Hidden dynamics of soccer leagues: the predictive ‘power’ of partial standings

Objectives Soccer leagues reflect the partial standings of the teams involved after each round of competition. However, the ability of partial league standings to predict end-of-season position has largely been ignored. Here we analyze historical partial standings from English soccer to understand the mathematics underpinning league performance and evaluate the predictive ‘power’ of partial standings. Methods Match data (1995-2017) from the four senior English leagues was analyzed, together with random match scores generated for hypothetical leagues of equivalent size. For each season the partial standings were computed and Kendall’s normalized tau-distance and Spearman r-values determined. Best-fit power-law and logarithmic functions were applied to the respective tau-distance and Spearman curves, with the ‘goodness-of-fit’ assessed using the R2 value. The predictive ability of the partial standings was evaluated by computing the transition probabilities between the standings at rounds 10, 20 and 30 and the final end-of-season standings for the 22 seasons. The impact of reordering match fixtures was also evaluated. Results All four English leagues behaved similarly, irrespective of the teams involved, with the tau-distance conforming closely to a power law (R2>0.80) and the Spearman r-value obeying a logarithmic function (R2>0.87). The randomized leagues also conformed to a power-law, but had a different shape. In the English leagues, team position relative to end-of-season standing became ‘fixed’ much earlier in the season than was the case with the randomized leagues. In the Premier League, 76.9% of the variance in the final standings was explained by round-10, 87.0% by round-20, and 93.9% by round-30. Reordering of match fixtures appeared to alter the shape of the tau-distance curves. Conclusions All soccer leagues appear to conform to mathematical laws, which constrain the league standings as the season progresses. This means that partial standings can be used to predict end-of-season league position with reasonable accuracy

Thyrotrophin-releasing hormone, vasoactive intestinal peptide, prolactin-releasing peptide and dopamine regulation of prolactin secretion by different lactotroph morphological subtypes in the rat.

In the male rat anterior pituitary, three morphological subtypes of cells secreting primarily prolactin (PRL) (lactotrophs) have been described. Type I contain predominantly large irregularly shaped granules, whereas type II and type III lactotrophs contain smaller spherical granules. We have previously shown that oestradiol and testosterone exert a rapid stimulatory effect selectively on type II lactotrophs but it is not known how the lactotroph subtypes respond to peptide secretagogues. We have therefore examined which cell subtype(s) release PRL in response to vasoactive intestinal peptide (VIP), thyrotrophin-releasing hormone (TRH) and prolactin-releasing peptide (PrRP-31). Pituitary segments were incubated in medium containing tannic acid (to capture exocytosis of secretory granules), either alone or with secretagogue peptide. VIP (1-10 nM), TRH (10 nM) and PrRP-31 (10 nM) all caused a significant increase (P < 0.05) in the amount of PRL granule exocytosis from type II and III lactotrophs, but had no effect on PRL exocytosis from type I. Dopamine (100 nM) inhibited basal exocytosis of immunoreactive (ir)-PRL from type I, II and III lactotrophs and PrRP-31-stimulated ir-PRL granule exocytosis from II and III lactotrophs. Treatment of lactating female rats with the dopamine D(2) receptor antagonist sulpiride (40 microg/kg) produced a significant increase (P < 0.05) in PRL granule exocytosis from type I and type III lactotrophs and a significant increase (P < 0.05) in the proportion of type I and II cells undergoing exocytosis of PRL. In conclusion, VIP, TRH and PrRP-31 selectively stimulate exocytosis from type II and III lactotrophs in the male rat, whereas all three lactotroph types are sensitive to dopamine inhibition of exocytosis in male and female rats

Generating parity check equations for bounded-distance iterative erasure decoding

iterative erasure decodin