Forecasting the long-term monthly variations of major floods

MARIO LEFEBVRE

Department of Mathematics and Industrial Engineering

Polytechnique Montréal

2500, chemin de Polytechnique, Montréal (Québec) H3T 1J4

CANADA

Abstract: The monthly variations of major floods are modelled as a discrete-time Markov chain. Based on this

stochastic process, it is possible, with the help of real-life data, to forecast the future variations of these events.

We are interested in the duration of the floods and in the area affected. By dividing the data set into two equal

parts, we can try to determine whether there are signs of the effects of climate change or global warming.

Key-Words: Markov chains, limiting probabilities, climate change.

Received: April 13, 2021. Revised: March 10, 2022. Accepted: April 13, 2022. Published: May 5, 2022.

1 Introduction

In [4], the author modelled the monthly variations of

major floods worldwide as a discrete-time Markov

chain having three possible states. Similarly, this sto-

chastic process was used as a model for the monthly or

yearly variations of earthquakes. In both cases, using

real-life data, it was found that the models that were

proposed were indeed appropriate to describe the evo-

lution of these events.

Moreover, the limiting probabilities of the Markov

chains were computed, in order to forecast the long-

term behaviour of the processes. Rather surprisingly,

the author concluded that the major floods were seem-

ingly occurring almost at random and did not show

signs of increase due to climate change, whereas

earthquakes, and especially major ones, were trend-

ing upwards.

Markov chains have been used by other authors

as models in various applications. In hydrology,

Avilés et al. [1] forecast drought events based on these

stochastic processes, while Matis et al. [5] used them

to forecast cotton yields. Drton et al. [2] proposed a

Markov chain to model tornadic activity.

Similarly, Markov or semi-Markov processes of-

ten served as models to forecast earthquakes; see, for

instance, Sadeghian [8] and Panorias et al. [6].

Now, in [4], in the case of major floods, the vari-

able of interest was their number per month. There are

however other variables that can be considered. In

the current paper, we will study two such variables,

namely the total duration of the floods and the total

area affected.

As in [4], the data set used will also be divided into

two equal parts to determine whether there have been

some significant changes in the variations of major

floods during the period considered.

In the next section, the mathematical background

will be presented. The model will then be imple-

mented for the total duration of the floods and the total

area affected in Sections 3 and 4, respectively.

2 Mathematical background

We will briefly recall the mathematical results needed

to carry out our study. See also [3] or [4].

A (time-homogeneous) discrete-time Markov

chain is a stochastic process {Xn, n =0,1,2, . . . }

such that

P[Xn+1 =j|Xn=i, Xn−1=in−1, . . . , X0=i0]

=P[Xn+1 =j|Xn=i] := pi,j

for all states i0, . . . , in−1, i, j in the state space Sand

for any n. In this paper, we will assume that the state

space of the process is the finite set S={0,1,2}.

Hence, we assume that for any n,Xnis equal to one

of the numbers 0, 1 or 2, which are actually a cod-

ing system. The matrix Pof the various pi,j ’s is the

transition matrix of the Markov chain.

In the case of a discrete-time Markov chain, the

states iand jcan be the same in pi,j . If we denote

by Kithe number of time units that the chain spends

in state ibefore moving to a different state, then (by

independence) we can write that

P[Ki=k] = (pi,i)k−1(1 −pi,i)

for k= 1,2,3, . . . That is, the random variable Ki

has a geometric distribution with parameter p:=

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Mario Lefebvre

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1−pi,i. Notice that the above probability is strictly

decreasing with k.

Next, we define the limiting probability that the

Markov chain will be in state iwhen it is in equilib-

rium:

πi=lim

n→∞

P[Xn=i].

Under some conditions that will clearly be fulfilled

in our case, we can show (see, for instance, [7]) that

the limiting probabilities exist and can be obtained by

solving the following system of linear equations:

π=π P ,(1)

where π:= (π0, π1, π2), subject to the condition

i=0

πi= 1.(2)

In the next section, the total duration of the ma-

jor floods that occurred during a given month will be

considered.

3 Total duration of the floods

Let Fnbe the number of floods during month n. In

[4], the author defined the following three states for

the variable Xn:

0 : if Fn−Fn−1<−2,

1 : if −2≤Fn−Fn−1≤2,

2 : if Fn−Fn−1>2.

Making use of the data set found on the site flood-

observatory.colorado.edu, which gives a list of large

flood events worldwide from 1985, it was found that

the stochastic process {Xn, n = 2,3, . . .}can be con-

sidered as a Markov chain.

The data for the years 2000 to 2016 were used in

the study. There are 2825 floods in the data set for

this period, so that the average number of floods per

month is 13,85.

For each flood, the data set provides the dates

when it began and ended, its magnitude, the number

of dead, the area affected, etc. The magnitude of a

flood is a number defined by

M=Log(Duration ×Severity ×Affected Area),

in which the Duration is in days, the Affected Area

is in square kilometres and the Severity is equal

to 1, 1,5 or 2 for large, very large and extreme

events, respectively. For the definition of the vari-

able Severity, see the site http://floodobservatory.col-

orado.edu/Archives/ArchiveNotes.html. A flood hav-

ing an Mgreater than 4 (respectively 6) is considered

as severe (respectively very severe). The vast major-

ity of the floods in the data set are at least severe.

The estimated transition matrix was found to be

P= 1/6 19/66 6/11

9/34 27/68 23/68

37/68 23/68 2/17 !,

from which we obtain the following limiting proba-

bilities:

π0=0,3257, π1=0,3420, π2=0,3324.

As mentioned above, we must therefore conclude

rather surprisingly that, in the long run, the three states

of the Markov chain are almost equally likely. Fur-

thermore, we find that the average value of the differ-

ences Fn−Fn−1is 0,0345. Thus, the monthly varia-

tions of the number of major floods do not show any

trend during the period 2000-2016. This conclusion

is strengthened when we divide the data set into two

parts (from 2000 to 2007, and from 2008 to 2016) and

we calculate the corresponding limiting probabilities;

see Table I.

Table I: Limiting probabilities calculated for the pe-

riods 2000-2007 and 2008-2016.

Period π0π1π2

2000-2007 0,3368 0,3263 0,3368

2008-2016 0,3149 0,3575 0,3275

Indeed, the πi’s did not change much between the two

time periods, and are consequently close to the values

obtained for the whole period. Actually, we see that

there are less variations during the period 2008-2016,

because state 1 then has the largest limiting probabil-

ity. This is confirmed by the fact that the standard de-

viation of the monthly variations decreased from 7,54

(in 2000-2007) to 5,90 (in 2008-2016). Finally, the

mean also decreased, from 0,116 to −0,037.

Now, although the number of monthly major

floods appears to be quite stable, there are other vari-

ables related to floods that are important. In this sec-

tion, we consider the total duration of the floods that

started during a given month.

Let Mnbe the total duration of the floods that

started during month n. As in the case of the number

of floods, we define three states for the stochastic pro-

cess {Xn, n = 2,3, . . .}. We write that Xnis equal

to 0 : if Mn−Mn−1<−50,

1 : if −50 ≤Mn−Mn−1≤50,

2 : if Mn−Mn−1>50.

Using the data for the whole time period 2000-2016,

we first obtain the histograms for the variables K0,K1

and K2defined above. These histograms are shown

in Figures 1 to 3, respectively.

As we can see, the histograms present approxi-

mately the exponential decrease that should be ob-

served if the random variable Ki, for i= 0,1,2, has a

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K_0

Frequency

543210

Mean 1,265

N49

Histogram of K_0_1

Exponential

Figure 1: Histogram of the variable K0in the case of

the total duration of the floods.

K_1

Frequency

86420

Mean 1,739

N46

Histogram of K_1_1

Exponential

Figure 2: Histogram of the variable K1in the case of

the total duration of the floods.

geometric distribution. Therefore, we may conclude

that assuming that {Xn, n = 2,3, . . .}is a Markov

chain is realistic.

Next, we can easily estimate the transition proba-

bilities pi,j , for i, j ∈ {0,1,2}. We find the following

estimated transition matrix:

P= 13/64 26/64 25/64

19/79 34/79 26/79

31/59 20/59 8/59 !.

Then, solving the system (1), (2), we obtain the lim-

iting probabilities:

π0=0,3120, π1=0,3962, π2=0,2918.

Moreover, we have the following descriptive statistics

of the 203 differences Mn−Mn−1:

¯x=−0,276 and s=160,4.

Hence, as in the case of the number of major floods,

we must come to the conclusion that the duration of

the floods is quite stable. There is in fact a very slight

decrease in the average total duration of the floods,

and π0is larger than π2.

K_2

Frequency

543210

Mean 1,160

N50

Histogram of K_2_1

Exponential

Figure 3: Histogram of the variable K2in the case of

the total duration of the floods.

To complete this section, we compute the limiting

probabilities for two equal subsets of the data set: first

from January 2000 to June 2008, and then from July

2008 to December 2016. The results are presented in

Table II.

Table II: Limiting probabilities for the total duration

of the floods calculated for the periods I: January 2000

to June 2008, and II: July 2008 to December 2016.

Period π0π1π2

I 0,2803 0,4098 0,3099

II 0,3431 0,3824 0,2745

Since the value of π0has increased very significantly

in the second time period considered, it is possible

to state that, not only the total duration of the major

floods shows no sign of increase, it actually seems to

be decreasing. However, the descriptive statistics of

the differences are

¯xI=−0,833 and sI=192,2,

and

¯xII =0,287 and sII =121,2.

Thus, the average difference increased slightly (about

1,12 days, or 26,88 hours), but the standard deviation

is much smaller during the second part of the period

considered.

In the next section, we will turn to the total area

affected by the floods.

4 Total area affected by the floods

Let Anbe the total area (in 106square kilometres)

affected by the floods during month n. We define the

following states for the random variable Xn:

0 : if An−An−1<−5,

1 : if −5≤An−An−1≤5,

2 : if An−An−1>5.

The histograms for the variables K0,K1and K2ob-

tained for the time period 2000-2016 are presented

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in Figures 4 to 6, respectively. Since the three ran-

dom variables behave approximately like a geometric

distribution, we can claim that the stochastic process

{Xn, n = 2,3, . . .}may be considered as a Markov

chain.

K_0

Frequency

543210

Mean 1,121

N58

Histogram of K_0

Exponential

Figure 4: Histogram of the variable K0in the case of

the total area affected by the floods.

K_1

Frequency

86420

Mean 1,756

N45

Histogram of K_1

Exponential

Figure 5: Histogram of the variable K1in the case of

the total area affected by the floods.

We find that the estimated transition matrix Pis

P= 7/62 32/62 23/62

17/82 37/82 28/82

38/58 12/58 8/58 !,

from which we estimate the limiting probabilities:

π0=0,3086, π1=0,4001, π2=0,2913.

We see that the limiting probabilities are very close

to the ones computed in the previous section. There-

fore, we must again conclude that there is no sign of

upward or downward trend for the monthly total area

affected by the floods. When we divide the data set

into two equal parts, we obtain the values in Table III.

Table III: Limiting probabilities for the total area af-

fected by the floods calculated for the periods I: Jan-

uary 2000 to June 2008, and II: July 2008 to Decem-

ber 2016.

K_2

Frequency

543210

Mean 1,16

N50

Histogram of K_2

Exponential

Figure 6: Histogram of the variable K2in the case of

the total area affected by the floods.

Period π0π1π2

I 0,2934 0,4171 0,2893

II 0,3235 0,3824 0,2941

We observe an increase (respectively a decrease) of π0

(respectively π1), and a slight increase of π2, which

implies that there are more variations in the second

part of the time period considered than in the first one.

However, the limiting probabilities are rather stable.

The main descriptive statistics of the differences

An−An−1are presented in Table IV.

Table IV: Descriptive statistics of the monthly dif-

ferences An−An−1.

Period ¯x s

01/01 – 12/16 0,0791 15,63

01/01 – 06/08 0,1380 15,87

07/08 – 12/16 0,0207 15,46

We see that the average and the standard deviation of

the monthly variations are quite stable, with a small

decrease in each case.

5 Conclusion

In this paper, we continued the study of the monthly

variations of the major floods worldwide that was

started in [4]. In the previous paper, it was found that

the number of major floods does not show any sign

of upward trend during the period 2000-2016. In the

current paper, we considered two important charac-

teristics of the floods, namely their duration and the

area affected. In both cases, the conclusion was the

same as in [4]. Indeed, we observe a slight increase

in the monthly variations, but the most likely state of

the Markov chain remains the one that corresponds

to small variations of the variable of interest. This

conclusion is strengthened when we compute the de-

scriptive statistics of the two variables. We find that

the mean of the observations is close to zero.

We also considered the number of people who died

because of the floods. This variable is more volatile,

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because it depends in particular on the countries that

were affected by the floods. At any rate, if we denote

by Dnthe total number of dead during month n, we

find that the average of the differences Dn−Dn−1

decreased during the period 2000-2016: it went from

2,35 between January 2000 to June 2008, to −10,60

between July 2008 to December 2016. Thus, again

we see no sign of upward trend.

Acknowledgments. This work was supported by the

Natural Sciences and Engineering Research Council

of Canada (NSERC).

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