Package 'spphpr'

Function for Implementing the Accumulated Degree Days Method Using Mean Daily Temperatures

Description

Estimates the starting date ($S$, in day-of-year) and base tempeature ($T_{0}$, in ${}^{\circ}$C) in the accumulated degree days method using mean daily air temperatures (Aono, 1993; Shi et al., 2017a, b).

Usage

ADD(S.pd = NULL, T0.arr, Year1, Time, Year2, DOY, Temp, DOY.ul = 120, 
    fig.opt = TRUE, S.def = 54, verbose = TRUE)

Arguments

S.pd	the pre-determined starting date for thermal accumulation (in day-of-year)
T0.arr	the candidate base temperatures (in ${}^{\circ}$C)
Year1	the vector of the years in which a particular phenological event was recorded
Time	the vector of the occurrence 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 determinations of the starting date and base temperature, and a comparison between the predicted and observed occurrence times
S.def	a mandatory defintion of the starting date when (i) S.pd is NULL and (ii) the minimum correlation coefficient method fails to find a suitable starting date
verbose	an optional argument allowing users to suppress the printing of computation progress

Details

The default of S.pd is NULL. In this case, the date associated with the minimum correlation coefficient [between the mean of the mean daily temperatures (from a candidate starting date to the observed occurrence time) and the observed occurrence time] will be determined to be the starting date on the condition that it is smaller than the mininum phenological occurrence time. If the determined date associated with the minimum correlation coefficient is greater than the mininum phenological occurrence time, S.def will be used as the starting date. If S.pd is not NULL, the starting date will be directly set as S.pd irrespective of the minimum correlation coefficient method and the value of S.def. This means that S.pd is superior to S.def in determining the starting date.

$\qquad$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.

Value

S.arr	the candidate starting dates (in day-of-year), whose default ranges from the minimum DOY to min(DOY.ul, the maximum DOY)
cor.coef.arr	the candidate correlation coefficients between the mean of the mean daily tempertures (from a candidate starting date to the observed occurrence time) and the observed occurrence time
cor.coef	the minimum correlation coefficient, i.e., min(cor.coef.arr)
search.failure	a value of 0 or 1 of showing whether the starting date is successfully determined by the minimum correlation coefficient method when S.pd = NULL, where 0 represents success and 1 represents failure
mAADD.arr	an vector saving the interannual mean of the annual acccumulated degree days (AADD) values for each of the candidate base temperatures
RMSE.arr	a vector saving the candidate root-mean-square errors (in days) between the observed and predicted occurrence times for each of the candidate base temperatures
AADD.arr	the annual accumulated degree days (AADD) values in different years
Year	The overlapping years between Year1 and Year2
Time	The observed occurrence times (day-of-year) in the overlapping years between Year1 and Year2
Time.pred	the predicted occurrence times in different years
S	the determined starting date (day-of-year)
T0	the determined base temperature (in ${}^{\circ}$C)
AADD	the expected annual accumulated degree days
RMSE	the smallest RMSE (in days) from the different candidate base temperatures
unused.years	the years that have phenological records but lack climate data

Note

The entire mean daily temperature data set for the spring of each year should be provided. AADD is represented by the mean of AADD.arr in the output. When the argument of S.pd is not NULL, the returned value of search.failure will be NA. When the argument of S.pd is NULL, and the minimum correlation coefficient method fails to find a suitable starting date, the argument of S.def is then defined as the determined starting date, i.e., the returned value of S. At the same time, the returned value of cor.coef is defined as NA.

Author(s)

Peijian Shi [email protected], Zhenghong Chen [email protected], Jing Tan [email protected], Brady K. Quinn [email protected].

References

Aono, Y. (1993) Climatological studies on blooming of cherry tree (Prunus yedoensis) by means of DTS method. Bulletin of the University of Osaka Prefecture. Ser. B, Agriculture and life sciences 45, 155$-$192 (in Japanese with English abstract).

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. 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. doi:10.1093/aesa/sax063

See Also

predADD

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
T0.arr0    <- seq(-5, 5, by = 0.1) 
S.pd0      <- NULL


  res1 <- ADD( S.pd = S.pd0, T0.arr = T0.arr0, Year1 = Year1.val, Time = Time.val, 
               Year2 = Year2.val, DOY = DOY.val, Temp = Temp.val, 
               DOY.ul = DOY.ul.val, fig.opt = TRUE, S.def=54, verbose = TRUE )
  res1

  S0 <- res1$S.arr 
  r0 <- res1$cor.coef.arr
  
  dev.new()
  par1 <- par(family="serif")
  par2 <- par(mar=c(5, 5, 2, 2))
  par3 <- par(mgp=c(3, 1, 0))
  plot( S0, r0, cex.lab = 1.5, cex.axis = 1.5, xlab = "Candidate starting date (day-of-year)", 
        ylab="Correlation coefficient between the mean temperature and FFD", type="l" )
  ind <- which.min(r0)
  points(S0[ind], r0[ind], cex = 1.5, pch = 16)
  text(S0[ind], r0[ind] + 0.1, bquote(paste(italic(S), " = ", .(S0[ind]), sep = "")), cex = 1.5)
  par(par1)
  par(par2)
  par(par3)

  resu1 <- ADD( S.pd = 65, T0.arr = seq(-10, 0, by = 0.1), Year1 = Year1.val, Time = Time.val, 
                Year2 = Year2.val, DOY = DOY.val, Temp = Temp.val, 
                DOY.ul = DOY.ul.val, fig.opt = TRUE, S.def = 54, verbose = TRUE )
  resu1

  # graphics.off()

---

Function for Implementing the Accumulated Degree Days Method Using Minimum and Maximum Daily Temperatures

Description

Estimates the starting date ($S$, in day-of-year) and base tempeature ($T_{0}$, in ${}^{\circ}$C) in the accumulated degree days method using minimum and maximum daily air temperatures (Aono, 1993; Shi et al., 2017a, b).

Usage

ADD2(S.pd = NULL, T0.arr, Year1, Time, Year2, DOY, Tmin, Tmax,  
     DOY.ul = 120, fig.opt = TRUE, S.def = 54, verbose = TRUE)

Arguments

S.pd	the pre-determined starting date for thermal accumulation (in day-of-year)
T0.arr	the candidate base temperatures (in ${}^{\circ}$C)
Year1	the vector of the years in which a particular phenological event was recorded
Time	the vector of the occurrence 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
Tmin	the minimum daily air temperature data (in ${}^{\circ}$C) corresponding to DOY
Tmax	the maximum 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 determinations of the starting date and base temperature, and a comparison between the predicted and observed occurrence times
S.def	a mandatory defintion of the starting date when (i) S.pd is NULL and (ii) the minimum correlation coefficient method fails to find a suitable starting date
verbose	an optional argument allowing users to suppress the printing of computation progress

Details

The default of S.pd is NULL. In this case, the date associated with the minimum correlation coefficient [between the mean of the (Tmin + Tmax)/2 values (from a candidate starting date to the observed occurrence time) and the observed occurrence time] will be determined to be the starting date on the condition that it is smaller than the mininum phenological occurrence time. If the determined date associated with the minimum correlation coefficient is greater than the mininum phenological occurrence time, S.def will be used as the starting date. If S.pd is not NULL, the starting date will be directly set as S.pd irrespective of the minimum correlation coefficient method and the value of S.def. This means that S.pd is superior to S.def in determining the starting date.

$\qquad$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.

Value

S.arr	the candidate starting dates (in day-of-year), whose default ranges from the minimum DOY to min(DOY.ul, the maximum DOY)
cor.coef.arr	the candidate correlation coefficients between the mean of the (Tmin + Tmax)/2 values (from a candidate starting date to the observed occurrence time) and the observed occurrence time
cor.coef	the minimum correlation coefficient, i.e., min(cor.coef.arr)
search.failure	a value of 0 or 1 of showing whether the starting date is successfully determined by the minimum correlation coefficient method when S.pd = NULL, where 0 represents success and 1 represents failure
mAADD.arr	an vector saving the interannual mean of the annual acccumulated degree days (AADD) values for each of the candidate base temperatures
RMSE.arr	a vector saving the candidate root-mean-square errors (in days) between the observed and predicted occurrence times for each of the candidate base temperatures
AADD.arr	the annual accumulated degree days (AADD) values in different years
Year	The overlapping years between Year1 and Year2
Time	The observed occurrence times (day-of-year) in the overlapping years between Year1 and Year2
Time.pred	the predicted occurrence times in different years
S	the determined starting date (day-of-year)
T0	the determined base temperature (in ${}^{\circ}$C)
AADD	the expected annual accumulated degree days
RMSE	the smallest RMSE (in days) from the different candidate base temperatures
unused.years	the years that have phenological records but lack climate data

Note

The entire minimum and maximum daily temperature data set for the spring of each year should be provided. AADD is represented by the mean of AADD.arr in the output. When the argument of S.pd is not NULL, the returned value of search.failure will be NA. When the argument of S.pd is NULL, and the minimum correlation coefficient method fails to find a suitable starting date, the argument of S.def is then defined as the determined starting date, i.e., the returned value of S. At the same time, the returned value of cor.coef is defined as NA.

Author(s)

Peijian Shi [email protected], Zhenghong Chen [email protected], Jing Tan [email protected], Brady K. Quinn [email protected].

References

Aono, Y. (1993) Climatological studies on blooming of cherry tree (Prunus yedoensis) by means of DTS method. Bulletin of the University of Osaka Prefecture. Ser. B, Agriculture and life sciences 45, 155$-$192 (in Japanese with English abstract).

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. 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. doi:10.1093/aesa/sax063

See Also

predADD2

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
Tmin.val   <- X2$MinDT
Tmax.val   <- X2$MaxDT
DOY.ul.val <- 120
T0.arr0    <- seq(3.5, 4, by = 0.1) 
S.pd0      <- NULL


  cand.res1 <- ADD2( S.pd = S.pd0, T0.arr = T0.arr0, Year1 = Year1.val, Time = Time.val, 
                     Year2 = Year2.val, DOY = DOY.val, Tmin = Tmin.val, Tmax = Tmax.val, 
                     DOY.ul = DOY.ul.val, fig.opt = TRUE, S.def=54, verbose = TRUE )
  cand.res1

  S0 <- cand.res1$S.arr 
  r0 <- cand.res1$cor.coef.arr
  
  dev.new()
  par1 <- par(family="serif")
  par2 <- par(mar=c(5, 5, 2, 2))
  par3 <- par(mgp=c(3, 1, 0))
  plot( S0, r0, cex.lab = 1.5, cex.axis = 1.5, xlab = "Candidate starting date (day-of-year)", 
        ylab="Correlation coefficient between the mean temperature and FFD", type="l" )
  ind <- which.min(r0)
  points(S0[ind], r0[ind], cex = 1.5, pch = 16)
  text(S0[ind], r0[ind] + 0.1, bquote(paste(italic(S), " = ", .(S0[ind]), sep = "")), cex = 1.5)
  par(par1)
  par(par2)
  par(par3)

  # graphics.off()

---

Function for Implementing the Accumulated Degree Days Method Using Mean Daily Temperatures for the Combinations of the Starting Date and Base Temperature

Description

Estimates the starting date ($S$, in day-of-year) and base temperature ($T_{0}$, in ${}^{\circ}$C) in the accumulated degree days (ADD) method using mean daily air temperatures (Konno and Sugihara, 1986; Aono, 1993; Shi et al., 2017a, b).

Usage

ADD3( S.arr, T0.arr, Year1, Time, Year2, DOY, Temp, DOY.ul = 120, 
      fig.opt = TRUE, verbose = TRUE )

Arguments

S.arr	the candidate starting dates for thermal accumulation (in day-of-year)
T0.arr	the candidate base temperatures (in ${}^{\circ}$C)
Year1	the vector of the years in which a particular phenological event was recorded
Time	the vector of the occurrence 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 determination of the combination the starting date and base temperature, and a comparison between the predicted and observed occurrence times
verbose	an optional argument allowing users to suppress the printing of computation progress

Details

When fig.opt is equal to TRUE, it will show the contours of the root-mean-square errors (RMSEs) based on different combinations of $S$ and $T_{0}$.

$\qquad$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.

Value

mAADD.mat	a matrix consisting of the means of the annual accumulated degree days (AADD) values from the combinations of $S$ and $T_{0}$
RMSE.mat	the matrix consisting of the RMSEs (in days) from different combinations of $S$ and $T_{0}$
AADD.arr	the AADD values in different years associated with the smallest value in RMSE.mat
Year	The overlapping years between Year1 and Year2
Time	The observed occurrence times (day-of-year) in the overlapping years between Year1 and Year2
Time.pred	the predicted occurrence times in different years
S	the determined starting date (day-of-year)
T0	the determined base temperature (in ${}^{\circ}$C)
AADD	the expected AADD
RMSE	the smallest RMSE (in days) in RMSE.mat from different combinations of $S$ and $T_{0}$
unused.years	the years that have phenological records but lack climate data

Note

The entire mean daily temperature data set for the spring of each year should be provided. AADD is represented by the mean of AADD.arr in the output.

Author(s)

Peijian Shi [email protected], Zhenghong Chen [email protected], Jing Tan [email protected], Brady K. Quinn [email protected].

References

Aono, Y. (1993) Climatological studies on blooming of cherry tree (Prunus yedoensis) by means of DTS method. Bulletin of the University of Osaka Prefecture. Ser. B, Agriculture and life sciences 45, 155$-$192 (in Japanese with English abstract).

Konno, T., Sugihara, S. (1986) Temperature index for characterizing biological activity in soil and its application to decomposition of soil organic matter. Bulletin of National Institute for Agro-Environmental Sciences 1, 51$-$68 (in Japanese with English abstract).

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. 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. doi:10.1093/aesa/sax063

See Also

predADD

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     <- seq(60, 70, by = 1)
T0.arr0    <- seq(-2, 5, by = 1)


  RES1 <- ADD3( S.arr = S.arr0, T0.arr = T0.arr0, Year1 = Year1.val, Time = Time.val, 
                Year2 = Year2.val, DOY = DOY.val, Temp = Temp.val, DOY.ul = DOY.ul.val, 
                fig.opt = TRUE, verbose = TRUE)
  RES1

  RMSE.mat0  <- RES1$RMSE.mat
  RMSE.range <- range(RMSE.mat0)

  dev.new()
  par1 <- par(family="serif")
  par2 <- par(mar=c(5, 5, 2, 2))
  par3 <- par(mgp=c(3, 1, 0))
  image( S.arr0, T0.arr0, RMSE.mat0, col = terrain.colors(200), axes = TRUE, 
         cex.axis = 1.5, cex.lab = 1.5, xlab = "Starting date (day-of-year)", 
         ylab = expression(paste("Base temperature (", degree, "C)", sep = "")))
  points( RES1$S, RES1$T0, cex = 1.5, pch = 16, col = 2 )
  contour( S.arr0, T0.arr0, RMSE.mat0, levels = round(seq(RMSE.range[1], 
           RMSE.range[2], len = 20), 4), add = TRUE, cex = 1.5, col = "#696969", labcex = 1.5)
  par(par1)
  par(par2)
  par(par3)

  # graphics.off()

---

Function for Implementing the Accumulated Degree Days Method Using Minimum and Maximum Daily Temperatures for the Combinations of the Starting Date and Base Temperature

Description

Estimates the starting date ($S$, in day-of-year) and base temperature ($T_{0}$, in ${}^{\circ}$C) in the accumulated degree days (ADD) method using minimum and maximum daily air temperatures (Konno and Sugihara, 1986; Aono, 1993; Shi et al., 2017a, b).

Usage

ADD4( S.arr, T0.arr, Year1, Time, Year2, DOY, Tmin, Tmax, DOY.ul = 120, 
      fig.opt = TRUE, verbose = TRUE )

Arguments

S.arr	the candidate starting dates for thermal accumulation (in day-of-year)
T0.arr	the candidate base temperatures (in ${}^{\circ}$C)
Year1	the vector of the years in which a particular phenological event was recorded
Time	the vector of the occurrence 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
Tmin	the minimum daily air temperature data (in ${}^{\circ}$C) corresponding to DOY
Tmax	the maximum 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 determination of the combination the starting date and base temperature, and a comparison between the predicted and observed occurrence times
verbose	an optional argument allowing users to suppress the printing of computation progress

Details

When fig.opt is equal to TRUE, it will show the contours of the root-mean-square errors (RMSEs) based on different combinations of $S$ and $T_{0}$.

$\qquad$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.

Value

mAADD.mat	a matrix consisting of the means of the annual accumulated degree days (AADD) values from the combinations of $S$ and $T_{0}$
RMSE.mat	the matrix consisting of the RMSEs (in days) from different combinations of $S$ and $T_{0}$
AADD.arr	the AADD values in different years associated with the smallest value in RMSE.mat
Year	The overlapping years between Year1 and Year2
Time	The observed occurrence times (day-of-year) in the overlapping years between Year1 and Year2
Time.pred	the predicted occurrence times in different years
S	the determined starting date (day-of-year)
T0	the determined base temperature (in ${}^{\circ}$C)
AADD	the expected AADD
RMSE	the smallest RMSE (in days) in RMSE.mat from different combinations of $S$ and $T_{0}$
unused.years	the years that have phenological records but lack climate data

Note

The entire mean daily temperature data set for the spring of each year should be provided. AADD is represented by the mean of AADD.arr in the output.

Author(s)

Peijian Shi [email protected], Zhenghong Chen [email protected], Jing Tan [email protected], Brady K. Quinn [email protected].

References

Aono, Y. (1993) Climatological studies on blooming of cherry tree (Prunus yedoensis) by means of DTS method. Bulletin of the University of Osaka Prefecture. Ser. B, Agriculture and life sciences 45, 155$-$192 (in Japanese with English abstract).

Konno, T., Sugihara, S. (1986) Temperature index for characterizing biological activity in soil and its application to decomposition of soil organic matter. Bulletin of National Institute for Agro-Environmental Sciences 1, 51$-$68 (in Japanese with English abstract).

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. 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. doi:10.1093/aesa/sax063

See Also

predADD2

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
Tmin.val   <- X2$MinDT
Tmax.val   <- X2$MaxDT
DOY.ul.val <- 120
S.arr0     <- seq(63, 66, by = 1)
T0.arr0    <- 3.8


  RES2 <- ADD4( S.arr = S.arr0, T0.arr = T0.arr0, Year1 = Year1.val, Time = Time.val, 
                Year2 = Year2.val, DOY = DOY.val, Tmin = Tmin.val, Tmax = Tmax.val, 
                DOY.ul = DOY.ul.val, fig.opt = TRUE, verbose = TRUE)
  RES2


  RESU2 <- ADD4( S.arr = 65, T0.arr = seq(2, 6, by = 1), Year1 = Year1.val, Time = Time.val, 
                 Year2 = Year2.val, DOY = DOY.val, Tmin = Tmin.val, Tmax = Tmax.val, 
                 DOY.ul = DOY.ul.val, fig.opt = TRUE, verbose = TRUE)
  RESU2

  # graphics.off()

---

Function for Implementing the Accumulated Developmental Progress Method Using Mean Daily Temperatures

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, b).

Usage

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

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 occurrence 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 comparison 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

Details

It is better not to set too many candidate 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. Let $\hat{E}_{i}$ represent the predicted occurrence time of the $i$th year. 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.

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 occurrence times (day-of-year) in the overlapping years between Year1 and Year2
Time.pred	the predicted occurrence 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

Note

The entire mean daily temperature data set for the spring of each year should be provided. In TDDR, the first column of Year saves the years, the second column of DOY saves the day-of-year values, the third column of Temperature saves the mean daily air temperatures calculated between the starting date to the occurrence times, and the fourth column of Rate saves the calculated developmental rates corresponding to the mean daily temperatures.

Author(s)

Peijian Shi [email protected], Zhenghong Chen [email protected], Jing Tan [email protected], Brady K. Quinn [email protected].

References

Nelder, J.A., Mead, R. (1965) A simplex method for function minimization. Computer Journal 7, 308$-$313. 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. 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. 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. 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. 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 )
##############################################################################


  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()

---

Function for Implementing the Accumulated Developmental Progress Method Using Minimum and Maximum Daily Temperatures

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 minimum and maximum daily air temperatures (Wagner et al., 1984; Shi et al., 2017a, b).

Usage

ADP2( S.arr, expr, ini.val, Year1, Time, Year2, DOY, Tmin, Tmax, 
      DOY.ul = 120, fig.opt = TRUE, control = list(), verbose = TRUE )

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 occurrence 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
Tmin	the minimum daily air temperature data (in ${}^{\circ}$C) corresponding to DOY
Tmax	the maximum 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 comparison 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

Details

It is better not to set too many candidate 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 ADTS2 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

$\mathrm{AADP}_{i} = \sum_{j=S}^{E_{i}}\sum_{w=1}^{24}\frac{r_{ijw}\left(\mathrm{\mathbf{P}}; T_{ijw}\right)}{24},$

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, $T_{ijw}$ represents the estimated mean hourly temperature of the $w$th hour of the $j$th day of the $i$th year (in ${}^{\circ}$C or K), and $r_{ijw}$ represents the developmental rate (per hour) at $T_{ijw}$, which is transferred to $r_{ij}$ (per day) by dividing 24. This packages takes the method proposed by Zohner et al. (2020) to estimate the mean hourly temperature for each of 24 hours:

$T_{w} = \frac{T_{\mathrm{max}} - T_{\mathrm{min}}}{2}\, \mathrm{sin}\left(\frac{w\pi}{12}- \frac{\pi}{2}\right)+\frac{T_{\mathrm{max}} + T_{\mathrm{min}}}{2},$

where $w$ represents the $w$th hour of a day, and $T_{\mathrm{min}}$ and $T_{\mathrm{max}}$ represent the minimum and maximum temperatures of the day, respectively.

$\qquad$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 $\sum_{j=S}^{F}\sum_{w=1}^{24}r_{ijw}/24 = 100\%$ (where $F \geq S$), it follows that $F$ is the predicted occurrence time; when $\sum_{j=S}^{F}\sum_{w=1}^{24}r_{ijw}/24 < 100\%$ and $\sum_{j=S}^{F+1}\sum_{w=1}^{24}r_{ijw}/24 > 100\%$, the trapezoid method (Ring and Harris, 1983) is used to determine the predicted occurrence time. Let $\hat{E}_{i}$ represent the predicted occurrence time of the $i$th year. 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.

Value

TDDR	the temperature-dependent developmental rate matrix consisting of the year, day-of-year, estimated mean daily temperature (= (Tmin + Tmax)/2) 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 occurrence times (day-of-year) in the overlapping years between Year1 and Year2
Time.pred	the predicted occurrence 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

Note

The entire minimum and maximum daily temperature data set for the spring of each year should be provided. In TDDR, the first column of Year saves the years, the second column of DOY saves the day-of-year values, the third column of Temperature saves the estimated mean daily air temperatures (= (Tmin + Tmax)/2) from the starting date to the occurrence times, and the fourth column of Rate saves the calculated developmental rates corresponding to the estimated mean daily temperatures.

Author(s)

Peijian Shi [email protected], Zhenghong Chen [email protected], Jing Tan [email protected], Brady K. Quinn [email protected].

References

Nelder, J.A., Mead, R. (1965) A simplex method for function minimization. Computer Journal 7, 308$-$313. 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. 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. 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. 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. doi:10.1093/aesa/77.2.208

Zohner, C.M., Mo, L., Sebald, V., Renner, S.S. (2020) Leaf-out in northern ecotypes of wide-ranging trees requires less spring warming, enhancing the risk of spring frost damage at cold limits. Global Ecology and Biogeography 29, 1056$-$1072. doi:10.1111/geb.13088

See Also

predADP2

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
Tmin.val   <- X2$MinDT
Tmax.val   <- X2$MaxDT
DOY.ul.val <- 120
S.arr0     <- 46

#### 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 )
##############################################################################

# The usage is similar to that of the "ADP" function. There is only a need to 
#   replace "Temp = Temp.val" with "Tmin = Tmin.val, Tmax = Tmax.val" when using 
#   the "ADP2" function.

---

Function for Implementing the Accumulated Days Transferred to a Standardized Temperature Method Using Mean Daily Temperatures

Description

Estimates the starting date ($S$, in day-of-year) and activation free energy ($E_{a}$, in kcal $\cdot$ mol${}^{-1}$) in the accumulated days transferred to a standardized temperature (ADTS) method using mean daily air temperatures (Konno and Sugihara, 1986; Aono, 1993; Shi et al., 2017a, b).

Usage

ADTS( S.arr, Ea.arr, Year1, Time, Year2, DOY, Temp, DOY.ul = 120, 
      fig.opt = TRUE, verbose = TRUE )

Arguments

S.arr	the candidate starting dates for thermal accumulation (in day-of-year)
Ea.arr	the candidate activation free energy values (in kcal $\cdot$ mol${}^{-1}$)
Year1	the vector of the years in which a particular phenological event was recorded
Time	the vector of the occurrence 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 determination of the combination the starting date and activation free energy, and a comparison between the predicted and observed occurrence times
verbose	an optional argument allowing users to suppress the printing of computation progress

Details

When fig.opt is equal to TRUE, it will show the contours of the root-mean-square errors (RMSEs) based on different combinations of $S$ and $E_{a}$.

$\qquad$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.

Value

mAADTS.mat	a matrix consisting of the means of the annual accumulated days transferred to a standardized temperature (AADTS) values from the combinations of $S$ and $E_{a}$
RMSE.mat	the matrix consisting of the RMSEs (in days) from different combinations of $S$ and $E_{a}$
AADTS.arr	the AADTS values in different years associated with the smallest value in RMSE.mat
Year	The overlapping years between Year1 and Year2
Time	The observed occurrence times (day-of-year) in the overlapping years between Year1 and Year2
Time.pred	the predicted occurrence times in different years
S	the determined starting date (day-of-year)
Ea	the determined activation free energy value (in kcal$\cdot$mol${}^{-1}$)
AADD	the expected AADTS
RMSE	the smallest RMSE (in days) in RMSE.mat from different combinations of $S$ and $E_{a}$
unused.years	the years that have phenological records but lack climate data

Note

The entire mean daily temperature data set for the spring of each year should be provided. AADTS is represented by the mean of AADTS.arr in the output.

Author(s)

Peijian Shi [email protected], Zhenghong Chen [email protected], Jing Tan [email protected], Brady K. Quinn [email protected].

References

Aono, Y. (1993) Climatological studies on blooming of cherry tree (Prunus yedoensis) by means of DTS method. Bulletin of the University of Osaka Prefecture. Ser. B, Agriculture and life sciences 45, 155$-$192 (in Japanese with English abstract).

Konno, T., Sugihara, S. (1986) Temperature index for characterizing biological activity in soil and its application to decomposition of soil organic matter. Bulletin of National Institute for Agro-Environmental Sciences 1, 51$-$68 (in Japanese with English abstract).

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. 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. doi:10.1093/aesa/sax063

See Also

predADTS

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     <- seq(40, 60, by = 1)
Ea.arr0    <- seq(10, 20, by = 1)


  res3 <- ADTS( S.arr = S.arr0, Ea.arr = Ea.arr0, Year1 = Year1.val, Time = Time.val, 
                Year2 = Year2.val, DOY = DOY.val, Temp = Temp.val, DOY.ul = DOY.ul.val, 
                fig.opt = TRUE, verbose = TRUE)
  res3

  RMSE.mat0  <- res3$RMSE.mat
  RMSE.range <- range(RMSE.mat0)

  dev.new()
  par1 <- par(family="serif")
  par2 <- par(mar=c(5, 5, 2, 2))
  par3 <- par(mgp=c(3, 1, 0))
  image( S.arr0, Ea.arr0, RMSE.mat0, col = terrain.colors(200), axes = TRUE, 
         cex.axis = 1.5, cex.lab = 1.5, xlab = "Starting date (day-of-year)", 
         ylab = expression(paste(italic(E["a"]), " (kcal" %.% "mol"^{"-1"}, ")", sep = "")))
  points( res3$S, res3$Ea, cex = 1.5, pch = 16, col = 2 )
  contour( S.arr0, Ea.arr0, RMSE.mat0, levels = round(seq(RMSE.range[1], 
           RMSE.range[2], len = 20), 4), add = TRUE, cex = 1.5, col = "#696969", labcex = 1.5)
  par(par1)
  par(par2)
  par(par3)

  resu3 <- ADTS( S.arr = 47, Ea.arr = seq(10, 20, by = 0.5), Year1 = Year1.val, Time = Time.val, 
                 Year2 = Year2.val, DOY = DOY.val, Temp = Temp.val, DOY.ul = DOY.ul.val, 
                 fig.opt = TRUE, verbose = TRUE)
  resu3

  # graphics.off()

---

Function for Implementing the Accumulated Days Transferred to a Standardized Temperature Method Using Minimum and Maximum Daily Temperatures

Description

Estimates the starting date ($S$, in day-of-year) and activation free energy ($E_{a}$, in kcal $\cdot$ mol${}^{-1}$) in the accumulated days transferred to a standardized temperature (ADTS) method using minimum and maximum daily air temperatures (Konno and Sugihara, 1986; Aono, 1993; Shi et al., 2017a, b).

Usage

ADTS2( S.arr, Ea.arr, Year1, Time, Year2, DOY, Tmin, Tmax, 
       DOY.ul = 120, fig.opt = TRUE, verbose = TRUE )

Arguments

S.arr	the candidate starting dates for thermal accumulation (in day-of-year)
Ea.arr	the candidate activation free energy values (in kcal $\cdot$ mol${}^{-1}$)
Year1	the vector of the years in which a particular phenological event was recorded
Time	the vector of the occurrence 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
Tmin	the minimum daily air temperature data (in ${}^{\circ}$C) corresponding to DOY
Tmax	the maximum 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 determination of the combination the starting date and activation free energy, and a comparison between the predicted and observed occurrence times
verbose	an optional argument allowing users to suppress the printing of computation progress

Details

When fig.opt is equal to TRUE, it will show the contours of the root-mean-square errors (RMSEs) based on different combinations of $S$ and $E_{a}$.

$\qquad$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.

Value

mAADTS.mat	a matrix consisting of the means of the annual accumulated days transferred to a standardized temperature (AADTS) values from the combinations of $S$ and $E_{a}$
RMSE.mat	the matrix consisting of the RMSEs (in days) from different combinations of $S$ and $E_{a}$
AADTS.arr	the AADTS values in different years associated with the smallest value in RMSE.mat
Year	The overlapping years between Year1 and Year2
Time	The observed occurrence times (day-of-year) in the overlapping years between Year1 and Year2
Time.pred	the predicted occurrence times in different years
S	the determined starting date (day-of-year)
Ea	the determined activation free energy value (in kcal$\cdot$mol${}^{-1}$)
AADD	the expected AADTS
RMSE	the smallest RMSE (in days) in RMSE.mat from different combinations of $S$ and $E_{a}$
unused.years	the years that have phenological records but lack climate data

Note

The entire minimum and maximum daily temperature data set for the spring of each year should be provided. AADTS is represented by the mean of AADTS.arr in the output.

Author(s)

Peijian Shi [email protected], Zhenghong Chen [email protected], Jing Tan [email protected], Brady K. Quinn [email protected].

References

Aono, Y. (1993) Climatological studies on blooming of cherry tree (Prunus yedoensis) by means of DTS method. Bulletin of the University of Osaka Prefecture. Ser. B, Agriculture and life sciences 45, 155$-$192 (in Japanese with English abstract).

Konno, T., Sugihara, S. (1986) Temperature index for characterizing biological activity in soil and its application to decomposition of soil organic matter. Bulletin of National Institute for Agro-Environmental Sciences 1, 51$-$68 (in Japanese with English abstract).

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. 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. doi:10.1093/aesa/sax063

See Also

predADTS2

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
Tmin.val   <- X2$MinDT
Tmax.val   <- X2$MaxDT
DOY.ul.val <- 120
S.arr0     <- seq(45, 47, by = 1)
Ea.arr0    <- seq(20, 24, by = 0.5)


  cand.res3 <- ADTS2( S.arr = S.arr0, Ea.arr = Ea.arr0, Year1 = Year1.val, Time = Time.val, 
                      Year2 = Year2.val, DOY = DOY.val, Tmin = Tmin.val, Tmax = Tmax.val,  
                      DOY.ul = DOY.ul.val, fig.opt = TRUE, verbose = TRUE)
  cand.res3

  RMSE.mat0  <- cand.res3$RMSE.mat
  RMSE.range <- range(RMSE.mat0)

  dev.new()
  par1 <- par(family="serif")
  par2 <- par(mar=c(5, 5, 2, 2))
  par3 <- par(mgp=c(3, 1, 0))
  image( S.arr0, Ea.arr0, RMSE.mat0, col = terrain.colors(200), axes = TRUE, 
         cex.axis = 1.5, cex.lab = 1.5, xlab = "Starting date (day-of-year)", 
         ylab = expression(paste(italic(E["a"]), " (kcal" %.% "mol"^{"-1"}, ")", sep = "")))
  points( cand.res3$S, cand.res3$Ea, cex = 1.5, pch = 16, col = 2 )
  contour( S.arr0, Ea.arr0, RMSE.mat0, levels = round(seq(RMSE.range[1], 
           RMSE.range[2], len = 20), 4), add = TRUE, cex = 1.5, col = "#696969", labcex = 1.5)
  par(par1)
  par(par2)
  par(par3)

  # graphics.off()

---

First flowering date records of Prunus armeniaca

Description

The data consist of the first flowering date records of Prunus armeniaca at the Summer Palace (39${}^{\circ}$54$'$38$''$ N, 116${}^{\circ}$8$'$28$''$ E, 50 m a.s.l.) in Beijing, China between 1963 and 2010 with the exception of 1969$-$1971, and 1997$-$2002. Data source: Chinese Phenological Observation Network (Guo et al., 2015).

Usage

data(apricotFFD)

Details

In the data set, there are two columns of vectors: Year and Time. Year saves the recording years; and Time saves the 1963$-$2010 first flowering dates of Prunus armeniaca (in day-of-year).

References

Guo, L., Xu, J., Dai, J., Cheng, J., Wu, H., Luedeling, E. (2015) Statistical identification of chilling and heat requirements for apricotflower buds in Beijing, China. Scientia Horticulturae 195, 138$-$144. doi:10.1016/j.scienta.2015.09.006

Examples

data(apricotFFD)
attach(apricotFFD)

dev.new()
par1 <- par(family="serif")
par2 <- par(mar=c(5, 5, 2, 2))
par3 <- par(mgp=c(3, 1, 0))
plot( Year, Time, asp = 1, cex.lab = 1.5, cex.axis = 1.5, 
      xlab = "Year", ylab = "First flowering date (day-of-year)" ) 
par(par1)
par(par2)
par(par3)

# graphics.off()

---

Daily Air Temperature Data of Beijing from 1952 to 2012.

Description

The data include the mean, minimum, and maximum daily temperatures (in ${}^{\circ}$C) of Beijing between 1952 and 2012. Data source: China Meteorological Data Service Centre (https://data.cma.cn/en).

Usage

data(BJDAT)

Details

In the data set, there are seven columns of vectors: Year, Month, Day, DOY, MDT, MinDT, and MaxDT. Year saves the recording years; Month saves the recording months; Day saves the recording days; DOY saves the dates in day-of-year; MDT saves the mean daily temperatures (in ${}^{\circ}$C) corresponding to DOY; MinDT saves the minimum daily temperatures (in ${}^{\circ}$C) corresponding to DOY; MaxDT saves the maximum daily temperatures (in ${}^{\circ}$C) corresponding to DOY.

References

Guo, L., Xu, J., Dai, J., Cheng, J., Wu, H., Luedeling, E. (2015) Statistical identification of chilling and heat requirements for apricotflower buds in Beijing, China. Scientia Horticulturae 195, 138$-$144. doi:10.1016/j.scienta.2015.09.006

Examples

data(BJDAT)
attach(BJDAT)

x    <- as.numeric( tapply(DOY, DOY, mean) )
y    <- as.numeric( tapply(MDT, DOY, mean) )
y.sd <- as.numeric( tapply(MDT, DOY, sd) )

dev.new()
par1 <- par(family="serif")
par2 <- par(mar=c(5, 5, 2, 2))
par3 <- par(mgp=c(3, 1, 0))
plot( x, y, cex = 1.5, xlim = c(0, 367), ylim = c(-10, 30), 
      cex.lab = 1.5, cex.axis = 1.5, type = "n", xlab = "Day-of-year", 
      ylab = expression(paste("Mean daily temperature (", degree, "C)", sep="")) )
for(i in 1:length(x)){
  lines(c(x[i], x[i]), c(y[i]-y.sd[i], y[i]+y.sd[i]), col=4)
}
points(x, y, cex = 1.5)
par(par1)
par(par2)
par(par3)

# graphics.off()

---

Prediction Function of the Accumulated Degree Days Method Using Mean Daily Temperatures

Description

Predicts the occurrence times using the accumulated degree days method based on observed or predicted mean daily air temperatures (Aono, 1993; Shi et al., 2017a, b).

Usage

predADD(S, T0, AADD, Year2, DOY, Temp, DOY.ul = 120)

Arguments

S	the starting date for thermal accumulation (in day-of-year)
T0	the base temperature (in ${}^{\circ}$C)
AADD	the expected annual accumulated degree days
Year2	the vector of the years recording the climate data for predicting 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

Details

In the accumulated degree days (ADD) method (Shi et al., 2017a, b), the starting date ($S$), the base temperature ($T_{0}$), and the annual accumulated degree days (AADD, which is denoted by $k$) are assumed to be constants across different years. Let $k_{i}$ denote the AADD of the $i$th year, which equals

$k_{i} = \sum_{j=S}^{E_{i}}\left(T_{ij}-T_{0}\right),$

where $E_{i}$ represents the ending date (in day-of-year), i.e., the occurrence time of a particular phenological event in the $i$th year, and $T_{ij}$ represents the mean daily temperature of the $j$th day of the $i$th year (in ${}^{\circ}$C). If $T_{ij} \le T_{0}$, $T_{ij} - T_{0}$ is defined to be zero. In theory, $k_{i} = k$, i.e., the AADD values of different years are a constant. However, in practice, there is a certain deviation of $k_{i}$ from $k$ that is estimated by $\overline{k}$ (i.e., the mean of the $k_{i}$ values). The following approach is used to determine the predicted occurrence time. When $\sum_{j=S}^{F}\left(T_{ij}-T_{0}\right) = \overline{k}$ (where $F \geq S$), it follows that $F$ is the predicted occurrence time; when $\sum_{j=S}^{F}\left(T_{ij}-T_{0}\right) < \overline{k}$ and $\sum_{j=S}^{F+1}\left(T_{ij}-T_{0}\right) > \overline{k}$, the trapezoid method (Ring and Harris, 1983) is used to determine the predicted occurrence time.

Value

Year	the years with climate data
Time.pred	the predicted occurrence times (day-of-year) in different years

Note

The entire mean daily temperature data set for the spring of each year should be provided.

Author(s)

Peijian Shi [email protected], Zhenghong Chen [email protected], Jing Tan [email protected], Brady K. Quinn [email protected].

References

Aono, Y. (1993) Climatological studies on blooming of cherry tree (Prunus yedoensis) by means of DTS method. Bulletin of the University of Osaka Prefecture. Ser. B, Agriculture and life sciences 45, 155$-$192 (in Japanese with English abstract).

Ring, D.R., Harris, M.K. (1983) Predicting pecan nut casebearer (Lepidoptera: Pyralidae) activity at College Station, Texas. Environmental Entomology 12, 482$-$486. 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. 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. doi:10.1093/aesa/sax063

See Also

ADD, ADD3

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.val      <- 65
T0.val     <- -0.5 
AADD.val   <- 235.5282

res2 <- predADD( S = S.val, T0 = T0.val, AADD = AADD.val, 
                 Year2 = Year2.val, DOY = DOY.val, Temp = Temp.val, 
                 DOY.ul = DOY.ul.val )
res2


ind1  <- res2$Year %in% intersect(res2$Year, Year1.val)
ind2  <- Year1.val %in% intersect(res2$Year, Year1.val)
RMSE1 <- sqrt( sum((Time.val[ind2]-res2$Time.pred[ind1])^2) / length(Time.val[ind2]) ) 
RMSE1

---

Prediction Function of the Accumulated Degree Days Method Using Minimum and Maximum Daily Temperatures

Description

Predicts the occurrence times using the accumulated degree days method based on observed or predicted minimum and maximum daily air temperatures (Aono, 1993; Shi et al., 2017a, b).

Usage

predADD2(S, T0, AADD, Year2, DOY, Tmin, Tmax, DOY.ul = 120)

Arguments

S	the starting date for thermal accumulation (in day-of-year)
T0	the base temperature (in ${}^{\circ}$C)
AADD	the expected annual accumulated degree days
Year2	the vector of the years recording the climate data for predicting the occurrence times
DOY	the vector of the dates (in day-of-year) for which climate data exist
Tmin	the minimum daily air temperature data (in ${}^{\circ}$C) corresponding to DOY
Tmax	the maximum daily air temperature data (in ${}^{\circ}$C) corresponding to DOY
DOY.ul	the upper limit of DOY used to predict the occurrence time

Details

In the accumulated degree days (ADD) method (Shi et al., 2017a, b), the starting date ($S$), the base temperature ($T_{0}$), and the annual accumulated degree days (AADD, which is denoted by $k$) are assumed to be constants across different years. Let $k_{i}$ denote the AADD of the $i$th year, which equals

$k_{i} = \sum_{j=S}^{E_{i}}\sum_{w=1}^{24}\frac{\left(T_{ijw}-T_{0}\right)}{24},$

where $E_{i}$ represents the ending date (in day-of-year), i.e., the occurrence time of a particular phenological event in the $i$th year, and $T_{ijw}$ represents the estimated mean hourly temperature of the $w$th hour of the $j$th day of the $i$th year (in ${}^{\circ}$C). If $T_{ijw} \le T_{0}$, $T_{ijw} - T_{0}$ is defined to be zero. This packages takes the method proposed by Zohner et al. (2020) to estimate the mean hourly temperature ($T_{w}$) for each of 24 hours:

$T_{w} = \frac{T_{\mathrm{max}} - T_{\mathrm{min}}}{2}\, \mathrm{sin}\left(\frac{w\pi}{12}- \frac{\pi}{2}\right)+\frac{T_{\mathrm{max}} + T_{\mathrm{min}}}{2},$

where $w$ represents the $w$th hour of a day, and $T_{\mathrm{min}}$ and $T_{\mathrm{max}}$ represent the minimum and maximum temperatures of the day, respectively.

$\qquad$In theory, $k_{i} = k$, i.e., the AADD values of different years are a constant. However, in practice, there is a certain deviation of $k_{i}$ from $k$ that is estimated by $\overline{k}$ (i.e., the mean of the $k_{i}$ values). The following approach is used to determine the predicted occurrence time. When $\sum_{j=S}^{F}\sum_{w=1}^{24}\left(T_{ijw}-T_{0}\right)/24 = \overline{k}$ (where $F \geq S$), it follows that $F$ is the predicted occurrence time; when $\sum_{j=S}^{F}\sum_{w=1}^{24}\left(T_{ijw}-T_{0}\right)/24 < \overline{k}$ and $\sum_{j=S}^{F+1}\sum_{w=1}^{24}\left(T_{ijw}-T_{0}\right)/24 > \overline{k}$, the trapezoid method (Ring and Harris, 1983) is used to determine the predicted occurrence time.

Value

Year	the years with climate data
Time.pred	the predicted occurrence times (day-of-year) in different years

Note

The entire minimum and maximum daily temperature data set for the spring of each year should be provided.

Author(s)

Peijian Shi [email protected], Zhenghong Chen [email protected], Jing Tan [email protected], Brady K. Quinn [email protected].

References

Aono, Y. (1993) Climatological studies on blooming of cherry tree (Prunus yedoensis) by means of DTS method. Bulletin of the University of Osaka Prefecture. Ser. B, Agriculture and life sciences 45, 155$-$192 (in Japanese with English abstract).

Ring, D.R., Harris, M.K. (1983) Predicting pecan nut casebearer (Lepidoptera: Pyralidae) activity at College Station, Texas. Environmental Entomology 12, 482$-$486. 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. 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. doi:10.1093/aesa/sax063

Zohner, C.M., Mo, L., Sebald, V., Renner, S.S. (2020) Leaf-out in northern ecotypes of wide-ranging trees requires less spring warming, enhancing the risk of spring frost damage at cold limits. Global Ecology and Biogeography 29, 1056$-$1072. doi:10.1111/geb.13088

See Also

ADD2, ADD4

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
Tmin.val   <- X2$MinDT
Tmax.val   <- X2$MaxDT
DOY.ul.val <- 120
S.val      <- 65
T0.val     <- 3.8
AADD.val   <- 136.5805


  cand.res2 <- predADD2( S = S.val, T0 = T0.val, AADD = AADD.val, Year2 = Year2.val, 
                         DOY = DOY.val, Tmin = Tmin.val, Tmax = Tmax.val, 
                         DOY.ul = DOY.ul.val )
  cand.res2


  ind1  <- cand.res2$Year %in% intersect(cand.res2$Year, Year1.val)
  ind2  <- Year1.val %in% intersect(cand.res2$Year, Year1.val)
  RMSE1 <- sqrt( sum((Time.val[ind2]-cand.res2$Time.pred[ind1])^2) / length(Time.val[ind2]) ) 
  RMSE1

---

Prediction Function of the Accumulated Developmental Progress Method Using Mean Daily Temperatures

Description

Predicts the occurrence times using the accumulated developmental progress (ADP) method based on observed or predicted mean daily air temperatures (Wagner et al., 1984; Shi et al., 2017a, b).

Usage

predADP(S, expr, theta, Year2, DOY, Temp, DOY.ul = 120)

Arguments

S	the starting date for thermal accumulation (in day-of-year)
expr	a user-defined model that is used in the accumulated developmental progress (ADP) method
theta	a vector saves the numerical values of the parameters in expr
Year2	the vector of the years recording the climate data for predicting 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

Details

Organisms exhibiting phenological events in early spring often experience several cold days during their development. In this case, Arrhenius' equation (Shi et al., 2017a, b, and references therein) has been recommended to describe the effect of the absolute temperature ($T$ in Kelvin [K]) on the developmental rate ($r$):

$r = \mathrm{exp}\left(B - \frac{E_{a}}{R\,T}\right),$

where $E_{a}$ represents the activation free energy (in kcal $\cdot$ mol${}^{-1}$); $R$ is the universal gas constant (= 1.987 cal $\cdot$ mol${}^{-1}$ $\cdot$ K${}^{-1}$); $B$ is a constant. To maintain consistency between the units used for $E_{a}$ and $R$, we need to re-assign $R$ to be 1.987$\times {10}^{-3}$, making its unit 1.987$\times {10}^{-3}$ kcal $\cdot$ mol${}^{-1}$ $\cdot$ K${}^{-1}$ in the above formula.

$\qquad$In the accumulated developmental progress (ADP) method, when the annual accumulated developmental progress (AADP) reaches 100%, the phenological event is predicted to occur for each year. Let $\mathrm{AADP}_{i}$ denote the AADP of the $i$th year, which equals

$\mathrm{AADP}_{i} = \sum_{j=S}^{E_{i}}r_{ij},$

where $E_{i}$ represents the ending date (in day-of-year), i.e., the occurrence time of a pariticular phenological event in the $i$th year. If the temperature-dependent developmental rate follows Arrhenius' equation, the AADP of the $i$th year is equal to

$\mathrm{AADP}_{i} = \sum_{j=S}^{E_{i}}\mathrm{exp}\left(B - \frac{E_{a}}{R\,T_{ij}}\right),$

where $T_{ij}$ represents the mean daily temperature of the $j$th day of the $i$th year (in 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.

$\qquad$The argument of 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.

Value

Year	the years with climate data
Time.pred	the predicted occurrence times (day-of-year) in different years

Note

The entire mean daily temperature data set for the spring of each year should be provided. It should be noted that the unit of Temp in Arguments is ${}^{\circ}$C, not K. In addition, when using Arrhenius' equation to describe $r$, to reduce the size of $B$ in this equation, Arrhenius' equation is multiplied by ${10}^{12}$ in calculating the AADP value for each year, i.e.,

$\mathrm{AADP}_{i} = \sum_{j=S}^{E_{i}}\left[{10}^{12} \cdot \mathrm{exp}\left(B - \frac{E_{a}}{R\,T_{ij}}\right)\right].$

Author(s)

Peijian Shi [email protected], Zhenghong Chen [email protected], Jing Tan [email protected], Brady K. Quinn [email protected].

References

Ring, D.R., Harris, M.K. (1983) Predicting pecan nut casebearer (Lepidoptera: Pyralidae) activity at College Station, Texas. Environmental Entomology 12, 482$-$486. 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. 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. 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. doi:10.1093/aesa/77.2.208

See Also

ADP

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.val      <- 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))
}

P0 <- c(-4.3787, 15.0431)
T2 <- seq(-10, 20, len = 2000)
r2 <- Arrhenius.eqn(P = P0, x = T2)

dev.new()
par1 <- par(family="serif")
par2 <- par(mar=c(5, 5, 2, 2))
par3 <- par(mgp=c(3, 1, 0))
plot( T2, r2, cex.lab = 1.5, cex.axis = 1.5, pch = 1, cex = 1.5, col = 2, type = "l", 
      xlab = expression(paste("Temperature (", degree, "C)", sep = "")), 
      ylab = expression(paste("Developmental rate (", {day}^{"-1"}, ")", sep="")) ) 
par(par1)
par(par2)
par(par3)

res6 <- predADP( S = S.val, expr = Arrhenius.eqn, theta = P0, Year2 = Year2.val, 
                 DOY = DOY.val, Temp = Temp.val, DOY.ul = DOY.ul.val )
res6

ind5 <- res6$Year %in% intersect(res6$Year, Year1.val)
ind6  <- Year1.val %in% intersect(res6$Year, Year1.val)
RMSE3 <- sqrt( sum((Time.val[ind6]-res6$Time.pred[ind5])^2) / length(Time.val[ind6]) ) 
RMSE3

---