 So, in the last video I talked about the simplest possible analytically tractable version of the forecast analysis cycle known as the Kalman filter. When I introduced the idea of the Kalman filter, I introduced it just in the simplest case of a univariate model. So a model that only has one variable, so there's only one slope, there's only one x that we're carried about. In many cases, for many problems, when we're doing data simulation, we have more than one variable that we're interested in. So instead of having just one x, I might have a whole vector of state variables. These could be different state variables at a given location, for example, different pools in a biogeochemical model, different stages or ages in an age-structured population model, different species in a community model. So it's very common that we're trying to make forecasts over multiple things at the same time. Otherwise, different x's could represent the same thing, say a population, but at different locations. So I might be trying to forecast different subpopulations of a single population or I might be trying to forecast a whole map. When I'm doing that, I now have a multivariate model. Now the math of the Kalman filter can be extended to the multivariate case through a bit of linear algebra. We can look at what that math looks like in the slides, but the take-home is that the underlying results are essentially the same. We still end up with the analysis mean being a weighted average of the forecast mean and the data. One of the real strengths of the multivariate version of the Kalman filter, though, is its ability to borrow strength across observations. So to take a simple case, imagine I'm making a forecast that has two variables, say x1 and x2. We have the fraction conifer on the landscape and the above-ground biomass that we observe in the landscape. And imagine also that there's some correlation between those forecasts. So for example, when I predict more conifers on the landscape, I'm predicting more biomass on the landscape. Now if I then observe one of those things, for example, I have a remote sensing observation of the fraction conifer or field survey or fossil pollen data in a paleoecological sense, I can use that new data about the fraction conifer to update that forecast to a new analysis posterior. Furthermore, because the fraction conifer is correlated with above-ground biomass, I can also update the latent variable, the above-ground biomass, which I don't observe directly, in proportion to the strength of its correlation with the thing that I do observe. So if the correlation between these two things is perfect, correlation coefficient of 1, all of the information about fraction conifer is transferred to updating above-ground biomass. If there's no correlation between these two, when I update the fraction conifer, it has no impact on above-ground biomass analysis. This idea of borrowing strength across variables works in both directions. So if I were able to observe the fraction conifer and the above-ground biomass, then my posterior estimates of both would involve both the direct constraint coming from their direct observations and then an indirect constraint due to their correlation with each other. When we do this process of updating variables by essentially combining multiple sources of information during the analysis step, I refer to this as thinking of models as scaffolds. So what we're essentially doing is we're engaging in a data-driven exercise of combining multiple sources of information, but we need some way of relating multiple sources of information to each other. So if I have observations of fraction conifer in the landscape and I have observations of biomass, I need some way of relating those to each other. And here what I'm going to do is I'm leveraging their covariance to do that update, and specifically leveraging the covariance that's coming from the model. So the model is essentially contributing its covariance structure in the forecast to this updating process. This behavior we see of multiple pieces of information being able to be synthesized and mutually updating each other during the multivariate analysis is directly analogous to things that many ecologists have seen before in spatial statistics and in time series analysis. So for example, in spatial statistics, we construct a covariance matrix that tells us how observations at different locations are related to each other. In a spatial statistical analysis, we typically make the assumption that the correlation between information at multiple locations is related to each other as a function of distance and there's some empirical distance decay function that we use to model how the correlation changes as a function of distance. So in a spatial statistics model, we would fill in this covariance matrix using this function that describes how information decays as a function of distance. By contrast in data assimilation, the math is actually completely identical. It's the exact same math. The only difference between a spatial statistics analysis and a data assimilation analysis is how we filled in that covariance matrix. So in a data assimilation example, we're filling in that covariance matrix based on our process model that we use in our forecast. So essentially we're using our hypotheses about how ecosystems work, our understanding that's built into our models and built into our forecasts. That is what's being used to generate this correlation structure between different variables when we do this data assimilation, whether they be different locations in space or correlations between different processes or different stages or different organisms in any sort of multivariate model.