In ggcostoya/tpcurves2: T-P Curves

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The influence of an environmental factor into an individual's fitness is defined by a curve. Since, empirically, fitness can only be aproximated through an individual's performance by measuring a performance-indicator trait (henceforth called P, for short), these curves are often called performance curves. Furthermore, if temperature (T) is the environmental factor of interest, these are called thermal-performance curves (TPC).

A TPC has a carachteristic shape in statistical terms; it has a unique maximum at a given P and T value descreasing at both sides of this point to a certain width. Lynch & Gabriel (1987, 1992) where the first to point out that this shape could be statistically analyzed using a Gaussian function similar to:

$$P = ae^{-0.5[(T-b)/{c}]^2} $$ A Gaussian function can account for the variabilily in the position of both the peak and the width of the TPC accurately, nonetheless, realistic TPCs are often skewed in nature since one of the temperature extremes might be more or less harmful than the other (Bulté & Blouin-Demers 2006). Therefore, instead of using a Gaussian function exclusively, many studies perform a model selection procedure (through the comparison of information criterion scores) to determine which function best defines the relationship between P and T, using expressions that can account for the potential skewness of their resulting TPCs such as the Quadratic, Modified Gaussian, Weibull, Exponentially Modified Gaussian etc. (Angiletta 2006, Shi & Ge 2010, Ashrafi et al. 2018). Following this procedure, once the best TPC-predictor function is found, the thermal performance traits (TPTs) such as the thermal optimum (Topt) or the critical thermal ranges (CTmax and CTmin) are determined. It is important to point out that the TPTs obtained from the TPC and not the parameters of the predictor function that generated the curve are the element of comparison since, following this method, each individual and/or population can be defined better by a different formula.

However, one factor that has not yet been considered is the potential kurtosis of a TPC. Kurtosis is actually of special biological relevance since both platykurtic (kurtosis < 0, resulting in a broad and flat TPC) and leptokurtic (kurtosis > 0, resulting in a narrow TPC) shapes can occur in nature. For example, a very stochastic thermal environment might drive adaptation towards semi-independence of P with respect to T. Such conditions might trnaslate into a platykurtic TPC with a flat section in which P remains equal for a given T range. Oppositely, a very stable thermal environment might result in selection towards a very narrow TPC, where P is optimized for a very small T range. [Here it would be cool to add examples where these two dynamics have happened]. Nonetheless, many of the most commonly used predictor functions are limited in the range of kurtosis values their resulting TPCs can assume, specially when these are closer to zero or even negative.

Increased thermal stochasticity has been proven to play a major selective role in natural populations (Sæther et al. 2003, Saltz et al. 2006, [Add more references]), an evolutionary dynamic that results into flat and consequently platykurtic TPCs [Expand a lot with examples here]. Consequently, predictor functions able to generate TPCs with a wider range of possible kurtosis values could be much more useful to analyze this kind of relationships between P and T, not only to describe existing scenarios, but also as a tool to model evolution driven by increasing thermal fluctuations caused by climate chane.

In these pages, we will present two candidate TPC-predictor functions that are known to account for a wider range of kurtosis in their resulting distributions, namely, a modification from Pearson IV (PIV) distribution's probability density function (PDF) (Pearson 1894) as presented by Nagahara (1999,2003), and a modification of the Extended Skewed Generalized Normal (ESGN) distribution's PDF as developed by Choudhury & Matin 2011 . To test the capabilities of PIV and ESGN as TPC-predictor functions, We will use them to fit TPCs on a compilation of thermal-performance datasets (TPDs) [For now these will be self-generated, but the end goal here is to use real-life data!], and assess their quality with respect to other more commonly used expressions through the comparison of their WAIC scores ([Citation]).

Our goal is to test wehther expressions that can result in TPCs with a wider range of kurtosis values are better to fit a wide range of relationships between P and T. Our hypothesis is that they can, that at the risk of overfitting caused by an increase in the number of parameters, either PIV and ESGN might be, in average, the best TPC-predictor function for the wider variety of TPC shapes.

[The goal can also be proving that the shape of the TPC is the ultimate TPT and that the parameters of the function and not the values extracted from the TPC's shape should be the TPTs that count and that can be generalized]

ggcostoya/tpcurves2 documentation built on Jan. 1, 2021, 2:19 a.m.