 What I'd like to discuss in this next series of videos is to think about a key part of how we actually perform ecological forecasts which is how do we project the uncertainties that we have right now into our forecasts so that we include uncertainties in our projections and predictions. So these lectures are going to focus on the kind of the methodological approaches that we use to propagate uncertainty into our forecasts. In looking at these approaches we're going to focus on two key sets of trade-offs. The first trade-off is going to be between methods that provide an analytical solution so there's a mathematical equation that describes in closed form how we propagate uncertainties into the future versus numerical methods those that rely on computer simulation to approximate the way that we propagate uncertainty into the future. At the same time we're going to consider another axis that distinguishes between methods that give us the full probability distribution of our forecast versus methods that approximate that in terms of just a key set of statistical moments such as the mean and standard deviation of the forecast. In general it's easier to just get estimates of means and standard deviations than it is to generate a full distribution and then between the analytical and numerical methods there's a trade-off between having to do hard math versus having to do a lot of computation. So let's start first by thinking about the analytical methods. So in one corner we have analytical methods that provide us with a full probability distribution. This is what I will take as the gold standard for doing this and the way that we propagate this uncertainty is to make use of statistical methods for transforming variability. So we have rules from statistics that can show us mathematically how to take a probability distribution and translate it through some function into a new probability distribution and so that probability distribution transformation is essentially the same problem as propagating into the future. The challenge with this gold standard is the math involved can get very hairy very quickly. So while it's a useful point of reference it's not actually a very practical one for most ecological forecasts because the math involved is very challenging and very rarely has a closed form solution that's actually helpful for us. So let's next think about analytical methods that just give us some key statistical moments. One important case of that are models that are linear. So models that just involve linear combinations of the variables we're interested in and their coefficients. It turns out that if all we're interested in is the means and variances or the means and standard deviations of our forecast then we can actually get an exact analytical solution for how to propagate uncertainty in linear models. And we do this by making use of a series of rules that describe how random variables interact with each other and how random variables interact with constants. One general pattern that we see when we do this is that a linear model that has multiple terms each term contributes specific components to the predictive uncertainty. So the first thing that you'll see is that there's a direct component contributed from each individual term where there's a variance in the thing that's an input. So for example there's a variance in a parameter, variance in a meteorological driver, variance in an initial condition and that is multiplied by the slope of that term. So in a linear model there's a slope and specifically it's the square of the slope. So the direct contribution of a parameter to the predictive uncertainty is scales as the square of the slope times the variance of that component. In addition to the direct effects of each term that we also need to include their interactions and in that case if I have two things interacting with each other I need to know about their covariance and I need to know the product of each of their individual slopes. So I have a slope of the first factor times slope of the second factor times the covariance between them and this is how we're capturing the interaction between variables when we propagate uncertainty. So to recap when we have linear models we have the capacity to get an exact solution to propagating uncertainty in terms of the means and variances but we're going to do moving forward is think about how we can translate that to more complex models that may be non-linear.