A Stochastic Model for the Impact of Climate Change

on Temperature and Precipitation

M$5,2 LEFEBVRE

Department of Mathematics and Industrial Engineering

Polytechnique Montréal

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

CANADA

Abstract: The variations from year to year of the monthly average temperatures are modelled as a discrete-time

Markov chain. By computing the limiting probabilities of the Markov chain, we can see the impact of climate

change on these temperatures. The same type of model is proposed for the variations of the monthly amounts of

precipitation. An application to Jordan is presented.

Key-Words: Markov chain, forecasting, limiting probabilities, geometric distribution, goodness-of-fit tests,

climate change.

Received: May 28, 2024. Revised: October 9, 2024. Accepted: November 10, 2024. Published: November 28, 2024.

1 Introduction

Because of the great importance of climate change,

there have been numerous papers written on

prediction models for this phenomenon; see, [1]

and the long list of references therein, [2] and, [3] for

a recent example of such a model.

The author has used stochastic processes, in

particular diffusion processes, as models in hydrology

and meteorology. For example, [4] used filtered

Poisson processes to forecast river flows. This

problem has also been addressed by various authors;

see, [5], [6], [7], [8], [9], [10], [11], [12] and, [13].

In, [14], the author considered the problem of

modelling and forecasting the variations from year

to year of the monthly average temperatures and

amounts of precipitation in Italy. In both cases, he

proposed a discrete-time Markov chain as a model.

After having justified the validity of the model by

making use of real-life data, he computed the limiting

time probabilities of the Markov chains, from which

one can forecast the long-term behaviour of the

chains. Then, he divided the data set into two

equal parts and computed the limiting probabilities

for each part in order to check whether they had

varied significantly in the time period considered.

He concluded that there were some signs of the

effects of climate change on the averages of interest,

particularly on the monthly average amounts of

precipitation. Related papers on this topic are, [15],

[16], [17], [18] and, [19].

In this paper, the aim is to carry out the same type

of analysis for Jordan, a country having a climate

which is quite different from that of Italy. The climate

in Jordan is not uniform, but it can be considered

in general to be subtropical arid. Moreover, most

of the time rainfall is scarce. According to the

experts, temperatures are likely to increase due to

climate change, while the total annual precipitation

will probably decrease. We will see whether the

model that we propose agrees with these statements.

Furthermore, we will be able to quantify the expected

changes.

Let {Xn, n = 0,1, . . .}be a discrete-time

stochastic process having state space S. We recall that

{Xn, n = 0,1, . . .}is a Markov chain if

pi,j (n) := P[Xn+1 =j|Xn=i, (1)

Xn−1=in−1, . . . , X0=i0]

=P[Xn+1 =j|Xn=i]

for all states i0, . . . , in−1, i, j in S, and for any n∈

{0,1, . . .}. In most applications, it is also assumed

that the Markov chain is time-homogeneous or

stationary, so that the one-step transition probabilities

pi,j (n)do not depend on n:

pi,j (n)≡pi,j := P[X1=j|X0=i](2)

for any states i, j ∈S. In practice, this property

may not hold for any n. For instance, in hydrology

the Markov chain might be almost stationary during a

given season, but not over the whole year. In such a

case, a different model may be used for each season.

The matrix P:= (pi,j )i,j∈Sis called the one-step

transition probability matrix. We define the n-step

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transition probability matrix

P(n)=(p(n)

i,j )i,j∈S,(3)

where

p(n)

i,j := P[Xn=j|X0=i].(4)

It can be shown that P(n)=Pn. An important

question is to determine whether the limit

limn→∞ P(n)exists and, if so, to compute the

limiting probabilities

πj:= lim

n→∞ P[Xn=j],(5)

which must not depend on the initial state i.

When the Markov chain is irreducible and ergodic

(see, [20], for example), it can be shown that the πj’s

exist and are the unique positive solution of the linear

system

π=πP,(6)

subject to

∞

∑

i=0

πi= 1,(7)

where

π:= (π0, π1, . . .).(8)

Remark. In theory, we can also obtain the limiting

probabilities by computing the limit limn→∞ P(n). If

the size of the state space Sis large, or even infinite,

this task is quite difficult. However, making use of

a mathematical software program, we can compute

P(n)for nlarge enough, in order to see how fast p(n)

i,j

converges to πj.

In the next section, we will propose a particular

Markov chain as a model for the variations from year

to year of the monthly average temperatures in Jordan.

Next, we will confirm the validity of the model by

making use of real-life data, and we will compute

the limiting probabilities of the chain in order to

forecast what is likely to happen when the process

stabilizes. The model will enable us to look for signs

of climate change in Jordan. Similarly, in Section 3 a

Markov chain will serve as a model for the variations

from year to year of the monthly average amounts of

precipitation. We will conclude this paper with a few

remarks in Section 4.

2 Variations of Monthly Average

Temperatures

We will define the state Xnof the stochastic process

{Xn, n = 0,1, . . .}after nmonths in terms of the

difference Dnbetween the average temperature (in

degrees Celsius) for this month and the corresponding

one of the previous year. Because the experts

consider that a one-degree change in the average

temperature is already significant, we consider the

following states:

Xn={1if Dn≤ −1,

2if Dn∈(−1,1),

3if Dn≥1.

(9)

In practice, we cannot check whether Eqs. (1)

and (2) are indeed satisfied for any states and any

n. Actually, in real-life applications, it is impossible

to exactly satisfy these properties. Our aim is to

propose a model that is at least realistic. To do so,

we can consider the random variable Kithat denotes

the number of months that the process {Xn, n =

0,1, . . .}spends in state ibefore making a transition

to another state. If {Xn, n = 0,1, . . .}is indeed a

Markov chain, then by independence we can write

that Kihas a geometric distribution with parameter

p:= 1 −pi,i, so that

P[Ki=k] = pk−1

i,i (1 −pi,i),for k= 1,2, . . .

(10)

When we have a set of real-life data, we can

perform Pearson’s goodness-of-fit statistical test to

check whether the geometric distribution is an

acceptable model for Ki,i= 1,2,3. More simply,

we can look at the histograms of the random variables

to see whether their form is that of a decreasing

exponential function, which should be the case if they

have a geometric distribution.

The historical monthly average temperatures

and rainfalls for all the countries in

the world can be found on the website

Climate Change Knowledge Portal (CCKP)

(https://climateknowledgeportal.worldbank.org/)

that was created by the World Bank. To estimate the

transition probabilities pi,j of the Markov chains, the

data for the years 1991-2016 will be used. Then, we

will compute the limiting probabilities πj, which will

enable us to forecast the long-term behaviour of the

variations of the monthly average temperatures and

rainfalls. Finally, we will divide the dataset into two

equal subsets: from 1991 to 2003, and from 2004

to 2016. We will compute the limiting probabilities

for each subset and compare them to see if there are

significant signs of climate change.

The differences in monthly average temperatures

seem to be quite random and symmetrical, as can

be seen in Figure 1 and Figure 2 produced by the

statistical software program Minitab. Moreover,

based on the Anderson-Darling normality test

performed by Minitab, we can easily accept that the

differences follow a Gaussian distribution, with a

p-value greater than 0.58; see Figure 3. Hence, it is a

good dataset.

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Index

Diff

3102792482171861551249362311

5,0

2,5

0,0

-2,5

-5,0

Time Series Plot of Diff

Fig1: Time series plot of the differences in

monthly average temperatures.

Diff

Frequency

4,53,01,50,0-1,5-3,0-4,5-6,0

Mean 0,05835

StDev 1,653

N 300

Histogram of Diff

Normal

Fig2: Histogram of the differences in monthly

average temperatures.

Using the whole dataset, Minitab produced the

histograms of the random variables K1,K2and K3

that are shown in Figure 4, Figure 5, and Figure 6,

together with an exponentially decreasing function.

Although K1and K3took only respectively two and

three different values, we can state that the form of

each histogram (especially that of K2, which took

7 different values) is in agreement with that of a

geometric distribution. Based on the histograms,

a Markov chain with the above states seems to be

appropriate to serve as an approximate model for

the variations of the monthly average temperatures

in Jordan between 1991 and 2016. To make this

statement more precise, we performed Pearson’s

chi-square statistical test to check the goodness of

fit of a geometric distribution to the data. The

statistics χ2and the p-values are given in Table 1.

From these values, one can indeed accept that each

random variable Kihas approximately a geometric

distribution.

Table 1Pearson’s chi-square statistical test for a

geometric distribution in the case of temperature

Variable χ2p-value

K10.60 0.74

K21.72 0.89

K30.02 0.99

Diff

Percent

5,02,50,0-2,5-5,0

99,9

0,1

Mean

0,582

0,05835

StDev 1,653

N 300

AD 0,299

P-Value

Probability Plot of Diff

Normal

Fig3: Probability plot of the differences in

monthly average temperatures.

K_1

Frequency

543210

Mean 1,086

N70

Histogram of K_1

Exponential

Fig4: Histogram of the random variable K1in the

case of temperature.

Remarks. (i) Because of the small number of values

taken by K1and K3, the p-values given in Table 1

are only approximate. Moreover, in order to perform

the tests, we assumed that the point estimate ˆpof

the parameter pof the geometric distribution was in

fact the known value of p(otherwise, the number of

degrees of freedom in the case of K1and K3would

be equal to 0). Nevertheless, the p-values in Table 1

are so high that the geometric distribution is surely a

valid model for the random variables Ki.

(ii) In the case when at least one of the variables Ki

cannot be assumed to have a geometric distribution,

we can try to define a new state space in order to

satisfy this fundamental property.

Next, we can estimate the transition probabilities

pi,j of the Markov chain. We find that the estimated

one-step transition probability matrix is given by

P=[6/74 27/74 41/74

27/137 81/137 29/137

39/77 28/77 10/77 ].(11)

Because pi,j = 0 for any pair of states i, j, the

Markov chain {Xn, n = 0,1, . . .}is both irreducible

and aperiodic. Moreover, the number of states being

finite, it is also positive recurrent, and thus ergodic.

Hence, the limiting probabilities exist. Solving the

system (6), (7), we get

π1=0.2530, π2=0.4712, π3=0.2758.(12)

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K_2

Frequency

1086420

Mean 2,306

N62

Histogram of K_2

Exponential

Fig.5: Histogram of the random variable K2in the

case of temperature.

K_3

Frequency

543210

Mean 1,143

N70

Histogram of K_3

Exponential

Fig.6: Histogram of the random variable K3in the

case of temperature.

According to the limiting probabilities, it would

be slightly more likely to observe in the future

an increase of the average temperature of at least

one degree during a given month, compared to the

previous year, than a decrease of at least one degree.

This is consistent with the fact that the average of the

300 monthly differences is equal to 0.0583. However,

the standard deviation is equal to 1.653, which is

relatively large and creates uncertainty.

Now, our aim is to detect signs of climate

change. Therefore, we performed the same analysis

as above for each of the two equal subsets of data.

The transition probability matrices for the periods

1991-2003 and 2004-2016 are respectively

P=[1/31 13/31 17/31

12/77 50/77 15/77

18/36 12/36 6/36 ](13)

and

P=[5/43 14/43 24/43

15/60 31/60 14/60

21/41 16/41 4/41 ].(14)

The resulting limiting probabilities are presented in

Table 2.

We see that the value of π3has increased by about

8.5% in the second time period considered, which is

significant, but the value of π1has increased by a

staggering 30.5%! The long-term probabilities that

the Markov chain will be in state 1 or in state 3 are

Table 2.Limiting probabilities for the time periods

1991-2016, 1991-2003 and 2004-2016 in the case of

monthly average temperatures

Period π1π2π3

1991-2016 0.2530 0.4712 0.2758

1991-2003 0.2199 0.5150 0.2651

2004-2016 0.2870 0.4255 0.2875

almost equal. These results are quite comparable

to the corresponding ones in Italy over the period

1991-2015. However, the changes observed in Jordan

are more pronounced.

According to the website CCKP, the mean annual

temperature has increased by 0.89 degrees Celsius

in Jordan since 1900. We can conclude from the

above data that the variations in monthly average

temperatures have greatly accelerated in recent years.

Thus, the impact of climate change is observable in

Jordan.

Remarks. (i) We find that if we raise the matrix P

given in Eq. (11) to the power 16, then its three lines

are practically equal to the limiting probabilities π1,

π2and π3. Therefore, these limiting probabilities

could be observed after 16 months. Hence, the

process is already almost in steady state. However,

the limiting probabilities should be updated when new

observations become available.

(ii) The time period considered, namely from 1991

to 2016, is relatively short. In order to make the

above conclusions more credible, we calculated the

differences in monthly average temperatures from

1901 onwards. The mean difference for the period

1901-1990 is equal to 0.0056, compared to 0.0583

between 1991 and 2016. It is therefore clear that

the increase in the average temperature has been felt

mainly since the 1990s. More precise results can

be found in Table 3. Notice the steady increase in

the average temperature. Finally, we also computed

Table 3.Differences in monthly average temperatures

for the time period 1901-2016

Period 1901-1930 1931-1960 1961-1975

Mean difference −0.0170 0.0133 0.0140

Period 1976-1990 1991-2003 2004-2016

Mean difference 0.0260 0.0570 0.0600

the probability that the average monthly temperature

will increase by at least 1 degree Celsius, compared

to the previous year, if we assume that the average

monthly difference follows a Gaussian distribution.

The various probabilities are given in Table 4. After

a marked increase between the years 1961 and 1990,

the probability of interest stabilized in the time period

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1991-2016. This probability for 1991-2016 is close

to the value of π3in Table 2. Thus, we may

conclude that these results are consistent with the

above conclusions.

Table 4.Probability that the monthly average

temperature will increase by at least 1 degree Celsius

for various time periods

Period 1931-1960 1961-1990 1991-2016

Probability 0.1897 0.2756 0.2845

(iii) Our aim was to determine whether we can expect

the average annual temperature to increase by at least

1 degree Celsius in the next few years. We can be

more precise and look for changes in the average

temperatures during the various seasons. Because

we need enough data to reach reliable conclusions,

we divided the data set into two equal parts: from

November to April (winter), and from May to October

(summer). Proceeding as above, we obtained the

following results:

Pwinter =[5/47 11/47 31/47

13/48 24/48 11/48

26/49 14/49 9/49 ](15)

and

Psummer =[1/28 18/28 9/28

13/87 54/87 20/87

15/29 13/29 1/29 ].(16)

The corresponding limiting probabilities are given in

Table 5. We see that it is actually during the winter

months that the average monthly temperature is more

likely to increase than to decrease by at least 1 degree

Celsius. The temperatures during the summer months

are expected to remain fairly stable.

Table 5.Limiting probabilities for the monthly

average temperatures during the winter and summer

months for the time period 1991-2016

Season π1π2π3

Winter 0.3099 0.3433 0.3468

Summer 0.2028 0.5894 0.2078

In the next section, the same analysis will be

performed to see whether the conclusions are similar

in the case of monthly average rainfalls in Jordan.

3 Variations of Monthly Average

Rainfalls

Contrary to Italy, there is very little precipitation in

Jordan from about April to October. In Figure 7, we

present the differences in monthly average rainfalls

in the period 1991-2016. We cannot model these

differences as a stationary Markov chain. Instead,

we will consider the months of interest, namely from

November to March. The new time series is shown

in Figure 8, as well as the corresponding histogram

and probability plot in Figure 9 and Figure 10. As

in the previous section, we can surely accept the

hypothesis that the differences follow a Gaussian

distribution; this time the p-value is greater than 0.63.

The mean difference is equal to −0.8434 millimetres

of precipitation, and the standard deviation to 13.46.

Therefore, precipitation has decreased during the

time period considered. On the website CCKP,

we can read that Global Historical Climatology

Network (GHCN) data for the country indicates a

2.92 mm/month per century reduction in average

annual precipitation since 1900. The majority of local

station records indicate that precipitation dropped

from 94 mm to 80 mm during the last 10 years for the

period 1937/38 to 2004/2005. The minimum and

Index

Diff

3102792482171861551249362311

-10

-20

-30

-40

Time Series Plot of Diff

Fig.7: Time series plot of the differences in

monthly average rainfalls

Index

Diff_1

130117104917865523926131

-10

-20

-30

-40

Time Series Plot of Diff_1

Fig.8: Time series plot of the differences

in monthly average rainfalls for the months from

November to March

maximum differences are equal to −34.30 and 35.79

respectively. The median difference is −0.28.

It is less obvious than in the case of temperatures

how we should define the states. We denote by D∗

nthe

difference between the average rainfall (in mm) for a

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Diff_1

Frequency

3020100-10-20-30

Mean -0,8434

StDev 13,46

N 125

Histogram of Diff_1

Normal

Fig.9: Histogram of the differences in monthly

average rainfalls for the months from November to

March.

Diff_1

Percent

403020100-10-20-30-40-50

99,9

0,1

Mean

0,636

-0,8434

StDev 13,46

N 125

AD 0,281

P-Value

Probability Plot of Diff_1

Normal

Fig.10: Probability plot of the differences in

monthly average rainfalls for the months from

November to March.

month of a given year and the corresponding month

of the previous year. We define

Xn={1if D∗

n≤ −5,

2if D∗

n∈(−5,5),

3if D∗

n≥5.

(17)

The histograms of the random variables K1,K2and

K3are presented in Figure 11, Figure 12 and Figure

13. We see that each variable took only two values,

namely 1 and 2. Therefore, the stochastic process

{Xn, n = 0,1, . . .}did not remain for more than

two months in the same state, which concurs with the

volatility observed in Figure 8 and the fact that the

standard deviation is large (13.46). Still, the required

exponential decrease can be observed in each case, so

that the fact that {Xn, n = 0,1, . . .}is approximately

a Markov chain is at least plausible. As in Section 2,

we performed Pearson’s chi-square statistical test

for a geometric distribution; see Table 6. The

(approximate) p-values are all large enough to indeed

consider {Xn, n = 0,1, . . .}as a Markov chain.

Next, with the help of our real-life dataset,

we calculated the estimated one-step transition

probability matrix:

P=[4/47 14/47 29/47

12/29 6/29 11/29

31/45 7/45 7/45 ].(18)

K_1

Frequency

543210

Mean 1,089

N45

Histogram of K_1

Exponential

Fig.11: Histogram of the random variable K1in

the case of monthly average rainfalls.

K_2

Frequency

6543210

Mean 1,318

N22

Histogram of K_2

Exponential

Fig.12: Histogram of the random variable K2in

the case of monthly average rainfalls.

As in the previous section, we can state that the

limiting probabilities exist. By solving the system (6),

(7), we find that

π1=0.3913, π2=0.2227, π3=0.3860.(19)

According to the limiting probabilities, it would be

slightly more likely to observe a drop of the monthly

average precipitation of at least 5 mm during a

given month, compared to the previous year, than an

increase of at least 5 mm. This is consistent with the

mean monthly difference which is equal to −0.8434

mm, thus pointing towards a small decrease.

Finally, for the periods 1991-2003 and 2004-2016

the transition probability matrices are respectively

P=[2/23 7/23 14/23

7/16 3/16 6/16

15/21 4/21 2/21 ](20)

and

P=[2/24 7/24 15/24

5/13 3/13 5/13

16/24 3/24 5/24 ].(21)

The limiting probabilities are given in Table 7.

We see that the value of π3has increased by

more than 10% in the second time period considered,

which is really significant, while the probability of

a difference of less than 5 mm in absolute value has

decreased also by more than 10%. Moreover, rather

unexpectedly, it is now slightly more likely to observe

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K_3

Frequency

543210

Mean 1,184

N38

Histogram of K_3

Exponential

Fig.13: Histogram of the random variable K3in

the case of monthly average rainfalls

Table 6.Pearson’s chi-square statistical test for a

geometric distribution in the case of precipitation

Variable χ2p-value

K10.42 0.81

K20.71 0.40

K31.76 0.41

in the future a monthly increase rather than a decrease

of at least 5 mm of precipitation. Remember however

that these conclusions are only valid for the months

from November to March. Even if there is a small

increase in precipitation during the winter, the total

precipitation for the whole year may still decrease.

Remarks. (i) As in the case of temperature, the three

rows of the matrix P16, where Pis defined in Eq. (18),

are almost equal to limiting probabilities for the time

period 1991-2016. Hence, we may state that our

forecasts should be valid after at most 16 months.

This is due to the fact that the Markov chain is almost

in steady state since 1991.

(ii) We computed the mean differences in monthly

average precipitation for various time periods from

1901 to 2016; see Table 8. After a steady

increase between 1901 and 1960, we observe that

the trend is reversed in the time period 1991-2016.

There is therefore more volatility than in the case

of temperature. Notice however than the mean

differences are rather small.

In Table 9, we present the probability that

the monthly average amount of precipitation will

decrease by at least 5 mm for various time periods,

assuming that the monthly average differences follow

a Gaussian distribution. Because the standard

deviation of the observations was larger in the time

periods 1931-1960 and 1961-1990 than between 1991

and 2016, we find that the probability of interest is

actually quite stable. This probability is close to the

value of π1in Table 7, which again strengthens the

validity of our forecasts.

Table 7.Limiting probabilities for the time periods

1991-2016, 1991-2003 and 2004-2016 in the case of

monthly average precipitation

Period π1π2π3

1991-2016 0.3913 0.2227 0.3860

1991-2003 0.3989 0.2352 0.3659

2004-2016 0.3834 0.2112 0.4053

Table 8.Mean differences in monthly average

precipitation between 1901 and 2016

Period 1901-1930 1931-1960

Mean difference 0.008 0.02

Period 1961-1990 1991-2016

Mean difference 0.21 −0.84

4 Conclusion

In this paper, we proposed a simple stochastic

model to forecast the impacts of climate change

in Jordan. We were interested in forecasting the

expected changes in monthly average temperature

and precipitation. Thanks to a website created

by the World Bank containing real-life data for all

the countries in the world, we were able to both

justify the validity of the Markov chains that we

put forward as models and to estimate their various

transition probabilities. Then, it was a simple

matter to compute the limiting probabilities of the

Markov chains, which enable us to forecast the

future variations of the variables of interest. These

limiting probabilities should be recalculated when

new observations become available.

Because of the scarcity of precipitation in Jordan,

especially in the summer, defining a state space of the

stochastic process {Xn, n = 0,1, . . .}such that this

process could indeed be considered as a Markov chain

was more difficult than in the case of temperature.

By restricting the period during which the model was

used, namely from November to March, we were able

to justify the validity of this assumption and then to

quantify the expected effects of climate change on

precipitation in Jordan.

Jordan is a small enough country for our study

to make sense. Indeed, the statistical data analysis

that we carried out would not be very useful for a

really large country like Russia or Canada. For large

countries, we would need data for their various parts,

which can also be found on the website of the World

Bank.

Table 9.Probability that the monthly average amount

of precipitation will decrease by at least 5 mm for

various time periods

Period 1931-1960 1961-1990 1991-2016

Probability 0.3791 0.3835 0.3786

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Transactions of the Royal Society A,

Vol. 366, 2008, pp. 4647–4664. DOI:

10.1098/rsta.2008.0169

[2] D. M. Wolock, G. J. McCabe Jr., G. D.

Tasker and M. E. Moss, Effects of climate

change on water resources in the Delaware River

basin, Journal of the American Water Resources

Association, Vol. 29, 1993, pp. 475–486. DOI:

10.1111/j.1752-1688.1993.tb03225.x

[3] S. B. Bruns, Z. Csereklyei and D. I. Stern,

A multicointegration model of global

climate change, Journal of Econometrics,

Vol. 214, No. 1, 2020, pp. 175–197. DOI:

10.1016/j.jeconom.2019.05.010

[4] M. Lefebvre and F. Bensalma, Modeling and

forecasting river flows by means of filtered

Poisson processes, Applied Mathematical

Modelling, Vol. 39, No. 1, 2015, pp. 230–243.

DOI: 10.1016/j.apm.2014.05.027 Appl. Math.

Model.

[5] G. Weiss, Shot noise models for the generation

of synthetic streamflow data, Water Resources

Research, Vol. 13, 1977, pp. 101–108. DOI:

10.1029/WR013i001p00101

[6] B. A. Bodo and T. E. Unny, Modèles linéaires

stochastiques théoriques pour la réponse des

petits bassins, Revue des sciences de l’eau

/ Journal of Water Science, Vol. 3, 1990,

pp. 151–182. DOI: 10.7202/705069ar

[7] A. Lawrance and N. Kottegoda, Stochastic

modelling of riverflow time series, Journal of

the Royal Statistical Society. Series A, Vol. 140,

1977, pp. 1–47. DOI: 10.2307/2344516

[8] J. Kelman, A stochastic model for

daily streamflow, Journal of Hydrology,

Vol. 47, 1980, pp. 235–249. DOI:

10.1016/0022-1694(80)90095-5

[9] R. W. Koch, A stochastic streamflow model

based on physical principles, Water Resources

Research, Vol. 21, 1985, pp. 545–553. DOI:

10.1029/WR021i004p00545

[10] F. Konecny, On the shot-noise streamflow

model and its applications, Stochastic

Hydrology and Hydraulics, Vol. 6, No. 4,

1992, pp. 289–303. DOI: 10.1007/BF01581622

[11] F. Murrone, F. Rossi and P. Claps,

Conceptually-based shot noise modeling of

streamflows at short time interval, Stochastic

Hydrology and Hydraulics, Vol. 11, No. 6,

1997, pp. 483–510. DOI: 10.1007/BF02428430

[12] M. L. Kavvas and J. W. Delleur, A

statistical analysis of the daily streamflow

hydrograph, Journal of Hydrology,

Vol. 71, No. 3, 1984, pp. 253–275. DOI:

10.1016/0022-1694(84)90100-8

[13] Y. Yin, Y. Li and W. Bulleit, Stochastic

modeling of snow loads using a filtered Poisson

process, Journal of Cold Regions Engineering,

Vol. 25, No. 1, 2011, pp. 16–36. DOI:

10.1061/(ASCE)CR.1943-5495.0000021

[14] M. Lefebvre, Modelling and forecasting

temperature and precipitation in Italy, Atti

della Accademia Peloritana dei Pericolanti

- Classe di Scienze Fisiche, Matematiche e

Naturali, Vol. 97, No. 2, A2, 2019, 9 pages.

DOI: 10.1478/AAPP.972A2

[15] A. Avilés, R. Célleri, A. Solera and J. Paredes,

Probabilistic forecasting of drought events using

Markov chain- and Bayesian network-based

models: A case study of an Andean regulated

river basin, Water, Vol. 8, No. 2, 2016, 16 pages.

DOI: 10.3390/w8020037

[16] J. E. Caskey Jr., A Markov chain model for

the probability of precipitation occurrence in

intervals of various length, Monthly Weather

Review, Vol. 91, 1963, pp. 298–301. DOI:

10.1175/1520-0493(1963)091<0298:AMCMF

T>2.3.CO;2

[17] M. Drton, C. Marzban, P. Guttorp and

J. T. Schaefer, A Markov chain model of

tornadic activity, Monthly Weather Review,

Vol. 131, 2003, pp. 2941–2953. DOI:

10.1175/1520-0493(2003)131<2941:AMCMO

T>2.0.CO;2

[18] J. H. Matis, T. Birkett and D. Boudreaux,

An application of the Markov chain

approach to forecasting cotton yields

from surveys, Agricultural Systems,

Vol. 29, No. 4, 1989, pp. 357–370. DOI:

10.1016/0308-521X(89)90097-8

References:

[1] R. Knutti, Should we believe model predictions

of future climate change?, Philosophical

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E-ISSN: 2224-3496

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[19] F. Tagliaferri, B. P. Hayes, I. M. Viola and

S. Z. Djokić, Wind modelling with nested

Markov chains, Journal of Wind Engineering

and Industrial Aerodynamics, Vol. 157, 2016,

pp. 118–124. DOI: 10.1016/j.jweia.2016.08.009

[20] S. M. Ross, Introduction to Probability Models,

12th ed., Elsevier/Academic Press, Amsterdam,

2019. DOI: 10.1016/C2017-0-01324-1

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