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DOI: 10.2151/sola. 2022-007.

J-STAGE Advance published date: Jan. 26, 2022

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SOLA, 2022, Vol. 18, 36-39(TBA), doi:10.2151/sola.2022-007 1

 1 Determining multiple thresholds for thermal health risk levels

 2 using the segmented Poisson regression model

 3 Ju-Young Shin1, Kyu Rang Kim1, and Yong Hee Lee1

 4 1
 High-Impact Weather Research Department, National Institute of Meteorological

 5 Sciences, South Korea

 6

 7 Corresponding author: Ju-Young Shin, High-Impact Weather Research Department,

 8 National Institute of Meteorological Sciences, South Korea. E-mail: jyshin83@korea.kr.

 9

10 Abstract

11 Determining the thresholds for risk assessment is critical for the successful

12 implementation of thermal health warning systems. A risk assessment methodology with

13 multiple thresholds must be developed to provide detailed warning information to the

14 public and decision makers. This study developed a new methodology to identify multiple

15 thresholds for different risk levels for heat or cold wave events by considering

16 simultaneously impact on public health. A new objective function was designed to

17 optimize segmented Poisson regression, which relates public health to temperature

18 indicators. Thresholds were identified based on the values of the objective functions for

19 all threshold candidates. A case study in identifying thresholds for cold and heat wave

20 events in Seoul, South Korea, from 2014 to 2018, was conducted to evaluate the

21 appropriateness of the proposed methodology. Daily minimum or maximum air

22 temperature, mortality, and morbidity data were used for threshold identification and

2 Shin et al., Determining multiple thresholds for thermal health risk levels

 1 evaluation. The proposed methodology can successfully identify multiple thresholds to

 2 simultaneously represent different risk levels. These thresholds show comparable

 3 performance to those using the relative frequency approach.

 4

 5 (Shin, J.-Y., K. R. Kim, and Y. H. Lee, 2021: Determining multiple thresholds for

 6 thermal health risk levels using the segmented Poisson regression model. SOLA, 2022,

 7 Vol. 18, 36-39(TBA), doi:10.2151/sola.2022-007.)

 8

 9 1. Introduction

10 There is consensus on the acknowledgement of global warming in scientific

11 communities (Trenberth and Josey, 2007), which supports the fact that the thermal

12 environment has changed worldwide (Chi et al., 2018; Shin et al., 2021a; Tzanis et al.,

13 2019). These changes have led to an increase in the magnitude and frequency of heat

14 wave events (Estrada et al., 2021; Shin et al., 2020b; Varquez et al., 2020). Although the

15 magnitudes of cold wave events have decreased due to global warming, this decrease is

16 smaller than the increase in heat wave events in South Korea (Shin et al., 2022). This

17 implies that both heat wave and cold wave events adversely impact public health in some

18 regions, regardless of global warming. To mitigate the impacts of extreme thermal events,

19 a thermal health early warning system has been developed (Ebi, 2007; Ebi and Schmier,

20 2005; Issa et al., 2021).

21 Two components, the selection of temperature indicators and the risk level threshold,

22 are critical for the successful development of thermal health early warning systems

23 (Robinson, 2001; Smith et al., 2013). Several temperature indicators for assessing heat-

SOLA, 2022, Vol. 18, 36-39(TBA), doi:10.2151/sola.2022-007 3

 1 related risks to public health have been suggested (Barnett et al., 2010; Heo et al., 2019),

 2 and their effectiveness has been evaluated (Xu et al., 2018). Recently, physiology-based

 3 thermal comfort indices have been proposed to assess the risk from heat wave events (Di

 4 Napoli et al., 2018; Kang et al., 2020). Many studies in various regions have attempted to

 5 identify thresholds using different temperature indicators to assess the risk level posed by

 6 extreme thermal events (Cheng et al., 2019; Pantavou et al., 2018). Because most of these

 7 studies identified only one threshold to define heat wave events, varying risk levels that

 8 depend on the magnitude of the temperature were not assessed. To identify multiple

 9 thresholds using previous methods, the hierarchical approach has been suggested

10 (Muggeo, 2008). Thus, there is a need for a simple and fundamental methodology to

11 simultaneously identify thresholds for different risk levels, which considers the adverse

12 impacts of both types of extreme thermal events on public health.

13 The current study aimed to develop a novel methodology to identify multiple

14 thresholds for different risk levels based on public health data for both types of extreme

15 thermal events. The relationship between the temperature indicators and public health

16 was modeled using the segmented Poisson regression. A new objective function was

17 designed to fit the regression with the given constraints. The thresholds were identified

18 based on the values of the objective functions. A case study for identifying thresholds for

19 cold wave events in Seoul, South Korea, from 2014 to 2018, was conducted to evaluate

20 the appropriateness of the proposed methodology. This methodology is a powerful tool to

21 provide thresholds for different risk levels for extreme thermal events that account for

22 impacts on public health. These thresholds will aid the public in taking mitigating actions

23 and aid policy makers in establishing mitigation strategies.

24

4 Shin et al., Determining multiple thresholds for thermal health risk levels

 1 2. Theoretical background

 2 A simple Poisson regression has been broadly adopted to represent the relationship

 3 between the temperature indicators and public health (Ha and Kim, 2013; Lee et al., 2016).

 4 This regression typically cannot represent changes in relationships owing to its linear

 5 assumptions and the fact that positive and negative slopes can simultaneously exist. The

 6 segmented Poisson regression model can overcome these drawbacks (Muggeo, 2008). In

 7 the current study, a modified version of the segmented Poisson regression model was

 8 designed to identify thresholds and different risk levels.

 9

10 2.1 Segmented Poisson regression model

11 The segmented Poisson regression model can be formulated using Eqs. (1) and (2), as

12 follows:

13 ( | ) = exp ��∑ =1 � + �, −1 ≤ < , 1 ≤ ≤ (1)

14 ( | ) = exp �(∑ +1
 =1 ) + +1 �, ≤ (2)

15 where Y, x, , , −1 ( 0 ), and m are the mortality, temperature indicator, partial

16 slope of the ith range, intercept of jth range, jth threshold of the separating range, lower

17 bound of the temperature indicator (or minimum value of the temperature indicator), and

18 the number of thresholds, respectively. The partial slope indicates an increment of the

19 slope at the current range from the slope of the previous range. To avoid local fluctuations

20 in slopes due to the existence of negative and positive partial slope values in the regression

21 model, is restricted to positive numbers. To make continuous lines in the segmented

22 regression, the intercepts are given by Eq. (3).

SOLA, 2022, Vol. 18, 36-39(TBA), doi:10.2151/sola.2022-007 5

 = �∑ =1 � −1 + −1 , = 2, … , + 1
 1 � (3)
 1 = 1 , = 1

 2 Except for 1 , all intercepts are determined by the partial slope ( ) and threshold ( )

 3 estimates.

 4

 5 2.2 Algorithm to identify thresholds using the segmented Poisson regression model

 6 To fit the proposed segmented Poisson regression, an optimization algorithm was

 7 proposed that represents the maximum likelihood. The log-likelihood (LL) function of

 8 the segmented Poisson regression proposed in this study is given by Eq. (4).

 ��∑ =1 � + �
 9 LL = ∑ 
 =1 ∑ −1 ≤ 

6 Shin et al., Determining multiple thresholds for thermal health risk levels

 1 optimal parameters for the fitted segmented Poisson regression for the sample data. The

 2 thresholds cannot be directly estimated from the objective function because they must be

 3 fixed to calculate it.

 4 To find the threshold values, the objective functions of all threshold candidates and

 5 their the and estimates were calculated. The thresholds and and estimates

 6 that provide the minimum value for objective function are the optimal parameters for the

 7 fitted segmented Poisson regression model. The thresholds indicate different risk levels

 8 for target hazardous thermal events, such as heat waves or cold waves. The formulas

 9 presented in this study were designed for heat waves. For cold waves, the temperature

10 indicator data should be multiplied by negative one before they are used for threshold

11 identification. To optimize the objective function given in this study, the Nelder and Mead

12 optimization method (Nelder and Mead, 1965), that is a heuristic optimization, was used.

13

14 3. Case study

15 A case study of cold and heat waves in Seoul was conducted to evaluate the

16 appropriateness of the proposed methodology. Thus, the thresholds for different risk

17 levels for cold and heat wave events in Seoul were identified using the proposed

18 methodology, which modelled the relationship between the mortality data and minimum

19 or maximum air temperature using the segmented Poisson regression model. The

20 appropriateness of the identified thresholds was evaluated using morbidity data for cold-

21 or heat-related diseases based on several evaluation metrics.

22

SOLA, 2022, Vol. 18, 36-39(TBA), doi:10.2151/sola.2022-007 7

 1 3.1 Data

 2 Seoul, the capital city of South Korea, was the study region for the case study.

 3 Approximately eight million people live in this city. The geographical location of Seoul

 4 is shown in Fig. 1. Observed daily minimum and maximum air temperate data were

 5 collected from the Seoul station of the automated surface observing system operated by

 6 the Korea Meteorological Administration (data.kma.go.kr). In the winter, from December

 7 to February, the mean and standard deviation of the daily minimum air temperature were

 8 -4.8 ℃ and 4.7 ℃, respectively. In summer, from May to September, the mean and

 9 standard deviation of the daily maximum air temperature were 28.5 ℃ and 3.9 ℃,

10 respectively. Mortality and morbidity (the number of daily patients for cold- or heat-

11 related diseases) datasets were used in this study to represent public health. Because

12 patients with cold- or heat-related diseases rarely appeared, the number of patient

13 appearances (i.e., hospital admissions) was small. Thus, this dataset is inappropriate for

14 fitting the segmented Poisson regression model. Hence, mortality data were used to

15 identify the thresholds instead, and morbidity data were used to evaluate the

16 appropriateness of the mortality-identified thresholds. The mortality data, which included

17 all causes of death, were collected from Statistics Korea (mdis.kostat.go.kr), and the

18 morbidity data, which recorded the number of the patients admitted to a hospital for cold-

19 or heat-related diseases, were collected from the Korea Disease Control and Prevention

20 Agency (www.kdca.go.kr). The period for all data sets was from 2014 to 2018, with daily

21 temporal resolutions.

22

23 3.2 Results

24 In the current study, three thresholds were selected because three levels were adopted

8 Shin et al., Determining multiple thresholds for thermal health risk levels

 1 for assessing thermal health risks of cold or heat wave events in impact-based forecasts

 2 operated by KMA. The risk level representing disastrous case is not considered in this

 3 study. The first, second, and third thresholds indicated low, medium, and high risks,

 4 respectively (hereafter denoted as levels 1, 2, and 3, respectively). Fig. 2 presents the

 5 fitted segmented Poisson regression model and identifies the three thresholds based on

 6 mortality and daily minimum air temperature. The thresholds for levels 1, 2, and 3 were

 7 -7 ℃, -10 ℃, and -12 ℃, respectively. As shown in Fig. 2, the fitted regression line

 8 successfully represents the relationship between mortality and daily minimum air

 9 temperature.

10 To evaluate the appropriateness of the identified thresholds, several metrics (Shin et al.,

11 2020a; Shin et al., 2021b) specifically accuracy, precision, recall, false alarm rate, and F1

12 score, were calculated based on the morbidity data. A relative frequency approach was

13 employed as a baseline method for this comparison, and 5%, 3%, and 1% were adopted

14 for the relative frequencies for levels 1, 2, and 3, respectively (Lee et al., 2018). The

15 relative frequency threshold values for levels 1, 2, and 3 based on daily minimum air

16 temperature data in Seoul were -9 ℃, -10 ℃, and -13 ℃, respectively. A binary

17 classification problem was assumed for the calculation of the evaluation metrics because

18 it was difficult to classify the low, medium, and high risks based on morbidity data. For

19 example, in the case of level 1, the total number of true positives indicated the total

20 number of days in which a hospital admission was reported for cold-related diseases when

21 the daily minimum air temperature was below -7 ℃. Hence, the evaluation metrics for

22 level 1 account for the range below -7 ℃ instead of a range between the thresholds of

23 levels 1 and 2 (-10 ℃< x ≤-7 ℃).

24

SOLA, 2022, Vol. 18, 36-39(TBA), doi:10.2151/sola.2022-007 9

 1

 2 Table 1 presents the evaluation metrics for the appropriateness of the identified

 3 thresholds as well as the annual frequency of warnings. However, it is difficult to

 4 determine the best threshold set based on the calculated evaluation metrics.

 5 To investigate the characteristics of the identified thresholds, a graphical

 6 investigation was conducted using the relationship between the daily minimum air

 7 temperature and the number of patients (morbidity); this is plotted in Fig. 3. The figure

 8 allows the appropriateness of the identified thresholds to be visually evaluated based on

 9 the range between them. As shown in Fig. 3 (a), the number of daily patients increased

10 past the level 1 threshold; thus, this range appears to successfully represent different risk

11 levels. This supports the hypothesis that the thresholds identified by the proposed

12 methodology successfully differentiate varied risk levels for cold waves. As shown in Fig.

13 3 (b), the level 2 and 3 thresholds using the relative frequency adequately represented the

14 risk levels for cold waves based on the morbidity data. Unlike these thresholds, the range

15 covered by the level 1 threshold using the relative frequency was too narrow to represent

16 low-risk cold waves. Because this threshold does not consider these events, the level 1

17 threshold using the proposed methodology may be more appropriate for low-risk events,

18 based on morbidity data. Therefore, the thresholds identified by the proposed

19 methodology would be a good option to provide thresholds for different risk levels for

20 cold waves in Seoul, South Korea. The time series of the number of daily patients and the

21 risk levels for the identified thresholds are presented in Fig. 3 (c) and (d) to facilitate

22 examination of the detailed threshold characteristics. Based on this information, the

23 period from December 1, 2017, to February 28, 2018, was employed for investigation.

24 Warnings based on the thresholds using the proposed method successfully provided risk

10 Shin et al., Determining multiple thresholds for thermal health risk levels

 1 information. However, some warnings were not perfectly matched to the hospital

 2 admission dates; for example, warnings were issued on January 3 and 4, 2018. Hospital

 3 admissions were not reported on those days but rather on January 2 and 5. For levels 2

 4 and 3, the thresholds using the relative frequency performed similarly to those using the

 5 proposed method. However, the thresholds using the relative frequency failed to provide

 6 some warnings, particularly on low-risk event days (i.e., level 1). During this period, the

 7 proposed method can provide more appropriate thresholds than the relative frequency for

 8 assessing the risks of cold waves in Seoul.

 9 For heat wave events in Seoul, thresholds of risk level 1, 2, and 3 identified by the

10 proposed method are 31 ℃, 34 ℃, and 35 ℃, respectively. Thresholds obtained by the

11 relative frequency are 32 ℃, 34 ℃, and 36 ℃, respectively. The evaluation metrics for

12 these thresholds are summarized in Table 1. The differences between thresholds from two

13 methods are occurred in level 1 and 3. Based on F1 score, the thresholds identified by the

14 proposed method lead to better performance than those by relative frequency. Fig. 4

15 presents the fitted segmented Poisson regression. The slopes within level 3 is much

16 steeper than that for level 1 and 2. This implies that the heat-related health risk become

17 very severe when the daily maximum temperature is higher than 35 ℃. To investigate the

18 characteristics of the identified thresholds, a graphical investigation was conducted using

19 the relationship between the daily maximum air temperature and the number of patients

20 (morbidity); this is plotted in Fig. 5 (a) and (b). It is difficult to compare performances for

21 two threshold sets from the two methods based on visual inspection. For threshold of level

22 1, both methods seem to be good enough to detect heat-related risk. For threshold of level

23 3, the proposed method seems to be better than the relative frequency because the

24 threshold by the relative frequency misses many patients (larger than 18 persons per day)

SOLA, 2022, Vol. 18, 36-39(TBA), doi:10.2151/sola.2022-007 11

 1 within 34 ℃ – 35 ℃. Results of evaluation metric also are evidence to support that the

 2 threshold of level 3 identified by proposed method is more appropriate than that by

 3 relative frequency. The time series of the number of daily patients and the risk levels for

 4 the identified thresholds are presented in Fig. 5 (c) and (d) to facilitate examination of the

 5 detailed threshold characteristics. Based on this information, the period from May 1, 2018,

 6 to September 31, 2018, was employed for investigation. Both methods successfully warn

 7 very high risk occurred in early August. The relative frequency leads to level 2 warning

 8 from mid-July to mid-August when the number of patients is higher than 18. These days

 9 seems to be warned for level 3 instead of level 2 based on the number of patients. Hence,

10 the threshold of level 3 identified by the proposed method would be good for warning

11 very high risk in Seoul.

12

13 4. Discussion and Conclusions

14 The current study proposed a novel methodology to identify multiple thresholds for

15 different risk levels for the adverse impacts of heat and cold wave events on public health.

16 In the proposed methodology, the relationship between public health data and temperature

17 indicators was modeled using a segmented Poisson regression model. The thresholds

18 were determined by examining all candidate threshold sets based on the values of the

19 proposed objective function. For evaluation, a case study of cold and heat waves in Seoul,

20 South Korea, was conducted. The appropriateness of the proposed methodology was

21 evaluated using morbidity data and compared to the thresholds that used a relative

22 frequency approach. The results of this case study show that the proposed methodology

23 can successfully provide multiple thresholds to simultaneously represent the risk levels

12 Shin et al., Determining multiple thresholds for thermal health risk levels

 1 of cold waves. Based on the results of the analysis for cold waves in Seoul, South Korea,

 2 these thresholds are more robust than the thresholds using the relative frequency.

 3 However, since the number of stations is small and period of used data set is short, there

 4 may be large uncertainties for appropriateness evaluation in other regions. To reduce these

 5 uncertainties, the appropriateness of the proposed methodology should be assessed using

 6 massive data sets in the future.

 7 Public health data, such as mortality and morbidity, must be used to implement the

 8 proposed methodology for identifying risk level thresholds. These datasets are often

 9 unavailable because of the lack of a monitoring system and privacy issues. Thus, when

10 public health data are not accessible, thresholds based on relative frequency would be a

11 good option. However, because the relative frequency cannot account for the relationship

12 between the temperature indicators and public health, the appropriateness of the

13 thresholds identified by the relative frequency should be carefully investigated.

14 Additionally, the methodology proposed in this study can identify threshold values, but it

15 cannot identify the optimal number of thresholds. Because a predefined number of

16 thresholds is employed, there is still uncertainty in identifying different risk level

17 thresholds. Longden (2018) suggested a methodology to find the optimal number of

18 thresholds based on the F-test to determine the thresholds for different risk levels for heat

19 waves. The suggested methodology cannot be directly employed in this study because the

20 formulation and basic assumptions proposed in this study differ. Although the number of

21 thresholds is often determined by political reasoning, the development of a methodology

22 to determine the optimal number of thresholds would be beneficial because it can provide

23 fundamental information for policymakers in establishing a thermal health warning

24 system. Thus, a methodology to determine the optimal number of thresholds should be

SOLA, 2022, Vol. 18, 36-39(TBA), doi:10.2151/sola.2022-007 13

1 developed in the future.

2

3 Acknowledgements

4 This work was funded by the Korea Meteorological Administration Research and

5 Development Program " Development of Production Techniques on User-Customized

6 Weather information" under Grant (KMA2018-00622).

7

14 Shin et al., Determining multiple thresholds for thermal health risk levels

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16

17 List of Figure Captions

18 Fig. 1. Geographical location of Seoul.

19 Fig. 2. The fitted segmented Poisson regression based on mortality and daily minimum air

20 temperature in Seoul. Green, yellow, red, and blue colored solid lines indicate fitted lines for

21 days with level 1, level 2, level 3, and without cold waves, respectively.

22 Fig. 3. Comparison of thresholds identified using the proposed method and relative frequency for

23 the number of daily patients of cold-related diseases in Seoul: (a) risk levels based on the

24 proposed method, (b) risk levels based on the relative frequency, (c) the time series from

18 Shin et al., Determining multiple thresholds for thermal health risk levels

 1 December 1, 2017 to February 28, 2018 with the risk levels based on the proposed method,

 2 and (d) the time series with the risk levels based on the relative frequency. Green, yellow,

 3 and red colored areas indicate daily minimum air temperature thresholds for level 1, 2, and

 4 3, respectively.

 5 Fig. 4. The fitted segmented Poisson regression based on mortality and daily maximum air

 6 temperature in Seoul. Green, yellow, red, and blue colored solid lines indicate fitted lines for

 7 days with level 1, level 2, level 3, and without cold waves, respectively.

 8 Fig. 5 Comparison of thresholds identified using the proposed method and relative frequency for

 9 the number of daily patients of heat-related diseases in Seoul: (a) risk levels based on the

10 proposed method, (b) risk levels based on the relative frequency, (c) the time series from

11 May 1 to September 31, 2018 with the risk levels based on the proposed method, and (d) the

12 time series with the risk levels based on the relative frequency. Green, yellow, and red

13 colored areas indicate daily minimum air temperature thresholds for level 1, 2, and 3,

14 respectively.

15

16 List of Table Captions

17 Table 1: Evaluation metrics for the appropriateness of the identified thresholds.

18

SOLA, 2022, Vol. 18, 36-39(TBA), doi:10.2151/sola.2022-007 19

 1

 2

 3

 4

 5

 6 Fig. 1. Geographical location of Seoul.
 7

 8

 9

10

20 Shin et al., Determining multiple thresholds for thermal health risk levels

1

2

3

4 Fig. 2. The fitted segmented Poisson regression based on mortality and daily minimum
5 air temperature in Seoul. Green, yellow, red, and blue colored solid lines indicate fitted
6 lines for days with level 1, level 2, level 3, and without cold waves, respectively.

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SOLA, 2022, Vol. 18, 36-39(TBA), doi:10.2151/sola.2022-007 21

 1

 2

 3 Fig. 3. Comparison of thresholds identified using the proposed method and relative
 4 frequency for the number of daily patients of cold-related diseases in Seoul: (a) risk levels
 5 based on the proposed method, (b) risk levels based on the relative frequency, (c) the time
 6 series from December 1, 2017 to February 28, 2018 with the risk levels based on the
 7 proposed method, and (d) the time series with the risk levels based on the relative
 8 frequency. Green, yellow, and red colored areas indicate daily minimum air temperature
 9 thresholds for level 1, 2, and 3, respectively.

10

22 Shin et al., Determining multiple thresholds for thermal health risk levels

1

2

3

4 Fig. 4. The fitted segmented Poisson regression based on mortality and daily maximum
5 air temperature in Seoul. Green, yellow, red, and blue colored solid lines indicate fitted
6 lines for days with level 1, level 2, level 3, and without heat waves, respectively.

7

SOLA, 2022, Vol. 18, 36-39(TBA), doi:10.2151/sola.2022-007 23

 1

 2

 3 Fig. 5. Comparison of thresholds identified using the proposed method and relative
 4 frequency for the number of daily patients of heat-related diseases in Seoul: (a) risk levels
 5 based on the proposed method, (b) risk levels based on the relative frequency, (c) the time
 6 series from May 1 to September 31, 2018 with the risk levels based on the proposed
 7 method, and (d) the time series with the risk levels based on the relative frequency. Green,
 8 yellow, and red colored areas indicate daily minimum air temperature thresholds for level
 9 1, 2, and 3, respectively.

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24 Shin et al., Determining multiple thresholds for thermal health risk levels

1

2

3 Table 1. Evaluation metrics for the appropriateness of the identified thresholds. Note
4 that ranges presented in this table used only for evaluation due to use of metrics for a
5 binary classification problem.
6
 Cold wave
 Segmented Poisson regression Relative frequency
 Name L1 L2 L3 L1 L2 L3
 (≤-7℃) (≤-10℃) (≤-12℃) (≤-9℃) (≤-10℃) (≤-13℃)
 Annual frequency
 29.2 10.6 4.6 15.6 10.6 3.2
 of warning (day)
 Accuracy 0.667 0.798 0.825 0.778 0.798 0.831
 Precision 0.240 0.340 0.391 0.333 0.340 0.438
 Recall 0.473 0.243 0.122 0.351 0.243 0.095
 False alarm rate 0.294 0.093 0.037 0.138 0.093 0.024
 F1 score 0.318 0.283 0.186 0.342 0.283 0.156
 Heat wave
 L1 L2 L3 L1 L2 L3
 Name
 (≥31℃) (≥34℃) (≥35℃) (≥32℃) (≥34℃) (≥36℃)
 Annual frequency
 24.6 5.8 7.0 14.0 9.0 3.8
 of warning (day)
 Accuracy 0.831 0.807 0.774 0.841 0.807 0.753
 Precision 0.711 0.969 1.000 0.821 0.969 1.000
 Recall 0.639 0.298 0.168 0.529 0.298 0.091
 False alarm rate 0.097 0.004 0.000 0.043 0.004 0.000
 F1 score 0.673 0.456 0.288 0.643 0.456 0.167
7