On Quantization Errors in Approximate and Sample Entropy

Approximate and sample entropies are acclaimed tools for quantifying the regularity and unpredictability of time series. This paper analyses the causes of their inconsistencies. It is shown that the major problem is a coarse quantization of matching probabilities, causing a large error between their estimated and true values. Error distribution is symmetric, so in sample entropy, where matching probabilities are directly summed, errors cancel each other. In approximate entropy, errors are accumulating, as sums involve logarithms of matching probabilities. Increasing the time series length increases the number of quantization levels, and errors in entropy disappear both in approximate and in sample entropies. The distribution of time series also affects the errors. If it is asymmetric, the matching probabilities are asymmetric as well, so the matching probability errors cease to be mutually canceled and cause a persistent entropy error. Despite the accepted opinion, the influence of self-matching is marginal as it just shifts the error distribution along the error axis by the matching probability quant. Artificial lengthening the time series by interpolation, on the other hand, induces large error as interpolated samples are statistically dependent and destroy the level of unpredictability that is inherent to the original signal.

Hydrological response of a peri-urban catchment exploiting conventional and unconventional rainfall observations: the case study of Lambro catchment

Commercial Microwave Links (CMLs) can be used as opportunistic and unconventional rainfall sensors by converting the received signal level into path-averaged rainfall intensity. Since in meteorology and hydrology the reliable reconstruction of the rainfall spatial distribution is still a challenging issue, there is a wide-spread interest in integrating the precipitation estimates gathered by the ubiquitous CMLs with the conventional rainfall sensors, i.e. rain gauges (RGs) and weather radars. Here we investigate the potential of a dense CML network, for the estimation of river discharges via a semi-distributed hydrological model. The analysis is conducted on Lambro, a peri-urban catchment located in northern Italy and covered by 50 links. A two-level comparison is made between CML- and RG-based outcomes, relying on 12 storm/flood events. First, rainfall data are spatially interpolated and assessed in a set of significant points of the catchment area. Rainfall depth values obtained from CMLs are definitively comparable with direct RG measurements, except for the spells of persistent light rain, due to limited sensitivity of CMLs caused by the coarse quantization step of raw power data. Moreover, it is showed that, when changing the type of rainfall input, a new calibration of model parameters is required. In fact, after the re-calibration of model parameters, CML-driven outputs performances are comparable with RG-driven ones, confirming that the exploitation of a CML network may lead to benefit in hydrological modelling.

Genetic Algorithms to Maximize the Relevant Mutual Information in Communication Receivers

The preservation of relevant mutual information under compression is the fundamental challenge of the information bottleneck method. It has many applications in machine learning and in communications. The recent literature describes successful applications of this concept in quantized detection and channel decoding schemes. The focal idea is to build receiver algorithms intended to preserve the maximum possible amount of relevant information, despite very coarse quantization. The existent literature shows that the resulting quantized receiver algorithms can achieve performance very close to that of conventional high-precision systems. Moreover, all demanding signal processing operations get replaced with lookup operations in the considered system design. In this paper, we develop the idea of maximizing the preserved relevant information in communication receivers further by considering parametrized systems. Such systems can help overcome the need of lookup tables in cases where their huge sizes make them impractical. We propose to apply genetic algorithms which are inspired from the natural evolution of the species for the problem of parameter optimization. We exemplarily investigate receiver-sided channel output quantization and demodulation to illustrate the notable performance and the flexibility of the proposed concept.