(5h) Fisher Information Theory for Environmental Sustainability

World commission on Environment and Development defines sustainability as ?development that meets the needs of the present without compromising the ability of future generations to meet their own needs'. The concept of Sustainability requires study of complex integrated systems such as environment, economy, industry, technology, ecology, society, law etc. Usually sustainability is studied through various indices which represent the key components of the system. However there is no general underlying mathematical theory embodying concepts from the different components of the general system. The conceptual difficulties arise from the fact that the area of sustainability cannot be successfully investigated within the confines of a single discipline, which leads to severe operational difficulties in cross-disciplinary approaches. The key to the study of sustainability lies in building a unified theory involving all the relevant disciplines.

We have developed such a unified theory for sustainability based on Fisher information theory. The sound logic behind using information theory is that since there is no general underlying similarity between the information or data from different disciplines (such as social and economic data) we need to device a mechanism where any type of data can be converted into information regardless of disciplinary origin, which is precisely what information theory addresses. Information therefore serves as common interdisciplinary bridge. Furthermore, Fisher information is a very fundamental quantity from which many other known laws of nature can be derived.

The Fisher theory can be used to study the sustainability of the system (e.g. ocean) regime and detect regime changes (e.g. interannual, interdecadal changes) inflicted to the system, both natural (e.g. cyclones) and anthropogenic (e.g. industrial green house emissions). As per the theory the state of a system is defined by its state variables i.e. if one knows the values of the state variables one knows the state of the system. The behavior of these variables with time determines the stability or sustainability of the system. The first and the most crucial step is the identification of the state variables representing the system (in most case these variables would cut across various disciplines). The variables are then mapped into n-dimensional (for ?n' variables) state-space trajectory which needs to be tracked with time. The calculation of fisher information involves binning of all the state points into an n-dimensional state grid. The size and shape of the n-dimensional grid is determined by the uncertainties associated with the measurement of the n-variables. In order to calculate the Fisher information it is necessary to determine a probability density function (PDF), which has to be extracted from the state-space mapping, for the system in question. The general idea is that the more time a system spends in a specific state, the more likely one is to find it in that state when sampling. When normalized over the entire space of possibilities, a PDF over the states of the system results.

Since the Fisher information contains the first derivative of the PDF it tracks the fluctuations in the slope of the PDF and hence it is a local measure as against other information such as Shannon information which is a global measure. Since fisher information tracks local fluctuations it is ideal for regime change studies.

This theory is employed to study the Pacific Ocean regime shifts that have occurred in the last century. Concrete evidence of climate-ocean variability and the ecosystem response in northern pacific are analyzed through fisher information. The study involves 65 environmental and biological variables of Pacific Ocean.