Korean Journal of Agricultural and Forest Meteorology, Vol. 16, No. 4, (2014), pp. 304~308
ⓒ Author(s) 2014. CC Attribution 3.0 License.
(2014년 09월 05일 접수; 2014년 11월 04일 수정; 2014년 11월 04일 수락)
Using Sequential Optimization
University of Washington, Seattle, WA 98195, USA
(Received September 05, 2014; Revised November 04, 2014; Accepted November 04, 2014)
Cherry blossoms are celebrated to signify the onset of spring in many countries and cultures around the world. Cities and towns organized cherry blossom festivals that are centered on peak bloom of their local cherry trees. Therefore, it is critical to predict peak bloom date of these trees accurately using local climate data. The phenology of flowering trees including dormancy release, bud break, floral development, and peak bloom is often described using thermal time accumulations. A two-step method was proposed to predict bud-burst after dormancy release, which was modeled with two consecutive steps of chill and heat unit accumulations (Cesaraccio et al., 2004). The model was later modified to predict flowering time, with an assumption that the flowering was also a consequence of extended heat accumulation (Yun, 2006).
While this approach was able to estimate peak bloom dates of flowering cherry trees in the Tidal Basin, Washington D.C., with reasonable accuracy (Chung et al., 2011), adapting the model for new location and variety required an arduous parameterization process through extensive grid search. This brute-force method has several shortcomings. Firstly, a large amount of computation time was required for the grid search when finding an optimal parameter set. Secondly, the parameter space was indeed not unimodal, leading to multiple solutions with a low cost of error. Thirdly, available observation records were not fully exploited, but only peak bloom dates were used. Intermediate development stages are often reported together and they could possibly improve parameterization.
In this study, we propose a sequential parameter calibration algorithm to address these issues. Each parameter is optimized individually with an additional constraint on heat accumulation between two development stages. When parameterized with the existing dataset from Tidal Basin, the result from the new approach was almost identical to the known solution found by brute-force method but with smaller errors and considerably reduced computation times (i.e., less than a minute compared to days).
The phenology of flowering is modeled with consecutive accumulation of two types of thermal units: chill days and anti-chill days. The unit is a degree day above 0 ℃ and partitioned with base temperature (Tc) and its relation to daily maximum and minimum temperatures (Cesaraccio et al., 2004). Heating unit (Ca) is calculated first and its difference with total degree day becomes chilling unit (Cd). Once a rest period of dormancy is initiated in October 1, the chilling units need to be accumulated up to chill requirement (Rc). It is then considered the dormancy has released and moved onto a quiescence period. From this point, the heating units are accumulated in the opposite direction to reach heat requirement (Rh) until peak bloom occurs. Consequently, an estimated peak bloom date (PB) ̂ of year y can be obtained by a function f of three parameters, Tc, Rc, and Rh, as shown in equation (1). Figure 1 illustrates an example run of the model with associated parameters and variables.
The U.S. National Park Service monitors flower development of cherry trees (e.g., Prunus × yedoensis) in the Tidal Basin, Washington D.C. in regards with five distinct stages: green color in buds (GC), florets visible (FV), extension of florets (EF), peduncle elongation (PE), puffy white (PW), and peak bloom (PB). The first stage is usually detected in mid February to early March, and then followed by next stages until peak bloom, which usually happens around late March or early April. As visible changes should be accompanied by bud burst after dormancy release (DR), noticing these stages presumably mean that it is in the second step of the model, where only heat accumulation is counted. Equation (2) states that the heat requirement δ for year y between any two stages, α and β, is simply a sum of heat accumulation Ca in the corresponding period p. Tc is required for driving Ca, but Rc is not, because only the relative difference is needed in this case.
Calibrating three parameters, Tc, Rc, and Rh, is an optimization problem of minimizing sum of squared differences between the estimated peak bloom dates ((PB) ̂) and the actual dates (PB) observed in the site. Instead of a time-consuming grid search fitting all parameters simultaneously (Chung et al., 2011), a sequential algorithm is applied for fitting each parameter separately, using other parameters discovered in the previous steps. Sequential parameter estimation is often preferred over simultaneous procedure when phenology parameters have no known interdependencies (Wallach et al., 2014).
Tc is the first parameter in a series as it controls the amount of thermal units, thus affecting how much degree Rc and Rh are being filled up. Assuming an equivalent heat requirement for each year y between the first and last stages, GC and PB, T ̂c is estimated by minimizing sum of annual variances as shown in equation (3).
Rc can be found in a similar manner. With a chilling unit (Cd) for the range determined by a given T ̂c, a dormancy release date (DR) now becomes a function of Rc. R ̂c is then estimated by minimizing sum of variances of the entire heat requirement δTc,DR,PB)=Rh-Rc) as shown in equation (4).
After two parameters revealed, Rh is solved by an original scheme shown in equation (5) that minimizes sum of differences between estimated and actual peak bloom dates, (PB) ̂ and PB, respectively.
Phenology observation records for cherry trees (Prunus × yedoensis) in the Tidal Basin, Washington D.C. were used for parameterization. The starting year was set to 1992 because the detailed records have been only available since then. The end year was chosen to be 2010 for comparison with previous research (Chung et al., 2011).
Our approach resulted in Tc estimate of 4.28 ℃, which is almost identical to the solution found in Chung et al. (2011) using a grid search method (Table 1). On the other hand, our Rc and Rh were slightly larger by 14.52 and 17.03 respectively than those from Chung et al. (2011). Interestingly, the entire heat requirements (Rh-Rc) were still very close to each other. It would suggest that our solution successfully captured a right amount of heat accumulation required for the phenology development.
As the performance of the previous algorithm was dependent on the grid size, it could easily become very slow. A fine-grained search whose Tc spanning from 0.0 ℃ to 10.0 ℃ by 0.1 ℃, Rc from -200 to 0 by 1, and Rh from 100 to 200 by 1, would generate a grid of 100∙200∙200=4,000,000 points, which may take up to several days to run. On the other hand, our algorithm had a constant running time, less than a minute, which would be almost negligible on regular circumstances.
Various metrics indicated that our solution was marginally better in terms of error, at least during the calibration period. Figure 2 shows a plot of peak bloom dates and corresponding errors from each parameter set.
The most important part of our sequential optimization algorithm is lying on the use of additional constraints formed by observing multiple phenological stages. Not only GB used in the proposed method, different stages could have been plugged into α for estimating Tc in equation (3). However, the profiling result shown in Figure 3 suggested that other constraints might not be sufficient to construct a stable constraint. Only GB and FV had approached a previously known value of Tc, 4.3 ℃, with 4.28 ℃ and 4.40 ℃ respectively, while others had kept decreasing like until PW reached 0.0001 ℃.
The heat variance is basically controlled by Tc in the middle balancing out an amount of heating and chilling units. Once Tc becomes smaller than minimum temperature, it can no longer help minimizing the variance. Tc of 0 ℃ is actually a boundary case by definition where no chilling unit can be generated at all. Therefore, later stages from much warm environments are apparently more susceptible to degeneracy.
A sequential algorithm could estimate a decent parameter set that almost matches up with one found from a brute-force method. The entire process took only a fraction of time compared to the previous method, while calibration error was even smaller. However, its own drawbacks were also revealed. Observation of other development stage is mandatory and requires extra attention on its selection. Temperature range between two stages should span around base temperature, Tc, to ensure valid thermal accumulation.
We propose a sequential parameter calibration algorithm for a cherry blossom phenology model. While shown its own strength in terms of speed and accuracy, it may not be always applicable because of the extra requirement. Additional constraints should be carefully chosen from intervening development stages that can balance out thermal unit generation. To apply this method successfully, it is advised to expand observations to include earlier phenological developments during heat accumulation step for avoiding the degeneracy issue on optimization.
벚꽃의 만개일은 관련 행사일정을 결정하는 중요한 요소로써 생육기간 중 기온에 따른 변화의 폭이 크다. 이를 예측하기 위한 방법으로는 벚꽃의 발달을 휴면기와 생장기의 2단계로 구분하여 저온(chill)과 고온(heat) 요구에 대한 온도시간(thermal time) 누적을 기술하는 모형이 개발되어 있다. 하지만 모수 추정시 모수공간내 일정 간격의 격자 전체를 계산하여 많은 시간을 소모한다는 단점이 있었다. 본 연구에서는 기존 모형이 고려하지 않던 벚꽃 발달의 중간단계 관측자료를 활용하여 고온요구에 대한 새로운 조건을 추가하고, 이를 기반으로 각 모수를 순차적으로 추정하여 최적화 시간을 단축하는 새로운 방법을 제안한다. 미국 워싱턴 DC 지역의 벚꽃개화 관측 자료를 기준으로 검증한 결과, 기존 모형에서 제안된 모수와 근사한 값을 단축된 시간 내에 계산해내는 것을 확인하였다.
This study was carried out with the support of “Cooperative Research Program for Agricultural Science & Technology Development (Project No. PJ01000707)”, Rural
Development Administration, Republic of Korea.
Cesaraccio, C., D. Spano, R. L. Snyder, and P. Duce, 2004: Chilling and forcing model to predict bud-burst of crop and forest species. Agricultural and Forest Meteorology 126(1-2), 1-13.
Chung, U., L. Mack, J. I. Yun, and S.-H. Kim, 2011: Predicting the timing of cherry blossoms in Washington, DC and Mid-Atlantic states in response to climate change. PloS one 6(11), e27439.
Wallach, D., D. Makowski, J. W. Jones, and F. Brun, 2013: Working with Dynamic Crop Models (2nd ed.). Academic Press, 247-248.
Yun, J. I., 2006: Climate change impact on the flowering season of Japanese cherry (prunus serrulata var. spontanea) in Korea during 1941-2100. Korean Journal of Agricultural and Forestry Meteorology 8(2), 68-76.