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Concept: Probability theory


Emotions are often felt in the body, and somatosensory feedback has been proposed to trigger conscious emotional experiences. Here we reveal maps of bodily sensations associated with different emotions using a unique topographical self-report method. In five experiments, participants (n = 701) were shown two silhouettes of bodies alongside emotional words, stories, movies, or facial expressions. They were asked to color the bodily regions whose activity they felt increasing or decreasing while viewing each stimulus. Different emotions were consistently associated with statistically separable bodily sensation maps across experiments. These maps were concordant across West European and East Asian samples. Statistical classifiers distinguished emotion-specific activation maps accurately, confirming independence of topographies across emotions. We propose that emotions are represented in the somatosensory system as culturally universal categorical somatotopic maps. Perception of these emotion-triggered bodily changes may play a key role in generating consciously felt emotions.

Concepts: Statistics, Sense, Mind, Consciousness, Probability theory, Qualia, Topography, Body


Politicians world-wide frequently promise a better life for their citizens. We find that the probability that a country will increase its per capita GDP (gdp) rank within a decade follows an exponential distribution with decay constant λ = 0.12. We use the Corruption Perceptions Index (CPI) and the Global Competitiveness Index (GCI) and find that the distribution of change in CPI (GCI) rank follows exponential functions with approximately the same exponent as λ, suggesting that the dynamics of gdp, CPI, and GCI may share the same origin. Using the GCI, we develop a new measure, which we call relative competitiveness, to evaluate an economy’s competitiveness relative to its gdp. For all European and EU countries during the 2008-2011 economic downturn we find that the drop in gdp in more competitve countries relative to gdp was substantially smaller than in relatively less competitive countries, which is valuable information for policymakers.

Concepts: European Union, Radioactive decay, Probability theory, Derivative, Exponential growth, Exponential function, Competitiveness, Global Competitiveness Report


Scoring goals in a soccer match can be interpreted as a stochastic process. In the most simple description of a soccer match one assumes that scoring goals follows from independent rate processes of both teams. This would imply simple Poissonian and Markovian behavior. Deviations from this behavior would imply that the previous course of the match has an impact on the present match behavior. Here a general framework for the identification of deviations from this behavior is presented. For this endeavor it is essential to formulate an a priori estimate of the expected number of goals per team in a specific match. This can be done based on our previous work on the estimation of team strengths. Furthermore, the well-known general increase of the number of the goals in the course of a soccer match has to be removed by appropriate normalization. In general, three different types of deviations from a simple rate process can exist. First, the goal rate may depend on the exact time of the previous goals. Second, it may be influenced by the time passed since the previous goal and, third, it may reflect the present score. We show that the Poissonian scenario is fulfilled quite well for the German Bundesliga. However, a detailed analysis reveals significant deviations for the second and third aspect. Dramatic effects are observed if the away team leads by one or two goals in the final part of the match. This analysis allows one to identify generic features about soccer matches and to learn about the hidden complexities behind scoring goals. Among others the reason for the fact that the number of draws is larger than statistically expected can be identified.

Concepts: Time, Scientific method, Statistics, Mathematics, Estimator, Probability theory, Probability, Estimation


We often make decisions with uncertain consequences. The outcomes of the choices we make are usually not perfectly predictable but probabilistic, and the probabilities can be known or unknown. Probability judgments, i.e., the assessment of unknown probabilities, can be influenced by evoked emotional states. This suggests that also the weighting of known probabilities in decision making under risk might be influenced by incidental emotions, i.e., emotions unrelated to the judgments and decisions at issue. Probability weighting describes the transformation of probabilities into subjective decision weights for outcomes and is one of the central components of cumulative prospect theory (CPT) that determine risk attitudes. We hypothesized that music-evoked emotions would modulate risk attitudes in the gain domain and in particular probability weighting. Our experiment featured a within-subject design consisting of four conditions in separate sessions. In each condition, the 41 participants listened to a different kind of music-happy, sad, or no music, or sequences of random tones-and performed a repeated pairwise lottery choice task. We found that participants chose the riskier lotteries significantly more often in the “happy” than in the “sad” and “random tones” conditions. Via structural regressions based on CPT, we found that the observed changes in participants' choices can be attributed to changes in the elevation parameter of the probability weighting function: in the “happy” condition, participants showed significantly higher decision weights associated with the larger payoffs than in the “sad” and “random tones” conditions. Moreover, elevation correlated positively with self-reported music-evoked happiness. Thus, our experimental results provide evidence in favor of a causal effect of incidental happiness on risk attitudes that can be explained by changes in probability weighting.

Concepts: Decision making, Risk, Decision theory, Probability theory, Probability, Theory, Choice, Prospect theory


The aim of this study was to test the possibility of the ultra-short-term lnRMSSD (measured in 1-min post-1-min stabilization period) to detect training induced adaptations in futsal players. Twenty-four elite futsal players underwent HRV assessments pre- and post-three or four weeks preseason training. From the 10-min HRV recording period, lnRMSSD was analyzed in the following time segments: 1) from 0-5 min (i.e., stabilization period); 2) from 0-1 min; 1-2 min; 2-3 min; 3-4 min; 4-5 min and; 3) from 5-10 min (i.e., criterion period). The lnRMSSD was almost certainly higher (100/00/00) using the magnitude-based inference in all periods at the post- moment. The correlation between changes in ultra-short-term lnRMSSD (i.e., 0-1 min; 1-2 min; 2-3 min; 3-4 min; 4-5 min) and lnRMSSDCriterion ranged between 0.45-0.75, with the highest value (p = 0.75; 90% CI: 0.55 - 0.85) found between ultra-short-term lnRMDSSD at 1-2 min and lnRMSSDCriterion. In conclusion, lnRMSSD determined in a short period of 1-min is sensitive to training induced changes in futsal players (based on the very large correlation to the criterion measure), and can be used to track cardiac autonomic adaptations. Key pointsThe ultra-short-term (1 min) natural log of the root-mean-square difference of successive normal RR intervals (lnRMSSD) is sensitive to training effects in futsal playersThe ultra-short-term lnRMSSD may simplify the assessment of the cardiac autonomic changes in the field compared to the traditional and lengthier (10 min duration) analysisCoaches are encouraged to implement the ultra-short-term heart rate variability in their routines to monitor team sports athletes.

Concepts: Time, Cardiology, Heart, Measurement, Assessment, Probability theory, The Criterion, Heart rate variability


Zero-determinant strategies are a new class of probabilistic and conditional strategies that are able to unilaterally set the expected payoff of an opponent in iterated plays of the Prisoner’s Dilemma irrespective of the opponent’s strategy (coercive strategies), or else to set the ratio between the player’s and their opponent’s expected payoff (extortionate strategies). Here we show that zero-determinant strategies are at most weakly dominant, are not evolutionarily stable, and will instead evolve into less coercive strategies. We show that zero-determinant strategies with an informational advantage over other players that allows them to recognize each other can be evolutionarily stable (and able to exploit other players). However, such an advantage is bound to be short-lived as opposing strategies evolve to counteract the recognition.

Concepts: Game theory, Evolution, Probability theory, Strategy, Players, Prisoner's dilemma, Nash equilibrium, Player


Tornadoes cause loss of life and damage to property each year in the United States and around the world. The largest impacts come from ‘outbreaks’ consisting of multiple tornadoes closely spaced in time. Here we find an upward trend in the annual mean number of tornadoes per US tornado outbreak for the period 1954-2014. Moreover, the variance of this quantity is increasing more than four times as fast as the mean. The mean and variance of the number of tornadoes per outbreak vary according to Taylor’s power law of fluctuation scaling (TL), with parameters that are consistent with multiplicative growth. Tornado-related atmospheric proxies show similar power-law scaling and multiplicative growth. Path-length-integrated tornado outbreak intensity also follows TL, but with parameters consistent with sampling variability. The observed TL power-law scaling of outbreak severity means that extreme outbreaks are more frequent than would be expected if mean and variance were independent or linearly related.

Concepts: Time, Arithmetic mean, Mean, Probability theory, Standard deviation, Cauchy distribution, Tornado, Means


Given small sample sizes, loss of animals in preclinical experiments can dramatically alter results. However, effects of attrition on distortion of results are unknown. We used a simulation study to analyze the effects of random and biased attrition. As expected, random loss of samples decreased statistical power, but biased removal, including that of outliers, dramatically increased probability of false positive results. Next, we performed a meta-analysis of animal reporting and attrition in stroke and cancer. Most papers did not adequately report attrition, and extrapolating from the results of the simulation data, we suggest that their effect sizes were likely overestimated.

Concepts: Statistics, Sample size, Statistical significance, Type I and type II errors, Experiment, Probability theory, Effect size, Theory


How rare are magic squares? So far, the exact number of magic squares of order n is only known for n ≤ 5. For larger squares, we need statistical approaches for estimating the number. For this purpose, we formulated the problem as a combinatorial optimization problem and applied the Multicanonical Monte Carlo method (MMC), which has been developed in the field of computational statistical physics. Among all the possible arrangements of the numbers 1; 2, …, n2 in an n × n square, the probability of finding a magic square decreases faster than the exponential of n. We estimated the number of magic squares for n ≤ 30. The number of magic squares for n = 30 was estimated to be 6.56(29) × 102056 and the corresponding probability is as small as 10-212. Thus the MMC is effective for counting very rare configurations.

Concepts: Statistics, Mathematics, Probability theory, Monte Carlo, Monte Carlo method, Optimization, Applied mathematics, Combinatorial optimization


The purpose of this study was to assess the effect of resistance training cessation on strength performance through a meta-analysis. Seven databases were searched from which 103 of 284 potential studies met inclusion criteria. Training status, sex, age, and the duration of training cessation were used as moderators. Standardized mean difference (SMD) in muscular performance was calculated and weighted by the inverse of variance to calculate an overall effect and its 95% confidence interval (CI). Results indicated a detrimental effect of resistance training cessation on all components of muscular performance: [submaximal strength; SMD (95% CI) = -0.62 (-0.80 to -0.45), P < 0.01], [maximal force; SMD (95% CI) = -0.46 (-0.54 to -0.37), P < 0.01], [maximal power; SMD (95% CI) = -0.20 (-0.28 to -0.13), P < 0.01]. A dose-response relationship between the amplitude of SMD and the duration of training cessation was identified. The effect of resistance training cessation was found to be larger in older people (> 65 years old). The effect was also larger in inactive people for maximal force and maximal power when compared with recreational athletes. Resistance training cessation decreases all components of muscular strength. The magnitude of the effect differs according to training status, age or the duration of training cessation.

Concepts: Old age, Probability theory, Strength training, Normal distribution, Effect size, Standard deviation, Isometric exercise, Dose-response relationship