I will now describe how to incorporate seasonal components in a normal dynamic linear model. What we will do is we will first talk about the so-called single Fourier component representation. Just in case you have a single frequency and how to incorporate that single frequency in your model for the seasonality. Then using the superposition principle, you can incorporate several frequencies or a single frequency and the corresponding harmonics in your model. Let's begin with that. There are other seasonal representations as well. I will just focus on the Fourier representation because it allows you to have a lot of flexibility without having a lot of parameters, in case you want to consider, for example, a fundamental frequency but you don't want all the harmonics of that frequency. The Fourier representation, if you happen to have a single frequency. We will discuss two cases. The case in which the frequency is between zero and Pi, so this is your Omega frequency, any frequency between zero and Pi. Then we will describe the case in which the frequency is exactly Pi, so the component representation is different. In the case of any frequency in this range, you're going to have dynamic linear model that has this structure. You will have the F vector is going to be a two-dimensional vector. The first component is one, the second component is zero. This is the usual 1,0 here. Then you're going to have this G, which I will describe in a minute, then you have your corresponding v_t and W_t, wherever you want to put in here. In the case of the G matrix, that's the matrix for your model, is going to have this form. It's a two-by-two matrix so your state parameter vector is also a state parameter vector of dimension 2. You're going to have the cosine of the frequency here, the sine of the frequency. That's your G matrix. If you think about the forecast function, h steps ahead, so you are at time t and you want to look for h steps ahead. If you remember, the way we work with this is going to be your E_2 transpose, then you have to take this G matrix, which is just this J_2^1 Omega, to the power of h, and then you have a vector, I'm going to call a_t and b_t, which is just going to be this vector value of your Theta t vector given the information up to the time t. It's going to have two components, I'm just going to generically call them a_t and b_t. When you take this to the power of h using just trigonometric results, you're going to get that J_2^1,Omega to the power of h is just going to give you cosine of Omega h sine of Omega h minus sine of Omega h cosine of Omega h. When you look at this expression, you get something that looks like this, and then you have, again, times these a_t, b_t. You're going to have the cosine and sine only multiplied by this. In the end, you're going to have something that looks like this. You have this sinusoidal form with the period Omega in your forecast function. You can also write this down in terms of an amplitude that I'm going to call A_t and then a phase that is B_t. Here again, you have your periodicity that appears in this cosine wave. This is again for the case in which you have a single frequency and the frequencies in this range. There was a second case that I mentioned, and that case is the case in which the Omega is exactly Pi. In this case, your Fourier representation is going to be your model that has a state vector that is just one dimensional. In the case where Omega is between zero and Pi, you have a two-dimensional state, vector here you're going to have a one-dimensional state vector. This is going to be your F and your G. Then you have again whatever you want to put here as your v_t and W_t. This gives me, if I think about the forecast function, h steps ahead is just going to be something that has the form minus 1^h times a_t. Now I have a single component here, is uni-dimensional. This is going to have an oscillatory behavior between a_t and minus a_t if I were to look h steps ahead forward when I'm at time t. These two forms give me the single component Fourier representation and using the superposition principle, we will see that we can combine a single frequency and the corresponding harmonics or several different frequencies just using the superposition principle in the normal dynamic linear model. You can also incorporate more than one component in a full Fourier representation. Usually the way this works is you have a fundamental period, let's say p. For example, if you are recording monthly data, p could be 12 and then you are going to incorporate in the model the fundamental frequency, and then all the harmonics that go with that fundamental frequency related to the period p. Here p, is the period and in this case, we are going to discuss essentially two different situations. One is when p is an odd number, the other one is when p is an even number. Let's begin with the case of p is odd and in this particular scenario, we can write down p as 2 times m minus 1 for some value of m. This gives me a period that is odd. How many frequencies I'm going to incorporate in this model? I'm going to be able to write down Omega j as 2 Pi times j over p, which is the fundamental period. j here goes from one all the way to m minus 1. Now we can use the superposition principle thinking we have a component DLM representation for each of these frequencies. They are all going to be between 0 and Pi. For each of them I'm going to have that two-dimensional DLM representation in terms of the state vector and then I can use the superposition principle to concatenate them all and get a model that has all these frequencies, the one related to the fundamental period and all the harmonics for that. Again, if I think about what is my F and my G here, I'm not writing down the t because both F and G are going to be constant over time. So my F is going to be again, I concatenate as many E_2 as I have frequencies in here. I'm going to have E_2 transpose and so on and I'm going to have m minus one of those. Times 2 gives me the dimension of Theta t. The vector here is 2 times m minus 1 dimensional vector. My G is going to have that block diagonal structure where we are going to just have all those J_2 1 Omega 1, all the way down to the last harmonic. Each of these blocks is a two-by-two matrix and I'm going to put them together in a block diagonal form. This gives me the representation when the period is odd, what is the structure of the forecast function? Again, using the superposition principle, the forecast function is going to be just the sum of m minus 1 components, where each of those components is going to have an individual forecast function that has that cosine wave representation that we discussed before. Again, if I think about the forecast function at time t h steps ahead, I will be able to write it down like this. This should be a B. B_t,j. Again here, I have an amplitude for each of the components and a phase for each of the components so it depends on time but does not depend on h. The h enters here, and this is my forecast function. In the case of P even the situation is slightly different. But again, it's the same in terms of using the superposition principle. In this case, we can write down P as 2 times m because it's an even number. Now I can write down these Omega j's as a function of the fundamental period. Again, this goes from 1 up to m minus 1. But there is a last frequency here. When j is equal to m, this simplifies to be the Nyquist frequency. In this case, I have my Omega is equal to Pi. In this particular case, when I concatenate everything, I'm going to have again an F and a G that look like this. Once again, I concatenate all of these up to the component m minus 1. Then I have this 1 for the last frequency. Then my G is going to be the block diagonal. For the last frequency I have that minus 1. This determines the dimension of the state vector, in this case I'm going to have 2 times m minus 1 plus 1. My f function, my forecast function, is again a function of the number of steps ahead. I'm going to have the same structure I had before for the m minus 1 components. Then I have to add one more component that corresponds to the frequency Pi. This one appears with the power of h. As you can see, I'm using once again the superposition principle to go from component representation to the full Fourier representation. In practice, once we set the period, we can use a model that has the fundamental period and all the harmonics related to that fundamental period. We could also use, discard some of those harmonics and use a subset of them. This is one of the things that the Fourier representation allows. It allows you to be flexible in terms of how many components you want to add in this model. There are other representations that are also used in practice. One of them is the seasonal factors representation. In that case, you're going to have a model in which the state vector has dimension p for a given period. It uses a G matrix that is a permutation matrix. There is a correspondence between this parameterization using the Fourier representation and that other parameterization. If you want to use that parameterization, the way to interpret the components of this state vector, since you have P of those, is going to be a representation in terms of factors. For example, if you think about monthly data, you will have the say January factor, February factor, March factor, and so on. You could think about those effects and do a correspondence with this particular model. We will always work in this class with these representations because it's more flexible. But again, you can go back and forth between one and the other.
