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FISHER INFORMATION AND ECOSYSTEM REGIME CHANGES
Citation:
CABEZAS, H. AND C. PAWLOWSKI. FISHER INFORMATION AND ECOSYSTEM REGIME CHANGES. Presented at CITSA 2005 Conference Session on Fisher Information Research in Science & Technology, Orlando, FL, July 14 - 17, 2005.
Impact/Purpose:
Presented paper
Description:
Following Fisher’s work, we propose two different expressions for the Fisher Information along with Shannon Information as a means of detecting and assessing shifts between alternative ecosystem regimes. Regime shifts are a consequence of bifurcations in the dynamics of an ecosystem and are important because much of environmental protection is really an effort to preserve through time particular ecological regimes. The transformation of a lake from oligotrophic (clear) to eutrophic (cloudy) is an example of a regime change. As another example, North Africa is thought to have changed regimes between a wet and forested area to the present Sahara desert. Such regime changes often occur without obvious warning and can have catastrophic consequences for humans and other species. For the case of the measurement of one variable x in an effort to evaluate a parameter θ, the Fisher Information is well defined in the literature. The expression for the Fisher Information is obtained by letting θ=so where so is a particular state of the system, and by letting x=s where s is an arbitrary state of the system. The probability density function p(s|so) is then the likelihood of observing a particular state of the system so while observing arbitrary states s. A state of the system is defined by specifying, within observational error, a value for each of the state variables of the system. The resulting Fisher Information expression is a measure of order, and we call it the Order Fisher Information. This is a measure of the Fisher Information that can be obtained from observing the patterns in the state variables of a dynamic system. High order leads to high Order Fisher Information and low order leads to low Order Fisher Information. Order in this case refers to the fact that orderly dynamic systems seem about the same with repeated observations. For example, a tree is a complex dynamic system. But it is still a tree from day to day, year to year. Hence, it possesses dynamic order because its measurable properties vary more or less regularly, not wildly from one observation to the next. This kind of order is a typical characteristic of well functioning biological systems. However, we hypothesize that, in general, different dynamic regimes of the same biological system have different degrees of dynamic order and, therefore, different Order Fisher Information. Therefore, the loss or complete loss of dynamic order in a biological system may signify a regime change or in the latter case a malfunction of the present regime to the extent that biological systems are orderly systems. A second expression for the Fisher Information is obtained by letting θ=t where t is time, and by letting x= s as before. It measures the information obtainable from observing changes with time in the distribution of states of the system. The probability density function p(s|t) is then the likelihood of observing a particular time t while observing arbitrary states s. Since there is a one to one correspondence between states of the system and time along the system trajectory in phase space, so is replaced by time t to obtain p(s|t) from p(s|so). We call this the Time-Shift Fisher Information. Observable states of well functioning biological systems, i.e., orderly dynamic systems, are not randomly distributed. Rather, a system shows a definite preference for particular states. This preference gives rise to a non-random distribution of states of the system. If the distribution of states of the system does not change with time, then the Time-Shift Fisher Information is zero; conversely, it is non-zero if the distribution of states of the system is changing with time. A logical consequence of the previous hypothesis on dynamic regime order is that different dynamic regimes in general have different distributions of states of the system. Hence, the Time-Shift Fisher Information will be zero for a system staying in any particular dynamic regime through time, and non-zero during the transition between dynamic regimes. The Time-Shift Fisher Information is, therefore, a very sensitive detector of regime change. Following the work of Claude Shannon, we have constructed an expression for the Shannon Information that is a measure of the observable entropy in the dynamics of the system. Entropy generally correlates with disorder. A logical consequence of the already discussed hypothesis is that the order and entropy of different dynamic regimes are different, and the Shannon Information is, therefore, also dissimilar for different dynamic regimes. Hence, the Shannon Information changes as a system undergoes a change in dynamic regime. Shannon Information is, however, a lower order and less sensitive measure of information than Fisher Information. It could, therefore, be more likely than Fisher Information to remain constant under dynamic regime changes. Detecting and assessing dynamic regime changes in ecosystems under actual field conditions is complicated by measurement error and noise in the data, and by the simple lack of understanding of the underlying system dynamics. For example, one does not in general have a complete model for real systems that is capable of giving a reasonably accurate picture of the behavior space of the system. Rather, all that is frequently available is a dataset for the system over a limited time-period. To effectively work under these conditions, we propose the use of the three aforementioned information measures jointly to detect and assess regime changes. In this fashion, some protection is afforded against the vagaries of the data giving rise to false regime change signals. We have obtained meaningful results from Fisher Information and the Shannon Information calculations using model generated and field data for many different systems. Here, we illustrate the theory with examples from (1) a model for a lake that shifts from oligotrophic to eutrophic with phosphorus input, and from (2) field data taken for the Pacific Ocean around the Bering Strait.