Bayesian Analysis of the Disaster Damage in Brazil

Abstract

Statistical modeling is a useful methodology to support the research on disaster risk reduction (DRR) and the development of strategies for sustainability. It has been established in at least two of the major United Nations Conferences and Summits which have laid the solid foundation for sustainable development: The 2030 Agenda and the Sendai Framework. International standards also indicate various statistical techniques and tools to support risk management, especially at the risk assessment process. The main goal of this chapter is to present the Bayesian inference framework as a promising tool for risk identification process in the context of DRR. For such, it was used the data collected from the S2ID Brazilian database, period 2003–2016. Considering the geographical complexity, this is only a diagnostic research, but it is a first step and a very interesting beginning because Brazilian complexity is a prominent condition to understand risks and disasters deep causes in this country, and an opportunity to strengthen mathematical and statistical solutions to sustainable development challenges.

1 Introduction

Disaster risk reduction (DRR) is a priority action for sustainable development, due to the disaster impacts on all societies; its study and understanding are a matter to several knowledge areas. Among the exact sciences, the statistical sciences play an important role in that sense. One of the implementation means of the sustainable development goals (SDG) [22] is to intensify our efforts to strengthen statistical capacities in developing countries. In fact, those goals and their targets must be followed up and reviewed by using a set of global indicators approved by a Statistical Commission and complemented by regional and national indicators [23].

Data is another sensible aspect of DRR and SDG strategies. The 2030 Agenda refers to data requirements needed to help with the measurement of SDGs progress, which will be paramount to decision-making: top quality, accessible, timely, and reliable, disjointed in relevant characteristics of national contexts. In the same way, the Sendai Framework [21] promotes collection, analysis, management and use of relevant, reliable and real-time access data to help global, regional, and national DRR programs. The disaster loss databases on detailed scale and relevant non-aggregated data are some useful supports to research disaster risk patterns, causes, and effects, as well as to strengthen disaster risk modeling, assessment, mapping, monitoring, and multi-hazard early warning systems [21]. The Sendai Framework also highlighted the importance of development and dissemination of science-based methodologies and tools to record and share related databases.

Statistical methodologies and tools play a relevant role for the organization and analysis of those data. The international, regional, and bilateral support to strengthen countries statistical capacity for DRR is one of the eight indicators of the Sendai Global Target F: it substantially enhances international cooperation to developing countries through adequate and sustainable support to complement their national actions for implementation of this framework by 2030. The development of all standards and metadata, methodologies, training and technical support, and technical material guidance for follow-up and operationalization of the Sendai indicators are relevant for DRR [24].

The exploratory statistical analysis of these data helps, first and foremost, in identifying variables and patterns that may be related to risk attributes, both qualitative and quantitative, and in simulating possible scenarios of risk. An example of spatial pattern could be the recurrent distribution of floods in certain territories; a time pattern may be their recurrence in the same month or season along many decades. On the other hand, disaster frequency may indicate levels of territory susceptibility and levels of population vulnerability.

This chapter presents a Bayesian inference statistical methodology applied to a disaster database to identify some impact patterns of natural phenomena in Brazil, in terms of their typology, frequency, and regional distribution (South, Southeast, Central West, North, and Northeast region). It is a diagnostic study that intends to help researches on socioeconomic component and disaster risk modeling. The systematic registry and organization of disaster occurrences, as well as the statistical processing of long-term data, allow the identification of disaster background causes in the country, as they may directly support monitoring and warning systems and, indirectly, the development of public policies to adapt to climate change and DRR.

This work is not intended to be an exhaustive study; it is also not the first work that presents a diagnosis of the distribution of risk and disaster patterns in Brazil. The focus of this work is to highlight the importance of investing in the exploration of statistical tools that are suitable for complex analysis of occurrence and disaster impact data, particularly in the Bayesian framework. It is important to draw attention to the importance of developing methodologies for statistical modeling capable of achieving results with the minimum of uncertainties in more detailed studies on risk and disasters.

2 Material and Methods

2.1 Field of Study

Brazil is one of the largest countries in the world, with more than eight million km² of land extension. In this territory, there is a great diversity of natural phenomena (i.e., hydrological, geological, meteorological, etc.) that can cause harm. Furthermore, according to the most up-to-date estimate projections by the national census authority [15], today, Brazil is the sixth most populous country worldwide, with more than 208 million of people. This geographical complexity is a prominent condition to understand the root causes of risks and disasters in this country, because its territory is especially susceptible to climate changes [6] and its population is notably vulnerable to the impact of climate-associated phenomena [12]. It is important to keep in mind that risk equation involves integrating natural and social forces; therefore, risk and disaster analysis must consider both natural hazards and social vulnerabilities [24].

To ease risk and disasters research at national level, it is advisable to consider some territorial zoning in sets of homogeneous regions, either physical or social. As a prior approach to achieve the objectives of this work, it was used, as a geographical reference, the officially recognized regional division of Brazil [15]: North, Northeast, Central West, South, and Southeast regions; note that each region is a group of several states (Fig. 1). Despite being a merely academic division (the regions do not have political autonomy), because it considers geographic, social, and economic factors, it is useful for statistical evaluations, public management, and economic and socio-environmental research.

Fig. 1

Regional division of Brazil. North States: Acre (AC), Amapá (AP), Amazonas (AM), Pará (PA), Rondônia (RO), Roraima (RR); Northeast States: Alagoas (AL), Bahia (BA), Ceará (CE), Maranhão (MA), Paraíba (PB), Pernambuco (PE), Piauí (PI), Rio Grande do Norte (RN), Sergipe (SE); Central West: Distrito Federal (DF), Goiás (GO), Mato Grosso (MT), Mato Grosso do Sul (MS); Southeast States: Espírito Santo (ES), Minas Gerais (MG), Rio de Janeiro (RJ), São Paulo (SP); South: Paraná (PR), Rio Grande do Sul (RS), Santa Catarina (SC)

2.1.1 Disaster Database

Disaster records for the 2003–2016 period were used as input data for this study. These data were obtained from the S2ID (Integrated Disaster Information System) database of the Secretaria Nacional de Proteção e Defesa Civil - SEDEC [4]. Those data are available to download from the SEDEC website; the records are arranged in a spreadsheet by date of occurrence, cities/state affected, and type of event or natural phenomena according to the Cobrade: Brazilian Code of Disasters [8]. Cobrade comprises geological, hydrological, meteorological, climatological, biological, and technological disasters. A spreadsheet with a total of 28,011 records was downloaded from the S2ID database; 51 of those were excluded from the studies because they did not belong to Cobrade. Then, 27,960 records were considered for this work, and they relate to the same number of documents about disasters caused by natural phenomena in Brazil between 2003 and 2016.

The S2ID database was established in 2012 by the national government [7] and nowadays, it is the national system officially recognized for supplying federal resources to cities and states affected by disasters [5]. The files accessed in the Historic Series queries, S2ID database, contain the main information on the Federal Emergency Situations and State of Calamity carried out by SEDEC. It is important to realize that the information registered on S2ID documents refers to Brazilian cities; however, as this is a country diagnostic study, those documents were grouped by states and regions.

2.2 Statistical Inference

Whenever it comes to quantitative analysis, many fields of science use Statistics for assertive and data-driven solutions. A most common interest in real-life problems is to properly assess some unknown quantities, as well as any “uncertainty” concerning this assessment. One could, for example, study the probability of raining in a particular forest, or the mean number of wildfires in some dry regions. We collected a sample of the experiment in study and, after observing it, derived conclusions about these unknown quantities. Since we have only one realization (sample) from many other that are theoretically possible, we must also account for this “randomness” sample in measuring our unknown quantity—or parameter—of interest. This is a very well-known area of statistics called inference.

Statistical inference is all about studying some parameters of interest based on a random sample from an allegedly infinite population. The probability of each sample drawn from all possible population outcomes is very likely to depend on our parameter of interest and can be expressed by analytical devices called probability distributions. An unobserved sample point will always have some probability distribution behind it, and it is also called as a random variable—let us depict it by an upper-case X. An observed sample point is no longer random (now we know its value) and it is denoted by a lower-case x. A variable can be classified into discrete, when its range of possible values is enumerable (generally integers), or continuous when it is not (every real number in range).

Let X be a random variable with probability distribution depending on θ, where θ is our parameter of interest. A probability distribution depends on one or more parameters and is well defined by any of the two following formulations: a probability density function (or a probability mass function when our variable is discrete) represented by p(x|θ), which yields probabilities for continuous ranges of possible values of X (or even for point values if X is discrete), and a cumulative distribution function, the probability of a value lower than or equal to x, denoted by F(x|θ).

As an example, let X denote the number of disasters in some regions. For one to properly investigate X, it makes sense to know more about how it is probabilistically distributed in that specific region. Since we are interested in a counting variable (thus, discrete), a very common probability distribution to explain such class of variables is the Poisson distribution, which depends on a parameter θ—the mean number of disasters in that region (which is unknown)—and has a probability mass function given by

$$\displaystyle \begin{aligned} \begin{array}{rcl} p(x|\theta) = P[X = x|\theta] = \frac{ \theta^{x} e^{-\theta}}{x!} {}\vspace{-3pt} \end{array} \end{aligned} $$

(1)

and cumulative distribution function denoted by

$$\displaystyle \begin{aligned} \begin{array}{rcl} F(x|\theta) = P[X \leq x |\theta] = \displaystyle\sum_{k=0}^{x}\frac{ \theta^{k} e^{-\theta}}{k!}. {} \end{array} \end{aligned} $$

(2)

The Poisson probability distribution is defined by only one parameter—the population mean (several distributions are indexed by two or more parameters). A random variable X with Poisson distribution is indexed by X ∼ Poisson(θ), where θ is the population mean.

Now, let X ₁, X ₂, …, X _n be a random sample of size n drawn from X. In our example, X ₁, X ₂, …, X _n stand for the yearly number of disasters in a given region collected for n years—X _i is the number of disasters at year i (i = 1, 2, …, n). If we can properly use the sample data to “guess” the value of θ in a reliable way, then we will be able to make probabilistic relevant statements about how disaster frequency behaves in that region—a common procedure called estimation.

The next step is to convert the sample data into relevant information to estimate θ; and two basic assumptions about our sample are essential to carry this on:

• 1. the data points must be independent from each other, or equivalently; the occurrence of some value for X _i can not impact the probabilities of possible values of X _j (i ≠ j). Usually, this is not the case for time-dependent sampling such as our example—temporal dependence matters, after all—but for the moment, let us assume independence for the sake of simplicity (see Section 4 for citations of time-dependent inference approaches);

• 2. all sample points (which are random variables) came from the same population; therefore, having the same probability distribution.

If X ₁, X ₂, …, X _n are independent random variables, it is possible to derive a joint probability distribution for all the sample data by multiplying their probability density (or mass) functions, in an analogous way of calculating joint probabilities of independent events in probability theory. Then, the joint probability density (mass) function is defined by $p(\mathbf {x}|\theta ) = \prod _{i=1}^{n} p(x_i|\theta )$ and can be used to derive probabilities of possible samples of size n.

This quantity can also be seen as a function of θ, in which its interpretation changes drastically. Now, this function shows how much a value of θ is “likely” to be the true θ under the sample information. In other words, it shows which values of θ would give our sample a high probability of occurrence (since we managed to collect it anyway). When interpreted in this way, it is denoted by L(θ|x) and called as a likelihood function of θ. If X ∼ Poisson(θ), the likelihood function of θ for a given sample x = (x ₁, x ₂, …, x _n) is

$$\displaystyle \begin{aligned} \begin{array}{rcl} L(\theta | \mathbf{x}) = \prod_{i=1}^{n} p(x_i|\theta) = \frac{\theta^{\sum_{i=1}^{n} x_i} e^{-n\theta} }{\prod_{i=1}^{n} x_i!}. {} \end{array} \end{aligned} $$

(3)

A very common estimation procedure is to find the value of θ that maximizes the likelihood function or, in other words, the one θ that optimizes our sample probability of occurrence. This estimator of θ is known as its maximum likelihood estimator and composes an integral part of the classical inference context. One main issue of it, though, is that we cannot include any subjective statement about θ in its estimation—it is solely carried out based on the sampled data. In fact, the θ parameter is a fixed, immutable point determined by Nature and no prior beliefs about it are allowed to support its estimation in the classic inferential procedure. On the other hand, Bayesian inference [2, 16, 18] presents an entirely different approach for the problem of inference about θ, by interpreting it as some measurable quantity via probability distributions. This context allows us to combine some prior belief about θ into the defined-by-data likelihood to assist the inferential process.

2.3 Bayesian Inference

2.3.1 Basic Concepts

The Bayesian methodology consists in specifying some probability distributions for the observed variables conditioned to one (or more) unknown parameter(s), denoted by θ for simplification purposes and constructing its likelihood function L(θ|x) based on sample data. It is assumed that θ is also a random variable with some prior probability function p(θ) defined before sample collection. The inference about the parameter is based on the posterior probability distribution obtained by the application of the well-known Bayes’ theorem [18],

$$\displaystyle \begin{aligned} \begin{array}{rcl} p(\theta|\mathbf{x}) = \frac{ L(\theta | \mathbf{x}) p(\theta)} {\int_{\varTheta}L(\theta | \mathbf{x}) p(\theta)} \\ \ \ \ = \frac{ L(\theta | \mathbf{x}) p(\theta)} {p(\mathbf{x}) }, {} \end{array} \end{aligned} $$

(4)

where Θ denotes the parametric space (the set of all possible values) of θ and p(x) is the normalizing constant of p(θ|x), (since p(θ|x) is a probability density and must be integrable to one) called the marginal distribution or predictive distribution of X [14]. Note that L(θ|x) is actually a function of the sample, which explains why the Bayes’ theorem applies. From (4), p(θ|x) is proportional to the multiplication of the likelihood and the prior function,

$$\displaystyle \begin{aligned} \begin{array}{rcl} p(\theta|\mathbf{x}) \propto L(\theta | \mathbf{x}) p(\theta). {} \end{array} \end{aligned} $$

(5)

Bayes’ theorem is one of the few results of Mathematics that proposes to characterize learning with experience. In Bayesian inference, each problem is unique and θ is a quantity of interest taken with different levels of knowledge that depends on the problem in hand and on who analyzes it. Then, for Bayesian, the probability distribution that captures this variability is based on a prior information and is subjective by nature [18].

One of the most important questions in Bayesian inference concerns prior distributions, which represents the knowledge about an uncertain parameter θ before observing the results of a new experiment. Being aware of which information is going into the prior distribution and the properties of the resulting posterior are crucial when setting up a prior distribution for a specific problem. An important note is that the prior distribution does not need to carry any information about θ at all, representing in this fashion our ignorance about its behavior. Examples of non-informative priors involve flat distributions for θ (such as Normal with $\sigma _\theta ^2 > 10^6$) and functional forms designed to intentionally maximize our ignorance (or entropy) about θ for a given dataset.

These priors are included in the class of objective priors, in a sense that their impact on the posterior distribution does not depend on individual and subjective beliefs, thus leading to “objective” results. On the other hand, there are situations in which may be desirable to add some prior knowledge onto the elicitation of the prior distribution. Objective informative priors are given when knowledge about θ comes from a quantifiable source, like historical data. When such information, however, comes from an expert observation, we are dealing with subjective information, since it is not quantifiable in any unique and well-defined way and brings to surface a class of prior distributions called subjective priors.

One problem in the implementation of Bayesian methodologies is the analytical intractability. The class of conjugate priors aims to get over this issue by formulating a prior distribution which has the same functional form of the posterior distribution, when combined with the data information expressed through the likelihood function. It deserves special attention when we want to sequentially update our inference about θ as a new data that becomes available over time.

If X ∼ Poisson(θ), we could consider a Gamma(α, β) prior distribution for θ, which depends on parameters α and β and has the following form:

$$\displaystyle \begin{aligned} \begin{array}{rcl} p(\theta|\alpha,\beta) = \frac{\beta^\alpha}{\varGamma(\alpha)}\theta^{\alpha - 1} e^{-\beta\theta}, {} \end{array} \end{aligned} $$

(6)

where Γ(α) is the Gamma function applied on α. When combined with the Poisson likelihood given in (3), the posterior probability density of θ is given by

$$\displaystyle \begin{aligned} p(\theta|\mathbf{x}) & \propto L(\theta | \mathbf{x})p(\theta|\alpha,\beta) \\ & = \frac{\theta^{\sum_{i=1}^{n} x_i} e^{-n\theta} }{\prod_{i=1}^{n} x_i!}\frac{\beta^\alpha}{\varGamma(\alpha)}\theta^{\alpha - 1} e^{-\beta\theta} \\ & \propto \theta^{\sum_{i=1}^{n} x_i + \alpha - 1} e^{-(n + \beta)\theta}, \end{aligned} $$

where constant quantities in θ were omitted (a useful practice when calculating posterior distributions). It turns out that the remaining expression is proportional to a Gamma distribution with parameters $\sum _{i=1}^{n} x_i + \alpha$ and n + β. This is an example of a conjugate prior: if X ∼ Poisson(θ) and θ ∼ Gamma (α, β), then $\theta |\mathbf {X} \sim Gamma~(\sum _{i=1}^{n} x_i + \alpha , n + \beta )$. Further details about conjugate priors are given in [16] and [18].

Posterior distributions can be used to provide point estimates or interval estimates (such as the highest posterior density ones [16]) of the parameter of interest, hypothesis testing, or to predict the value of future observations. It provides a unified set of Bayesian solutions to the conventional problems of scientific inference. The choice of the Bayesian point estimates of θ depends on the form of the posterior distribution, as well as the objectives of its use. The most used estimates are the posterior mode, posterior mean, and posterior median:

• Posterior mode: $\hat {\theta }$ such that $\max \limits _{\theta \in \varTheta } p(\theta |\mathbf {x})$;

• Posterior mean: $\hat {\theta }= E[\theta |\mathbf {x}]$;

• Posterior median: $\hat {\theta }$ such that $P[\theta \geq \hat {\theta } ] \geq 1/2$ and $P[\theta \leq \hat {\theta } ] \geq 1/2$.

Bayesian approach can be used to sequentially update the information about the parameter as new data become available. Suppose we formulate a prior for the parameter θ and observe a random sample x ₁. Then the posterior is

$$\displaystyle \begin{aligned}p(\theta|{\mathbf{x}}_1) \propto L(\theta | {\mathbf{x}}_1 )p(\theta).\end{aligned}$$

If a new sample x ₂ is observed, we can use the previous posterior as the new prior and derive a new posterior,

$$\displaystyle \begin{aligned}p(\theta|{\mathbf{x}}_2) \propto L(\theta | {\mathbf{x}}_2 )p(\theta|{\mathbf{x}}_1).\end{aligned}$$

This “sequential updating” process can continue indefinitely in the Bayesian setup.

The Bayesian operation is often difficult to execute and requires the use of numerical methods and the approximate Monte Carlo simulation method via Markov Chains. All results presented from this point on were produced with the statistical software R, version 3.3.3 [10].

2.3.2 Bayesian Inference for Rate of Disasters Occurrence

Then, let X be a random variable related to the number of disasters occurring in a given period. We have that X given θ has a Poisson distribution with parameter θ, the disaster occurrence rate in the selected period. Consider x = (x ₁, x ₂, …, x _n) an observed random sample of the random variable X. With likelihood function given by (3) and prior function for θ given by (6), that is, a Gamma(α, β) conjugate prior, the posterior distribution of θ is $\theta | \mathbf {X} \sim Gamma(\sum _{i=1}^{n}x_{i} + \alpha , n+\beta )$, with posterior mean given by

$$\displaystyle \begin{aligned} \begin{array}{rcl} E[\theta | X] = \frac{\sum_{i=1}^{n}x_{i} + \alpha}{n+\beta}. {} \end{array} \end{aligned} $$

(7)

and posterior variance

$$\displaystyle \begin{aligned} \begin{array}{rcl} Var[\theta | X] = \frac{\sum_{i=1}^{n}x_{i} + \alpha}{ (n+\beta)^2 }. {} \end{array} \end{aligned} $$

(8)

Figure 2 represents the described Bayesian framework for some simulated data. It shows the comparison of the prior distribution with the posterior distribution to assess to what extent the experimental information can alter the initial belief.

Fig. 2

Illustration of prior distribution, likelihood function, and posterior distribution, considering simulated data

2.3.3 Bayesian Prediction

Professionals from a multitude of fields often need to make some realistic statements about the likely outcome of a future “experiment of interest” based on the distribution of previously conducted related experiments. In the Bayesian framework, given an observed quantity X related to an unobserved parameter θ through a posterior distribution p ₁(θ|x), our interest relies on making inference about another random quantity Y  related to X and θ through p ₂(Y |θ, X).

After observing a random sample x = {x ₁, x ₂, …, x _n}, why not just predict some new observation x _n+1 by plugging into the distribution of X some posterior point estimate $\hat {\theta }$? It turns out that we must account for the uncertainty about θ in our prediction, otherwise, new observations would have underestimated variability. Since this uncertainty about θ is reflected on its posterior, averaging the distribution of x _n+1 over it would yield

$$\displaystyle \begin{aligned} p(x_{n+1} | \mathbf{x}) = \displaystyle\int p(x_{n+1}|\theta)p(\theta|\mathbf{x}) d\theta. {} \end{aligned} $$

(9)

If X follows a Poisson distribution with parameter θ having a Gamma(α, β) conjugate prior, it was shown that $\theta | \mathbf {X} \sim Gamma \left (\sum _{i=1}^{n} x_i + \alpha ; n + \beta \right )$. If we are interested in making inference about some unobserved value x _n+1 from the same population of {x ₁, x ₂, …, x _n} and θ-conditionally independent from them, the predictive distribution is given by

$$\displaystyle \begin{aligned} p(x_{n+1} | \mathbf{x}) & = \int_{-\infty}^{\infty} p(x_{n+1}|\theta)p(\theta|\mathbf{x}) d\theta \\ & = \int_{-\infty}^{\infty} \frac{\theta^{x_{n+1}} e^{-\theta}}{x_{n+1}!} \frac{(n + \beta)^{\sum_{i=1}^{n} x_i + \alpha}}{\varGamma\left(\sum_{i=1}^{n} x_i + \alpha \right)}\theta^{\sum_{i=1}^{n} x_i + \alpha - 1} e^{-(n + \beta)\theta} d\theta \\ & = \frac{(n + \beta)^{\sum_{i=1}^{n} x_i + \alpha}}{\varGamma\left(\sum_{i=1}^{n} x_i + \alpha \right) x_{n+1}!} \int_{-\infty}^{\infty} \theta^{\sum_{i=1}^{n+1} x_i + \alpha - 1} e^{-(n + \beta + 1)\theta} d\theta \\ & = \frac{(n + \beta)^{\sum_{i=1}^{n} x_i + \alpha}}{\varGamma\left(\sum_{i=1}^{n} x_i + \alpha \right) x_{n+1}!} \frac{\varGamma\left(\sum_{i=1}^{n+1} x_i + \alpha \right)}{(n + \beta + 1)^{\sum_{i=1}^{n+1} x_i + \alpha}}, \end{aligned} $$

since $\theta ^{\sum _{i=1}^{n+1} x_i + \alpha - 1} e^{-(n + \beta + 1)\theta }$ is proportional to .

Note that (9) is completely general, in that p can be any density or even an extremely complex model, and θ can represent one, two, or even thousands of unknown parameters. In practice, this can be a difficult calculation. So, most of the time, we simulate values from the predictive density, rather than getting the exact analytical solution:

• 1. Draw a sample from the posterior density of θ, p(θ|x).

• 2. Then plug it into p(x _n+1|θ) and draw an x _n+1 observation from it.

• 3. Repeat steps 1 and 2 many times (at least 1000). Since a different value of θ is used in each time, we automatically restore the uncertainty missing when we plug in just a single value of θ.

2.3.4 Mixture of Distributions

An alternative way to evaluate the distribution of the number of disasters in each Brazilian state can be made possible after the idea of mixture of distributions, which is described by [17]. If p ₀(x), p ₁(x), …, p _k(x) is a sequence of either all discrete probability mass functions or all probability density functions, and ω ₀, ω ₁, …, ω _k are a sequence of weights satisfying ω _i > 0 and $\sum _{i=1}^{k} \omega _i = 1$, then $\sum _{i=1}^{k} \omega _i p_i(x)$ is also a probability mass/density function, which is called a mixture of distributions.

As an example, consider just one state of Brazil for some period of J years, and that we want to study its mean number of disasters θ, θ ∈ Θ. For the j-th year (j = 1, …, J), let X _j be a random variable representing the number of occurred disasters at year j, with probability density function given by p(x _j|θ). The likelihood function for the j-th year is $L_{j}(\theta | \boldsymbol {x}_{j})= \prod _{i=1}^{n_{j}} p(x_{ij}|\theta ) = p(x_{j}|\theta )$, since we have only one observation for each year in this case (n _j = 1).

Under the Bayesian context, we derive the prior distribution of θ, p(θ), and the posterior distribution of θ for each year given their respective data, p _j(θ|x _j). Therefore, for the chosen state, the posterior distribution of θ is given by the mixture of J posterior distributions of θ. Then, the mixture of posterior distributions is given by

$$\displaystyle \begin{aligned} p_{M}(\theta|\boldsymbol{x}) = \displaystyle\sum_{j=1}^{J} \omega_{j} p_{j} (\theta|\boldsymbol{x}_{j}) {} \end{aligned} $$

(10)

where ω _j is the weight of the j-th year.

3 Results

The Brazilian country has 5570 municipalities, from which 75% of them (4168) have declared an emergency situation at least once during the study period; for the remaining 25%, data is not available. Among the first 75%, some cities stand out for the number of times where an emergency situation was declared in Ceara State (CE): Irauçuba (28 occurrences), Caridade (27), Tauá, (27), and Pedra Branca (26); in Pernambuco State (PE): Lagoa Grande (26), and Santa Cruz (26). The most affected states of the country in the same period were Paraíba (PB) and Rio Grande do Sul (RS), with 3286 and 3215 emergency situations reported, respectively.

The map of Fig. 3 shows the spatial distribution of disasters in Brazil between 2003 and 2016. The map revels that six states (RR, AP, AC, RO, TO, GO) and Distrito Federal (DF), each one, had less than 200 emergency situations declared, a very low number if compared to the others. PB and RS are the most affected states, registering more than 3000 occurrences, and six other states (BA, CE, MG, SC, PE, PI) had between 2000 and 3000 occurrences in the same period. There are many factors that can explain these numbers, like frequency, magnitude, or intensity of hazardous events or the exposure level of municipalities to them, but this type of analysis is out of scope from this study.

Fig. 3

Spatial distribution of disasters in Brazil. It refers to S2ID data from 2003 to 2016

It is advisable to approach risk and disaster research by classifying the territory in homogeneous regions of environmental determinants such as topography, climate, geological substrate, soils, hydrology, or vegetation because of their correlation with natural threats. In spite of that, another useful approach is by identifying the impacts and the level of hazards damage on the population; this information is related to the spatial distribution of natural phenomena that can cause damage, but also can reflect social asymmetries that are predominant in each social-physical region. For example, Fig. 3 shows that the greater frequencies of disaster occurrences are in the set of the most densely populated states, instead of those exposed to multi-hazards. However, this study does not consider if there is or not a civil defense in the municipalities responsible for generating S2ID documents.

Figure 4 shows a predominance of the flood event in much of the territory in the Northern Region: (AC, AM, AP, and PA states); dryness is spread to all Brazilian Regions: North (RO, RR, TO), Northeast (BA, CE, MA, PB, PE, PI), Central West (MS), Southeast (MG and RJ), and South (PR and RS); drought stands out in AL, SE, and RN states of Northeast; higher frequency of flash floods predominates in GO (Central West-CO), SP and ES (Southeast), and SC (South); MT and CO stand out due to intense frequency of rainfalls. A small portion of Federal District (DF) records is related to wildfires.

Fig. 4

Spatial distribution of disaster occurrences by cause typology in Brazil. It refers to S2ID data from 2003 to 2016

Although the spatial distribution of the number of occurrences is not related with the regional division of Brazil (Fig. 1), the typology has some correlation: the Northern region is more susceptible to floods; Central West region is more susceptible to intense rainfall and flash floods; Northeast is more susceptible to dryness and drought; Southeast and South regions cope with multiple hazards, standing out among those flooding, dryness, and flash floods.

Dryness and drought were the two more frequent types of threat that generate impacts in the country (2003 to 2012), encompassing 56 and 14% (respectively) over all records used for the study. On the other hand, 2013 was the year with the highest number of disaster occurrences reported on Brazilian S2ID database (Fig. 5), 3743 in total; among those, the most frequent event was dryness (64%), followed by drought (23%). Those results are in agreement with the previous Brazilian studies: dryness and droughts, which are directly related to the reduction of rainfall and to the water deficit, respectively, were the most frequent types of hazard in Brazil period 1991–2012, and are considered the major national problems related to natural threats in this country, like documented on the Atlas Nacional de Desastres Naturais.

Fig. 5

Temporal distribution of the number of disasters per year, Brazil, 2003–2016

To perform our analysis in the Bayesian framework, we only considered the dryness or drought data of Alagoas, Northeast Region, in the study period (Table 1), as the procedure is analogous to the other states. Figure 6 illustrates the sequential updating procedure to derive Bayesian inference and shows how the posterior distribution evolves across the years (it “walks” over the θ axis) as more data is available to update it. In this context, the prior for θ in 2003 is a conjugate, non-informative one (Gamma with α = 0.5 and β = 0.0001, resulting in a large variance) and the data from 2003 (Poisson with parameter θ) combines to form the posterior distribution of θ in 2003, also a Gamma distribution. For the subsequent years, the posterior from the previous year is used as the prior distribution and combined with the current year data to form the updated posterior distribution of θ (note that these posterior distributions are always Gamma).

Fig. 6

Updating procedure per year, Alagoas, Brazil, 2003–2015

Table 1 Series of the number of disasters (dryness and droughts) for the State of Alagoas, Brazil (2003–2016)

It is noticeable how little the posterior evolves at the last years; in fact, it has already absorbed so much past information, especially the 2004–2012 period, that makes new data struggle to change our belief about θ, regardless of how impactful they are. The posterior distribution of θ in 2015 tells us that having 60 or more disasters in Alagoas is highly unlikely (almost zero probability), even though 97 disasters were reported in 2013, just 2 years before.

For comparison purposes, Fig. 7 shows the mixed posterior distribution of θ considering the period from 2003 to 2015 (2016 was left out to compare predictions from both approaches). Instead of sequential updating, here all the year samples are being observed at the same time. The weight of each sample is proportional to how fresh is the data: newer data has more weight (as its information is more up-to-date), which declines proportionally for older data. Since we have a 13-year time series for analysis (2003–2015), the most recent year of 2015 has a weight of 13∕(1 + 2 + … + 13) = 13∕91. The year of 2014 has a weight of 12∕91 and so on until 2003, with a weight of 1∕91. In this way, information loses value as it grows too old and no longer reflects the current reality.

Fig. 7

Mixture of posterior distributions of θ, Alagoas, Brazil, 2003–2015

The likelihood of each year is combined with a non-informative prior distribution that is the same for all years (again, Gamma(0.5, 0.0001)). Then, the calculated posteriors for each year are combined with the described weights, leading to a mixed posterior distribution that considers all the different information gathered at the period of 2003–2015, giving more weight to recent events. As 2013 showed 97 disasters, this is why low values (< 50) are still predominant, but higher values (> 80) are somehow feasible as well. Table 2 compares both posterior distributions and leads to the same conclusion: even though location measures are similar for both, the 95% HPD credibility interval includes a much wider range of values.

Table 2 Summary of posterior distributions of θ, Alagoas, Brazil, 2003–2015

After calculating the posterior for the year 2015 in both ways (sequential updating and mixing), Fig. 8 shows the predictive distributions attempting to predict the number of disasters in 2016, in the state of Alagoas. It is clear to see that the predictive distribution obtained by the sequentially updated posterior fails to capture the true number of disasters in that state in 2016 (81), as it is located in a region of lower counts and lacks the needed variability to, at least, consider the true 2016 number as possible—a very informative distribution indeed as it has low variance, but misses its target entirely ($P\left [ X_{2016} \geq 81 \right | \theta ]$ = 0). The predictive given by the mixed posterior, in contrast, is much less informative as it has greater variance, but that is exactly a trait of Alagoas series: high variability. The “mixed” predictive distribution carried this trait on and, as a result, was able to capture the true value of 2016 with some degree of certainty ($P\left [ X_{2016} \geq 81 \right | \theta ]$ = 0.17).

Fig. 8

Predictive distribution of x (number of disasters—dryness and droughts) for 2016, Alagoas, Brazil

4 Discussion and Conclusion

This work aimed to summarize some spatio-temporal data of disasters in Brazil while proposing a comprehensive, didactic methodology to study their behavior over time. Among the main results that we observed, in agreement with previous studies [9], it is important to highlight dryness and drought among the natural hazards that cause the most disasters in Brazil. These phenomena corresponded to 51.3% of the documents reviewed in [9] and to 70% of the documents analyzed in this book chapter.

According to [9], the number of disasters in Brazil increases over time, but these numbers alone are not strong enough to assure that disasters are really happening more often. The increase in the number of occurrences recorded in S2ID can be related to several variables; it is not necessarily due to the increase of disasters or intensification of climate change effects, but it may also be related to the new risk management policies established by the Brazilian government as a result of the national system restructuring [7], in which competent official institutions are required to record the damage to the municipality for declaration of emergency or state of public calamity. It is possible to spot a rise in the numbers for the last 7 years of the study period (2003–2016) compared to the earlier ones, but as much as it can represent an actual rise, an improvement of data collection and upkeep is a feasible hypothesis as well.

We presented a Bayesian approach for inference about the number of disasters in Brazil, picking the 2003–2015 period to construct our posterior model and using it to make predictions for 2016. The mixture of distributions is a very simple and intuitive procedure, as it managed to derive some fairly assertive conclusions regarding the prediction year when the traditional sequential update could not. The resulting mixed posterior distribution of θ revealed itself as a much better summary of θ—being non-unimodal and asymmetrical, it is safe to say that it succeeded to capture the variability trait from the series of Alagoas.

There are more sophisticated, suitable ways to extract information from the time series. Some elementary readings in time series model construction are given by [3] and [20], while [1] presents some ways to deal with time series modeling in a Bayesian framework; and a very broad paperwork on Bayesian dynamic models can be found in [11]. To sum it up, some disaster related works include [19], who proposes a generalized Pareto distribution to model rare event occurrences for small samples, and [13] developed a social vulnerability index adaptation for Brazilian reality.

To present a didactic way of data analysis, we limited our study only to the number of disasters, region, and year. Weather and social/geographic exposure-related features would be very interesting to enrich our study, though we are still in the development of a more appropriate national database for making it possible. This is a very important matter, since it would allow us to understand, among other things, some causes of disaster occurrences and their impact on the population. Such a source of information would enable statistical studies to plan and foresight against disasters, followed by risk management to lower population exposure and would improve their response time in face of an occurrence.