In ggcostoya/tpcurves2: T-P Curves

knitr::opts_chunk$set(echo = TRUE)
devtools::load_all()

In this page I'll show how to analyze TPD using the tools from this package. By analyze I am referring to the extraction of both a prediction of a thermal performance curve (TPC) and thermal performance traits (TPTs) from a data set with temperature ($T_m$) and PIT equivalent to performance ($P$). I'll first show a basic individual-level example and later I'll move on to more complex population and multipopulation level cases.

Analyze Individual TPD

a. Get TPC

To get a TPC from an individual's TPD you first have to fit a polynomial regression between $Tm$ & $P$. A polynomial regression is defined by it's degree which is "the highest of the degrees of the polynomial's monomials (individual terms) with non-zero coefficients". In other words, the highest power it has. For example, the mathematical expression for a polynomial regression of degree 3 would look like:

$$y = \alpha + \beta_{1} x + \beta_{2} x^{2} + \beta_{3} x^{3}$$ Being "3" the highest power in the expression. The key point with a polynomial regression's degree is the shape of the resulting function. The most interesting polynomial functions to fit a TPC are the quadratic (Degree 2), cubic (Degree 3) and quadratic (Degree 4) functions. A quadratic regression fits a symmetric curve, a cubic function (if the upwards bump region is considered) can fit an asymmetric curve and a quadratic function can fit an asymmetric and non-steadily declining curve.

Which curve is best for a given data? The answer lies on their AIC scores. For a given data set, the regression that achieves the lowest AIC score will be the one best representing the model (Here I would get into a technical discussion on how the AIC works and why the WAIC is better but I'll keep that for myself).

As part of this package, I have developed a function called get_tpc (get thermal performance curve) which automates the process. As arguments, it takes:

Tm: A vector of temperature values.
P: A vector of performance values.
Degree: The degree of the polynomial regression to fit the data with. This parameter is optional, if it is not imputed the function will automatically test which degree has the lowest AIC score and choose it for you.

Using this information the get_tpc returns another data set with four columns:

Tm: A vector of placeholder temperature value.
P: A vector of predicted performance values for each placeholder temperature value.
L & U: Vectors for the Lower and Upper boundaries of the 95% Confidence Interval. Very useful when plotting and compare it to other curves.

Below I generate a mock TPD for an individual and I draw it's TPC using the get_tpc function.

# generate data
ind_tpdata <- gen_tpd( Topt = 25, CTmax = 28, CTmin = 22, Pmax = 22, Pmin = 0, Error = 2, Samples = 20, Degree = 3)

# get curve
ind_curve <- get_tpc( Tm = ind_tpdata$Tm, P = ind_tpdata$P, Degree = 3, Pmin = 0)
head(ind_curve)

ggplot( data = ind_curve, aes(x = Tm, y = P)) + 
  geom_ribbon( aes(ymin = L, ymax = U), fill = "grey92") +
  geom_line( col = "red", size = 2) +
  geom_point( data = ind_tpdata, aes(x = Tm, y = P), size = 2) +
  theme_classic() +
  labs( x = "Temperature", y = "Performance") + 
  geom_hline( aes( yintercept = 0))  
  #ylim(0, max(ind_tpdata$P))

b. Get TPTs

Complementary to the TPC another interesting piece of information to extract from TP Data are its TPTs. According to Logan & Cox 2020 a TPC is defined by 4 TPTs:

$T_{opt}$: The thermal optimum, i.e. the temperature at which maximal performance capacity ($P_{max}$) is reached, the peak of the TPC.
$CT_{max}$ & $CT_{min}$: The critical thermal max & min, i.e. the temperature extremes at which performance is minimal. It is important to point out that these two values can be different since a TPC doesn't need to be symmetric around the optimum.
$T_br$: The temperature breath of the TPC, i.e. the difference between CTmax - CTmin.
$P_{max}$: The maximal performance capacity, i.e. the performance at the peak of the TPC.

As part of this package I have also developed the get_tpts function which, from a TPC can extract all these traits for a given TP data. Let me show an example:

ind_tpts <- get_tpts(Tm = ind_tpdata$Tm, P = ind_tpdata$P, Degree = 4, Pmin = 0)
ind_tpts

Analyze Population TPD

The same process can be applied to a whole population both by drawing a curve for the entire population and by getting each individuals (or the whole populations) TPTs. As an example below, I generate some population TP data and from it I extract the TPC and the TPTs.

# generate population data 
pop_tpd <- gen_pop_tpd(N = 15, Topt = 25, CTmax = 27, CTmin = 23, Pmax = 10, Pmin = 0, Error = 2, Samples = 10, Degree = 3)

# get the TPC 
pop_tpc <- get_tpc(Tm = pop_tpd$Tm, P = pop_tpd$P, Degree = 3, Pmin = 0)

# get the TPTs 
pop_tpts <- get_pop_tpts(TPD = pop_tpd, Degree = 3, Pmin = 0)

# small sample of the TPTs data
head(pop_tpts)

ggplot( data = pop_tpc, aes(x = Tm, y = P)) + 
  geom_ribbon( aes(ymin = L, ymax = U), fill = "grey92") +
  geom_line( col = "red", size = 2) +
  geom_point( data = pop_tpd, aes(x = Tm, y = P, colour = factor(ID)), size = 2) +
  theme_classic() +
  labs( x = "Temperature", y = "Performance", colour = "Individual ID") + 
  geom_hline( aes( yintercept = 0)) +
  theme(legend.position = "right")