then the G, you can create it just taking a block diagonal structure, G_1 and G_2. You just put together, the first one is 1, 1, 0, 1 and then I concatenate this one as a block diagonal. This should be one. This gives me the full G function for the model. Now a model with this F_t and this G that is constant over time will give me this particular forecast function. I'm using the superposition principle to build this model. If you want additional components, we will learn how to incorporate seasonal components, regression components, trend components. You can build a fairly sophisticated model with different structures into this particular model using the superposition principle.
