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DOI 10.1007/s11269-006-9062-y

ORIGINAL ARTICLE

Drought forecasting using the Standardized Precipitation

Index

A. Cancelliere · G. Di Mauro · B. Bonaccorso · G. Rossi

Received: 24 January 2006 / Accepted: 7 June 2006

C ?Springer Science+Business Media B.V. 2006

Abstract Unlike other natural disasters, drought events evolve slowly in time and their

impactsgenerallyspanalongperiodoftime.Suchfeaturesdomakepossibleamoreeffective

drought mitigation of the most adverse effects, provided a timely monitoring of an incoming

drought is available.

Among the several proposed drought monitoring indices, the Standardized Precipitation

Index (SPI) has found widespread application for describing and comparing droughts among

differenttimeperiodsandregionswithdifferentclimaticconditions.However,limitedefforts

have been made to analyze the role of the SPI for drought forecasting.

The aim of the paper is to provide two methodologies for the seasonal forecasting of

SPI, under the hypothesis of uncorrelated and normally distributed monthly precipitation

aggregated at various time scales k. In the first methodology, the auto-covariance matrix of

SPI values is analytically derived, as a function of the statistics of the underlying monthly

precipitation process, in order to compute the transition probabilities from a current drought

condition to another in the future. The proposed analytical approach appears particularly

valuable from a practical stand point in light of the difficulties of applying a frequency

approach due to the limited number of transitions generally observed even on relatively long

SPI records. Also, an analysis of the applicability of a Markov chain model has revealed the

inadequacyofsuchanapproach,sinceitleadstosignificanterrorsinthetransitionprobability

as shown in the paper. In the second methodology, SPI forecasts at a generic time horizon M

are analytically determined, in terms of conditional expectation, as a function of past values

of monthly precipitation. Forecasting accuracy is estimated through an expression of the

Mean Square Error, which allows one to derive confidence intervals of prediction. Validation

of the derived expressions is carried out by comparing theoretical forecasts and observed SPI

values by means of a moving window technique. Results seem to confirm the reliability of

the proposed methodologies, which therefore can find useful application within a drought

monitoring system.

A. Cancelliere · G. Di Mauro · B. Bonaccorso (?) · G. Rossi

Department of Civil and Environmental Engineering, University of Catania, V.le A. Doria, 6-95125

Catania, Italy

e-mail: bbonacco@dica.unict.it

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Keywords Drought.Precipitation.SPI.Stochastic techniques.Transition probabilities.

Forecast

Introduction

Due to a slow evolution in time, drought is a phenomenon whose consequences take a

significant amount of time with respect to its inception in order to be perceived by the socio-

economic systems. Taking advantage of this feature, an effective mitigation of the most

adverse drought impacts is possible, more than in the case of other extreme hydrological

events such as floods, earthquakes, hurricanes, etc., provided a drought monitoring system,

able to promptly warn of the onset of a drought and to follow its evolution in space and time,

is in operation (Rossi, 2003). To this end, an accurate selection of indices for drought identi-

fication, providing a synthetic and objective description of drought conditions, represents a

key point for the implementation of an efficient drought watch system.

Among the several proposed indices for drought monitoring, the Standardized Precipita-

tion Index (SPI) has found widespread application (McKee et al., 1993; Heim, 2000; Wilhite

etal.,2000;RossiandCancelliere,2002).Guttman(1998)andHayesetal.(1999)compared

SPI with Palmer Drought Severity Index (PDSI) and concluded that the SPI has advantages

of statistical consistency, and the ability to describe both short-term and long-term drought

impacts through the different time scales of precipitation anomalies. Also, due to its intrin-

sic probabilistic nature, the SPI is the ideal candidate for carrying out drought risk analysis

(Guttmann,1999).Anevaluationofcommonindicators,accordingtosixweightedevaluation

criteria of performance (robustness, tractability, transparency, sophistication, extendability,

and dimensionality), indicates strengths of the SPI and Deciles over the PDSI (Keyantash

and Dracup, 2002).

Although most of the indices have been developed with the intent to monitor current

drought conditions, nevertheless some of them can be used to forecast the possible evolution

of an ongoing drought, in order to adopt appropriate mitigation measures and policies for

water resources management. Within this framework, Karl et al. (1986) assessed the amount

of precipitation needed to restore normal conditions after a drought event, with reference

to the Palmer Hydrologic Drought Index (PHDI). Cancelliere et al. (1996) proposed a pro-

cedure for short-middle term forecasting of the Palmer Index and tested its applicability to

Mediterranean regions, by computing the probability that an ongoing drought will end in

the following months. Other authors (Lohani et al., 1998) proposed a forecasting procedure

of the Palmer index based on the first-order Markov chains, which enables one to forecast

drought conditions for future months, based on the current drought class described by the

PHDI values. Recently, Bordi et al. (2005) compared two stochastic techniques, namely an

autoregressive model and a novel method called Gamma Highest Probability (GAHP), for

forecasting SPI series at lag 1. The latter method forecasts precipitation of the next month as

the mode of a Gamma distribution fitted to the observed precipitation series. They concluded

that the GAHP performs better, especially in spring and summer months.

In the present paper, a seasonal forecast of the SPI is addressed by means of stochastic

techniques. In particular, transition probabilities from a drought class to another at different

time horizons are analytically derived as a function of the statistical properties of the un-

derlying monthly precipitation. The usefulness of such analytical derivation for estimating

transition probabilities is evident in light of the fact that a Markov chain approach is not ad-

equate to model SPI series, as it is demonstrated in a following section. Also, the analytical

approach enables one to overcome the difficulties related to a frequency approach, whose

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Table 1 Wet and drought period classification according to the SPI index

Index valueClass Probability

?P

Non DroughtSPI ≥ 2.00

1.50 ≤ SPI < 2.00

1.00 ≤ SPI < 1.50

−1.00 ≤ SPI < 1.00

−1.50 ≤ SPI < −1.00

−2.00 ≤ SPI < −1.50

SPI < −2.00

Extremely wet

Very wet

Moderately wet

Near normal

0.977–1.000

0.933–0.977

0.841–0.933

0.159–0.841

0.023

0.044

0.092

0.682

Moderate drought

Severe drought

Extreme drought

0.067–0.159

0.023–0.067

0.000–0.023

0.092

0.044

0.023

reliability may be hindered by the generally limited sample size of the available precipitation

series.

The spatial variability of transition probabilities in Sicily is analyzed in order to provide

indications about the different behaviour of drought phenomena in different areas of the

island.

Also a model to evaluate SPI forecast on the basis of past values of precipitation has been

developed.Morespecifically,analyticalexpressionsofshort-middletermforecastsoftheSPI

arederivedastheexpectationoffutureSPIvaluesconditionedonpastmonthlyprecipitation,

under the hypothesis of uncorrelated and normally distributed precipitation aggregated at

different time scales k. The accuracy of the model is evaluated in terms of the Mean Square

Error (MSE) of prediction (Brockwell and Davis, 1996), which allows confidence intervals

for forecasted values to be computed. Forecasting future values in terms of conditional

expectation ensures that the corresponding forecasts will have minimum MSE. Validation of

the model is carried out based on the historical series observed at 43 precipitation stations in

Sicily (Italy), making use of a moving window scheme for parameters estimation.

The standardized precipitation index

TheSPIisabletotakeintoaccountthedifferenttimescalesatwhichthedroughtphenomenon

occursand,becauseofitsstandardization,isparticularlysuitedtocomparedroughtconditions

among different time periods and regions with different climatic conditions (Bonaccorso

et al., 2003).

The index is based on an equi-probability transformation of aggregated monthly precipi-

tation into a standard normal variable. In practice, computation of the index requires fitting

a probability distribution to aggregated monthly precipitation series (e.g. k = 3, 6, 12, 24

months, etc.), computing the non-exceedence probability related to such aggregated values

and defining the corresponding standard normal quantile as the SPI. McKee et al. (1993)

assumed an aggregated precipitation gamma distributed and used a maximum likelihood

method to estimate the parameters of the distribution.

Although McKee et al. (1993) originally proposed a classification restricted only to

drought periods, it has become customary to use the index to classify wet periods as well.

Table 1 reports the climatic classification according to the SPI, provided by the National

Drought Mitigation Center (NDMC, http://drought.unl.edu). Also, the probabilities ?P, that

the index lies within each class are listed. Since our present work focuses on forecasting

drought conditions, the near normal and wet classes have been grouped into one class termed

“Non-drought”.

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Analytical derivation of transition probabilities of drought classes

Let Z(k)

time scale k of monthly precipitation. Also, let’s indicate by Cithe generic drought class, for

instance C1= Extreme, C2= Severe, C3= Moderate, C4= Non-drought. The probability

that the SPI value after M months lies within a class Cjgiven that the SPI value at the current

month lies within a class Ci, can be expressed as (Mood et al., 1974):

ν,τindicate the SPI value at year ν and month τ = 1, 2, ... 12, for an aggregation

P?Z(k)

ν,τ+M∈ Cj

??Z(k)

ν,τ∈ Ci

?=

??

ci,cjfZ(k)

ν,τ,Z(k)

CifZ(k)

ν,τ+M(t,s) · dt · ds

ν,τ(t) · dt

?

(1)

where fZ(k)

density function of Z(k)

extended to the range of each drought class.

Since, by definition, SPI is marginally distributed as a standard normal variable, it is fair

to assume the joint density function in Equation (1) to be bivariate normal, namely :

ν,τ,Z(k)

ν,τ+M(·) is the joint density function of Z(k)

ν,τ, t and s are integration dummy variables, and the integrals are

ν,τand Z(k)

ν,τ+M, fZ(k)

ν,τ(·) is the marginal

fZ(k)

ν,τ,Z(k)

ν,τ+M(t,s) =

1

2π|?|· exp

?

−1

2XT?−1X

?

(2)

where X = [t, s]T, and ? represents the variance-covariance matrix:

?

? =

1 cov?Z(k)

ν,τ, Z(k)

1

ν,τ+M

?

cov?Z(k)

ν,τ, Z(k)

ν,τ+M

?

?

(3)

Thus,thecomputationoftransitionprobabilitiesinEquation(1)requiresthedetermination

of the autocovariance at lag M of Z(k)

Although in principle, such autocovariance could be estimated from an available sample,

it is of interest here to derive its expression as a function of the statistics of the underlying

precipitation. In general terms, such derivation is not straightforward, because of the equi-

probability transformation underlying the SPI computation. However, under the hypothesis

of monthly precipitation aggregated at time scale k normally distributed, the corresponding

value of SPI can be computed through a simple standardization procedure:

ν,τ+Mnamely cov [Z(k)

ν,τ, Z(k)

ν,τ+M].

Z(k)

ν,τ=Y(k)

ν,τ− μ(k)

σ(k)

τ

τ

(4)

with Y(k)

By assuming precipitation at month τ with mean μτ, the mean of the corresponding

aggregated precipitation Y(k)

ν,τ=?k−1

i=0Xν,τ−iaggregated precipitation at k months.

ν,τwill be respectively:

μ(k)

τ =

k−1

?

i=0

μτ−i

(5a)

Also, if σ2

tion values uncorrelated in time, the standard deviation of the corresponding aggregated

τis the variance of precipitation at month τ, under the hypothesis of precipita-

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precipitation Y(k)

ν,τwill be:

σ(k)

τ

=

?

?

?

?

k−1

?

i=0

σ2

τ−i(x) (5b)

Substituting, Equation (4) becomes:

Z(k)

ν,τ=

?k−1

i=0Xν,τ−i−?k−1

i=0μτ−i

??k−1

i=0σ2

τ−i

(6)

Therefore, the autocovariance can be expressed as:

cov?Z(k)

ν,τ+M, Z(k)

ν,τ

?=

1

??k−1

??k−1

i=0σ2

τ+M−i

?k−1

j=0σ2

τ−j

·

k−1

?

i=0

k−1

?

j=0

cov[Xν,τ+M−j, Xν,τ−i]

=

1

i=0σ2

τ+M−i

?k−1

j=0σ2

τ−j

·

k−M−1

?

i=0

σ2

τ−i

(7)

By substituting Equation (7) in the variance-covariance matrix ?, it follows:

? =

⎡

⎢

⎢

⎣

1

?k−M−1

τ+M−i

i=0

σ2

?k−1

τ−i

??k−1

i=0σ2

j=0σ2

τ−j

?k−M−1

τ+M−i

i=0

σ2

?k−1

τ−i

??k−1

i=0σ2

j=oσ2

τ−j

1

⎤

⎥

⎥

⎦

(8)

Finally, by combining Equation (8) with Equations (1) and (2), it is possible to express

the SPI transition probabilities, in terms of the variances of monthly precipitation. Although

the hypothesis of normality for aggregated monthly precipitation may appear restrictive, it is

worth observing that it can be justified, especially for higher values of the aggregation time

scale k, as a consequence of the central limit theorem.

SPI forecasting

From a stochastic point of view, the problem of forecasting future values of a random vari-

able is equivalent to the determination of the probability density function of future values

conditioned by past observations. Once the conditional distribution is known, the forecast

is usually defined as the expected value or a quantile of such distribution, and confidence

intervals of the forecast values can be computed.

In practice, however, the derivation of the conditional probability distribution of future

valuescanbecumbersomeinmostcases;therefore,usuallyafunctionofthepastobservations

that forecasts future values is sought instead. More formally, let’s consider a sequence of

random variables Y1, Y2, ..., Yt. The interest lies in determining a function f(Y1, Y2, ..., Yt)

that forecast a future value Yt+Mwith minimum error. The latter is usually expressed as the

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Mean Square Error (MSE) of prediction, defined as (Brockwell and Davis, 1996):

MSE = E[(Yt+M− f (Y1,Y2,...,Yt))2](9)

It can be shown, that the function f(•) that minimizes the MSE is the expected value of

Yt+Mconditioned on Y1, Y2, ..., Yt, i.e.:

f (Y1, Y2,...,Yt) = E[Yt+M|Y1, Y2,...,Yt] (10)

The above property allows one to derive the “best” forecast (in MSE sense), provided

the conditional expectation can be computed. Also, it may be worthwhile to note that if

Yt+Mis independent of Y1, Y2, ..., Yt, the best predictor of Yt+Mis its expected value, and

furthermore, the MSE of prediction is just the variance of Yt+M.

Under the hypothesis of aggregated monthly precipitation normally distributed, the best

predictor of the SPI M months ahead˜Z(k)

ν,τ+M, given observations up to month τ will be:

ν,τ−1,...?

i=0(Xν,τ+M−i− μτ+M−i)

σ(k)

τ+M

˜Z(k)

ν,τ+M= E?Z(k)

= E

ν,τ+M

??Z(k)

ν,τ, Z(k)

??k−1

????

k−1

?

i=0

Xν,τ−i,

k−1

?

i=0

Xν,τ−1−i,...

?

(11)

Moreover, since conditioning on the sequence of aggregated precipitation values is equiv-

alent to conditioning on single precipitation values, Equation(11) can be written as:

˜Z(k)

ν,τ+M= E

??k−1

i=0(Xν,τ+M−i− μτ+M−i)

σ(k)

τ+M

|Xν,τ= xν,τ, Xν,τ−1= xν,τ−1,...

?

(12)

Previous equation can be expressed as the sum of two components, namely:

˜Z(k)

ν,τ+M=

?k−M−1

??M

i=0

(xν,τ−i− μτ−i)

σ(k)

τ+M

i=1(Xν,τ+i− μτ+i)

σ(k)

τ+M

+E

|Xν,τ= xν,τ, Xν,τ−1= xν,τ−1,...

?

(13)

where the first term on the right hand side is referred to monthly precipitation observed in the

past (i.e., from τ − k + M + 1 up to the current month τ), while the second one corresponds

to future values from τ + 1 up to month τ + M.

By assuming monthly precipitation serially independent, the above expression simplifies

as:

˜Z(k)

ν,τ+M=

?k−M−1

i=0

(xν,τ−i− μτ−i)

σk

τ+M

(14)

Note that the predictor in Equation (14) is unbiased, as can be easily verified by taking

expectations on both sides.

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The corresponding MSE can be computed as:

MSE(k)

τ+M= E??˜Z(k)

ν,τ+M− Z(k)

ν,τ+M

?2?

(15)

which, upon substitution of Equations (6) and (14) into (15), after some algebra becomes:

MSE(k)

τ+M= E

⎡

⎣

??M−1

?2

i=0(Xν,τ+M−i− μτ+M−i)

σ(k)

τ+M

⎡

i=0

?2⎤

⎦

=

?

1

σ(k)

τ+M

· E

⎣

?M−1

?

(Xν,τ+M−i− μτ+M−i)

?2⎤

⎦

(16)

Since E[?M−1

⎡

i=0(Xν,τ+M−i− μτ+M−i)] = 0, then:

E

⎣

?M−1

i=0

?

(Xν,τ+M−i− μτ+M−i)

?2⎤

⎦= var

= var

?M−1

i=0

?M−1

i=0

?

?

(Xν,τ+M−i− μτ+M−i)

?

(Xν,τ+M−i)

?

= σ2(M)

τ+M

(17)

Thus, by making a substitution, Equation (16) simplifies as:

MSE(k)

τ+M=

?

σ(M)

τ+M

σ(k)

τ+M

?2

(18)

whereσ(M)

observed at the M months preceding month τ + M.

Besides MSE, a practical way of quantifying the accuracy of the forecast is by estimating

the confidence interval of prediction, i.e. an interval that contains the future observed value

withafixedprobabilityα(e.g.95%).Obviously,thewidertheinterval,thelessistheaccuracy

of the forecast and vice-versa. Confidence intervals of prediction for SPI can be estimated by

capitalizing on the intrinsic normality of the index and by observing that, since the predictor

isunbiased,itsvariancecoincideswiththeMSE.Thus,theupperandlowerconfidencelimits

Z1,2of fixed probability α can be computed as:

τ+Misthestandarddeviationofmonthlyprecipitationaggregatedbasedonthevalues

Z1,2=ˆZ ±

√MSE · u

?1 − α

2

?

(19)

where,forthesakeofbrevity,ˆZ representsthegenericforecastandu(·)isthestandardnormal

quantile corresponding to the considered probability.

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Applications to precipitation series observed in Sicily

Theproposedmethodologieshavebeenappliedtomonthlyprecipitationobservedfrom1921

until 2003, for 43 precipitation stations in Sicily. The selected stations are included in the

droughtmonitoringbulletinpublishedontheweb-siteoftheSicilianRegionalHydrographic

Office (Rossi and Cancelliere, 2002, http://www.uirsicilia.it).

Evaluation of transition probabilities

By fixing several combinations of forecasting time horizon M (months) and aggregation

time scales k, for each station the transition probabilities from a class of SPI at month τ

to another one at month τ + M have been computed by Equation (1), for every month. In

order to compute the double integral in Equation (1), which expresses the normal joint cdf

(cumulative distribution function), the algorithm MULNOR has been adopted (Schervish,

1984).

Themeanvaluesoftransitionprobabilitiescorrespondingtothe43stations,for M = 6and

k = 24,arepresentedinFigure1,asafunctionofthecurrentmonthτ.Inparticular,transition

probabilitiesrelatedtodifferentcombinationsofinitialandfinaldroughtconditions(extreme

Ex, severe Se, moderate Mo, non-drought N) have been considered. In order to show also the

variabilitytransitionprobabilityamongdifferentstations,inthesameplots,thelimitsrelated

to ±1 standard deviation are indicated by dashed lines. It can be observed that transition

probabilities vary from one month to another, and for some transitions, also from one station

to another (as indicated by the width of the limits). In particular, the mean probability value,

indicated by the continuous line, of remaining in the extreme class (Ex/Ex) ranges from 60%

(forFebruary–March)tolessthan25%(forAugust–September).Themeanprobabilityvalue

of remaining in the non-drought class (N/N) presents a limited variability across the months,

since it ranges from 95% (for February–March) to about 90% (for August–September). The

transition probabilities from one class to another, for instance from extreme to non-drought

or vice-versa, are very low, at least for the considered time horizon, namely M = 6 months.

In general, it can be concluded that starting from a wet month there is an higher probability

to remain in the same drought class M months ahead (plots along the diagonal), than when

starting from dry months, and conversely a lower probability to return to normal conditions.

Such different behavior can be justified by considering that starting from a wet month and

considering a 6 months time horizon, the occurrence probability of precipitation events, able

to restore normal conditions, is very low. On the contrary, starting from a dry month, there

is a high chance to observe values of precipitation such to modify drought conditions during

the next 6 months.

In order to analyze the effects of the aggregation time scale k and of the forecasting

time horizon M, transition probabilities computed for the 43 stations have been averaged and

representedona3D-plotforaspecificstartingmonthandclass,asafunctionofthefinalclass

and of the M values. In Figure 2 the case corresponding to extreme drought as starting class,

August as starting month and k = 6, 12 and 24 is illustrated. In Figure 3 similar 3D-plots for

February as starting month are shown.

It can be seen that probabilities to remain in the same class generally decrease as the

forecasting time horizon M increases, while transition probabilities to non-drought condition

show an opposite behaviour. Further, as the aggregation time scale k increases, the prob-

abilities of remaining in the same class increase, whereas transition probabilities to return

to non-drought condition generally decrease. Finally, by comparing transition probabilities

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Fig. 1 Mean of transition probabilities (continuous line) for M = 6 and k = 24 computed on 43 stations, and

limits corresponding to ±1 standard deviation (dashed line)

Fig. 2 Mean transition probabilities (computed on the 43 stations) as a function of forecasting time horizon

M (starting class: extreme drought (Ex), starting month: August)

related to August to those corresponding to February, it can be inferred that it is generally

easier to recover from drought conditions starting from August than starting from February.

Finally, an analysis of the spatial variability of transition probabilities in Sicily, has been

also carried out for the 43 stations, by considering an aggregation time scale k = 12 months

and M = 3, 6 and 9 months, with reference to a typical wet month, February, and a dry

month, August. In particular, by interpolating the values of transition probabilities derived

for each station by means of the IDW method (Inverse Distance Weighted), maps of the

spatial distribution of transition probabilities over Sicily region have been obtained.

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Fig. 3 Mean transition probabilities (computed on the 43 stations) as a function of forecasting time horizon

M (starting class: extreme drought (Ex), starting month: February)

Fig. 4 Spatial distribution of transition probabilities from a drought class to a non-drought class (Starting

month: August)

In Figures 4 and 5, maps related respectively to August and February, as starting months,

are shown. It can be concluded that the probability to return to a normal condition increases

as the forecasting time horizon M increases and the starting drought condition decreases.

For the case of August, it can be observed that the north-central part of Sicily is generally

characterized by lower values of probabilities for returning to normal condition with respect

to the rest of the island, while, for the case of February, the same area is characterized by

higher values than anywhere else. Such an opposite behaviour can be partially explained by

the different pluviometric regime in the different areas, which affects the covariance term in

Equation (7) and in turn, transition probability values.

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Fig. 5 Spatial distribution of transition probabilities from a drought class to a non-drought class (Starting

month: February)

Need of analytical approach for deriving transition probabilities

Despite the apparent complexity of the proposed analytical approach for estimation of tran-

sition probabilities, it should be pointed out that the methodology yields results that can

be considered in general more reliable than those obtained by alternative approaches, such

as frequency analysis of observed transitions in an historical sample, or application of a

Markov chain scheme. With regard to the frequency approach, it should be underlined that

the limited sample size of observed transitions among different SPI classes, even in the case

of rather long records of monthly precipitation, hinders the possibility of reliable frequency

estimates.

As an example, Table 2 reports the number of transitions, observed from February to

August, among SPI classes computed on precipitation series observed for Caltanissetta ag-

gregated at k = 3, 6, 9, 12 and 24 months (80 years). It can be inferred that the number of

observed transitions is generally not sufficient to compute reliable frequency estimates. Fur-

thermore, the lack of observed transitions in some cases (e.g. going from Extreme conditions

in February to any other conditions in August) would lead to the misleading conclusion that

suchtransitionshavezerooccurrenceprobability,whichisobviouslynotcorrect.Application

of the analytical approach on the other hand, allows one to always estimate transition proba-

bilities, even from relatively short records, since the whole available precipitation series, and

not just the few observed transitions, are utilized.

Regarding the applicability of Markov chain hypothesis to model transitions of SPI val-

ues from one drought class to another, it should be mentioned that such an assumption

may not be valid in general. Indeed, under the non-homogeneous lag-1 Markov hypothe-

sis, the lag-M transition probability matrix ?(M)τ, whose generic element (i, j) is given by

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Table 2 Number of transitions between drought/non-drought classes starting in February, ending

in August, computed from SPI series ( k = 3, 6, 9, 12, 24 months) of Caltagirone station, Sicily (80

years)

Drought class in August

Drought class in February Extreme SevereModerate Non droughtTotal

k = 3Extreme

Severe

Moderate

Non drought

Extreme

Severe

Moderate

Non drought

Extreme

Severe

Moderate

Non drought

Extreme

Severe

Moderate

Non drought

Extreme

Severe

Moderate

Non drought

0

0

0

0

0

0

1

1

0

0

1

0

0

0

0

0

0

0

1

0

0

0

0

4

0

0

0

0

0

1

0

0

0

1

2

3

0

5

0

0

0

0

0

3

2

4

3

2

4

105771

k = 60

0

1

1

3

6

1

3

8

1057 68

k = 90

1

3

9

1

0

0

5

0

3

2

3

0

1

6

0

3

10

6758

k = 120

3

5

1

4

7

6068

k = 240

2

0

0

10

3

6764

P[Z(k)

ν,τ+M∈ Cj|Z(k)

ν,τ∈ Ci], can be written, for fixed k, as (Bremaud, 1999):

?(M)τ= ?τ?τ+1...?τ+M−1

(20)

where ?τis the lag 1 transition probability matrix whose generic element (i, j) is given by

P[Z(k)

Note that the non homogenous formulation is required for the SPI, since the homogenous

one would obviously not be able to model the general strong seasonal pattern observed in

transition probabilities (see Figure 1).

In order to verify whether the SPI can be modeled by a Markov chain, the (exact) lag-M

transition probability matrix ?(M)τ computed by means of Equation (1), has been com-

pared with the one obtained, under the Markov hypothesis, by Equation (20). As an example,

in Tables 3 and 4, the percentage differences between each element of the two matrices

are shown with reference to Caltagirone station and to the transitions from February and

August with time horizon M = 3, 6 months, and at k = 9, 12, 24 months. From the ta-

ble, it can be inferred that the differences between the two approaches are generally not

negligible. Furthermore, since the analytical approach can be considered exact, such dif-

ferences can be interpreted as percentage errors in computing transition probabilities when

a Markov chain model is adopted. For instance, under the Markov hypothesis, the per-

centage error in the probabilities of transition from Extreme condition in February to Ex-

treme condition in May are generally in the order of 15%. In other cases, the errors appear

ν,τ+1∈ Cj|Z(k)

ν,τ∈ Ci].

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Fig. 6 Theoretical values of SPI’s MSE for different k and M and different ratios M/k, under the hypothesis

of normally distributed aggregated precipitation series

much larger, although the errors related to the Normal-Normal transition are always very

low.

SimilarresultshavealsobeenobtainedbyconsideringothervaluesofMandkanddifferent

stations.

The above errors, coupled with the unreliability of the frequency estimation of transition

probabilities, clearly indicates the practical value of the proposed analytical approach, which

can find application even when relatively short records of precipitation are available.

SPI forecasts

The forecasting model proposed has been applied by considering several aggregation time

scales, k = 6, 9, 12 and 24 months, as well as different forecasting time horizons M = 3, 6,

9 and 12 months.

First, theoretical MSE values (see Equation (18)) have been computed for all 43 stations.

Figure 6 illustrates the boxplots of monthly MSE values for different aggregation time scales

kandtimehorizon M = 3months.Theoverallheightofeachboxplotindicatesthevariability

of MSE among the different stations. As expected, the performance of the forecasting model

gets better (lower MSE’s) as the ratio of M over k decreases. The effect of seasonality is

evident in the considered cases, although with different patterns according to the scale of

aggregation k. Clearly such an effect disappears if k and M are multiples or equal to 12

months, so that MSE’s remain more or less the same, regardless of the forecasting month

(see Figure 7).

The forecasting model has been validated by comparing observed and forecasted SPI

(computedbyEquation14)duringaperioddifferentfromthatusedforparametersestimation.

Such validation is usually carried out by splitting the available sample into two sub-samples

to be used for parameter estimation and model validation respectively (Klemes, 1986). Here

a slightly different approach is proposed, where the generic SPI value at a given time interval

is compared with the corresponding forecast, computed by estimating the parameters on the

previousNyears.Thus,aperiodofpastNyearsisconsideredforparametersestimationevery

year. This is consistent with the fact that when the model is applied for real time forecast, its

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Water Resour Manage

Table 3 PercentagedifferencesoftransitionprobabilitymatricescomputedexactlybyEquation(1)andunderthelag-1Markovhypothesis

by Equation (20) for different k and M, for Caltagirone station. Starting month: February

k = 9

k = 12

k = 24

Ex

Se

Mo

N

Ex

Se

Mo

N

Ex

Se

Mo

N

M = 3

Ex

15.8

−8.0

−22.7

−78.7

15.1

−6.5

−9.9

−42.1

15.4

−9.6

−50.7

−206.7

Se

−7.5

12.7

10.4

−35.6

−5.7

9.6

12.7

−22.7

−9.5

15.3

6.5

−64.2

Mo

−24.1

9.8

26.4

−21.4

−11.6

11.7

26.5

−18.2

−51.3

6.3

24.3

−24.5

N

−77.9

−34.0

−21.7

1.6

−41.4

−21.0

−18.7

1.5

−206.0

−63.5

−24.6

1.6

M = 6

Ex

2.2

−9.0

3.8

1.2

26.4

0.3

−3.6

−70.6

27.9

−5.2

−52.6

−309.5

Se

−11.3

−1.0

15.2

−3.6

1.7

16.3

25.4

−34.6

−4.7

24.1

19.1

−90.1

Mo

8.2

18.9

27.6

−11.2

−6.9

23.4

38.1

−23.4

−54.0

18.6

38.7

−34.9

N

−0.3

−5.5

−10.3

1.0

−68.4

−32.2

−24.3

2.4

−307.8

−88.6

−35.2

2.8

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Water Resour Manage

Table 4 Percentage differences of transition probability matrices computed exactly by Equation (1) and under the lag-1 Markov

hypothesis by Equation (20) for different k and M, for Caltagirone station. Starting month: August

k = 9

k = 12

k = 24

Ex

Se

Mo

N

Ex

Se

Mo

N

Ex

Se

Mo

N

M = 3

Ex

−8.8

−10.1

3.1

3.7

8.0

−4.8

1.5

−4.9

15.0

−5.6

−11.5

−42.0

Se

−11.1

−3.1

9.6

−0.8

−7.1

1.6

12.2

−5.2

−6.4

9.6

11.6

−21.0

Mo

4.6

11.1

17.8

−5.5

4.6

15.1

24.3

−11.5

−9.9

12.5

26.2

−18.5

N

3.4

−1.4

−5.2

0.5

−5.5

−7.3

−10.7

1.0

−42.7

−22.6

−18.1

1.5

M = 6

Ex

−86.6

−51.8

−14.2

9.9

−40.5

−29.2

−0.2

8.9

20.7

1.0

8.1

−27.0

Se

−51.4

−28.9

−6.0

4.8

−30.2

−15.7

5.8

2.8

0.9

10.4

24.7

−17.0

Mo

−17.2

−7.9

1.4

0.9

0.2

6.5

14.3

−3.0

8.1

24.7

35.4

−17.3

N

10.3

5.1

0.7

−0.6

9.0

2.6

−2.9

0.0

−26.9

−17.0

−17.3

2.0

Springer