 So state space model is a framework for modeling a dynamic process, a process where you have latent variables that are connected or linked in time or space, a time series. When you've got these latent variables that are themselves observed with some sort of error and the error can be random noise or it can itself be a process that can change in time or change with observer identity, you've got these latent variables and they're not independent, right? So if you have a time series, then you know that times that are more closely together are more likely to be similar. So you need to have a way to model that dependent structure. A state space framework is really powerful in the sense that it can take those observations, if we call them y, it can take y at time t and make an estimation of a true state. So y is the observation, the true state called x at time t and the y's at the different times are themselves conditionally independent, so because they depend on x and the x's, that process state is where the dependent structure is modeled and so you don't have to be beholden to the assumption that your data have independent errors because you're modeling that in the state or process stage of the model framework. A more specific example, if we're thinking about a state space model in time, a hidden Markov process is a model where the true state at time t, so at now, is dependent on the previous time step. So a Markov process is when you have a lag of one, the current state depends on the last state. So a simple hidden Markov model would have the true state dependent on the state at the previous time step and those states would themselves be informed by the observations y at t and t minus one, but the y's, the observations and the observation error would themselves be independent. For example, you may go out and count birds at time t and you know you go out and you do a point count of birds, you're getting an estimate of the true population but it's not necessarily the true population and we often use a Poisson distribution to describe a count and so your observation model would be y is distributed as Poisson with some mean and that mean would be equal to the x, the true state of the population at time t. The observed count at time t and t minus one would themselves be treated as conditionally independent observations given the true state and so the true state at time t, which is the mean of the Poisson, could then itself be modeled as a function of temperature or region, you could build all of the model structure into that state variable and that itself would be informed by the state at the previous time step. State space models can also be in space so they can be spatial and in that sense the current or the site of interest is assumed to be the true state at site i is assumed to be a function of the true state at the sites around i and so you have to define the adjacency matrix which could be a grid most closely associated with grid point i but you have a lot of flexibility to define your adjacency neighborhood and even evaluate it again within the probabilistic framework of the state space approach. One of the reasons this is a really powerful and important tool for forecasting is that the state space approach allows us both to separate observation error from process error which in itself is very important. So observation error isn't going to propagate out over time. Process error is going to increase over time which is obviously of importance if you're trying to forecast out in time or it's going to increase the further you get away from whatever the last data point you actually have. So when you have observations you have more information about the true state at that time step. When you're trying to forecast the true state at a future time step or at a different site the further away you are in a year or in one neighborhood over the further away you get the bigger the uncertainty gets and that's intuitive but it's not actually what always happens if you're not partitioning uncertainty into observation and process error. The classic assumption of anything we can't explain in the model gets kind of lumped into the observation error term when you're forecasting out beyond the data. If you're not pulling from that observation error and you haven't described it accurately then you're likely to be overly confident in your forecast and more likely to miss a true future state.