ecoqu EconoQuantum EconoQuantum 1870-6622 2007-9869 Universidad de Guadalajara 10.18381/eq.v18i2.7225 Artículos Hierarchical forecasts of Diabetes mortality in Mexico by marginalization and sex to establish resource allocation Pronósticos jerárquicos de mortalidad por diabetes en México por marginación y sexo para establecer asignación de recursos 0000-0003-0499-0446 Silva Eliud * 0000-0003-4289-0075 Sparks Corey ** Universidad Anáhuac, México. México. Correo electrónico: jose.silva@anahuac.mx. Universidad Anáhuac Universidad Anáhuac México Mexico jose.silva@anahuac.mx The University of Texas at San Antonio. USA. Correo electrónico: corey.sparks@utsa.edu. University of Texas The University of Texas USA corey.sparks@utsa.edu 30 07 2021 Jul-Dec 2021 18 02 82 98 25 10 2020 10 03 2021 This is an open-access article distributed under the terms of the Creative Commons Attribution License Abstract: <italic>Objective:</italic>

The mexican population has experimented an astounding rise in type II Diabetes mortality as well as a growing trend for the economic burden in the recent years. The paper’s purpose is to propose an approach to establish a distribution of resource allocation objectively to face the future economic burden.

 <italic>Methodology:</italic>

Hierarchical forecasts of Diabetes mortality to 2030 by sub-domains of the population are estimated based on marginalization and sex.

 <italic>Results:</italic>

The forecasts confirm that differences related to sub-domains will be significant. In fact, the rates will increase most notably both in low and high marginalized.

 <italic>Limitations</italic>:

The hierarchical method just provide point forecast without prediction intervals.

 <italic>Originality</italic>:

There is not a similar application for mexican data to do that objectively.

 <italic>Conclusions</italic>:

The most recommendable budget distribution should be mainly addressed among the low and high levels. Implications of these estimates should support unpostponable health policy in general and for the mentioned sub-domains in particular.

 Resumen: <italic>Objetivo:</italic>

La población mexicana ha experimentado un aumento asombroso en la mortalidad por Diabetes tipo II, así como una tendencia creciente de su carga económica recientemente. El propósito del trabajo es proponer un enfoque para establecer una distribución de la asignación objetiva de recursos para enfrentar su carga futura.

<italic>Metodología:</italic>

Se estiman pronósticos jerárquicos de mortalidad por Diabetes tipo II al 2030 por subdominios de la población por marginación y sexo.

 <italic>Resultados:</italic>

Se confirma que las diferencias de los subdominios serán significativas. De hecho, las tasas aumentarán de manera más notable en los niveles bajo y alto. Limitaciones: El método solo proporciona pronósticos puntuales sin intervalos de predicción.

 <italic>Originalidad:</italic>

No existe una aplicación similar para datos mexicanos que permita hacer objetivamente tales estimaciones.

 <italic>Conclusiones:</italic>

La distribución presupuestaria más recomendable debe abordarse principalmente entre los niveles bajo y alto. Sus implicaciones deberían respaldar la política de salud impostergable en general y para los subdominios mencionados en particular.

 Keywords: Diabetes mortality hierarchical forecasts marginalization resource allocation JEL Classification: C32 C53 C46 Palabras clave: Diabetes mortalidad pronósticos jerárquicos marginación asignación de recursos Clasificación JEL: C32 C53 C46 Introduction

Over the past decades, the Mexican population has undergone the epidemiological transition, from a primarily preventable causes of death due to infection and other preventable diseases, to the emergence of an increasing degenerative causes of death. Along these lines, one primary concern is the rise in obesity and type II Diabetes (Diabetes from now on). Associated with these increasing morbidity rates, Mexico has experienced a significant expansion of the Diabetes mortality rates (Barquera, Campos-Nonato, & Hernández-Barrera, 2013).

Diabetes prevalence has increased among adults of every age group and it has been one of the most important causes of death in Mexico since 2000, being at least 1.6 times as often the underlying cause of death (Bustamante-Montes, Lezama-Fernández, Fernández-De Hoyos, Villa-Romero, & Borja-Aburto, 1990). These changes have had significant negative effects on life expectancy in Mexico (Agudelo-Botero & Dávila-Cervantes, 2015; Dávila-Cervantes & Pardo, 2014; Palloni, Beltrán-Sánchez, Novak, Pinto, & Wong, 2015).

The Diabetes prevalence pattern in Mexico has been extremely heterogeneous; unlike the United States, it has shown that the epidemiological transition across states has not occurred simultaneously. A group of Mexican researchers called this phenomenon the “polarization of the transition” (Frenk, Bobadilla, Sepúlveda, & López, 1989) where different regions of the country are experiencing the epidemiological transition in different ways (Frenk & Chacón, 1991a, 1991b).

There is evidence of the recent trends in the prevalence of Diabetes and its risk factors in national health surveys (Villalpando, Shamah-Levy, Rojas, & Aguilar-Salinas, 2010). From 1993 to 2006, the prevalence of Diabetes increased from 6.7% to 14.4%; to note, the relative change in mortality for the period 1980-2000 shows that the most significant increase has been mainly in the southern central and Mexico City regions (Barquera, Tovar-Guzmán, Campos-Nonato, González-Villalpando, & Rivera-Dommarco, 2003). This dynamic predicts larger increments in the near future for Diabetes morbidity and mortality.

At national level, the adults aged 20 years and above, with overweight and obesity was 75.2% (39.1% overweight and 36.1% obesity) in 2018, compared to 71.3% in 2012. Likewise, the states that presented the higher percentages were Campeche, Tamaulipas, Hidalgo, Mexico City and Nuevo Leon. The percentage of the population aged 20 years and above with a diagnosis of Diabetes in 2012 was 9.2% (6.4 million); by sex, 9.7% were women and 8.6% men; in 2018, it was 10.3% (8.6 million), 11.4% females and 9.1% males (Instituto Nacional de Estadística y Geografía [INEGI], Instituto Nacional de Salud Pública [INSP], & Secretaría de Salud [SS], 2019).

The purpose of the current study is to propose an objective approach to allocate resources, based on hierarchical forecasts of Diabetes mortality to 2030, using the Hyndman and Athanasopoulos (2018) method. The hierarchical time series model allows to forecast the number of deaths by Diabetes based on sub-domains of the population. It is important to recognize that the hierarchical structure is absolutely flexible and it depends on the analyst’s criteria and the goal pursued.

This paper is organized as follows. The next section is devoted to the economic burden of Diabetes and different estimates made for the mexican case. In the next section, the methodology is presented, describing how data is handled and how the model is established, in accordance with the proposed hierarchical structure. Then, the results of the forecasts are exposed from the first to the last level of the mentioned structure. Here, it is evident the coherence among the forecasts. Afterwards, the main conclusions are described and justified to highlight a new public health policy.

 Estimates economic burden of Diabetes

Diabetes has become one of the leading public health challenges of the twenty-first century due to the large economic burden and its adverse impact on the overall health of the population. This emerging morbid condition has placed increasing costs on the Mexican healthcare system. Its economic burden affects a wide range of variables, including economic and human development, as well as the conditions of equity and poverty (Barquera et al., 2013). In other words, its economic impact encompasses the direct costs associated with spending on health care, that is medical services and drugs, and indirect costs of the disease that relate to the effect of premature mortality and disability of a person to participate effectively in the labor market.

The 2013 estimates suggest that the economic burden of Diabetes was about 2.25% of the mexican GDP. This amount is greater than the actual annual growth of the Mexican economy of 2.1%, registered by INEGI at the end of 2014, and the one projected for 2021. The direct costs of Diabetes were estimated at $179,495.3 million pesos in 2013 (Barraza-Lloréns et al., 2015). Likewise, financially speaking, comparing the economic impact in 2012 versus 2010 Arredondo and Reyes (2013) estimated a 33% increase in costs associated to this public affection. Thus, they posited the need to review the current organization of the mexican health system, to move from a curative health care mode to preventive models that enable a better way to deal with the expected challenges.

Most recently, Arredondo, Orozco, Alcalde-Rabanal, Navarro, and Azar (2018) estimate the economic burden of the health services demand due to Diabetes and hypertension for the mexican insured and uninsured population in five regions. According to their results, between 2013 and 2018, the economic burden of both diseases increased between 58%-66%. They also argue that, based on their forecasts, on each of the analyzed regions, to address these diseases, the authorities will require between 13% and 15% of the public health budget for the uninsured population and between 15% and 17% for the insured population.

To determine trends related to the economic burden from Diabetes, Arredondo et al. (2019) developed a longitudinal analysis. This analysis generated two key findings: there was a 26% increase in the economic burden from incidence from 2016 to 2018, and the total amount allocated to treat Diabetes in 2017 was $9 684 780 574 (us dollars). Thus, they suggest reviewing and rethinking strategies of prevention, planning, organization and resource allocation.

The same analysis of economic burden of Diabetes in the elderly was made by Arredondo (2020). He compared the economic burden for 2020 versus 2022 and concluded that the increase was estimated at 29%. He also pointed that amid the coronavirus 2019 pandemic, there is a serious complication to achieve the scope of universal coverage for diabetics in Mexico. It is worth mentioning that based on the aforementioned literature, there is no proposal for the objective allocation of resources to face the economic burden, based on the deaths that occur by sub-domains of the population, and which is considered a relevant contribution of this paper.

 Methodology <italic>Data</italic>

Data was extracted from two sources. First, the mortality data are taken from the INEGI and the SS (1985-2017); individual level micro-data available at https://cutt.ly/8wMbkfu. Second, official population estimates are obtained from the Mexican population council (Consejo Nacional de Población [CONAPO], 2019) available at https://cutt.ly/JwMn6p0. From the INEGI data, data counts of deaths by Diabetes-related causes (ICD-10 codes) were obtained for years 1985 to 2017 (the most recent data available). The (mid-year) anual population estimates data from CONAPO for the 1985 to 2030 period were considered. All these data were classified by marginalization and sex. The datasets analyzed and generated during the current study are available in Table 1.

Deaths by Diabetes and Population
Deaths (thousands) Population (millions)
Year Total HI LO ME VH VL HIM HIW LOM LOW MEM MEW VHM VHW VLM VLW Total HI LO ME VH VL HIM HIW LOM LOW MEM MEW VHM VHW VLM VLW
1985 20.41 4.26 5.66 2.77 1.04 6.68 1.87 2.39 2.43 3.23 1.23 1.54 0.44 0.60 3.00 3.68 76.03 18.55 22.22 12.44 7.98 14.84 9.22 9.33 11.04 11.18 6.20 6.23 3.99 3.99 7.27 7.57
1986 23.03 5.05 6.58 3.10 1.10 7.20 2.18 2.87 2.86 3.72 1.35 1.74 0.45 0.65 3.03 4.17 77.69 18.94 22.80 12.77 8.24 14.95 9.40 9.54 11.32 11.48 6.36 6.41 4.11 4.13 7.33 7.62
1987 23.97 5.44 6.50 3.21 1.12 7.71 2.33 3.11 2.76 3.74 1.38 1.83 0.49 0.63 3.29 4.42 79.34 19.32 23.37 13.10 8.50 15.04 9.58 9.74 11.60 11.77 6.52 6.58 4.24 4.26 7.38 7.67
1988 24.98 5.43 7.26 3.55 1.23 7.50 2.40 3.04 3.19 4.07 1.56 1.99 0.53 0.70 3.20 4.30 80.97 19.70 23.94 13.44 8.76 15.13 9.76 9.94 11.88 12.07 6.68 6.76 4.36 4.40 7.42 7.70
1989 25.57 5.46 7.45 3.68 1.29 7.68 2.41 3.05 3.23 4.22 1.61 2.07 0.55 0.74 3.39 4.29 82.58 20.07 24.52 13.77 9.02 15.20 9.93 10.14 12.16 12.36 6.83 6.93 4.48 4.53 7.46 7.74
1990 25.71 5.34 7.71 3.61 1.25 7.79 2.29 3.05 3.34 4.37 1.64 1.97 0.53 0.72 3.38 4.42 84.17 20.43 25.11 14.07 9.22 15.35 10.10 10.33 12.45 12.66 6.98 7.09 4.58 4.64 7.54 7.81
1991 27.07 5.76 8.15 4.00 1.35 7.80 2.54 3.23 3.59 4.56 1.80 2.20 0.57 0.78 3.46 4.34 85.75 20.75 25.73 14.34 9.35 15.57 10.25 10.50 12.75 12.98 7.11 7.23 4.64 4.71 7.65 7.93
1992 28.26 5.98 8.48 4.24 1.44 8.11 2.63 3.35 3.78 4.71 1.90 2.35 0.63 0.81 3.62 4.49 87.31 21.08 26.35 14.61 9.48 15.79 10.40 10.67 13.05 13.30 7.24 7.37 4.70 4.78 7.75 8.04
1993 29.52 6.30 8.85 4.59 1.62 8.15 2.69 3.61 3.91 4.94 1.99 2.60 0.68 0.94 3.58 4.58 88.85 21.39 26.97 14.87 9.61 16.01 10.55 10.84 13.35 13.62 7.36 7.51 4.76 4.85 7.86 8.15
1994 30.26 6.64 8.98 4.45 1.84 8.36 2.84 3.79 4.00 4.98 1.94 2.51 0.79 1.06 3.69 4.67 90.36 21.70 27.59 15.13 9.73 16.22 10.69 11.00 13.65 13.94 7.49 7.64 4.82 4.91 7.96 8.26
1995 33.25 7.17 10.11 5.11 1.99 8.88 3.15 4.02 4.48 5.63 2.27 2.84 0.86 1.13 3.89 4.99 91.84 21.99 28.20 15.38 9.85 16.42 10.83 11.16 13.95 14.25 7.61 7.77 4.87 4.98 8.05 8.36
1996 34.80 7.79 10.52 5.32 2.12 9.05 3.45 4.34 4.74 5.77 2.38 2.94 0.91 1.21 3.95 5.10 93.29 22.27 28.81 15.62 9.99 16.61 10.95 11.31 14.25 14.57 7.72 7.90 4.94 5.05 8.15 8.46
1997 35.97 7.71 11.25 5.54 2.31 9.16 3.30 4.41 5.07 6.18 2.41 3.13 0.98 1.33 4.10 5.07 94.72 22.52 29.42 15.84 10.15 16.80 11.06 11.46 14.54 14.87 7.82 8.02 5.01 5.14 8.25 8.55
1998 41.78 8.81 12.86 6.69 2.63 10.79 3.84 4.97 5.73 7.13 3.02 3.67 1.08 1.55 4.91 5.88 96.12 22.76 30.02 16.05 10.31 16.98 11.15 11.61 14.83 15.18 7.91 8.13 5.08 5.23 8.34 8.64
1999 45.59 9.92 14.13 7.11 2.93 11.50 4.25 5.67 6.34 7.79 3.15 3.96 1.26 1.67 5.24 6.26 97.48 23.00 30.61 16.25 10.47 17.15 11.24 11.76 15.12 15.49 8.00 8.25 5.15 5.32 8.43 8.72
2000 46.55 10.55 13.97 7.27 3.08 11.69 4.54 6.01 6.43 7.54 3.25 4.02 1.31 1.77 5.31 6.38 98.79 23.23 31.15 16.45 10.60 17.35 11.33 11.89 15.38 15.77 8.09 8.36 5.20 5.40 8.53 8.82
2001 49.87 11.71 15.00 7.72 3.40 12.04 5.11 6.60 6.80 8.20 3.47 4.26 1.47 1.93 5.47 6.56 100.11 23.46 31.66 16.66 10.72 17.60 11.43 12.03 15.62 16.04 8.19 8.47 5.25 5.47 8.65 8.95
2002 54.85 12.66 16.50 8.69 3.94 13.06 5.71 6.95 7.72 8.78 3.90 4.79 1.75 2.19 6.07 6.98 101.49 23.71 32.20 16.88 10.84 17.86 11.54 12.17 15.88 16.32 8.29 8.59 5.30 5.54 8.78 9.08
2003 59.14 13.66 17.84 9.24 4.18 14.22 6.01 7.65 8.24 9.60 4.15 5.10 1.84 2.35 6.53 7.68 102.89 23.97 32.74 17.10 10.97 18.12 11.65 12.32 16.14 16.60 8.39 8.71 5.36 5.61 8.91 9.21
2004 62.23 14.51 18.54 9.92 4.39 14.88 6.42 8.10 8.60 9.93 4.50 5.42 1.89 2.51 6.98 7.89 104.27 24.21 33.28 17.32 11.09 18.37 11.75 12.46 16.40 16.88 8.49 8.83 5.41 5.68 9.03 9.34
2005 67.15 15.86 19.89 10.54 4.92 15.94 7.11 8.75 9.27 10.62 4.79 5.75 2.20 2.72 7.50 8.44 105.67 24.46 33.82 17.54 11.21 18.63 11.86 12.60 16.66 17.16 8.59 8.96 5.46 5.75 9.16 9.47
2006 68.42 16.13 20.56 11.00 4.86 15.87 7.26 8.86 9.67 10.90 5.17 5.83 2.17 2.69 7.64 8.23 107.16 24.75 34.36 17.82 11.36 18.86 11.99 12.76 16.92 17.44 8.72 9.10 5.53 5.84 9.27 9.59
2007 70.51 16.48 21.24 11.32 5.27 16.21 7.59 8.89 10.23 11.00 5.34 5.98 2.37 2.91 7.79 8.42 108.74 25.08 34.91 18.15 11.55 19.04 12.15 12.93 17.18 17.73 8.89 9.27 5.62 5.93 9.36 9.69
2008 75.64 17.71 22.66 12.55 5.84 16.88 8.14 9.57 10.91 11.75 5.91 6.64 2.63 3.21 8.10 8.78 110.41 25.43 35.49 18.50 11.75 19.24 12.32 13.11 17.46 18.02 9.06 9.44 5.72 6.04 9.45 9.79
2009 77.69 18.35 23.49 12.46 6.16 17.24 8.50 9.84 11.40 12.09 5.87 6.59 2.79 3.37 8.44 8.80 112.10 25.78 36.07 18.86 11.95 19.44 12.50 13.28 17.75 18.32 9.24 9.61 5.81 6.14 9.55 9.89
2010 82.96 20.15 24.91 13.36 6.85 17.69 9.36 10.79 12.19 12.72 6.33 7.03 3.10 3.75 8.72 8.97 113.75 26.14 36.63 19.18 12.13 19.66 12.68 13.46 18.03 18.60 9.41 9.77 5.90 6.23 9.66 10.01
2011 80.79 19.69 24.58 12.75 6.80 16.97 9.16 10.54 12.03 12.55 6.13 6.62 3.09 3.72 8.47 8.50 115.37 26.51 37.18 19.49 12.30 19.89 12.86 13.65 18.30 18.88 9.56 9.92 5.98 6.32 9.77 10.13
2012 85.05 20.89 25.50 13.73 7.30 17.63 9.73 11.17 12.62 12.88 6.70 7.03 3.25 4.05 8.94 8.69 116.94 26.87 37.72 19.80 12.46 20.09 13.04 13.83 18.57 19.15 9.72 10.08 6.06 6.40 9.86 10.23
2013 89.46 22.26 26.56 14.69 7.71 18.24 10.44 11.82 13.22 13.34 7.08 7.61 3.59 4.12 9.05 9.19 118.45 27.21 38.24 20.11 12.61 20.28 13.21 14.00 18.82 19.42 9.88 10.23 6.13 6.48 9.95 10.33
2014 94.00 23.37 28.18 15.41 8.18 18.86 11.07 12.29 14.06 14.12 7.49 7.93 3.63 4.55 9.51 9.35 119.94 27.55 38.76 20.41 12.76 20.46 13.37 14.17 19.08 19.68 10.04 10.38 6.20 6.56 10.03 10.42
2015 98.41 24.53 29.28 15.91 9.68 19.02 11.50 13.03 14.62 14.66 7.70 8.21 4.34 5.34 9.60 9.42 121.35 27.86 39.26 20.70 12.90 20.62 13.53 14.33 19.32 19.93 10.18 10.52 6.26 6.64 10.11 10.51
2016 105.57 26.68 31.16 17.53 9.89 20.31 12.62 14.06 15.67 15.50 8.49 9.04 4.57 5.32 10.38 9.93 122.72 28.16 39.75 20.97 13.04 20.79 13.68 14.48 19.57 20.19 10.32 10.65 6.33 6.71 10.20 10.60
2017 106.28 26.63 31.23 17.87 10.29 20.26 12.59 14.04 15.69 15.54 8.70 9.17 4.83 5.46 10.38 9.89 124.04 28.43 40.25 21.22 13.17 20.98 13.81 14.62 19.82 20.44 10.44 10.78 6.39 6.78 10.29 10.69
2018 110.61 28.03 32.39 18.63 10.87 20.69 13.51 14.52 16.47 15.92 9.22 9.41 5.17 5.70 10.61 10.08 125.33 28.69 40.74 21.46 13.29 21.15 13.93 14.75 20.05 20.69 10.56 10.90 6.45 6.84 10.37 10.78
2019 114.29 29.14 33.38 19.21 11.44 21.11 14.14 15.00 17.07 16.31 9.57 9.65 5.51 5.93 10.84 10.27 126.58 28.94 41.21 21.70 13.41 21.32 14.05 14.88 20.29 20.93 10.68 11.02 6.51 6.91 10.45 10.87
2020 118.06 30.26 34.36 19.90 12.01 21.53 14.78 15.48 17.67 16.69 10.01 9.88 5.85 6.17 11.07 10.47 127.79 29.18 41.67 21.93 13.53 21.48 14.17 15.01 20.51 21.16 10.79 11.14 6.56 6.97 10.53 10.95
2021 121.78 31.37 35.34 20.53 12.58 21.96 15.41 15.96 18.27 17.08 10.40 10.12 6.18 6.40 11.30 10.66 128.97 29.42 42.11 22.15 13.65 21.64 14.28 15.13 20.73 21.39 10.90 11.25 6.62 7.03 10.61 11.03
2022 125.53 32.48 36.33 21.19 13.15 22.38 16.04 16.44 18.86 17.46 10.83 10.36 6.52 6.63 11.53 10.86 130.12 29.65 42.54 22.37 13.76 21.79 14.39 15.25 20.94 21.61 11.01 11.36 6.67 7.09 10.68 11.11
2023 129.26 33.59 37.31 21.83 13.72 22.81 16.68 16.92 19.46 17.85 11.23 10.60 6.85 6.87 11.76 11.05 131.23 29.87 42.96 22.58 13.87 21.94 14.50 15.37 21.14 21.82 11.11 11.47 6.72 7.15 10.75 11.19
2024 133.00 34.70 38.29 22.48 14.29 23.23 17.31 17.40 20.06 18.23 11.64 10.84 7.19 7.10 11.99 11.24 132.31 30.09 43.37 22.79 13.98 22.09 14.60 15.48 21.34 22.03 11.21 11.58 6.77 7.20 10.82 11.26
2025 136.73 35.82 39.27 23.13 14.86 23.66 17.94 17.87 20.66 18.62 12.05 11.08 7.52 7.34 12.22 11.44 133.35 30.29 43.76 22.99 14.08 22.23 14.70 15.59 21.53 22.23 11.31 11.68 6.82 7.26 10.89 11.34
2026 140.47 36.93 40.26 23.77 15.43 24.08 18.58 18.35 21.26 19.00 12.46 11.31 7.86 7.57 12.45 11.63 134.36 30.49 44.14 23.19 14.19 22.36 14.79 15.70 21.71 22.43 11.40 11.78 6.87 7.32 10.95 11.41
2027 144.21 38.04 41.24 24.42 16.00 24.50 19.21 18.83 21.86 19.39 12.87 11.55 8.20 7.81 12.68 11.82 135.34 30.69 44.50 23.38 14.29 22.49 14.89 15.80 21.89 22.62 11.49 11.88 6.92 7.37 11.01 11.47
2028 147.95 39.15 42.22 25.07 16.57 24.93 19.84 19.31 22.45 19.77 13.28 11.79 8.53 8.04 12.91 12.02 136.28 30.87 44.85 23.56 14.38 22.61 14.97 15.90 22.05 22.80 11.58 11.98 6.96 7.42 11.07 11.54
2029 151.69 40.26 43.21 25.72 17.14 25.35 20.47 19.79 23.05 20.15 13.69 12.03 8.87 8.27 13.14 12.21 137.19 31.05 45.19 23.74 14.48 22.73 15.06 16.00 22.22 22.98 11.67 12.07 7.01 7.47 11.13 11.60
2030 155.42 41.38 44.19 26.37 17.71 25.78 21.11 20.27 23.65 20.54 14.10 12.27 9.20 8.51 13.37 12.41 138.07 31.23 45.52 23.91 14.57 22.84 15.14 16.09 22.37 23.15 11.75 12.16 7.05 7.52 11.18 11.66

Source: The observed deaths (1985-2017) are taken from INEGI & SS (1985-2017) and the rest are forecasted (2018-2030); for Population all the date are coming from CONAPO (2019).

The state level information was aggregated, based on the marginalization index (shown in Figure 1) created by CONAPO in 2015 (CONAPO, 2016). The index was constructed by means of Principal Components Analysis using the following variables from the Intercensal Survey (EIC) 2015 (INEGI, 2015): Illiteracy rate, % of the population without primary school completed, % of household without adequate toilet facilities, % of households without electricity, % of households without an external water source (municipal water), % of overcrowded households, % of households with dirt floor, % of the population living in rural areas, % of the population with low wages (measured as twice the minimum wage).

Mexican states by marginalization level 2015 Source: CONAPO (2015).

In short, we divide the Mexican population into two sub-domains: marginalization level and sex. Since previous work has documented the differences in Diabetes mortality by gender, forecasts must not assume a common rate for men and women. Likewise, variation by socioeconomic status of residence has also been linked to variation in Diabetes mortality within Mexico (Flores, Sparks, & Silva, 2016), so the forecast methodology will separately forecast the Diabetes deaths by level of marginalization. Figure 1 shows the five levels subdivision of Mexican states based on the marginalization index. Very High and High levels of marginalization is typically linked to high poverty, poor housing conditions, small communities, and low socioeconomic status.

The importance of looking at regional differences on Diabetes patterns consist of the relationship with local socioeconomic status conditions. Research on the topic suggests that Diabetes is not only associated with socioeconomic characteristics at the individual level, but also at the regional level. Socioeconomic determinants such as income, education, housing, and access to nutritious food are central to the development and progression of Diabetes. Moreover, the incidence and prevalence of Diabetes appear to be socially graded, as individuals with lower income and less education are 2 to 4 times more likely to develop Diabetes than more advantaged individuals (Hill et al., 2013). In this sense, poverty and material deprivation, defined as a lack of resources to meet the prerequisites for health, play a key role.

 Hierarchical time series model

A hierarchical time series is a collection of several time series that are linked together in a hierarchical structure (Hyndman & Athanasopoulos, 2018). In our case, the hierarchical structure to forecast is identified through Figure 2. The same structure, both for total deaths and for total population is used. Then, the mortality rate by Diabetes per 100 000 with these forecasts is calculated considering the denominators from CONAPO’S projections. According to Figure 2, it is necessary to construct the time series at the bottom-level, where marginalization and sex are employed.

Hierarchical time series Structure of Diabetes deaths to forecast

The marginalization index is classified as follows, in decreasing order of severity: Very High (VH) (Chiapas, Guerrero and Oaxaca), High (H) (Campeche, Hidalgo, Michoacan, Puebla, San Luis Potosi, Veracruz and Yucatan), Medium (ME) (Durango, Guanajuato, Morelos, Nayarit, Quintana Roo, Sinaloa, Tabasco, Tlaxcala and Zacatecas), Low (L) (Aguascalientes, Baja California Sur, Chihuahua, Colima, Jalisco, Mexico, Queretaro, Sonora and Tamaulipas) and Very Low (VL) (Baja California, Ciudad de Mexico, Coahuila and Nuevo Leon). In this way, the number of time series al bottom-level is 10 (2 x 5), that is the levels of marginalization by two levels sex men (M) and woman (W) respectability. So, the total number of time series to forecast is 16 (10+5+1).

To forecast the hierarchical time series, it is necessary to establish at least some restrictions, such as

Y ^ n h = Y ^ V H , n h + Y ^ H , n h + Y ^ M E , n h + Y ^ L , n h + Y ^ V L , n h

Where

Y ^ V H , n h = Y ^ V H M , n h + Y ^ V H W , n h

Y ^ H , n h = Y ^ H M , n h + Y ^ H W , n h

Y ^ M E , n h = Y ^ M E M , n h + Y ^ M E W , n h

Y ^ L , n h = Y ^ L M , n h + Y ^ L W , n h

Y ^ V L , n h = Y ^ V L M , n h + Y ^ V L W , n h

and Y ^ j , n h is the vector of initial h-step forecast, made at time n for the time series j. In particular Y ^ n h is stacked in same order as Yt (see below).

One possibility to forecast the hierarchical time series is to use ARIMA models or smoothing techniques. However, the sum of the respective forecasts may not add up. In other words, the above restrictions, that give coherent forecasts, can be unsatisfied. In matrix notation a generalization of (1) - (6) (Hyndman & Athanasopoulos, 2018), where, in addition the individual time series considered, can be written as

Y t = Y t Y V H , t Y H , t Y M E , t Y L , t Y V L , t Y V H M , t Y V H W , t Y H M , t Y H W , t Y M E M , t Y M E W , t Y L M , t Y L W , t Y V L M , t Y V L W , t = 1111111111 1100000000 0011000000 0000110000 0000001100 0000000011 I 10 ⏟ S Y V H M , t Y V H W , t Y H M , t Y H W , t Y M E M , t Y M E W , t Y L M , t Y L W , t Y V L M , t Y V L W , t ⏟ B t

where I10 is the identity matrix of size 10x10. In hierarchical terms, let Y ~ n h be the forecast given by

Y ~ n h = S P Y ^ n h

for some matrix P that extract and combine base or bottom-level forecasts Y ^ n h and S that adds them up. That is, Y ~ n h are the revised reconciled forecasts. There are three generic methods to estimate the forecast: bottom-up, top-down and middle-out. To obtain the best estimate, Generalized Least Squares is employed (Hyndman, Ahmed, Athanasopoulos, & Shang, 2011). It can be seen that the forecasts are aggregated consistently, unbiased and have minimum variance.

To choose the best forecasts the hts library (Hyndman, Lee, Wang, & Wickramasuriya, 2018) in R 4.0.1 is used (R Core Team, 2019), the forecasting methods ETS (Exponential Smoothing), ARIMA (Autoregressive Integrated Moving Average) and RW (Random walk) are explored. Forecasts are distributed in the hierarchy using optimal combination method (comb), bottom-up (bu), middle-out (mo), top-down (three methods: the two Gross-Sohl methods -tdgsa and tdgsf- and the forecast-proportion approach -tdfp-) (see Hyndman et al., 2011).

Finally, the statistical criteria giving forecast accuracy measures used are (Hyndman & Koehler, 2006): ME (Mean Error), RMSE (Root Mean Square Error), MAE (Mean Absolute Error), MAPE (Mean Absolute Percentage Error), MPE (Mean Percentage Error) and MASE (Mean Absolute Scaled Error). Based on their mean at the bottom-level time series, the best forecast is chosen.

One limitation of these estimates is that the method does not generate prediction intervals. However, for our objective, the point forecasts are sufficient because we assume that propose an approach to establish a distribution of resource allocation objectively can be made based on this kind of previsions. The imposed forecast horizon is h = 13. Other limitation is the assumption that the marginalization level is the same for all the forecast horizon. The libraries used for R were: forecast (Hyndman et al., 2019; Hyndman & Khandakar, 2008), data.table (Dowle & Srinivasan 2019) and zoom (Barbu, 2013). The code is available upon request to the authors.

 Results

It is found that the best forecast was the obtained through ARIMA based on the mean of several statistical criteria at the bottom-level time series (see Appendix). We present results from our analysis in both tabular and graphical form. Table 1 shows the observed and forecast numbers of death from Diabetes in each of the hierarchical levels of the forecast. To understand, it the column labeled Total is the entire expected deaths, the next five columns represent the second level of the hierarchy, based on level of marginalization (High, Low, Medium, Very High and Very Low). The next ten columns show the data by the combination of marginalization and sex, with the last character of the column label indicating if the forecast is for men (M) or women (W).

Figure 3 and Figure 4 show graphically the forecast for each level of the hierarchy used in the forecasting methodology. Figure 3 (top) corresponds to the national level forecast of the Diabetes mortality rate. It clearly increases, following the prevailing trend in the country over the past years. Figure 3 (bottom) shows the estimated Diabetes mortality rate for the second level of the hierarchy, based on the level of marginalization. Up until 2017, the highest level of Diabetes mortality was in areas of Very Low marginalization, suggesting that Diabetes is a disease of the affluent people in Mexico, which corresponds to work on Diabetes and obesity in the country (Sparks & Sparks, 2012); however, the gap between areas of High and Low marginalization was shrinking later in the observed data. In the forecast, the areas of High (H) and Very High (VH) marginalization show the greatest increases in Diabetes mortality.

Forecast of Diabetes mortality rate at the national level (top) and based on level of marginalization (bottom) to 2030

Forecast of Diabetes mortality rate by marginalization and sex (top) and enlarged (bottom) to 2030

Figure 4 presents the observed and forecast rates of mortality separately by marginalization level and sex. In the observed data prior to 2017, women faced a larger burden of mortality related to Diabetes than men (Flores et al., 2016), but again this gap has been shrinking in recent years. In the forecast, many of the ongoing trends in male versus female Diabetes mortality experience a cross over, where men begin to experience higher Diabetes mortality in some sub-areas of the country than women. Similar to what was shown in Figure 3, Figure 4 shows that in areas of High and Very High marginalization, men show forecast rates of Diabetes mortality that cross over female rates and become higher over time.

According to Table 2, the hierarchical forecasts point that the appropriate distribution of the resource allocation should considers mainly the High and Low levels, given that they accumulate more than half of the future deaths by Diabetes in Mexico for the next years. In fact, in each one, the number of male deaths will be greater than female deaths for both levels. The maximum expected of deaths will be for male population at Low level. This scenario highlights the mentioned polarization in Mexico and it also suggest that a preventive health policy should be applied as soon as possible.

Percentage of deaths by marginalization and sex: Observed (1985-2017) and forecasted (2018-2030)
Economic Burden distribution (%)
Year Total Total
First level HI LO ME VH VL Second level HIM HIW LOM LOW MEM MEW VHM VHW VLM VLW
1985 100.00 20.87 27.72 13.56 5.11 32.73 100.00 9.18 11.70 11.89 15.83 6.01 7.56 2.17 2.94 14.69 18.04
1986 100.00 21.94 28.56 13.45 4.78 31.28 100.00 9.48 12.46 12.43 16.14 5.88 7.57 1.97 2.81 13.16 18.12
1987 100.00 22.68 27.11 13.41 4.66 32.15 100.00 9.71 12.97 11.52 15.59 5.76 7.65 2.04 2.62 13.71 18.44
1988 100.00 21.75 29.07 14.23 4.92 30.02 100.00 9.59 12.16 12.78 16.29 6.25 7.98 2.13 2.79 12.80 17.22
1989 100.00 21.37 29.16 14.38 5.04 30.06 100.00 9.44 11.93 12.64 16.52 6.30 8.08 2.13 2.91 13.26 16.80
1990 100.00 20.79 29.99 14.06 4.85 30.32 100.00 8.91 11.87 12.98 17.01 6.38 7.67 2.05 2.80 13.14 17.18
1991 100.00 21.28 30.11 14.78 4.99 28.83 100.00 9.36 11.92 13.28 16.84 6.65 8.14 2.12 2.88 12.79 16.04
1992 100.00 21.16 30.02 15.02 5.08 28.72 100.00 9.29 11.87 13.36 16.65 6.72 8.30 2.22 2.86 12.82 15.89
1993 100.00 21.35 29.97 15.56 5.50 27.62 100.00 9.11 12.24 13.24 16.73 6.74 8.82 2.31 3.19 12.12 15.51
1994 100.00 21.93 29.67 14.71 6.08 27.61 100.00 9.39 12.53 13.23 16.44 6.41 8.30 2.59 3.49 12.18 15.43
1995 100.00 21.55 30.41 15.37 5.97 26.70 100.00 9.46 12.09 13.47 16.94 6.84 8.53 2.58 3.39 11.68 15.01
1996 100.00 22.39 30.22 15.29 6.09 26.00 100.00 9.92 12.48 13.63 16.59 6.84 8.45 2.61 3.49 11.35 14.65
1997 100.00 21.43 31.28 15.40 6.41 25.47 100.00 9.16 12.27 14.11 17.18 6.71 8.70 2.72 3.69 11.39 14.09
1998 100.00 21.09 30.78 16.01 6.29 25.83 100.00 9.20 11.89 13.72 17.06 7.22 8.79 2.59 3.70 11.76 14.07
1999 100.00 21.75 31.00 15.60 6.43 25.22 100.00 9.31 12.44 13.92 17.08 6.91 8.69 2.77 3.66 11.49 13.73
2000 100.00 22.66 30.00 15.61 6.62 25.10 100.00 9.74 12.92 13.81 16.20 6.98 8.63 2.82 3.81 11.40 13.71
2001 100.00 23.49 30.09 15.48 6.81 24.13 100.00 10.25 13.23 13.64 16.44 6.95 8.54 2.95 3.86 10.97 13.16
2002 100.00 23.08 30.08 15.84 7.19 23.80 100.00 10.42 12.67 14.07 16.01 7.11 8.73 3.20 3.99 11.07 12.73
2003 100.00 23.10 30.16 15.63 7.07 24.04 100.00 10.16 12.93 13.93 16.23 7.01 8.62 3.10 3.97 11.05 12.99
2004 100.00 23.32 29.79 15.94 7.06 23.90 100.00 10.31 13.01 13.83 15.96 7.23 8.71 3.03 4.03 11.22 12.68
2005 100.00 23.61 29.63 15.70 7.33 23.74 100.00 10.59 13.02 13.81 15.82 7.14 8.56 3.28 4.05 11.17 12.57
2006 100.00 23.57 30.05 16.08 7.10 23.20 100.00 10.61 12.96 14.13 15.92 7.55 8.53 3.17 3.93 11.17 12.03
2007 100.00 23.37 30.12 16.05 7.48 22.99 100.00 10.76 12.61 14.51 15.61 7.57 8.48 3.36 4.12 11.05 11.94
2008 100.00 23.42 29.96 16.59 7.72 22.32 100.00 10.77 12.65 14.43 15.53 7.82 8.78 3.48 4.24 10.71 11.61
2009 100.00 23.61 30.23 16.04 7.93 22.19 100.00 10.94 12.67 14.67 15.56 7.56 8.48 3.59 4.34 10.86 11.33
2010 100.00 24.29 30.03 16.11 8.25 21.32 100.00 11.28 13.01 14.69 15.33 7.63 8.47 3.73 4.52 10.51 10.82
2011 100.00 24.37 30.43 15.78 8.42 21.00 100.00 11.33 13.04 14.89 15.54 7.59 8.20 3.82 4.60 10.48 10.52
2012 100.00 24.56 29.98 16.14 8.59 20.73 100.00 11.43 13.13 14.84 15.15 7.87 8.27 3.82 4.76 10.51 10.21
2013 100.00 24.88 29.68 16.42 8.62 20.39 100.00 11.67 13.21 14.78 14.91 7.92 8.51 4.02 4.61 10.11 10.28
2014 100.00 24.86 29.98 16.39 8.71 20.06 100.00 11.78 13.08 14.96 15.02 7.96 8.43 3.86 4.84 10.12 9.95
2015 100.00 24.92 29.75 16.16 9.83 19.33 100.00 11.69 13.24 14.86 14.90 7.82 8.34 4.41 5.43 9.76 9.57
2016 100.00 25.27 29.52 16.60 9.37 19.24 100.00 11.95 13.32 14.84 14.68 8.04 8.56 4.33 5.04 9.83 9.41
2017 100.00 25.06 29.38 16.81 9.68 19.06 100.00 11.85 13.21 14.76 14.62 8.19 8.63 4.54 5.14 9.76 9.30
2018 100.00 25.34 29.29 16.84 9.83 18.70 100.00 12.22 13.13 14.89 14.40 8.34 8.50 4.68 5.15 9.59 9.11
2019 100.00 25.50 29.20 16.81 10.01 18.47 100.00 12.38 13.13 14.93 14.27 8.37 8.44 4.82 5.19 9.48 8.99
2020 100.00 25.63 29.10 16.85 10.17 18.24 100.00 12.52 13.11 14.96 14.14 8.48 8.37 4.95 5.22 9.37 8.87
2021 100.00 25.76 29.02 16.86 10.33 18.03 100.00 12.65 13.10 15.00 14.02 8.54 8.31 5.08 5.26 9.28 8.75
2022 100.00 25.88 28.94 16.88 10.48 17.83 100.00 12.78 13.09 15.03 13.91 8.62 8.25 5.19 5.28 9.18 8.65
2023 100.00 25.99 28.86 16.89 10.62 17.64 100.00 12.90 13.09 15.06 13.81 8.69 8.20 5.30 5.31 9.10 8.55
2024 100.00 26.09 28.79 16.90 10.75 17.47 100.00 13.01 13.08 15.08 13.71 8.75 8.15 5.41 5.34 9.01 8.45
2025 100.00 26.19 28.72 16.91 10.87 17.30 100.00 13.12 13.07 15.11 13.61 8.81 8.10 5.50 5.37 8.94 8.36
2026 100.00 26.29 28.66 16.92 10.99 17.14 100.00 13.22 13.07 15.13 13.53 8.87 8.05 5.60 5.39 8.86 8.28
2027 100.00 26.38 28.60 16.93 11.10 16.99 100.00 13.32 13.06 15.16 13.44 8.92 8.01 5.68 5.41 8.79 8.20
2028 100.00 26.46 28.54 16.95 11.20 16.85 100.00 13.41 13.05 15.18 13.36 8.98 7.97 5.77 5.43 8.73 8.12
2029 100.00 26.54 28.48 16.95 11.30 16.71 100.00 13.50 13.05 15.20 13.29 9.02 7.93 5.85 5.46 8.66 8.05
2030 100.00 26.62 28.43 16.96 11.40 16.59 100.00 13.58 13.04 15.22 13.21 9.07 7.89 5.92 5.47 8.60 7.98

It is also important to recognize that the Medium and Very Low levels will jointly concentrate 1 of every 3 deaths from Diabetes in Mexico (see Table 2). It is also observed that the deaths per sex for both levels will be greater for male population. The smallest resource is required for the Very High level where the poverty and its consequences are their main characteristics. In other words, there is not a direct relationship among the marginalization and the expected percentage of deaths by Diabetes for the mexican case.

 Conclusion

This analysis presents results of an analysis of hierarchical forecasts applied to the problem of distribution of resources to address the burden that Diabetes mortality is expected for Mexico by 2030. Overall, the forecasts estimated here show significant differences based on levels of marginalization and sex. Likewise, the gaps between men and women are notorious as well as between the levels of marginalization. Thereby, there is enough evidence that is not a good idea to consider a uniform distribution to face the economic burden caused by the Diabetes.

The hierarchical forecasts show that if the current trends continue, there will be a divergence by marginalization and sex in mortality in Mexico. Even worse, these forecasts indicate that the rates will increase the most simultaneously at high income level and some poor areas of the country. Through this statistical tool, it was possible to show not only the long-term trends in Diabetes mortality, but how the trends vary among areas of the country and by subpopulations.

The phenomenon presents notable differences by marginalization and sex; in fact, the increase in the mortality rate is clearly differentiated. This suggest that it could be necessary to implement preventive and specific health policies based on geographic region. It is also evident that the current preventive health policies need be rethought to reduce mortality from Diabetes in Mexico.

 Acknowledgments

The authors gratefully acknowledge the comments and suggestions from two anonymous reviewers and the editor of this journal. Eliud Silva dedicates this article to the memory of his always dear uncle Soco as well as Professor Oscar Vargas for his worth comments.

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Accurate statistics by forecast method and hierarchical perspective for individual series
Time Series
Statistics HIM HIW LOM LOW MEM MEW VHM VHW VLM VLW Mean
ETS, comb
ME 37.7280 20.9346 52.1806 7.0980 38.8140 44.3801 29.0629 23.8024 29.0113 -12.3165 27.0695
RMSE 263.0375 314.9066 287.3520 291.1597 188.0735 238.1917 123.2657 159.0585 231.4569 267.1759 236.3678
MAE 210.4737 242.9141 230.2858 243.6743 137.5848 177.2446 80.7564 100.1290 171.3872 209.2680 180.3718
MAPE 4.1668 3.5896 3.3616 3.2021 3.5449 3.8632 4.3238 4.0088 2.9432 3.4557 3.6460
MPE 0.5617 0.1644 0.7385 -0.4666 0.7515 0.6808 1.5426 1.2997 0.4383 -0.6069 0.5104
MASE 0.5737 0.6297 0.5344 0.5935 0.5465 0.6366 0.5743 0.6250 0.6705 0.8239 0.6208
ARIMA, comb
ME 35.1087 39.0870 55.5419 8.5916 48.9606 6.8383 35.4474 38.1102 0.6655 0.8089 26.9160
RMSE 255.3413 297.8797 270.8985 300.3741 182.9644 232.6668 124.6221 155.5576 235.7998 259.4100 231.5514
MAE 211.8437 232.8921 204.3192 248.1299 135.8920 172.0523 80.9180 102.7285 185.9218 193.3167 176.8014
MAPE 4.2070 3.4570 2.9708 3.2492 3.6458 3.9463 4.1979 4.1838 3.3392 3.0002 3.6197
MPE 0.3659 0.2072 0.6667 -0.4430 1.2207 -0.8101 1.8972 1.7926 -0.6429 -0.1264 0.4128
MASE 0.5774 0.6037 0.4742 0.6044 0.5398 0.6180 0.5754 0.6412 0.7273 0.7611 0.6122
RW, comb
ME 334.9375 364.2188 414.4688 384.6250 233.5625 238.3438 137.0625 151.9063 230.5000 193.8750 268.3500
RMSE 448.3285 476.3709 507.0032 488.9572 317.4291 336.7042 200.5305 223.0612 331.8646 326.3546 365.6604
MAE 366.8750 385.7813 430.9063 410.5625 251.7500 278.4063 140.6250 160.2188 255.6250 254.0000 293.4750
MAPE 6.4502 5.6048 5.8927 5.0026 6.2288 6.2597 7.3851 7.1162 4.1920 3.8217 5.7954
MPE 5.6667 5.2956 5.5798 4.7142 5.8203 5.3017 7.0772 6.5489 3.7268 2.9557 5.2687
MASE 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
ETS, bu
ME 37.5357 20.6606 45.9435 -0.1015 39.3016 45.1547 29.9351 25.2182 31.0995 -9.6158 26.5131
RMSE 261.3285 311.9974 275.7980 296.3138 188.4151 237.4663 123.4239 157.2529 235.4252 267.7355 235.5157
MAE 209.8246 241.2964 219.1647 249.9734 137.2041 175.8403 81.0389 99.1561 173.0626 209.6145 179.6175
MAPE 4.1571 3.5724 3.1802 3.3696 3.5165 3.8186 4.3042 3.9854 2.9542 3.4182 3.6277
MPE 0.5439 0.1453 0.4835 -0.6908 0.7746 0.7098 1.5627 1.3263 0.5827 -0.4759 0.4962
MASE 0.5719 0.6255 0.5086 0.6089 0.5450 0.6316 0.5763 0.6189 0.6770 0.8253 0.6189
ARIMA, bu
ME 37.2668 42.1604 48.7640 0.0863 44.8959 0.0395 35.2197 37.7325 0.0839 0.1057 24.6355
RMSE 252.5123 301.3408 265.3871 297.2897 181.0829 234.1966 122.0932 157.2513 235.1100 258.5183 230.4782
MAE 207.2960 232.6407 201.9412 250.1241 135.0178 172.4941 78.5704 103.4700 186.1748 193.2875 176.1017
MAPE 4.0760 3.3826 2.9047 3.3366 3.6178 4.0923 4.1268 4.1772 3.3553 3.0087 3.6078
MPE 0.3722 0.2185 0.4822 -0.6259 0.9484 -1.1655 1.8402 1.7232 -0.6631 -0.1451 0.2985
MASE 0.5650 0.6030 0.4686 0.6092 0.5363 0.6196 0.5587 0.6458 0.7283 0.7610 0.6096
RW, bu
ME 334.9375 364.2188 414.4688 384.6250 233.5625 238.3438 137.0625 151.9063 230.5000 193.8750 268.3500
RMSE 448.3285 476.3709 507.0032 488.9572 317.4291 336.7042 200.5305 223.0612 331.8646 326.3546 365.6604
MAE 366.8750 385.7813 430.9063 410.5625 251.7500 278.4063 140.6250 160.2188 255.6250 254.0000 293.4750
MAPE 6.4502 5.6048 5.8927 5.0026 6.2288 6.2597 7.3851 7.1162 4.1920 3.8217 5.7954
MPE 5.6667 5.2956 5.5798 4.7142 5.8203 5.3017 7.0772 6.5489 3.7268 2.9557 5.2687
MASE 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
ETS, mo
ME 37.1645 20.0731 60.5508 23.1506 37.1676 42.4592 26.9522 21.4561 26.8075 -20.0193 27.5762
RMSE 267.6493 322.9059 311.2819 293.7698 188.1703 237.3293 123.4091 161.8518 229.6268 276.9614 241.2955
MAE 211.5615 249.6546 250.5701 235.5260 138.2907 176.3267 80.1206 101.9027 171.7625 222.5084 183.8224
MAPE 4.1236 3.7094 3.7228 2.9759 3.5728 3.8337 4.3057 4.0181 2.9792 3.7091 3.6950
MPE 0.5301 0.1265 1.1146 -0.0104 0.6422 0.5774 1.4265 1.1873 0.1819 -0.8945 0.4882
MASE 0.5767 0.6471 0.5815 0.5737 0.5493 0.6333 0.5697 0.6360 0.6719 0.8760 0.6315
ARIMA, mo
ME 28.2180 31.9275 65.9611 23.2750 56.7039 16.6656 35.1854 38.2809 0.0839 0.1057 29.6407
RMSE 263.8437 292.8569 282.7252 307.8060 189.9409 232.0011 130.9082 153.5401 235.1100 258.5183 234.7250
MAE 222.3334 230.4395 210.5765 251.2278 140.0138 171.5694 86.3714 100.4632 186.1748 193.2875 179.2457
MAPE 4.4595 3.5475 3.1301 3.2156 3.8157 3.7694 4.3394 4.1116 3.3553 3.0087 3.6753
MPE 0.2404 0.1032 0.9441 -0.1451 1.7571 -0.3097 1.9073 1.8110 -0.6631 -0.1451 0.5500
MASE 0.6060 0.5973 0.4887 0.6119 0.5562 0.6163 0.6142 0.6270 0.7283 0.7610 0.6207
RW, mo
ME 334.9375 364.2188 414.4688 384.6250 233.5625 238.3438 137.0625 151.9063 230.5000 193.8750 268.3500
RMSE 448.3285 476.3709 507.0032 488.9572 317.4291 336.7042 200.5305 223.0612 331.8646 326.3546 365.6604
MAE 366.8750 385.7813 430.9063 410.5625 251.7500 278.4063 140.6250 160.2188 255.6250 254.0000 293.4750
MAPE 6.4502 5.6048 5.8927 5.0026 6.2288 6.2597 7.3851 7.1162 4.1920 3.8217 5.7954
MPE 5.6667 5.2956 5.5798 4.7142 5.8203 5.3017 7.0772 6.5489 3.7268 2.9557 5.2687
MASE 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
ETS, tdgsa
ME 275.5167 141.6664 250.0393 -115.4341 186.1718 66.5740 206.8873 208.6223 -256.0069 -665.8175 29.8219
RMSE 761.9894 402.9312 630.7999 585.7394 475.8387 241.5756 566.1257 560.7515 719.5201 1774.3676 671.9639
MAE 501.1943 285.8825 453.9422 420.8711 336.5558 198.6868 377.5481 372.4879 549.7590 1331.7306 482.8658
MAPE 7.8631 4.1187 6.1334 4.7679 8.5177 5.1145 22.1688 17.4824 8.8026 18.3863 10.3355
MPE -0.3781 0.2999 -0.0091 0.2249 -0.4608 0.2118 -5.8654 -3.7036 -0.5187 -3.7119 -1.3911
MASE 1.3661 0.7410 1.0535 1.0251 1.3369 0.7137 2.6848 2.3249 2.1506 5.2430 1.8640
ARIMA, tdgsa
ME 279.1943 146.1993 254.9955 -109.6870 188.7125 69.5835 207.9697 209.9981 -251.8725 -660.9766 33.4117
RMSE 747.6509 396.0418 619.4915 612.0923 472.9785 254.1954 561.8917 555.6817 734.6556 1787.8297 674.2509
MAE 516.3520 300.0356 463.6866 436.6220 342.3574 196.0412 379.6023 378.5918 526.4393 1299.9160 483.9644
MAPE 8.3415 4.2700 6.4776 4.7261 8.8917 5.1716 22.5948 18.1198 8.2482 17.8265 10.4668
MPE -0.3952 0.2986 -0.0443 0.2066 -0.5187 0.1829 -5.9856 -3.8234 -0.5254 -3.6850 -1.4289
MASE 1.4074 0.7777 1.0761 1.0635 1.3599 0.7042 2.6994 2.3630 2.0594 5.1178 1.8628
RW, tdgsa
ME 534.3479 452.0581 598.1072 262.4844 366.9778 273.1297 290.4123 311.8705 -10.6198 -395.2680 268.3500
RMSE 931.8876 626.9853 872.8975 582.1246 601.3943 377.1402 617.9916 625.4077 604.8573 1617.5420 745.8228
MAE 636.6406 474.8413 661.7605 468.2631 432.3602 313.5352 412.8759 426.8633 468.6928 1241.0736 553.6906
MAPE 8.2768 5.8092 7.1233 6.2025 8.7725 6.4197 20.7062 17.1214 8.3658 17.7603 10.6558
MPE 4.5459 5.0764 5.0510 4.7656 4.6696 5.0906 0.0657 1.8801 3.2879 -0.0321 3.4401
MASE 1.7353 1.2309 1.5357 1.1405 1.7174 1.1262 2.9360 2.6643 1.8335 4.8861 2.0806
ETS, tdgsf
ME 31.8729 38.2180 42.3003 46.8638 21.9968 25.2257 10.0593 12.4935 32.7809 36.4082 29.8219
RMSE 611.4952 348.3501 502.6745 519.8036 370.2743 228.8135 441.2936 436.7877 543.0443 1316.0607 531.8597
MAE 465.2538 250.0464 389.5916 385.5247 293.2781 190.4671 344.2115 331.3603 437.0435 1121.2255 420.8003
MAPE 9.2470 3.9456 6.3044 4.8805 9.2010 5.0788 27.4402 20.6450 8.4517 17.8489 11.3043
MPE -4.7188 -1.1853 -2.7452 2.0641 -4.6981 -0.6831 -18.4311 -13.3528 4.0642 6.1080 -3.3578
MASE 1.2682 0.6482 0.9041 0.9390 1.1650 0.6841 2.4477 2.0682 1.7097 4.4143 1.6248
ARIMA, tdgsf
ME 35.7095 42.8184 47.3920 52.5049 24.6446 28.2622 11.2702 13.9973 36.7268 40.7907 33.4117
RMSE 592.0496 339.6220 487.0588 549.5315 365.9135 241.9874 434.9571 429.3455 562.9431 1333.4672 533.6876
MAE 457.6052 275.8887 388.2488 409.9439 292.5223 188.7163 335.8340 321.0625 440.7904 1134.2067 424.4819
MAPE 9.3461 4.3841 6.5111 4.9074 9.4473 5.1649 27.4003 20.5980 8.3302 17.9397 11.4029
MPE -4.7367 -1.1866 -2.7814 2.0461 -4.7584 -0.7123 -18.5655 -13.4838 4.0579 6.1324 -3.3988
MASE 1.2473 0.7151 0.9010 0.9985 1.1620 0.6778 2.3882 2.0039 1.7244 4.4654 1.6284
RW, tdgsf
ME 296.4418 351.0459 395.2603 420.9602 206.6690 232.7552 98.2195 120.3605 271.3672 290.4205 268.3500
RMSE 724.0261 530.8234 678.2329 634.3233 456.4934 342.9909 470.5942 473.6675 557.3712 1288.7213 615.7244
MAE 502.2276 397.5933 506.9187 525.6359 332.6845 281.6424 341.1810 331.1626 501.2453 1143.7272 486.4018
MAPE 7.6478 4.9877 5.9036 7.2455 7.9651 5.8607 23.5224 17.5450 10.0814 18.5800 10.9339
MPE 0.4181 3.6624 2.4533 6.5211 0.6487 4.2395 -11.7960 -7.2496 7.6972 9.4394 1.6034
MASE 1.3689 1.0306 1.1764 1.2803 1.3215 1.0116 2.4262 2.0669 1.9609 4.5029 1.8146
ETS, tdfp
ME 39.3703 23.2284 62.0201 26.7724 38.3127 43.3736 27.3562 21.9250 30.2254 -14.3647 29.8219
RMSE 270.6147 318.5305 343.6103 306.2313 188.6614 248.8523 123.9756 163.0204 223.5041 255.3214 244.2322
MAE 218.3813 247.5655 286.9094 241.8558 140.2955 189.8175 80.2305 101.2012 176.7088 206.8442 188.9810
MAPE 4.3596 3.6548 4.2403 2.9599 3.7413 4.2463 4.5456 4.1594 3.0882 3.4444 3.8440
MPE 0.7041 0.3148 1.2645 0.1697 0.8160 0.7337 1.5863 1.3566 0.3643 -0.7032 0.6607
MASE 0.5952 0.6417 0.6658 0.5891 0.5573 0.6818 0.5705 0.6316 0.6913 0.8143 0.6439
ARIMA, tdfp
ME 31.8071 36.4986 70.3861 29.7926 58.9079 19.7280 35.7390 39.0220 5.1623 7.0731 33.4117
RMSE 260.2963 303.4112 306.0749 335.2375 199.1667 244.2644 130.7689 158.0069 250.6704 274.7725 246.2670
MAE 215.7857 239.3780 231.3405 265.0061 147.0085 179.3341 86.5814 106.7968 186.5863 207.5034 186.5321
MAPE 4.3941 3.6336 3.3753 3.3224 3.9653 3.9211 4.3648 4.4058 3.2700 3.1147 3.7767
MPE 0.4081 0.2632 1.0890 0.0026 1.9020 -0.1570 2.0692 1.9577 -0.5096 0.0108 0.7036
MASE 0.5882 0.6205 0.5369 0.6455 0.5839 0.6441 0.6157 0.6666 0.7299 0.8169 0.6448
RW, tdfp
ME 334.9375 364.2188 414.4688 384.6250 233.5625 238.3438 137.0625 151.9063 230.5000 193.8750 268.3500
RMSE 448.3285 476.3709 507.0032 488.9572 317.4291 336.7042 200.5305 223.0612 331.8646 326.3546 365.6604
MAE 366.8750 385.7813 430.9063 410.5625 251.7500 278.4063 140.6250 160.2188 255.6250 254.0000 293.4750
MAPE 6.4502 5.6048 5.8927 5.0026 6.2288 6.2597 7.3851 7.1162 4.1920 3.8217 5.7954
MPE 5.6667 5.2956 5.5798 4.7142 5.8203 5.3017 7.0772 6.5489 3.7268 2.9557 5.2687
MASE 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

Note: ETS (Exponential Smoothing), ARIMA (Autoregressive Integrated Moving Average), RW (Random walk), comb (optimal combination method), bu (bottom-up), mo (middle-out), tdgsa and tdgsf (Gross-Sohl methods top-down) and tdfp (forecast-proportion top-down approach).