# Prediction Function of the Accumulated Days Transferred to a Standardized Temperature Method Using Minimum and Maximum Daily Temperatures Predicts the occurrence times using the accumulated days transferred to a standardized temperature (ADTS) method based on observed or predicted minimum and maximum daily air temperatures (Konno and Sugihara, 1986; Aono, 1993; Shi et al., 2017a, b). ```r predADTS2(S, Ea, AADTS, Year2, DOY, Tmin, Tmax, DOY.ul = 120) ``` ## Arguments - `S`: the starting date for thermal accumulation (in day-of-year) - `Ea`: the activation free energy (in kcal $\cdot$ mol${}^{-1}$) - `AADTS`: the expected annual accumulated days transferred to a standardized temperature - `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 consistence 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. $\qquad$According to the definition of the developmental rate ($r$), it is the developmental progress per unit time (e.g., per day, per hour), which equals the reciprocal of the developmental duration $D$, i.e., $r = 1/D$. Let $T_{s}$ represent the standard temperature (in K), and $r_{s}$ represent the developmental rate at $T_{s}$. Let $r_{j}$ represent the developmental rate at $T_{j}$, an arbitrary temperature (in K). It is apparent that $D_{s}r_{s} = D_{j}r_{j} = 1$. It follows that $$ \frac{D_{s}}{D_{j}} = \frac{r_{j}}{r_{s}} =\mathrm{exp}\left[\frac{E_{a}\left(T_{j}-T_{s}\right)}{R\,T_{j}\,T_{s}}\right], $$ where $D_{s}/D_{j}$ is referred to as the number of days transferred to a standardized temperature (DTS) (Konno and Sugihara, 1986; Aono, 1993). $\qquad$In the accumulated days transferred to a standardized temperature (ADTS) method, the annual accumulated days transferred to a standardized temperature (AADTS) is assumed to be a constant. Let $\mathrm{AADTS}_{i}$ denote the AADTS of the $i$th year, which equals $$ \mathrm{AADTS}_{i} = \sum_{j=S}^{E_{i}}\sum_{w=1}^{24}\left\{\frac{1}{24}\,\mathrm{exp}\left[\frac{E_{a}\left(T_{ijw}-T_{s}\right)}{R\,T_{ijw}\,T_{s}}\right]\right\}, $$ 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, 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 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, $\mathrm{AADTS}_{i} = \mathrm{AADTS}$, i.e., the AADTS values of different years are a constant. However, in practice, there is a certain deviation of $\mathrm{AADTS}_{i}$ from $\mathrm{AADTS}$ that is estimated by $\overline{\mathrm{AADTS}}$ (i.e., the mean of the $\mathrm{AADTS}_{i}$ values). The following approach is used to determine the predicted occurrence time. When c("$\\sum_{j=S}^{F}\\sum_{w=1}^{24}\\left\\{\\frac{1}{24}\\,\\mathrm{exp}\\left[\\frac{E_{a}\\left(T_{ijw}-T_{s}\\right)}\n$", "${R\\,T_{ijw}\\,T_{s}}\\right]\\right\\} = \\overline{\\mathrm{AADTS}}$") (where $F \geq S$), it follows that $F$ is the predicted occurrence time; when c("$\\sum_{j=S}^{F}\\sum_{w=1}^{24}\\left\\{\\frac{1}{24}\\,\\mathrm{exp}\\left[\n$", "$\\frac{E_{a}\\left(T_{ijw}-T_{s}\\right)}{R\\,T_{ijw}\\,T_{s}}\\right]\\right\\} < \\overline{\\mathrm{AADTS}}$") and c("$\\sum_{j=S}^{F+1}\\sum_{w=1}^{24}\\left\\{\\frac{1}{24}\\,\\mathrm{exp}\\left[\\frac{E_{a}\\left(T_{ijw}-T_{s}\\right)}\n$", "${R\\,T_{ijw}\\,T_{s}}\\right]\\right\\} > \\overline{\\mathrm{AADTS}}$"), the trapezoid method (Ring and Harris, 1983) is used to determine the predicted occurrence time. ## Returns - **Year**: the years with climate data - **Time.pred**: the predicted occurrence times (day-of-year) in different years ## 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$-$192 (in Japanese with English abstract). Konno, T., Sugihara, S. (1986) Temperature index for characterizing biological activity in soil and its application to decomposition of soil organic matter. **Bulletin of National Institute for Agro-Environmental Sciences** 1, 51$-$68 (in Japanese with English abstract). Ring, D.R., Harris, M.K. (1983) Predicting pecan nut casebearer (Lepidoptera: Pyralidae) activity at College Station, Texas. **Environmental Entomology** 12, 482$-$486. tools:::Rd_expr_doi("10.1093/ee/12.2.482") Shi, P., Chen, Z., Reddy, G.V.P., Hui, C., Huang, J., Xiao, M. (2017a) Timing of cherry tree blooming: Contrasting effects of rising winter low temperatures and early spring temperatures. **Agricultural and Forest Meteorology** 240$-$241, 78$-$89. tools:::Rd_expr_doi("10.1016/j.agrformet.2017.04.001") Shi, P., Fan, M., Reddy, G.V.P. (2017b) Comparison of thermal performance equations in describing temperature-dependent developmental rates of insects: (III) Phenological applications. **Annals of the Entomological Society of America** 110, 558$-$564. tools:::Rd_expr_doi("10.1093/aesa/sax063") Zohner, C.M., Mo, L., Sebald, V., Renner, S.S. (2020) Leaf-out in northern ecotypes of wide-ranging trees requires less spring warming, enhancing the risk of spring frost damage at cold limits. **Global Ecology and Biogeography** 29, 1056$-$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. ## See Also `ADTS2` ## 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$MinDT Tmax.val <- X2$MaxD DOY.ul.val <- 120 S.val <- 46 Ea.val <- 22.3 AADTS.val <- 4.911035 cand.res4 <- predADTS2( S = S.val, Ea = Ea.val, AADTS = AADTS.val, Year2 = Year2.val, DOY = DOY.val, Tmin = Tmin.val, Tmax = Tmax.val, DOY.ul = DOY.ul.val ) cand.res4 ind3 <- cand.res4$Year %in% intersect(cand.res4$Year, Year1.val) ind4 <- Year1.val %in% intersect(cand.res4$Year, Year1.val) RMSE2 <- sqrt( sum((Time.val[ind4]-cand.res4$Time.pred[ind3])^2) / length(Time.val[ind4]) ) RMSE2 ```

predADTS2 function