Model This and Model That

Continuing from my previous post where we looked at ENSO forecasts into the spring of 2016, we shall now look at some of the different models used. First though, what is a model? Succinctly put, a model is a simplified version of a complex reality. Ok, good start, but let’s go deeper within this field and try to learn what statistical models, and dynamical models, are.

                                                                                  

Statistical Models

A statistical model allows us to infer things about a process from its observed data, and is represented through a set of variables. The model is constructed through 1 dependent variable, and at least 1 independent variable. The dependent variable (aka the response/outcome), is what is being studied and is the result of a combination of the independent (aka explanatory) variables. In a simplified case, this may mean that each independent variable has no relationship with any of the other independent variables.  Or, is more often the case, there are relationships between them, and these interactions are statistically known as correlations. Both dependent and independent variables can be observed and recorded, and what with a set of assumptions about the data, parameters (the unknown constants in the equation) can be estimated. An important part of a statistical model to be aware of is that it is non-deterministic. What this means is that some of the variables are stochastic, i.e. essentially random. Hence, a statistical model uses random variables to model the components of the process that are not currently fully understood scientifically.

Let’s take a look at 3 of the statistical models used by the IRI as shown in my previous post:

CPC MRKOV – Is the National Centres for Environmental Prediction/Climate Prediction Centre (NCEP/CPC) Markov Model. It is a linear statistical model with three multivariate empirical orthogonal functions (EOF), of observed sea surface temperature, surface wind stress, and sea level. EOF analysis is the decomposition of a signal in terms of both temporal and spatial patterns. 

CPC CCA – Is the Climate Prediction Centre Canonical Correction Analysis Model. It is a multivariate linear statistical model with predictors mean sea level pressure and sea surface temperature. Canonical correction analysis is a method to find the maximum correlation of a linear relationship between two multidimensional variables.

FSU REGR – Is the Florida State University Regression Statistical Model. It is a multiple linear regression model with predictor variables of upper ocean heat content, wind stress, and sea surface temperatures. 

Dynamical Models

A dynamical model, as with a statistical model, represents the relationship between a set of variables, however it uses functions to explain how the variables change over time given their current state. These functions which are related to their derivatives are known as differential equations, and can be thought of as the rate of change.

An extremely famous example of a non-linear dynamical model is the Lorenz model. Edward Lorenz took the Navier-Stokes (fluid dynamics) equations and simplified them to get 3 differential equations, with only 3 variables (I won’t go into the equations now, but take a look here if you wish to learn more about them). However, the point is they seemingly look very simple to solve, but plotting the solutions in 3D leads to an image such as the following:

Source: LorenzAttractor

If you were to place your finger at any point on the line in the above plot, and wanted to move say 1 position to the left (i.e. only 1 of your 3 variables in the equations change by 1 unit) it would take a lot longer than expected. You can’t jump over the white space between the lines; you have to trace your finger round the line until you reach your ‘destination’. Now, you can see that this is going to take a lot longer than you would have thought! This is a system of chaotic behaviour, and is commonly known as the butterfly effect. A small change in one variable can result in massive changes in a later state.

Now let’s take a look at 3 of the dynamical models used by the IRI as shown in my previous post:

NASA GMAO – Is the NASA Global Modelling and Assimilation Office, Goddard Earth Observing System Model (GEOS-5). It is constructed from an atmospheric model, catchment land surface model, and an ocean model, which are all coupled together by the use of the Earth System Modelling Framework.

UKMO – Is the UK Met Office General Circulation Dynamical Model. It is a coupled ocean-atmosphere GCM known as the GloSea (Global Seasonal) model. It is comprised of 3 models - an atmosphere, an ocean and a land surface model.

LDEO – Is the Lamont-Doherty Earth Observatory Model. It is an improved version of the original simple coupled ocean-atmosphere dynamical model by Zebiak and Cane, 1987.

Statistical-Dynamical Models

There also exist statistical-dynamical models, which, as is in the name, use a combination of both statistical and dynamical methods for different components. They usually take the statistical approach for parameters such as wind speed and direction, whilst using a dynamical method for modelling Newton’s Laws of Motion for energy diffusion.

Why all the models?

You may wonder why there are so many different approaches to modelling ENSO, and even within the different approaches, why so many different models exist. Well, as can be seen from looking at just a few of the different types of models above, there are numerous combinations of predictor variables are used, and all models incorporate different assumptions. Depending on the modellers’ knowledge, experience, and what their aim of modelling is, depends on what these variables and assumptions are. Hence, there exists such a variety of models. Models cannot be categorised as right or wrong, however they can be shown to be more or less predictive in comparison to other models. But even this may hold true for only a certain period of time.