# Prediction Function of the Accumulated Developmental Progress Method Using Minimum and Maximum Daily Temperatures Predicts the occurrence times using the accumulated developmental progress (ADP) method based on observed or predicted minimum and maximum daily air temperatures (Wagner et al., 1984; Shi et al., 2017a, b). ```r predADP2(S, expr, theta, Year2, DOY, Tmin, Tmax, 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 - `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 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 $$ \mathrm{AADP}_{i} = \sum_{j=S}^{E_{i}}\sum_{w=1}^{24}\frac{r_{ijw}}{24}, $$ 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. $r_{ijw}$ is the developmental rate (per hour), which is transferred to $r_{ij}$ (per day) by dividing 24. If the temperature-dependent developmental rate follows Arrhenius' equation, the AADP of the $i$th year is equal to $$ \mathrm{AADP}_{i} = \sum_{j=S}^{E_{i}}\sum_{w=1}^{24}\left\{\frac{1}{24}\mathrm{exp}\left(B - \frac{E_{a}}{R\,T_{ijw}}\right)\right\}, $$ where $T_{ijw}$ represents the estimated mean hourly temperature of the $w$th hour of the $j$th day of the $i$th year (in K). 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, $\mathrm{AADP}_{i} = 100\%$, 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 $\sum_{j=S}^{F}\sum_{w=1}^{24}\left(r_{ijw}/24\right) = 100\%$ (where $F \geq S$), it follows that $F$ is the predicted occurrence time; when $\sum_{j=S}^{F}\sum_{w=1}^{24}\left(r_{ijw}/24\right) < 100\%$ and $\sum_{j=S}^{F+1}\sum_{w=1}^{24}\left(r_{ijw}/24\right) > 100\%$, 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. ## Returns - **Year**: the years with climate data - **Time.pred**: the predicted occurrence times (day-of-year) in different years ## References Ring, D.R., Harris, M.K. (1983) Predicting pecan nut casebearer (Lepidoptera: Pyralidae) activity at College Station, Texas. **Environmental Entomology** 12, 482$-$486. tools:::Rd_expr_doi("10.1093/ee/12.2.482") Shi, P., Chen, Z., Reddy, G.V.P., Hui, C., Huang, J., Xiao, M. (2017a) Timing of cherry tree blooming: Contrasting effects of rising winter low temperatures and early spring temperatures. **Agricultural and Forest Meteorology** 240$-$241, 78$-$89. tools:::Rd_expr_doi("10.1016/j.agrformet.2017.04.001") Shi, P., Fan, M., Reddy, G.V.P. (2017b) Comparison of thermal performance equations in describing temperature-dependent developmental rates of insects: (III) Phenological applications. **Annals of the Entomological Society of America** 110, 558$-$564. tools:::Rd_expr_doi("10.1093/aesa/sax063") Wagner, T.L., Wu, H.-I., Sharpe, P.J.H., Shcoolfield, R.M., Coulson, R.N. (1984) Modelling insect development rates: a literature review and application of a biophysical model. **Annals of the Entomological Society of America** 77, 208$-$225. tools:::Rd_expr_doi("10.1093/aesa/77.2.208") Zohner, C.M., Mo, L., Sebald, V., Renner, S.S. (2020) Leaf-out in northern ecotypes of wide-ranging trees requires less spring warming, enhancing the risk of spring frost damage at cold limits. **Global Ecology and Biogeography** 29, 1056$-$1072. tools:::Rd_expr_doi("10.1111/geb.13088") ## Author(s) Peijian Shi pjshi@njfu.edu.cn , Zhenghong Chen chenzh64@126.com , Jing Tan jmjwyb@163.com , Brady K. Quinn Brady.Quinn@dfo-mpo.gc.ca . ## Note The entire minimum and maximum daily temperature data set for the spring of each year should be provided. It should be noted that the unit of `Tmin` and `Tmax` 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., $$ \mathrm{AADP}_{i} = \sum_{j=S}^{E_{i}}\sum_{w=1}^{24}\left[{10}^{12} \cdot \frac{1}{24} \cdot \mathrm{exp}\left(B - \frac{E_{a}}{R\,T_{ijw}}\right)\right]. $$ ## See Also `ADP2` ## Examples ```r 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 <- 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)) } P0 <- c(8.220327, 22.185942) 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) cand.res6 <- predADP2( S = S.val, expr = Arrhenius.eqn, theta = P0, Year2 = Year2.val, DOY = DOY.val, Tmin = Tmin.val, Tmax = Tmax.val, DOY.ul = DOY.ul.val ) cand.res6 ind5 <- cand.res6$Year %in% intersect(cand.res6$Year, Year1.val) ind6 <- Year1.val %in% intersect(cand.res6$Year, Year1.val) RMSE3 <- sqrt( sum((Time.val[ind6]-cand.res6$Time.pred[ind5])^2) / length(Time.val[ind6]) ) RMSE3 ```

predADP2 function