Posts

Showing posts from 2022

Metastability and Eye Movements: A Dynamical‑Systems Interpretation

-
  1. Why metastability is a natural language for gaze behavior The idea that perception, attention, and neural activity evolve through transiently stable states has deep roots in cognitive science and neuroscience. Concepts such as attractors , basins , noise‑induced transitions , and escape times appear in work on perceptual switching, decision‑making, and neural population dynamics (Kelso, 1995; Rabinovich et al., 2008; Deco & Jirsa, 2012). Eye movements, especially the alternation between fixations and saccades ,   fit remarkably well into this metastable picture. The Freidlin–Wentzell theory of rare events (Freidlin & Wentzell, 2012) provides a rigorous mathematical language for these intuitions. 2. Fixations as metastable states  Consider the gaze position X t as the state of a stochastic dynamical system. During a fixation the  gaze remains confined to a small region, microsaccades and noise generate small fluctuations, and the gaze tends to return...
Post a Comment
Read more

Adding gaussian noise to break the tied values

-
Image
  In statistics, the " rank " represents the indexed position of the numerical values in ascending (or descending) order measured on a random sample. So, by ranking, we transform the data and get another random variable. Numerically, ranking a random sample of size n corresponds to making a random permutation of the integers from 1 to n. Importantly, since each of the n! possible permutations have the same probability of occurrence, we can calculate the means of these random variables which are invariant with respect to any monotonic transformation of the original data. There are many statistics based on ranks, among which: Wilcoxon-Mann-Whitney test (also known as Wilcoxon Rank Sum Test ), Wilcoxon signed-rank test, Kruskal-Wallis test, Friedman test, Spearman’s rank correlation coefficient. The rank-based tests are generally more robust compared to their homologous parametric tests, enable to deal easily with ordinal data, and the asymptotic approximations of the rank sta...
Post a Comment
Read more

Popular posts from this blog

Understanding Anaerobic Threshold (VT2) and VO2 Max in Endurance Training

-
Image
  Introduction: The Science Behind Ventilatory Thresholds VT1 vs. VT2: Defining Energy Transitions p (Aerobic Threshold) : The transition from lipid metawwwpp (fat-burning) to a mixed metabolism, where both carbohydrates and fats fuel muscle activity. V0T2 (Anaerobic Threshold) : The point where the  shifts to a predominantly carbohydrate-based wmetabolism , leading to rapid lactate waccumpulation and increased reliance on anaerobic energy pathways. www w In p VT1 and VT2 Wwww ewAgep & Sex Younger athletes generally exhibit higher VT2 values , while aging naturally reduces aerobicw capacity. Men tend to have a higher VO2 Max due to greater lung capacity and muscle mass, but women can achieve similar endurance levels through optimized training. Body Composition & Health Status High body fat percentage may reduce VT2 efficiency, as excess weight increases ...
Read more

Diagnosing Singular Fits in Linear Mixed Models: Practical Examples, Code, and Alternatives

-
Image
  Introduction This post is the practical follow‑up to When Linear Mixed Models Don’t Converge , where I discussed the theoretical reasons why convergence issues often arise in linear mixed models. Here, we move from theory to practice: simulated examples, diagnostic tools, interpretation of singular fits, and practical modeling alternatives. If you haven’t read the first part yet, you may want to start there for the conceptual foundations. Figure. Conceptual map showing how model complexity and available information interact Legend of Figure 1. Green = identifiable models; yellow = risk of singularity; red = non‑identifiable models. A Concrete Example Using R: When a Random-Slope Model Becomes Singular Let’s simulate a simple dataset with only 8 groups and a very small random‑slope variance. This is a classic scenario where the model converges numerically but is statistically non‑identifiable. ```r set.seed(123) n_groups <- 8 n_per_group <- 20 group <- factor(rep(1:n_gro...
Read more

Owen's Function: A Simple Solution to Complex Problems

-
Definition   If you are interested in statistics or probability, you may have encountered the Owen's function , a special function that arises in various applications involving bivariate normal distributions. Let’s explain what Owen's function is, how it is defined and notated, and why it is useful.  The Owen's function is defined as follows: for any real numbers h and a, Owen's function T(h,a) is given by T ( h , a ) = 1 2 π · ∫ 0 a exp ( - h 2 2 · ( 1 ...
Read more