Cílem práce bylo zkoumat vztah mezi zobecněnou entropií diskrétní náhodné veličiny (tzv. f-entropií, třídou funkcí, zahrnující řadu indexů používaných pří měření biodiverzity) a minimální Bayesovskou pravděpodobností chyby při odhadu hodnoty této náhodné veličiny. Zejména studovat těsnost jejich vztahu. Morales a Vajda [1] nedávno zavedli míru zvanou průměrná nepřesnost (average inaccuracy), která kvantifikuje těsnost vztahu mezi aposteriorní Bayesovskou pravděpodobností chyby a mocninnými entropiemi. Tato míra je definována jako normalizovaný průměrný rozdíl horní a dolní meze aposteriorní Bayesovské pravděpodobnosti chyby za dané entropie. Tuto míru je možno zobecnit na jakoukoli striktně konkávní f-entropii a použít ji k vyjádření těsnosti vztahu této f-entropie k aposteriorní Bayesovské pravděpodobnosti chyby. Získaný vztah je však většinou poměrně složitý a není možné snadno analyticky porovnat průměrné nepřesnosti různých f-entropií. Navrhujeme hladkou aproximaci dolní meze aposteriorní Bayesovské pravděpodobnosti chyby za dané f-entropie, jejíž použití zjednoduší formuli průměrné nepřesnosti. Ukázali jsme, že při použití této aproximace má kvadratická entropie nejtěsnější vztah k aposteriorní Bayesovské pravděpodobnosti chyby mezi f-entropiemi. Kvadratická entropie má těsnější vztah k Bayesovské pravděpodobnosti chyby (ve smyslu popsaném v článku) než Shannonova entropie a další funkce příslušící do třídy f-entropií, jako např. Emlenův index, Ferreriho index, Goodův index a další.
We deal with the relation between the generalized entropy (f-entropy, a family of functions that include several biodiversity measures) of a discrete random variable and the minimal probability of error (Bayes error) when the value of this random variable is estimated. Namely the tightness of their relation is studied. Morales and Vajda [1] recently introduced a measure called average inaccuracy that aims to quantify the tightness of the relation between the posterior Bayes error and the power entropies. It is defined as a standardized average difference between the upper and the lower bound for the posterior Bayes error under given entropy. Their concept can be generalized to any strictly concave f-entropy and used to evaluate its relation to the Bayes probability of error. However, due to a complex form of the formula of the average inaccuracy, it is difficult to compare the average inaccuracies of most f-entropies analytically. We propose a smooth approximation of the lower bound for the posterior Bayes error under given f-entropy that simplifies the formula of the average inaccuracy. We show that under this approximation, the quadratic entropy has the tightest relation to the posterior Bayes error among f-entropies. The quadratic entropy has the tightest relation to the posterior Bayes error (in the sense described in this paper) than the Shannon’s entropy and other functions that belong to the family of f-entropies, like Emlen’s index, Ferreri’s index and Good’s index.
OBJECTIVES: General information-theoretic concepts such as f-divergence, f-information and f-entropy are applied to the genetic models where genes are characterized by randomly distributed alleles. The paper thus presents an information-theoretic background for measuring genetic distances between populations, genetic information in various observations on individuals about their alleles and, finally, genetic diversities in various populations. METHODS: Genetic distances were derived as divergences between frequencies of alleles representing a gene in two different populations. Genetic information was derived as a measure of statistical association between the observations taken on individuals and the alleles of these individuals. Genetic diversities were derived from divergences and information. RESULTS: The concept of genetic f-information introduced in the paper seems to be new. We show that the measures of genetic distance and diversity used in the previous literature are special cases of the genetic f-divergence and f-diversity introduced in the paper and illustrated by examples. We also display intimate connections between the genetic f-information and the genetic f-divergence on one side and genetic f-diversity on the other side. The examples at the same time also illustrate practical computations and applications of the important concepts of quantitative genetics introduced in the paper. CONCLUSIONS: We discussed a general class of f- divergence measures that are suitable measures of genetic distance between populations characterized by concrete frequencies of alleles. We have shown that a wide class of genetic information, called f-information, can be obtained from f-divergences and that a wide class of measures of genetic diversity, called f-diversities, can be obtained from the f-divergences and f-information.
BACKGROUND: Mood regulation is a complex and poorly understood process. In this study, we aimed to analyze the underlying dynamics of mood regulation in unaffected first degree relatives of patients diagnosed with bipolar disorder using time-series analysis. METHODS: We recruited 30 unaffected first-degree relatives of bipolar disorder patients. Participants rated their mood, anxiety and energy levels using a paper-based visual analog scale; they recorded their sleep and life events as well. Participants provided information on these variables over a three month period, twice per day. We compared their data using Box-Jenkins time series analysis with data from 30 healthy controls (HC) and 30 euthymic bipolar patients (BD) to obtain information on the autocorrelation and cross-correlation of the series, and calculated entropy for mood, anxiety and energy series. RESULTS: We analyzed 14,980 data points: 5200 in the healthy control group; 4970 in the bipolar group and 4810 in the unaffected relatives group. There were no significant differences between groups in terms of age, sex or education levels. Using Kolmogorov-Smirnov test, we found that individual measures were normally distributed in the whole sample (D = 0.23, p > 0.1). Autocorrelation functions for mood in all groups are governed by the ARIMA (1,1,0) model, which means that current values in the series are related to one previous point only. In terms of entropy for the mood series, unaffected relatives and bipolar patients showed lower values [mean (SD) : 1.028 ± 0.679; 1.042 ± 0.680], respectively, compared to healthy controls [(1.476 ± 0.33); F (2,74) = 4.39, p < 0.01]. The same case was seen in the energy series, with lower values in the unaffected relatives and bipolar patient groups [mean (SD) : 1.644 ± 0.566; 1.511 ± 0.879], respectively, compared to healthy controls [2.230 ± 0.531; F(2, 75) = 7.89, p < 0.001]. LIMITATIONS: Low resolution for the visual analog scale. CONCLUSIONS: Using nonlinear analyses, we found that the underlying structure of mood regulation in unaffected relatives is undistinguishable from the one found in bipolar patients. Compared to healthy controls, both bipolar patients and their unaffected relatives showed lower entropy levels, which is in keeping with a more rigid system, not as flexible to cope with the demands of a changing environment.
- MeSH
- Affect * MeSH
- Bipolar Disorder diagnosis psychology MeSH
- Cyclothymic Disorder psychology MeSH
- Adult MeSH
- Middle Aged MeSH
- Humans MeSH
- Nonlinear Dynamics MeSH
- Self-Control psychology MeSH
- Case-Control Studies MeSH
- Anxiety psychology MeSH
- Visual Analog Scale MeSH
- Check Tag
- Adult MeSH
- Middle Aged MeSH
- Humans MeSH
- Male MeSH
- Female MeSH
- Publication type
- Journal Article MeSH
- Research Support, Non-U.S. Gov't MeSH
BACKGROUND: Analysis of the heart activity is one of the important areas of research in biomedical science and engineering. For this purpose, scientists analyze the activity of the heart in various conditions. Since the brain controls the heart's activity, a relationship should exist among their activities. OBJECTIVE: In this research, for the first time the coupling between heart and brain activities was analyzed by information-based analysis. METHODS: Considering Shannon entropy as the indicator of the information of a system, we recorded electroencephalogram (EEG) and electrocardiogram (ECG) signals of 13 participants (7 M, 6 F, 18-22 years old) in different external stimulations (using pineapple, banana, vanilla, and lemon flavors as olfactory stimuli) and evaluated how the information of EEG signals and R-R time series (as heart rate variability (HRV)) are linked. RESULTS: The results indicate that the changes in the information of the R-R time series and EEG signals are strongly correlated (ρ=-0.9566). CONCLUSION: We conclude that heart and brain activities are related.
- MeSH
- Adult MeSH
- Electroencephalography * MeSH
- Electrocardiography MeSH
- Humans MeSH
- Adolescent MeSH
- Young Adult MeSH
- Brain MeSH
- Heart * MeSH
- Heart Rate physiology MeSH
- Check Tag
- Adult MeSH
- Humans MeSH
- Adolescent MeSH
- Young Adult MeSH
- Publication type
- Journal Article MeSH
