Analysis of SPI drought class transitions using loglinear models

Journal of Hydrology (2006) 331, 349– 359

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Analysis of SPI drought class transitions using loglinear models Elsa E. Moreira

a,*

˜o T. Mexia , Ana A. Paulo b, Luı´s S. Pereira b, Joa

a

a

´tica e Aplicac¸o ˜es, Faculdade de Cie ˆncias e Tecnologia, Universidade Nova de Lisboa, Centro de Matema 2829-516 Caparica, Portugal b ´cnica de Lisboa, Centro de Estudos de Engenharia Rural, Instituto Superior de Agronomia, Universidade Te Tapada da Ajuda, 1349-017 Lisboa, Portugal Received 20 January 2006; received in revised form 17 May 2006; accepted 17 May 2006

KEYWORDS

Summary A total period of 67 years of standardized precipitation index (SPI) data sets were divided into three periods of 22/23 years and a loglinear modeling approach has been used to investigate differences relative to drought class transitions among these three periods. The study was applied to several locations in Alentejo region, southern Portugal, and four drought severity classes were considered. The drought class transitions were computed for the three periods to form a 3-dimensional contingency table. The application of loglinear modeling to these data allowed the comparison of the three periods in terms of probabilities of transition between drought classes in order to detect a possible trend in time evolution of droughts which could be related to climate change. Results show that the drought behavior for the first and last periods is similar, both showing worse drought events than the second. If just the second and third periods were compared one could conclude that droughts were aggravating and easily this behavior could be attributed to climate change, supporting the common assumption that a trend for progressive aggravation of drought occurrence exists. Therefore, results are more consistent with the existence of a long-term natural periodicity; however, this hypothesis should be tested using longer time series. c 2006 Elsevier B.V. All rights reserved.

Standardized precipitation index; 3-Dimension loglinear models; Drought class transitions; Odds; Impacts of climate change



Introduction

* Corresponding author. Tel.: +351 936282422. E-mail addresses: [email protected] (E.E. Moreira), apaulo@ isa.utl.pt (A.A. Paulo), [email protected] (L.S. Pereira), jtm@ fct.unl.pt (J.T. Mexia).



The less predictable characteristics of droughts with respect to their initiation and termination, frequency and severity make drought both a hazard and a disaster: a hazard because it is a natural accident of unpredictable occurrence but of recognizable recurrence; a disaster because it corresponds to the failure of the precipitation

0022-1694/$ - see front matter c 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.jhydrol.2006.05.022

350 regime, causing the disruption of the water supply to the natural and agricultural ecosystems as well as to other human activities (Pereira et al., 2002). Forecasting of when a drought is likely to begin or to come to an end is extremely difficult. A better understanding of droughts is essential to develop tools for prediction or forecasting of drought initiation and ending, so that these occurrences may be clearly recognized (Sharma, 1997; Chiew et al., 1998). This is essential for timely and appropriate implementation of measures to cope with a drought. This is particularly important in agriculture. Solving such difficulties requires appropriate monitoring and prediction tools, which make that drought warning becomes possible (Sivakumar and Wilhite, 2002; Rossi, 2003). Several drought indices have already been used in Portugal, particularly in recent studies applied to the Alentejo region. Comparisons among drought indices show the appropriateness for using the standardized precipitation index (SPI) to characterize droughts in Alentejo (Paulo et al., 2003; Paulo and Pereira, 2006a) and the capabilities to use the stochastic properties of the SPI time series for predicting drought class transitions (Paulo et al., 2005; Paulo and Pereira, 2006b). It is common in our time the idea that water resources have been decreasing in consequence of several causes, mainly due to less precipitation in certain regions of the planet like the Mediterranean basin, as a result of climatic changes. In fact, it is often said that drought events are becoming more frequent and/or more severe due to climate change (Brunetti et al., 2004; Huntington, 2006). This idea could be supported by several studies using data from the last fifty years such as through the use of trend analysis and principal component analysis in precipitation and SPI time series (Bordi and Sutera, 2001; Bordi and Sutera, 2002; Bonaccorso et al., 2003). These studies revealed in several locations a decrease in annual rainfall and a trend toward drier conditions since the seventies; however, it was highlighted by these authors that such a linear trend can represent a descending part of a long-term natural periodicity, which is not visible because of the shortness of data sets used. When considered a longer time span other possibilities arise. Piervitali and Colacino (2001), using liturgical data for Sicily from 1565 to 1915, concluded that droughts have been comparatively more frequent during the 16th and 17th centuries than during the 18th and 19th centuries. Balling (1996) using data from 1895 to 1995 for the USA, found a non-significant trend in drought severity, away from increased drought conditions. The study by Moonen et al. (2002), concerning the period 1878–1999 of 122 years in Italy, has evidenced a ‘‘shift towards more extremely low rainfall events, but this shift is too low to have any consequence in agriculture [. . .]. Despite of a decrease in rainfall, there is no increased drought risk’’. In addition, the authors affirm that ‘‘the estimated climate changes are too small to confidently allow the rejection of the hypothesis as they could merely be a reflection of sampling fluctuations of a natural variability of climatic parameters in a stationary climate’’. Thus, two scenarios may be considered, either there being a trend toward worsening of droughts or there being long term cycles.

E.E. Moreira et al. Aiming to uncover drought behavior in the region of Alentejo in southern Portugal and contribute to answer the dilemma above, it was decided to analyze the monthly transitions between the SPI drought classes. Thus, a different statistical tool from trend analysis and principal component analyses had to be used since instead of analyzing time series, the object of this study concerns drought class transitions. Loglinear models (Nelder, 1974; Agresti, 1990) were considered to be a more adequate tool to perform this analysis because they have shown to be adequate to perform a monthly prediction of SPI drought class transitions (Paulo et al., 2005). As a result, the SPI with the 12-month time scale was analyzed through adjusting loglinear models to the probabilities of transitions between the SPI drought classes. The results are discussed in this paper.

Data used, SPI and drought classes Input data to this study consists of SPI monthly values, computed in a 12-month time scale, for the period September 1932 to September 1999, for 49 rainfall sites in Alentejo, a water scarce region in southern of Portugal. In SPI computation, the entire period of 67 years has been used to fit the best set of parameters. The SPI time series derived from monthly rainfall relative to six sites were used for a detailed analysis of drought class transitions. The six sites were selected according to its location, aiming to have a balanced spatial coverage of the region and high quality in the goodness of fit of the Gama distribution used for SPI computations. These sites are identified in Fig. 1: Portalegre, ´vora, Beja, Barrancos and Almodo Elvas, E ˆvar. The SPI was developed by McKee et al. (1993, 1995) for the identification of drought events and to evaluate its severity. Multiple time scales, from 3-month to 24-month, may be used. The drought monthly severity adopted in this study is defined in Table 1, where the severe and extremely severe drought classes are grouped. The methods used to compute the SPI and the data quality tests performed before using the precipitation data sets are reported by Paulo et al. (2003, 2005). Annual precipitation data sets were investigated for randomness, homogeneity and absence of trends using the autocorrelation test (Kendall s), the Mann–Kendall trend test and the homogeneity tests of Mann–Whitney for the mean and the variance (Helsel and Hirsch, 1992). In order to compare the time evolution in terms of probabilities of transition between drought classes, the total 67 years period was divided into three periods: • 1st period: from 1932/33 to 1954/55 with 23 years; • 2nd period: from 1955/56 to 1976/77 with 22 years; • 3rd period: from 1977/78 to 1998/99 with 22 years. In a previous study, the same 67 years data sets were divided into two periods only and the results did not show evidence of differences between the two periods, which led to divide it into three periods, given that 3 is the minimum number of points necessary to find a curvature. For each site and for these three time periods, the monthly drought classes were calculated based on Table 1;

Analysis of SPI drought class transitions using loglinear models

351

Site

Latitude Longitude Height (North) (West) (m) 39º17’ 7º25’ 596 38º53’ 7º09’ 208 38º34’ 7º54’ 309 38º01’ 7º52’ 246 38º08’ 6º58’ 380 37º57’ 8º24’ 61 37º31’ 8º04’ 270

Portalegre Elvas Evora Beja Barrancos Alvalade Almodovar

Figure 1

The Alentejo region with identification of the rainfall sites utilized in the study.

then, aiming at obtaining the observed frequency counts for building a 3-dimension contingency table, the monthly transitions between all drought classes were computed. The statistical mode for the 49 sites in Alentejo of the drought class in each month, i.e., the more frequent drought class in each month for all 49 sites, was computed in order to have a global vision of the temporal evolution of the drought classes for the entire Alentejo region. These results are presented in Fig. 2 for the three periods.

Table 1 Drought class classification of SPI (modified from McKee et al., 1993) Code

Drought classes

SPI values

1 2 3 4

Non-drought Near normal Moderate Severe/Extreme

SPI P 0 1 < SPI < 0 1.5 < SPI 6 1 SPI 6 1.5

st

4 3 2

Oct/1953

Oct/1954

Oct/1975

Oct/1976

Oct/1997

Oct/1998

Oct/1952 Oct/1974 Oct/1996

Oct/1951 Oct/1973 Oct/1995

Oct/1950

Oct/1949

Oct/1972 Oct/1994

Oct/1948 Oct/1971

Oct/1993

Oct/1947 Oct/1970 Oct/1992

Oct/1946 Oct/1969 Oct/1991

Oct/1945 Oct/1968

Oct/1944

Oct/1990

nd

Oct/1967

2

Oct/1989

Oct/1943

Oct/1942

Oct/1941

Oct/1940

Oct/1939

Oct/1938

Oct/1937

Oct/1936

Oct/1935

Oct/1934

Oct/1933

1 Oct/1932

Drought classes

1 period

period

3 2

3

rd

Oct/1966

Oct/1965

Oct/1964

Oct/1963

Oct/1962

Oct/1961

Oct/1960

Oct/1959

Oct/1958

Oct/1957

Oct/1956

1 Oct/1955

Drought classes

4

period

3 2

Oct/1988

Oct/1987

Oct/1986

Oct/1985

Oct/1984

Oct/1983

Oct/1982

Oct/1981

Oct/1980

Oct/1979

Oct/1978

1 Oct/1977

Drought classes

4

Figure 2 More frequent drought classes in each month in the Alentejo region for the 3 time periods. Drought classes: 1 – Nondrought; 2 – near normal drought; 3 – moderate drought; 4 –severe/extreme drought.

352

E.E. Moreira et al. Portalegre 3 2 1 0 -1 -2 -3

1932

1943

1954

1966

1977

1988

1999

1977

1988

1999

1977

1988

1999

1977

1988

1999

1977

1988

1999

1977

1988

1999

Elvas 3 2 1 0 -1 -2 -3 1932

1943

1954

1966

Évora 3 2 1 0 -1 -2 -3 1932

1943

1954

1966

Beja 3 2 1 0 -1 -2 -3 1932

1943

1954

1966

Barrancos 3 2 1 0 -1 -2 -3 1932

1943

1954

1966

Almodôvar 3 2 1 0 -1 -2 -3 1932

1943

Figure 3

1954

1966 Year

SPI values on a 12-month time scale, from 1932 to 1999, for the six selected sites.

Analysis of SPI drought class transitions using loglinear models

353 level of the category B, kCk is the parameter associated with kth level of the category C, ui and vj is the ith level of category A, jth level of category B scores (usually we take ui = i and vj = j), b is the linear association parameter, di are the parameters associated with the ith diagonal element, I(condition) is the indicator function defined as  0 if condition ¼ true IðconditionÞ ¼ : 1 if condition ¼ false

The time variability of the drought occurrences and the respective severity is obvious. To provide an insight into the long-term trend characterizing the SPI signals in the six selected sites, Fig. 3 presents the respective 12-month SPI values for the full period under analysis, i.e., from 1932 to 1999. The results show that the occurrence of drought and wet periods for the six sites is nearly coincident but the drought severities vary from one location to the other.

Adjustment of the models

Loglinear models with 3-dimensions

The loglinear models are considered to have error following the Poisson distribution and the residual deviance of a loglinear model following the Chi-Square distribution with the same degrees of freedom of the residual deviance. To evaluate the fitting of a loglinear model, a test was used (Nelder, 1974; Agresti, 1990). Are not rejected in the test those models which the residual deviance does not exceed the quantile of the Chi-square variable for a probability of 1  a = 0.95, with the same degrees of freedom of the residual deviance, i.e. are considered well adjusted all the models presenting a test p-value superior to the chosen significance level of a = 0.05. Only the quasi-association(QA) and the quasi-symmetry models, were not rejected confirming the results by Paulo et al. (2005). However, the QA model (1) proved to be the most adequate in the six sites. For each site, the respective degrees of freedom, residual deviance and p-values are presented in Table 3. A backward elimination method was applied to the QA models adjusted to each site in order to reduce the number of parameters without significant loss of information. The backward elimination allows the selection of an alternative submodel, by the elimination of parameters of the initial model. All possible submodels were rejected for four sites, thus the initial QA model (1) was kept for this sites. However, for Barrancos and Almodo ˆvar a submodel was selected. This submodel and the respective degrees of freedom, residual deviance and p-values are presented in Table 4 relative to these sites. Once again, for the fitted submodels the test p-values are greater than a = 0.05. The parameters of the QA model or submodel were then estimated and the expected frequencies for each cell were obtained. As an example, for the site Almodo ˆvar in Table 5 are presented the results of the observed and expected frequencies. The results for other sites are similar and, thus not shown in this paper.

Model presentation Loglinear models describe association patterns among categorical variables. Modelling is done upon the cell counts in contingency tables. The aim for using the 3-dimensional loglinear models (Agresti, 1990) is to fit the observed frequencies of drought class transitions, denoted as Oijk, and to model the corresponding expected frequencies, denoted as Eijk, for each cell of a 3-dimension contingency table (Table 2). This 3-dimension contingency table has three categories (A, B and C) with levels i,j and k (i = 1, . . ., 4), (j = 1, . . ., 4) and (k = 1, . . ., 3), respectively. Category A refers to drought classes at instant t, category B refer to drought classes at instant t + 1 (in this case the instant correspond to the month). The levels 1,. . .,4 are associated to drought classes 1, 2, 3 and 4. The category C represents the time period and levels 1,2,3 are associated to the 1st, 2nd and 3rd time periods defined above. The observed frequencies (Oijk) refer to number of transitions between drought class i at month t and drought class j at month t + 1 in each period k. For example, the observed O111 is the number of times that a given rainfall site stays for two consecutive months in drought class 1 (‘Nondrought’) during the 1st period. Several models for 3-dimensional contingency tables were fitted to the observed frequencies but the quasi-association model was the one that better fitted the observed frequencies. The quasi-association model is defined by: Log E ijk ¼ k þ kAi þ kBj þ kCk þ bui v j þ di Iði ¼ jÞ   þ kCk b ui v j þ kCk di Iði ¼ jÞ

ð1Þ

with i = 1, . . ., 4; j = 1, . . ., 4 and k = 1,2,3, where k is the constant term, kAi is the parameter associated with ith level of the category A, kBj is the parameter associated with jth

Table 2

Three-dimension contingency table of transitions between drought classes relative to the three periods 1st period

2nd period

3rd period

Drought class at time t + 1

Drought class at time t + 1

Drought class at time t + 1

Drought class at time t

1

2

3

4

1

2

3

4

1

2

3

4

1 2 3 4

O111 O211 O311 O411

O121 O221 O321 O421

O131 O231 O331 O431

O141 O241 O341 O441

O112 O212 O312 O412

O122 O222 O322 O422

O132 O232 O332 O432

O142 O242 O342 O442

O113 O213 O313 O413

O123 O223 O323 O423

O133 O233 O333 O433

O143 O243 O343 O443

354 Table 3

E.E. Moreira et al. Selected loglinear models for each site

Site

Selected model

Degrees of freedom

Residual deviance

p-value

Portalegre Elvas ´vora E Beja Barrancos Almodovar

Quasi-association Quasi-association Quasi-association Quasi-association Quasi-association Quasi-association

24 24 24 24 24 24

16.69 19.10 11.31 26.90 20.18 13.65

0.8616 0.7466 0.9866 0.3091 0.6865 0.9542

Table 4

Loglinear 3D submodels for Barrancos and Almodo ˆvar

Site

Selected submodel

D.F.

Residual deviance

p-value

Barrancos Almodo ˆvar

Quasi-association Log E ijk ¼ k þ kAi þ kBj þ kCk þ bui v j þ di Iði ¼ jÞ þ kCk di Iði ¼ jÞ

26 26

24.13 14.32

0.5685 0.9685

Table 5

Drought class transitions from time t to time t + 1: observed versus expected frequencies in Almodo ˆvar

Almodo ˆvar

1st period

2nd period

3rd period

Drought class at time t + 1

Drought class at time t + 1

Drought class at time t + 1

Drought Class at time t

1

2

3

4

1

2

3

4

1

2

3

4

Observed frequencies 1 2 3 4

112 14 0 0

13 87 6 2

0 5 3 4

0 2 3 25

134 9 0 0

8 79 8 1

1 8 11 2

0 0 3 0

111 11 0 0

12 72 7 1

0 9 9 4

0 0 6 21

Expected frequencies 1 2 3 4

111.94 11.81 0.17 0.00

11.81 87.01 7.56 1.06

0.17 7.56 2.98 3.88

0.00 1.06 3.88 25.18

133.89 9.64 0.14 0.00

9.64 78.96 6.17 0.86

0.14 6.17 10.92 3.17

0.00 0.86 3.17 0.00

110.94 12.05 0.17 0.00

12.05 72.02 7.71 1.08

0.17 7.71 8.94 3.96

0.00 1.08 3.96 21.14

*

Drought classes: 1 – Non-drought; 2 – near normal; 3 – moderate drought; 4 – severe/extreme drought.

Odds An odds is a ratio of expected frequencies, ranges from 0 to +1, and represents the number of times that it is more (less) probable the occurrence of a certain event instead of another event different from the first one. The selected odds for the 3-dimension models were defined by Xkljij ¼ E ijk =E ijl ; k 6¼ l

ð2Þ

The selection of these odds allows the comparison of the probabilities that, one month from now, the site will be in drought class j given that at present it is in class i, when comparing two among the three time periods. The odds X12jij compare the 1st time period with the 2nd time period, the odds X23jij compare the 2nd time period with 3rd time period and the odds X13jij compare the 1st time period with the 3rd time period. For example, X12j34 = 6.169 means that it is 6.169 times more probable the transition from ‘moderate drought’ class (i = 3) to ‘severe/extreme

drought’ class (j = 4) in the 1st period (k = 1) than in the 2nd period (l = 2). The estimates of the corresponding odds are thus obtained by exponentiation of the result obtained by replacE ing the parameters in Log Xkljij ¼ Log Eijk ¼ Log E ijk  Log E ijl ijl by their estimates obtained from the fit of the loglinear model. In these expressions i,j = 1,2,3,4; k,l = 1,2,3 with k 5 l. In the quasi-association model the logarithm of the odds may be written as Log Xkljij ¼ Log E ijk  Log E ijl ¼ kCk  kCl     þ kCk b ui v j  kCl b ui v j þ kCk di  Iði ¼ jÞ  kCl di Iði ¼ jÞ

ð3Þ

Asymptotic confidence intervals associated with a probability of 1  a = 0.95, are then obtained for these odds by exponentiation of the corresponding asymptotic confidence intervals for the logarithm of the odds

Comparing the 1st and 2nd periods: estimates of X12jij = Eij1/Eij2,i, j = 1,2,3,4 and its confidence intervals for the six sites studied Drought class at instant t + 1

Drought class at instant t Portalegre 1 2 3 4 ´vora E 1 2 3 4 Beja 1 2 3 4

Drought Class at instant t + 1

1

2

3

4

0.7388 0.5696 1.1813 0.6614 1.3936 0.7803 1.6441 0.9205

1.1813 0.6614 0.9401 0.5653 2.2883 1.2812 3.1848 1.7831

1.3936 0.7803 2.2883 1.2812 0.8893 0.2111 6.1694 3.4542

1.6441 0.9205 2.9365 3.1848 1.7831 5.6884 6.1694 3.4542 11.0190 53146.0998 0.0000 8.0322E + 28

1.0619 0.8343 1.6196 0.8958 1.6214 0.8968 1.6232 0.8978 0.7788 0.6047 0.9003 0.4958 1.0377 0.5715 1.1960 0.6587

0.9583 2.1099 2.4891 2.9365

1.3517 2.9283 2.9315 2.9348

1.0030 1.6348 1.8843 2.1718

1.6196 0.8958 0.5807 0.3600 1.6268 0.8998 1.6304 0.9017 0.9003 0.4958 0.9871 0.5937 1.5888 0.8750 2.1107 1.1624

2.1099 1.5633 4.0870 5.6884

2.9283 0.9366 2.9412 2.9477

1.6348 1.6411 2.8851 3.8326

1.6214 0.8968 1.6268 0.8998 1.4289 0.4837 1.6375 0.9057 1.0377 0.5715 1.5888 0.8750 5.4959 1.2069 3.7248 2.0513

2.4891 4.0870 3.7466 11.0190

2.9315 2.9412 4.2214 2.9607

1.8843 2.8851 25.0277 6.7635

1.6232 0.8978 2.9348 1.6304 0.9017 2.9477 1.6375 0.9057 2.9607 40038.6293 0.0000 2.7626E + 44 1.1960 0.6587 2.1718 2.1107 1.1624 3.8326 3.7248 2.0513 6.7635 29231.4365 0.0000 4.4200E + 28

Drought class at instant t Elvas 1 2 3 4 Barrancos 1 2 3 4 Almodovar 1 2 3 4

1

0.7640 0.5938 0.8217 0.4240 1.1167 0.5762 1.5177 0.7831 0.9675 0.7501 1.2202 0.8392 1.2202 0.8392 1.2202 0.8392 0.8361 0.6502 1.2251 0.8070 1.2251 0.8070 1.2251 0.8070

2

0.9829 1.5924 2.1642 2.9413

1.2480 1.7742 1.7742 1.7742

1.0752 1.8598 1.8598 1.8598

0.8217 0.4240 1.0002 0.5359 2.8033 1.4465 5.1779 2.6718 1.2202 0.8392 1.1853 0.8630 1.2202 0.8392 1.2202 0.8392 1.2251 0.8070 1.1019 0.8123 1.2251 0.8070 1.2251 0.8070

1.5924 1.8666 5.4327 10.0346

1.7742 1.6279 1.7742 1.7742

1.8598 1.4946 1.8598 1.8598

3

4

1.1167 0.5762 2.1642 2.8033 1.4465 5.4327 6.7544 1.1201 40.7303 17.6653 9.1153 34.2348

1.5177 0.7831 2.9413 5.1779 2.6718 10.0346 17.6653 9.1153 34.2348 2205.6996 0.0000 3.3598E + 27

1.2202 0.8392 1.2202 0.8392 0.2001 0.0684 1.2202 0.8392

1.2202 0.8392 1.2202 0.8392 1.2202 0.8392 3.6656 1.0220

1.2251 0.8070 1.2251 0.8070 0.2728 0.0762 1.2251 0.8070

1.7742 1.7742 0.5854 1.7742

1.8598 1.8598 0.9773 1.8598

1.7742

Analysis of SPI drought class transitions using loglinear models

Table 6

1.7742 1.7742 1.3147E + 01

1.2251 0.8070 1.8598 1.2251 0.8070 1.8598 1.2251 0.8070 1.8598 147709.0885 0.0000 1.0119E + 45

(In each cell the upper value is the odds estimates and the lower values are the odds confidence intervals. In italics, the cases when the confidence interval includes the value 1.)

355

356

Table 7

Comparing the 1st and 3rd periods: estimates of X13jij = Eij1/Eij3,i, j = 1,2,3,4 and its confidence intervals for the six sites studied Drought class at instant t + 1

Drought class at instant t Portalegre 1 2 3 4 ´vora E 1 2 3 4 Beja 1 2 3 4

Drought class at instant t + 1

1

2

3

4

0.8761 0.6690 1.2146 0.7018 1.2094 0.6988 1.2042 0.6958

1.2146 0.7018 1.2006 0.7778 1.1939 0.6898 1.1836 0.6839

1.2094 0.6988 1.1939 0.6898 0.3636 0.1204 1.1635 0.6722

1.2042 0.6958 1.1836 0.6839 1.1635 0.6722 3.4288 0.6623

1.3564 1.0491 2.0454 1.1093 1.8228 0.9886 1.6245 0.8811 0.9730 0.7444 1.0610 0.5899 1.0542 0.5861 1.0475 0.5824

1.1473 2.1021 2.0931 2.0841

1.7536 3.7714 3.3610 2.9953

1.2717 1.9084 1.8962 1.8841

2.0454 1.1093 0.6909 0.4336 1.2902 0.6997 1.0247 0.5557 1.0610 0.5899 1.2466 0.7916 1.0342 0.5749 1.0210 0.5676

2.1021 1.8531 2.0663 2.0486

3.7714 1.1008 2.3789 1.8894

1.9084 1.9630 1.8602 1.8365

1.8228 0.9886 1.2902 0.6997 1.3327 0.4882 0.6464 0.3506 1.0542 0.5861 1.0342 0.5749 1.6928 0.5709 0.9952 0.5533

2.0931 2.0663 1.0978 2.0137

3.3610 2.3789 3.6382 1.1918

1.8962 1.8602 5.0194 1.7901

1.6245 0.8811 1.0247 0.5557 0.6464 0.3506 0.2500 0.0491 1.0475 0.5824 1.0210 0.5676 0.9952 0.5533 0.5602 0.1012

Drought class at instant t Elvas 1 2.0841 2 2.0486 3 2.0137 4 17.7506 Barrancos 1 2.9953 2 1.8894 3 1.1918 4 1.2741 Almodovar 1 1.8841 2 1.8365 3 1.7901 4 3.1028

1

1.1032 0.8382 1.3464 0.7089 1.2682 0.6677 1.1946 0.6289 1.0736 0.8274 1.0171 0.7120 1.0171 0.7120 1.0171 0.7120 1.0090 0.7756 0.9802 0.6610 0.9802 0.6610 0.9802 0.6610

2

1.4519 2.5572 2.4088 2.2689

1.3930 1.4531 1.4531 1.4531

1.3128 1.4535 1.4535 1.4535

1.3464 0.7089 1.1341 0.7322 1.0599 0.5580 0.9404 0.4951 1.0171 0.7120 1.5373 1.0917 1.0171 0.7120 1.0171 0.7120 0.9802 0.6610 1.2080 0.8847 0.9802 0.6610 0.9802 0.6610

3

2.5572 1.7564 2.0132 1.7862

1.4531 2.1646 1.4531 1.4531

1.4535 1.6495 1.4535 1.4535

1.2682 0.6677 1.0599 0.5580 1.1745 0.4349 0.7404 0.3898 1.0171 0.7120 1.0171 0.7120 0.2001 0.0684 1.0171 0.7120 0.9802 0.6610 0.9802 0.6610 0.3335 0.0904 0.9802 0.6610

4

2.4088 2.0132 3.1719 1.4062

1.4531 1.4531 0.5853 1.4531

1.4535 1.4535 1.2311 1.4535

1.1946 0.6289 0.9404 0.4951 0.7404 0.3898 0.0666 0.0053 1.0171 0.7120 1.0171 0.7120 1.0171 0.7120 0.5505 0.2638 0.9802 0.6610 0.9802 0.6610 0.9802 0.6610 1.1912 0.6667

2.2689 1.7862 1.4062 0.8426

1.4531 1.4531 1.4531 1.1486

1.4535 1.4535 1.4535 2.1284

E.E. Moreira et al.

(In each cell the upper value are the odds estimates and the lower values are the odds confidence intervals. In italics, the cases when the confidence interval includes the value 1.)

Comparing the 2nd and 3rd periods: estimates of X23jij = Eij2/Eij3, i, j = 1,2,3,4 and its confidence intervals for the 6 sites studied Drought class at instant t + 1

Drought class at instant t Portalegre 1 2 3 4 ´vora E 1 2 3 4 Beja 1 2 3 4

Drought class at instant t + 1

1

2

3

4

1.1858 0.9234 1.0282 0.5617 0.8678 0.4741 0.7324 0.4001

1.0282 0.5617 1.2771 0.7560 0.5217 0.2850 0.3717 0.2030

0.8678 0.4741 0.5217 0.2850 0.4089 0.1064 0.1886 0.1030

0.7324 0.4001 0.3717 0.2030 0.1886 0.1030 0.0001 0.0000

1.2772 0.9835 1.2629 0.6418 1.1242 0.5714 1.0008 0.5086 1.2493 0.9702 1.1784 0.6441 1.0159 0.5553 0.8758 0.4787

1.5227 1.8820 1.5884 1.3406

1.6587 2.4848 2.2120 1.9692

1.6087 2.1560 1.8586 1.6023

1.2629 0.6418 1.1898 0.7551 0.7931 0.4031 0.6285 0.3194 1.1784 0.6441 1.2629 0.7499 0.6509 0.3558 0.4837 0.2644

1.8820 2.1574 0.9550 0.6803

2.4848 1.8748 1.5605 1.2367

2.1560 2.1267 1.1908 0.8850

1.1242 0.5714 0.7931 0.4031 0.9327 0.3034 0.3947 0.2006 1.0159 0.5553 0.6509 0.3558 0.3080 0.0648 0.2672 0.1460

1.5884 0.9550 1.5705 0.3452

2.2120 1.5605 2.8674 0.7766

1.8586 1.1908 1.4634 0.4888

1.0008 0.5086 0.6285 0.3194 0.3947 0.2006 0.0000 0.0000 0.8758 0.4787 0.4837 0.2644 0.2672 0.1460 0.0000 0.0000

Drought class at instant t Elvas 1 1.3406 2 0.6803 3 0.3452 4 9.7991E + 19 Barrancos 1 1.9692 2 1.2367 3 0.7766 4 4.30141E + 34 Almodovar 1 1.6023 2 0.8850 3 0.4888 4 2.8962E + 19

1

1.4440 1.1147 1.6385 0.8106 1.1356 0.5618 0.7871 0.3894 1.1096 0.8570 0.8336 0.5726 0.8336 0.5726 0.8336 0.5726 1.2068 0.9385 0.8001 0.5279 0.8001 0.5279 0.8001 0.5279

2

1.8706 3.3121 2.2955 1.5910

1.4366 1.2136 1.2136 1.12136

1.5519 1.2127 1.2127 1.2127

1.6385 0.8106 1.1338 0.6013 0.3781 0.1871 0.1816 0.0899 0.8336 0.5726 1.2969 0.9099 0.8336 0.5726 0.8336 05726 0.8001 0.5279 1.0964 0.7972 0.8001 0.5279 0.8001 0.5279

3

3.3121 2.1381 0.7643 0.3671

1.2136 1.8486 1.2136 1.2136

1.2127 1.5078 1.2127 1.2127

1.1356 0.5618 0.3781 0.1871 0.1739 0.0282 0.0419 0.0207 0.8336 0.5726 0.8336 0.5726 1.0000 0.5384 0.8336 0.5726 0.8001 0.5279 0.8001 0.5279 1.2226 0.5065 0.8001 0.5279

4

2.2955 0.7643 1.0704 0.0847

1.2136 1.2136 1.8573 1.2136

1.2127 1.2127 2.9512 1.2127

0.7871 0.3894 0.1816 0.0899 0.0419 0.0207 0.0000 0.0000 0.8336 0.5726 0.8336 0.5726 0.8336 0.5726 0.1502 0.0446 0.8001 0.5279 0.8001 0.5279 0.8001 0.5279 0.0000 0.0000

1.5910 0.3671 0.0847 4.4539E + 19

1.2136

Analysis of SPI drought class transitions using loglinear models

Table 8

1.2136 1.2136 0.5056

1.2127 1.2127 1.2127 5.5240E + 34

(In each cell the upper value are the odds estimates and the lower values are the odds confidence intervals. In italics, the cases when the confidence interval includes the value 1.)

357

358 

E.E. Moreira et al.

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dkljij Þ; LogX dkljij Þ dkljij þ z1a=2 Varð LogX dkljij  z1a=2 Varð LogX LogX

ð4Þ where z1a/2 is the 1  a/2 quantile of a standard normal random variable. When the confidence intervals for a given odds include the value 1 it means that the drought transitions from class i to class j in time period k and in time period l, are not significantly different, i.e., the transitions from class i to class j are equally probable in time periods k and l. If the value 1 is not included in the confidence interval of a given odds, it means that the transition is more or less probable in one time period than in the other, according to the situation. For example, the estimate for the odds X12j11 = E111/ E112 = 0.7388 in Portalegre (Table 6) means that it is 0.7388 times less probable that the site will stay 2 months in the ‘Non-drought’ (i = 1, j = 1) class in time period 1 (k = 1) than in time period 2 (k = 2). All the odds may be interpreted as presented above.

could be related to a long-term natural variability. Confirming these hypotheses, it can be observed from Fig. 2 that the major periods of severe/extreme drought (class 4) occurred in the 1st and in the 3rd time periods. Therefore, it can be concluded that, with the available data, there is no evidence of a growing tendency for an increase of droughts and drought severity in Alentejo, which could be attributed to a climate change due to man-induced effects. However, this conclusion would require a better confirmation if longer, near 100 years time series, could be available. These results are not in disagreement with the works by Bordi et al. (2004) and Bordi and Sutera (2001), which reveal a linear trend characterizing the climatic signals since the seventies in regions like the Mediterranean basin and central Europe. But, as it was underscored by those authors, this linear trend can be just a part of a long-term periodicity because of the limited length of the data and a longer time series should clarify the origin of the revealed trend.

Conclusions Results and discussion Results in Tables 6–8 concern these odds estimates and the respective confidence intervals for each site. Relative to the odds X12jij (Table 6), it can be observed in general terms that there are a significative number of odds with value greater than 1 and confidence interval not including the value 1, with predominance for the transitions to the highest drought class (class 4). So it can be said with a probability of 0.95 that those odds really are greater than 1. This means that during the period 1932/33 to 1954/55 there are more frequent severe/extreme droughts than during the next 22 years period. For X13jij (Table 7), it can be observed for almost of the sites that odds confidence intervals include the value 1 for almost all of the transitions between drought classes. So it can be said with a probability of 0.95 that those odds really are equal to 1. This may be interpreted as not existing significant differences among frequencies of drought events, including severe/extreme droughts when comparing the period 1932/33 to 1954/55 with that for 1977/78 to 1998/99. From Table 8, relative X23jij, it can be observed in general terms that there are a significative number of odds with value smaller than 1 and confidence interval not including the value 1, with predominance for the transitions to the highest drought class (class 4). So it can be said with a probability of 0.95 that those odds really are smaller than 1, i.e. there are less transitions to severe/extreme drought classes during the intermediate period, 1955/56 to 1976/77, than during the last 22 years. These results mean that the first and third time periods considered show a similar behavior in which concerns drought class transitions but there are significant differences when comparing the second period with both the first and the second periods. If a shorter time series would be used and only the second and third periods were compared one could conclude that droughts were aggravating and easily this behavior could be attributed to climate change. Contrarily, looking at the three periods, the results indicate that presently we may be in a cycle that

The loglinear models show to be a powerful tool to compare drought class transitions among different time periods through 3-dimension contingency tables. Results from comparing the 1st with 3rd period, show that almost all odds confidence intervals include the value 1, which indicates a great similarity between these periods, in terms of probabilities of transition between drought classes. When comparing the 1st with the 2nd period, and the 2nd with the 3rd period, there is no evidence of similarity between the 2nd and both the 1st and 3rd periods; in fact odds results indicate that, mainly for the transitions to the highest severity drought class 4, the probabilities of transition are greater in the 1st period than in the 2nd and are smaller in the 2nd period than in the 3rd. In addition, it could be observed that the major periods of severe/extreme drought (class 4) occurred in the 1st and in the 3rd time periods, therefore, confirming the conclusions derived from loglinear modeling. Hence, it can be concluded that, with the available data, there is no evidence of a trend for increased drought frequency and severity in Alentejo that could be attributed to climate change. However, analyzing data from an extended time period is needed to confirm this conclusion and hopefully to highlight a possible long term climate variability.

Acknowledgements Data used in this study were made available by the Institute for Water (INAG), Portugal. This study was first funded through the national project PEDIZA 1999.64.006326.1, and is now part of the research contract INTERREG III B MEDOC 2002-02-4.4-1-084.

References Agresti, A., 1990. Categorical Data Analysis. J. Wiley & Sons, New York.

Analysis of SPI drought class transitions using loglinear models Balling Jr., Robert C., 1996. Century-long variations in United States drought severity. Agricultural and Forest Meteorology 82, 293– 299. Bonaccorso, B., Bordi, I., Cancelliere, A., Rossi, G., Sutera, A., 2003. Spatial variability of drought: an analysis of the SPI in Sicily. Water Resources Management 17, 273–296. Bordi, I., Sutera, A., 2001. Fifty years of precipitation: some spatially remote teleconnections. Water Resources Management 15, 247–280. Bordi, I., Fraedrich, K., Gerstengarbe, F.-W., Werner, P.C., Sutera, A., 2004. Potential predictability of dry and wet periods: Sicily and Elbe-Basin (Germany). Theoretical and Applied Climatology 77, 125–138. Bordi, I., Sutera, A., 2002. An analysis of drought in Italy in the last fifty Years. Nuovo Cimento 25 (2), 185–206. Brunetti, M., Buffoni, L., Mangianti, F., Maugeri, M., Nanni, T., 2004. Temperature, precipitation and extreme events during the last century in Italy. Global and Planetary Change 40, 141–149. Chiew, F.H.S., Piechota, T.C., Dracup, J.A., McMahon, T.A., 1998. El Nino/Southern Oscillation and Australian rainfall, stream flow and drought: links and potential for forecasting. Journal of Hydrology 204, 138–149. Helsel, D.R., Hirsch, R.M., 1992. Statistical Methods in Water Resources. Elsevier, Amsterdam, 522 pp. Huntington, T.G., 2006. Evidence for intensification of the global water cycle: review and synthesis. Journal of Hydrology 319, 83–95. McKee, T.B., Doesken, N.J., Kleist, J., 1993. The relationship of drought frequency and duration to time scales. In: Proceedings of the Eighth Conference on Applied Climatology. American Meteorological Society, Boston, pp. 179–184. McKee, T.B., Doesken, N.J., Kleist, J., 1995. Drought monitoring with multiple time scales. In: Proceedings of the Nineth Conference on Applied Climatology. American Meteorological Society, Boston, pp. 233–236. Moonen, A.C., Ercoli, L., Mariotti, M., Masoni, A., 2002. Climate change in Italy indicated by agrometeorological indices over 122 years. Agricultural and Forest Meteorology 111, 13–27.

359 Nelder, J.A., 1974. Loglinear models for contingency tables: a generalization of classical least squares. Applied Statistics 23, 323–329. Paulo, A.A., Pereira, L.S., 2006a. Drought concepts and characterization. Comparing drought indices applied at local and regional scales. Water International 31, 37–49. Paulo, A.A., Pereira, L.S., 2006b. Prediction of SPI drought class transitions using Markov chains. Water International 31 (in press). Paulo, A.A., Pereira, L.S., Matias, P.G., 2003. Analysis of local and regional droughts in southern Portugal using the theory of runs and the Standardized Precipitation Index. In: Rossi, G., Cancelliere, A., Pereira, L.S., Oweis, T., Shatanawi, M., Zairi, A. (Eds.), Tools for Drought Mitigation in Mediterranean Regions. Kluwer, Dordrecht, pp. 55–78. Paulo, A.A., Ferreira, E., Coelho, C., Pereira, L.S., 2005. Drought class transition analysis through Markov and Loglinear models, an approach to early warning. Agricultural Water Management 77, 59–81. Pereira, L.S., Cordery, I., Iacovides, I., 2002. Coping with Water Scarcity. UNESCO IHP VI, Technical Documents in Hydrology No. 58, UNESCO, Paris, 267 pp. (available on line: http://unesdoc.unesco.org/images/0012/001278/127846e.pdf). Piervitali, E., Colacino, M., 2001. Evidence of drought in western Sicily during the period 1565–1915 from liturgical offices. Climatic Change 49, 225–238. Rossi, G., 2003. Requisites for a drought watch system. In: Rossi, G., Cancellieri, A., Pereira, L.S., Oweis, T., Shatanawi, M., Zairi, A. (Eds.), Tools for Drought Mitigation Mediterranean Regions. Kluwer, Dordrecht, pp. 147–157. Sharma, T.C., 1997. Estimation of drought severity on independent and dependent hydrologic series. Water Resources Management 11, 35–50. Sivakumar, M.V.K., Wilhite, D.A., 2002. Drought preparedness and drought management. In: Drought Mitigation and Prevention of Land Desertification (Proceedings of the International Conference, Bled, Slovenia), UNESCO and Slov. Nat. Com. ICID, Ljubljana, CD-ROM Paper 2.