dimensions. Policy optimization introduces the complex interaction between observing
the value of a suboptimal policy, and finding better policies.
There is a vast array of applications in the analysis of energy systems, economics and
policy which require the tools of statistics and machine learning to infer relationships
from observational data. We may need to understand the performance of different
molecular compounds in terms of converting solar energy, the status of different
components in the grid for a utility, the response of households to changes in electricity
prices, or the energy from sun or wind. Statistics arises within algorithms, where we
may have to estimate the value of being in a state and following a policy.
Statistical challenges come in many forms. For many spatial applications, we
need to estimate fine-grained behavior from coarse-grained observations. For example,
how can we predict the anticipated energy production from a particular wind farm using
observations from weather stations? We may have to estimate high-dimensional
functions (such as the price of energy at a node as a function of supplies, demands and
weather around the network), in some cases with relatively little data (the big p, small n
problem). We often have to estimate functions with complex structures, such as the
amount of energy to put into storage from wind as a function of wind, demand, prices
and their histories. Machine learning researchers use a variety of statistical learning
methods, e.g., support vector machines (SVMs) and boosting, together with kernel
transformation methods, but this is an active research area with good opportunities for
mathematical contributions. A popular area of research in statistics is in the general area
known as nonparametric statistics and locally polynomial regression. Such methods
typically approximate functions using a weighted sum of observations, where the weights
are given by kernel functions which put a higher weight on closer observations. Such
strategies suffer from the curse of dimensionality, since the likelihood of having a
reasonable number of observations within close proximity drops very quickly as the
dimensionality of the observation space grows.
There are many problems where observations are expensive, and we have to
collect information efficiently, an area that falls under names such as active learning and
optimal learning. Given limited resources, when and where should we measure wind
velocities, ocean temperatures, or test the performance of an energy saving technology
in a building? The problem of finding optimal information collection policies is
computationally intractable, and as a result research is needed to test the efficiency and
accuracy of different approximations. Optimal learning policies need to be developed
that work well in the context of the characteristics of the problem (dimensionality of the
explanatory variables, nature of the belief structure).
A continuing problem in statistics is the vast plethora of models and statistical estimation
techniques, without a single, dominant method (see Hastie, et al, 2009). Scientists have
developed methods such as ensemble models and boosting to combine the best results
from different models. Energy scientists need robust methods to solve their learning
problems to avoid turning every statistical estimation problem into a research project.
Simulation of Complex Systems
Simulation is an important technology for studying energy and many other
systems. Here we discuss the relationship between the structure of a complex system
and its dynamics.
There are many aspects of this question that form whole research areas by
themselves. Spatial pattern formation in fluids, reaction-diffusion systems and similar