spphpr (version 1.0.0)

ADP: Function for Implementing the Accumulated Developmental Progress Method

Description

Estimates the starting date (\(S\), in day-of-year) and the parameters of a developmental rate model in the accumulated developmental progress (ADP) method using mean daily air temperatures (Wagner et al., 1984; Shi et al., 2017a, 2017b).

Usage

ADP( S.arr, expr, ini.val, Year1, Time, Year2, DOY, Temp, DOY.ul = 120,
     fig.opt = TRUE, control = list(), verbose = TRUE )

Value

TDDR

the temperature-dependent developmental rate matrix consisting of the year, day-of-year, mean daily temperature and developmental rate columns

MAT

a matrix consisting of the candidate starting dates and the estimates of candidate model parameters with the corresponding RMSEs

Dev.accum

the calculated annual accumulated developmental progresses in different years

Year

The overlapping years between Year1 and Year2

Time

The observed occurence times (day-of-year) in the overlapping years between Year1 and Year2

Time.pred

the predicted occurence times in different years

S

the determined starting date (day-of-year)

par

the estimates of model parameters

RMSE

the RMSE (in days) between the observed and predicted occurrence times

unused.years

the years that have phenological records but lack climate data

Arguments

S.arr

the candidate starting dates for thermal accumulation (in day-of-year)

expr

a user-defined model that is used in the accumulated developmental progress (ADP) method

ini.val

a vector or a list that saves the initial values of the parameters in expr

Year1

the vector of the years in which a particular phenological event was recorded

Time

the vector of the occurence times (in day-of-year) of a particular phenological event across many years

Year2

the vector of the years recording the climate data corresponding to the occurrence times

DOY

the vector of the dates (in day-of-year) for which climate data exist

Temp

the mean daily air temperature data (in \({}^{\circ}\)C) corresponding to DOY

DOY.ul

the upper limit of DOY used to predict the occurrence time

fig.opt

an optional argument to draw the figures associated with the temperature-dependent developmental rate curve, the mean daily temperatures versus years, and a comparision between the predicted and observed occurrence times

control

the list of control parameters for using the optim function in the stats package

verbose

an optional argument allowing users to suppress the printing of computation progress

Author

Peijian Shi , Zhenghong Chen , Brady K. Quinn .

Details

It is better not to set too many candiate starting dates, as doing so will be time-consuming. If expr is selected as Arrhenius' equation, S.arr can be selected as the S obtained from the output of the ADTS function. Here, expr can be other nonlinear temperature-dependent developmental rate functions (see Shi et al. [2017b] for details). Further, expr can be any an arbitrary user-defined temperature-dependent developmental rate function, e.g., a function named myfun, but it needs to take the form of myfun <- function(P, x){...}, where P is the vector of the model parameter(s), and x is the vector of the predictor variable, i.e., the temperature variable.

\(\qquad\)The function does not require that Year1 is the same as unique(Year2), and the intersection of the two vectors of years will be kept. The unused years that have phenological records but lack climate data will be showed in unused.years in the returned list.

\(\qquad\)The numerical value of DOY.ul should be greater than or equal to the maximum Time.

\(\qquad\) Let \(r\) represent the temperature-dependent developmental rate, i.e., the reciprocal of the developmental duration required for completing a particular phenological event, at a constant temperature. In the accumulated developmental progress (ADP) method, when the annual accumulated developmental progress (AADP) reaches 100%, the phenological event is predicted to occurr for each year. Let \(\mathrm{AADP}_{i}\) denote the AADP of the \(i\)th year, which equals

$$\mathrm{AADP}_{i} = \sum_{j=S}^{E_{i}}r_{ij}\left(\mathrm{\mathbf{P}}; T_{ij}\right),$$

where \(S\) represents the starting date (in day-of-year), \(E_{i}\) represents the ending date (in day-of-year), i.e., the occurrence time of a pariticular phenological event in the \(i\)th year, \(\mathrm{\mathbf{P}}\) is the vector of the model parameters in expr, and \(T_{ij}\) represents the mean daily temperature of the \(j\)th day of the \(i\)th year (in \({}^{\circ}\)C or K). In theory, \(\mathrm{AADP}_{i} = 100\%\), i.e., the AADP values of different years are a constant 100%. However, in practice, there is a certain deviation of \(\mathrm{AADP}_{i}\) from 100%. The following approach is used to determine the predicted occurrence time. When \(\sum_{j=S}^{F}r_{ij} = 100\%\) (where \(F \geq S\)), it follows that \(F\) is the predicted occurrence time; when \(\sum_{j=S}^{F}r_{ij} < 100\%\) and \(\sum_{j=S}^{F+1}r_{ij} > 100\%\), the trapezoid method (Ring and Harris, 1983) is used to determine the predicted occurrence time. Assume that there are \(n\)-year phenological records. When the starting date \(S\) and the temperature-dependent developmental rate model are known, the model parameters can be estimated using the Nelder-Mead optiminization method (Nelder and Mead, 1965) to minimize the root-mean-square error (RMSE) between the observed and predicted occurrence times, i.e.,

$$\mathrm{\mathbf{\hat{P}}} = \mathrm{arg}\,\mathop{\mathrm{min}}\limits_{ \mathrm{\mathbf{P}}}\left\{\mathrm{RMSE}\right\} = \mathrm{arg}\,\mathop{\mathrm{min}}\limits_{ \mathrm{\mathbf{P}}}\sqrt{\frac{\sum_{i=1}^{n}\left(E_{i}-\hat{E}_i\right){}^{2}}{n}}.$$

Because \(S\) is not determined, a group of candidate \(S\) values (in day-of-year) need to be provided. Assume that there are \(m\) candidate \(S\) values, i.e., \(S_{1}, S_{2}, S_{3}, \cdots, S_{m}\). For each \(S_{q}\) (where \(q\) ranges between 1 and \(m\)), we can obtain a vector of the estimated model parameters, \(\mathrm{\mathbf{\hat{P}}}_{q}\), by minimizing \(\mathrm{RMSE}_{q}\) using the Nelder-Mead optiminization method. Then we finally selected \(\mathrm{\mathbf{\hat{P}}}\) associated with \(\mathrm{min}\left\{\mathrm{RMSE}_{1}, \mathrm{RMSE}_{2}, \mathrm{RMSE}_{3}, \cdots, \mathrm{RMSE}_{m}\right\}\) as the target parameter vector.

References

Nelder, J.A., Mead, R. (1965) A simplex method for function minimization. Computer Journal 7, 308\(-\)313. tools:::Rd_expr_doi("10.1093/comjnl/7.4.308")

Ring, D.R., Harris, M.K. (1983) Predicting pecan nut casebearer (Lepidoptera: Pyralidae) activity at College Station, Texas. Environmental Entomology 12, 482\(-\)486. tools:::Rd_expr_doi("10.1093/ee/12.2.482")

Shi, P., Chen, Z., Reddy, G.V.P., Hui, C., Huang, J., Xiao, M. (2017a) Timing of cherry tree blooming: Contrasting effects of rising winter low temperatures and early spring temperatures. Agricultural and Forest Meteorology 240\(-\)241, 78\(-\)89. tools:::Rd_expr_doi("10.1016/j.agrformet.2017.04.001")

Shi, P., Fan, M., Reddy, G.V.P. (2017b) Comparison of thermal performance equations in describing temperature-dependent developmental rates of insects: (III) Phenological applications. Annals of the Entomological Society of America 110, 558\(-\)564. tools:::Rd_expr_doi("10.1093/aesa/sax063")

Wagner, T.L., Wu, H.-I., Sharpe, P.J.H., Shcoolfield, R.M., Coulson, R.N. (1984) Modelling insect development rates: a literature review and application of a biophysical model. Annals of the Entomological Society of America 77, 208\(-\)225. tools:::Rd_expr_doi("10.1093/aesa/77.2.208")

See Also

predADP

Examples

data(apricotFFD)
data(BJDAT)
X1 <- apricotFFD
X2 <- BJDAT

Year1.val  <- X1$Year
Time.val   <- X1$Time
Year2.val  <- X2$Year
DOY.val    <- X2$DOY
Temp.val   <- X2$MDT
DOY.ul.val <- 120
S.arr0     <- 47

#### Defines a re-parameterized Arrhenius' equation ########################### 
Arrhenius.eqn <- function(P, x){
  B  <- P[1]
  Ea <- P[2]
  R  <- 1.987 * 10^(-3)
  x  <- x + 273.15
  10^12*exp(B-Ea/(R*x))
}
##############################################################################

#### Provides the initial values of the parameter of Arrhenius' equation #####
ini.val0 <- list( B = 20, Ea = 14 )
##############################################################################

# \donttest{
  res5 <- ADP( S.arr = S.arr0, expr = Arrhenius.eqn, ini.val = ini.val0, Year1 = Year1.val, 
               Time = Time.val, Year2 = Year2.val, DOY = DOY.val, Temp = Temp.val, 
               DOY.ul = DOY.ul.val, fig.opt = TRUE, control = list(trace = FALSE, 
               reltol = 1e-12, maxit = 5000), verbose = TRUE )
  res5

  TDDR <- res5$TDDR
  T    <- TDDR$Temperature
  r    <- TDDR$Rate
  Y    <- res5$Year
  DP   <- res5$Dev.accum

  dev.new()
  par1 <- par(family="serif")
  par2 <- par(mar=c(5, 5, 2, 2))
  par3 <- par(mgp=c(3, 1, 0))
  Ind <- sort(T, index.return=TRUE)$ix 
  T1  <- T[Ind]
  r1  <- r[Ind]
  plot( T1, r1, cex.lab = 1.5, cex.axis = 1.5, pch = 1, cex = 1.5, col = 2, type = "l", 
        xlab = expression(paste("Mean daily temperature (", degree, "C)", sep = "")), 
        ylab = expression(paste("Calculated developmental rate (", {day}^{"-1"}, ")", sep = "")) ) 
  par(par1)
  par(par2)
  par(par3)

  dev.new()
  par1 <- par(family="serif")
  par2 <- par(mar=c(5, 5, 2, 2))
  par3 <- par(mgp=c(3, 1, 0))
  plot( Y, DP * 100, xlab = "Year", 
        ylab = "Accumulated developmental progress (%)", 
        ylim = c(50, 150), cex.lab=1.5, cex.axis = 1.5, cex = 1.5 )
  abline( h = 1 * 100, lwd = 1, col = 4, lty = 2 ) 
  par(par1)
  par(par2)
  par(par3)

  # graphics.off()
# }