A numerical study of the Kullback-Leibler distance in functional magnetic resonance imaging

Abstract

The Kullback-Leibler distance (or relative entropy) is applied in the analysis of functional magnetic resonance (fMRI) data series. Our study is designed for event-related (ER) experiments, where a brief stimulus is presented and a long period of rest is followed. In particular, this relative entropy is used as a measure of the "distance" between the probability distributions p 1 and p 2 of the signal levels related to stimulus and non-stimulus. In order to avoid undesirable divergences of the Kullback-Leibler distance, a small positive parameter δ is introduced in the definition of the probability functions in such a way that it does not bias the comparison between both distributions. Numerical simulations are performed so as to determine the probability densities of the mean Kullback-Leibler distance D (throughout the N epochs of the whole experiment). For small values of N (N < 30), such probability densities f (D) are found to be fitted very well by Gamma distributions (χ 2 < 0.0009). The sensitivity and specificity of the method are evaluated by construction of the receiver operating characteristic (ROC) curves for some values of signal-to-noise ratio (SNR). The functional maps corresponding to real data series from an asymptomatic volunteer submitted to an ER motor stimulus is obtained by using the proposed technique. The maps present activation in primary and secondary motor brain areas. Both simulated and real data analyses indicate that the relative entropy can be useful for fMRI analysis in the information measure scenario.

Generalized relative entropy in functional magnetic resonance imaging

“…However, we should also mention that the same kind of analysis has been already performed with the usual Kullback-Leibler entropy D (which corresponds to the case q = 1 and so is subject to divergence) [12]. But in this study, the probabilities of the signal levels were subtly redefined as p ij = (n ij + δ)/(n i + Lδ), 0 < δ ≤ 1, in order to prevent the occurrence of divergent values of D. In fact, this latter definition ensures that p ij is non-null and tends to recover the usual meaning that p ij is the relative frequency of points within window W i and level I j for small values of δ (that is, p ij → n ij /n i as δ → 0 + ).…”

“…(1) might tend to zero if p 1j → 0 (p 2j = 0) and might diverge if p 2j → 0 (p 1j = 0). In a previous study [12], the undesirable divergence of Kullback-Leibler distance (D) was prevented by an appropriate definition of the set of probabilities p ij . Within a nonextensive thermostatistical formalism, the generalized Kullback-Leibler distance D q has been derived as the q-average of the generalized change of information ∆σ q,j = k [14], that is,…”

“…Here we should note that results corresponding to q ≥ 1 were calculated by using the same strategy as adopted in Ref. [12] (with δ = 10 −5 ) so as to prevent any divergence. Figs.…”

“…Recently, the relative entropy or Kullback-Leibler distance D has also been proven useful in the analysis of fMRI data series [12]. Within the context of information theory, we now propose to analyze time series data by the use of the generalized Kullback-Leibler distance D q , where q is the Tsallis parameter [13,14].…”

Structural changes in data communication in wireless sensor networks

“…The K L diverges for (ξ) = 0 and (ξ) = 0, as defined in equation (1). In order to avoid the division by zero, a small positive constant δ = 0 001 was added to each bin, and then the histogram is normalized to add 1 [7,24]. The original histogram is used to compute the Hellinger distance once it does not impose the positivity restriction on the probabilities.…”