Spatiotemporal Information Measures

In previous work we have discussed and analyzed how Fisher Information can be used to quantify the performance of an electronic nose (Sanchez-Montands and Pearce 2001, Pearce and Sdnchez-Montanes 2003). Basically, the Fisher Information Matrix (FIM) F is a square and symmetric matrix of i x i components, where s is the number of individual compounds whose concentration we are interested to estimate. In order to calculate F we should first calculate the individual FIMs for each sensor f. 

The usefulness of Fisher Information is given by the important property that the best square error across all unbiased techniques that use the noisy array responses to estimate the stimulus is (see Sdnchez-Montands and Pearce 2001 for discussion) 

Importantly, the Fisher Information matrix F is closely related to the discrimination ability of the system, which is why we consider it in this context. For instance, it can be demonstrated that in a two-alternative forced choice discrimination between two stimuli (i.e. the system has to determine which of two possible complex odours ci and C2 is being presented), the optimal probability of error P(e) that can be achieved using linear sensors is P(e) = 0.5-[l-erf( 0.5 )] with k = Vi- 5c - F -5c and 5c = C2-C1. 

In our previous work we have discussed how to calculate in practice this quantity when the temporal patterns of the responses of the individual sensors are not taken into account (corresponding to the first mechanism for discrimination identified in Section 1). Here we extend and calculate the Fisher Information for the spatio-temporal case which includes the role of time in the responses. The first step is to model the noise in the sensors, which will determine p (Yj I c). 

Let us define Yj. as the noisy temporal response (time series) of sensor j to stimulus 

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