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Analysis and modeling of Palmer's drought index series
of Hydrology, 68 (1984) 211-229 Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands
Journal
211
ANALYSIS AND MODELING OF PALMER’s DROUGHT INDEX SERIES
A. RAMACHANDRA
RAO and G. PADMANABHAN
School of Civil Engineering, Purdue University, Department of Civil Engineering, North Dakota
West Lafayette, IN 47907 (U.S.A.) State University, Fargo, ND 58105
(U.S.A.)
(Revised and accepted for publication
November 2, 1982)
ABSTRACT Rao, A.R. and Padmanabhan, G., 1984. Analysis and modeling of Palmer’s drought index series. In: G.E. Stout and G.H. Davis (Editors), Global Water: Science and Engineering -The Ven Te Chow Memorial Volume. J. Hydrol., 68: 211-229.
The objective of the present study is to investigate the stochastic nature of yearly and monthly Palmer’s drought index (PDI) series and to characterize them via valid stochastic models which may be used to forecast and to simulate the PDI series. The monthly and annual PDI series for Iowa (1930-1962) and Kansas (1887-1962) are analyzed in the present study. The period of data includes the severe droughts experienced in U.S.A. in the 1930’s. Valid autoregressive models are fitted to these time series and the final models are selected on the basis of class selection rules. Statistical characteristics of the observed data and those generated by the selected models are compared. The selected models are also tested for their forecasting ability. Results of these analyses and the modeling effort demonstrate that PDI series can be forecast with reasonable accuracy, one to several months ahead. The yearly forecasts are much less accurate. The models can also be used to generate synthetic data which preserve the important statistical characteristics such as the long-term oscillations of the original data.
INTRODUCTION
Because droughts cause serious economic as well as social impacts, it has been investigated by hydrologists (Stockton and Boggess, 1979) and others (Rosenberg, 1978; Ausuabel and Biswas, 1980; Hecht, 1981; Sen, 1982). Due to the random nature of the contributing factors, occurrence and severity of droughts are stochastic in nature. An important and often discussed characteristic of droughts and their indices is their apparent periodic behavior. Thomas (1962) has pointed out the 20-22 yr. periodicity in drought in the Great Plains region. Recently, Mitchell et al. (1979) have shown by an analysis of data reconstructed from tree rings, that the spatial extent of drought also follows an approximate 22-yr. cycle. Vines (1982) has analyzed rainfall data west of Mississippi River and has concluded that the rainfall has
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o 1984 Elsevier Science Publishers B.V.
212
both a 22-yr. and a 19-yr. (lunar) cycle. The evidence of a 22-yr. cycle in droughts gives rise to speculations about solar influence on droughts because the annual sunspot number series coded for polarity also has a period of 22yr. In spite of comments about the periodicity in the occurrence of droughts, the stochastic structure of time series of drought indicators does not appear to have been analyzed to establish and understand the nature of these series. Consequently, the basic objective of the present paper is to investigate the stochastic characteristics of Palmer’s drought index (PDI) series. One of the basic deficiencies in mitigating the effects of droughts is the inability to predict drought conditions accurately for months or years in advance (King et al., 1974). Accurate drought forecasts would enable optimal operation of irrigation systems. The data generated from valid models of drought indicator time series are useful in agricultural planning. Therefore, there is a clear need to examine the stochastic characteristics of drought indicators with special reference to long-term oscillations and to develop simple and effective models of these indicators which may be used for prediction of droughts and to generate synthetic drought index time series. There are many definitions and indices of drought, some of which are based only on rainfall, others on both rainfall and temperature, and still others which are based on related factors. Although the main cause of droughts is the lack of rainfall, the manifestations of droughts are varied. These include low streamflows, lower ground water-table levels, low reservoir fills and prolonged periods of soil-moisture deficiency with their attendant effect on agriculture. One of the best known indices of droughts is that proposed by Palmer (1965) and this index is therefore analyzed in this study. Palmer’s drought index is an indicator of drought severity which is being extensively and routinely used in the U.S.A. and many other countries of the world for agricultural planning, forecasting crop yields and other related purposes. The index is based on a definition of drought as a prolonged period of abnormal moisture deficiency, and that drought severity is a function of moisture demand as well as supply. The scale of severity of droughts in terms of Palmer’s drought index is given in Table I. The PDI is computed by using observed precipitation and temperature data and a moisture accounting procedure (Palmer, 1965). The PDI of various areas in the U.S.A. are routinely published in the Week@ Weather and Crop Bulletin. The main objective of the present study is to investigate the stochastic nature of yearly and monthly time PDI series and to characterize them via valid stochastic models which may be used to forecast and simulate the PDI series. The monthly and annual PDI series for Iowa (1930-1962) and Kansas (1887-1962), computed by Palmer (1965), are used in the present study, and include the severe droughts experienced in the U.S.A. in the 1930’s. The preliminary data analyses discussed herein give an insight to the stochastic nature of the data series. Valid autoregressive models are fitted
213 TABLE I Drought classification by Palmer’s drought index Palmer’s index
Degree of drought
PDSI < - 4.0 < PDSI < - 3.0 < PDSI < -2.O
- 4.0 - 3.0 - 2.0
extremely dry severely dry moderately dry mildly dry near normal mildly wet moderately wet severely wet extremely wet
+ 2.0 + 3.0 + 4.0
to these time series and final models are selected on the basis of class selection rules. Statistical characteristics of the original and synthetic data generated by the selected models are compared next. The selected models are next tested for their forecasting ability. Results of these analyses and modeling effort demonstrate that the PDI series can be forecast with reasonable accuracy, one to several months ahead. These models can also be used to generate synthetic data which preserve the important statistical characteristics, including the long-term oscillations, of the original data.
CHARACTERISTICS
OF THE DATA USED IN THE STUDY
The monthly PDI series are shown in Fig. 1. No obvious periodicity is evident in the plot of monthly PDI series of either Kansas or Iowa. In fact the data appear to be random. The elementary statistics of the data are given
,OqD
-10
IOWA
DROUGHT
oo/ 00
1000
MOO MONTHS
@9 10001
INDEX
300.0
4000
00
KANSAS
2000
DROUGHT
4000
INDEX
6000
a000
MONTHS
Fig. 1. Monthly Palmer’s drought index series for (A) Iowa and (B) Kansas.
700
214 TABLE
II
Elementary
statistics
of the data series
No.
Series
Mean
Standard deviation
Coefficient of variation
Skewness coefficient
Kurtosis
Hurst’s H
1 2 3 4
KAN(Y) IOW(Y) KAN(M) IOW(M)
- 0.004 0.353 - 0.004 0.353
2.32 2.02 2.71 2.51
579.5 5.7 678.0 7.11
-0.133 - 0.424 0.159 - 0.465
2.286 2.788 2.466 2.369
0.67 0.75 0.79 0.89
KAN(Y), IOW(Y) are yearly data of Kansas and Iowa, respectively; monthly data of Kansas and Iowa, respectively.
KAN(M),
IOW(M) are
in Table II. The mean values of monthly and annual PDI are close to zero for Kansas data and they are significantly greater than zero for Iowa data. The standard deviations of both Kansas and Iowa PDI series are large, which indicate a rather wide variability in drought conditions. The data are not normally distributed according to the skewness coefficient and kurtosis values given in Table II. Histograms of the data shown in Fig. 2 also support these observations. The monthly means, standard deviations and skewness coefficients of the data series (Fig. 3) indicate some periodicity in the monthly means of Kansas but not in Iowa data. The monthly standard deviations and skewness coefficients do not have any periodicity and show rather wide variations from one month to the next. The correlograms and power spectral density functions of the data estimated by using standard methods (Jenkins and Watts, 1968) and the Hamming window are shown in Figs. 4 and 5. Also shown in Figs. 4 and 5 are the correlograms and spectral densities of simulated data generated by using valid models and these are discussed later. Although long-period oscillations can be seen in the correlograms they are not very pronounced. The two-standard error (2~) limits are also shown on the correlograms. The power spectra obtained by using standard methods for the yearly and monthly series shown in Figs. 4 and 5 clearly indicate the presence of longterm oscillations. The spectra of these series are also estimated by using the high-resolution maximum-entropy spectral method (Padmanabhan and Rao, 1980) and these are shown in Fig. 6. These maximum-entropy spectra confirm the presence of long-term oscillations in the PDI series of approximate periods ranging from 13 to 20 yr. in the data. Resealed range-lag characteristics of data provide an important statistic which may be used in investigating long-term oscillations (Mandelbrot and Wallis, 1969; Kashyap and Rao, 1976). The statistic H, derived from log-log plots of resealed range [R(s)/o(s)] against lag s, indicates the nonstationarity and persistence in the series. If there is a predominant periodic component in the data series the log [R(s)/o(s)] vs. log (s) plot of the data will deviate from
215 @
04
TNSAS
SIMULATED
~:
SERIES,
02 5
ORIGINAL
DATA
,
02
01
ifI 2 i$
rSAS
:, 0’
5
00 -583
3
00
-5 17
u.
-524
4 81
u-
s
IOWA SIMULATED
9 ~
SERIES
o4iOWA
ORIGINAL
DATA
DROUGHT
INDEX
,
02
2
04 00 m -2 6
4 26 DROUGHT
@.mj~~s~~
INDEX
SIMULATED
KANSAS
SERIES,
ORIGINAL
DATA
.3oy1 20
>
s Y 8
E w 2
IOWA
SIMULATED
.I0
0 -7.05
e w .30
SERIES
3O
IOWA
926 ORIGINAL
DATA
? J K .I0
IO t
200 m -6.65
1.72 DROUGHT
INDEX
2 .200 t;-_l -6.70
7.72 DROUGHT
INDEX
Fig. 2. Histograms of original (right) and simulated (left) data of (A) yearly and (B) monthly drought index series of Iowa and Kansas.
being a straight line around lags corresponding to this periodicity. The resealed ranges of PDI series are plotted against the corresponding lags on log-log paper and are shown in Fig. 7. These plots again indicate the presence of long-term oscillations in the data around 13-20 yr. Hurst’s Hvalues given in Table II for the data indicate high persistence in the index and hence in droughts. The results of data analysis discussed above clearly point to the fact that the drought index series are highly persistent and have strong long-term oscillations. Evidence of nearly significant to significant long-term oscillations in PDI series has been reported in other countries also (Rao et al., 1973). Therefore, models proposed for drought index data should be capable of preserving these long-term oscillations.
216
‘U
0.: 03 w y 02
06
9 $01 *
0
JFMAMJJASOND MONTHS
0 1’11-11 JFMAMJ KANSAS
JASOND MONTHS
IOWA
Fig. 3. Monthly mean, standard deviation series of Kansas (left) and Iowa (right). SELECTION
OF CANDIDATE
and
skewness
coefficient
of drought
index
MODELS
Based on the data analysis discussed in the previous section, it is quite evident that the PDI series do not have the annual and semi-annual periods which are usually present in hydrologic time series. Long-term periodicities, however, are present in these data. Therefore, models belonging to the autoregressive moving average (ARMA) class are selected to model the PDI series. AR models of different orders m, such as those in the following equation were fitted to the PDI values which are denoted by y(t): y(t)
= ol,+a,y(t-l)+a,
+...+o,y(t-m)+W(t)
(1)
In many of these models of monthly data the value of o1 are found to be close to unity which indicates that the data are highly correlated. The reduction in variance of residuals w(t) from AR(l) to AR(6) models of monthly data is also very small. However, for models of annual series, residual variance decreases considerably with increasing model orders. In order to select the best models from these candidate models, the statistics Ui(W) and hi(W) given in eqs. 2 and 3, respectively, are used. These statistics developed by Akaike (1974) (eq. 2) and Kashyap (1977) (eq. 3) are given by: Ui(W) = n In@,) + 2pi hi(W) =
-(n
-mi)
In pi + [(p -Pi)/n]
(2) ‘-pi In(n) -mi
In(p) (3)
IOWR SIMLILRTION 1.33. IBWR DROUGHT
-Am ‘.w.m
INDEX
LRG IN YERRS
IOWR DROUGHT
INDEX
IOWR SIMIJLRTION B.on
""1
~
.ccol
q,,
;*‘""
DROUGHT
j
INDEX
.20x
.Bmo
.Yrml
FREQUENCY IN CYCLES PER YERR KANS
.a1303
SIHULFiTIBN
"m/
I
I
.w
,o.m
KRNSAS
DROUGHT
2o.m
LAG IN YEARS
INDEX
30.m
I
ro.m
-.m
( .m
KRNS
lOLo
2o.m
LRG IN YERRS
mm
i
w.03
SIMULRTI3N
Fig. 4. Correlograms and power spectra of original (left) drought index series of (A) Iowa and (B) Kansas.
and simulated
(right) yearly
i! b
RUTOCCIRRELRTION
$
RUTOCORRELRTION
8
J
SPECTRAL
SPECTRRL
DENSITY
DENSITY
RUTOCORRELAT
RUlCIC(1RRELFITION
rClN
@
KAN
ANNU%
DF: INDEX
I51
e.goo 1
KRNSRS
fiONTHLY OR INDEX
e0.m
Fig. 6. Maximum-entropy and (B) spectra of monthly
(601
0
61
spectra of the data series.
data
series:
(A)
spectra
of yearly
data
series;
In eqs. 2 and 3, n is the number of observations used in fitting the model; pi is the number of parameters; mi the maximum lag in the ith model; pi the residual variance of the ith model; and p the variance of the PDI series y(t). The model which gives the minimum ai or the maximum hi(W) is accepted as the best model for the series. The decision rule based on hi(W) is called the posterior probability rule and that based on ai( the AIC rule. These statistics ai and hi(w) are given in Tables III and IV. Akaike’s criterion ai indicates that the AR(3) and AR(5) models are the best models for monthly PDI series of Kansas and Iowa, respectively. It must be emphasized that Akaike’s criterion is not consistent (Kashyap, 1980). The posterior probability criterion based on the statistic hi(W), in which the principle of parsimony of parameters is emphasized more, indicates that the AR(l) model is the best model for both the monthly Kansas and Iowa series. However, as AR(l) models do not account for the long-term periodicities present in the data, the AR(3) and AR(5) models are selected for monthly PDI series. For annual PDI series both criteria indicate the AR(5) and AR(4) models for Kansas and Iowa data, respectively. However, for Iowa yearly data, the posterior probability rule decisively picks up AR(4),
40 60 s (YEARS)
LAG
LAG
100
s (YEARS)
30 a0 R(s) 2. IO
. . . OBSERVED
3 2 I
IO
DATA
* * * GENERATED
DATA
” CONFIDENCE
BAND
20
100
50 LAG
s (MONTHS)
200
1 400
IO
I
:fyf$2yf
20 LAG
50
100
s( MONTHS
200
‘
1
IOWA
KANSAS
Fig. 7. R(s)/o(s) characteristics of original drought index series of Kansas and Iowa.
and simulated
(A) yearly
and (B) monthly
whereas the AIC rule shows only a local minimum at AR(4) and later continues to decrease with increasing order of models (Table IV). One of the advantages in using the posterior probability rule (Kashyap, 1977) is that the probability of a model in correctly representing a system can be estimated by using it. Examples of these probabilities are shown in Fig. 8, for annual PDI series. The posterior probability rule clearly shows the superiority of AR(5) and AR(4) models for the Kansas and Iowa data, respectively. The AIC, on the other hand, does not give these probabilities.
VALIDATION
OF MODELS
AND SIMULATION
OF DROUGHT
INDEX
SERIES
The models selected by methods discussed in the previous section are validated by testing their residuals for whiteness and presence of periodicities. The residuals from the selected models of yearly data, their
221 TABLE III Selection of monthly models i
1 2 3 4 5 6
ai(w
hi(w)
Y
Kansas
Iowa
Kansas
Iowa
-45.16 -- 48.20 -51.63* -50.70 --49.20 - 50.81
-9.41 -7.94 -11.00 -11.64 -14.53* -12.74
28.34* 24.87 22.88 15.57 8.18 - 28.30
-4.91* -11.23 -13.12 -17.42 - 19.54 - 26.20
* Selected model. TABLE IV Selection of yearly models i
1 2 3 4 5 6
ai(w
)
hi(w
1
Kansas
Iowa
Kansas
Iowa
112.45 113.64 113.35 110.49 107.97* 109.73
38.96 39.02 35.41 22.25* 22.45 22.14
-115.20 - 116.55 -116.56 -114.42 -112.80* -114.90
- 39.10 -40.59 - 37.93 - 23.63* - 30.86 - 32.31
* Selected model.
correlograms, power spectra, and cumulative periodograms are shown in Fig. 9. The results of tests on residuals from monthly models are shown in Fig. 10. These results in Figs. 9 and 10 show that the residuals are white and have no periodicity. Residual histograms are also tested to determine whether they are normally distributed by using the Kolmogorov-Smirnov test. The results of this test indicated that they may be approximated by normal distribution. The normal distributions fitted to the histograms of residuals are shown in Fig. 11. The mean and standard deviation of the residuals are used to simulate y(t)-values. Normally distributed random numbers with their respective mean and variance values are generated and used with the AR models to synthetically generate drought index values. The statistical characteristics of the synthetic data are compared with those of observed data. Histograms of original and simulated data are given in Fig. 3, the correlograms and the power spectra in Figs. 4 and 5, and the resealed range-lag characteristics in Fig. 7. The elementary statistics and Hurst coefficients of the original and simulated data are given in Table V.
KANSAS
d----L--
Order of the AR Model
Fig. 8. Probability and (B) Iowa.
that the selected orders are correct
2
3
4
5
6
7
Order of the AR Model
for yearly models:
(A) Kansas
These results indicate that the characteristics of generated data correspond very well to those of original data. The models thus “preserve” the correlation, resealed range and probability distribution of the original data. Because a cycle of - 13-l 8 yr. was evident from the spectral characteristics of Kansas data, another experiment was undertaken to investigate noncontiguous AR models for Kansas yearly data. The noncontiguous terms of the lag close to the period of the cycle were introduced along with AR terms up to order 5, which were selected by using the selection criteria described in the previous section. Terms with lags varying from 13.---I8 (i = 12,14, . . . ,18) were used in: y(t)
= 010 + o!,y(t - 1) + ac,y(t - 2) + cX,y(t - 3) + a,y(t - 4) i = 13,14.. . . ,18 + a,y(t - 5) + cr,y(t - i),
(4)
The models and corresponding residual variances are given in Table VI. Of all the noncontiguous terms, the one with a lag of 16yr. appears to bring about the maximum reduction of - 2% in residual variance from that of AR(5) model already selected. This is not very significant considering that the percentage reduction is variance brought about by the AR(5) model is - 30%. Consequently, the noncontiguous models are not investigated further. FORECASTING
EXPERIMENTS
The forecasting ability of the AR models fitted to the monthly and annual data are tested by using split samples. The available observations are split in half, the first half of which is used to estimate the model parameters and the second half to check the forecasting accuracy. This one-step-ahead forecast is referred to herein as FCl. However, conducting such a forecasting experiment is not possible for the yearly Iowa data since
224
KRNSAS RESIDUALS
KANSRS RESIDURLS ““7
CENTRRL IGWR
IOWR AESIDL~ALS
RESlOURLS
IW s 1.m
,’
E ,’ 2
.m
z k 5.m F&I
,’
,,,,y:,.,. I
,’ lzl
’ IiT%LES
,emmi
Fig. 10. Residual tests for monthly models for: (A) Kansas and (B) Iowa drought index series.
225 KANSAS
YEARLY
MODEL
AR(5)
KANSAS
0
3.84
RESIDUALS If
0.6
w
IOWA
YEARLY
MODEL
IOWA
AR(3)
MODEL
@
AR (51 MODEL
AR(4)
0.2 0i -6.38
6.52 RESIDUALS
-3.47
3.44
RESIDUALS
Fig. 11. Histograms of residuals of selected models for Kansas and Iowa: (A) yearly and (B) monthly drought index series and the fitted normal distributions.
only 33 observations are available in this case. Therefore, the residual variance is used as an estimate of the forecast accuracy of the model of annual Iowa data. Finally, the parameters in the models are recursively estimated and the updated parameters are used to forecast PDI values one step ahead and these are designated FC2. Results of forecast analyses are presented in Table VII. The one-step-ahead forecasts FCl of the monthly models for Iowa [AR(5)] and Kansas [AR(3)] are also shown in Fig. 12. Mean square error of forecasts (FCl) for monthly models in Table VII are much smaller than the mean square PDI values (shown as MSS in Table VII). Consequently, these models fitted to monthly data may be used to TABLE V Statistics of original
and simulated
Yearly
data Monthly
data
Kansas
Iowa
original
simulated
original
Mean Variance
-0.003 5.373
-0.009 3.806
0.353 4.072
Skewness coefficient Kurtosis Hurst’s H
-0.134 2.286 0.672
0.042 3.023 0.651
-0.424 2.788 0.751
simulated
data
_ Iowa
Kansas
original
_
original
simulated
simulated
0.334 2.610
-0.004 7.359
-0.005 6.037
0.353 6.316
0.408 4.330
0.076 2.444 0.518
0.159 2.466 0.790
~ 0.044 2.704 0.807
-0.465 2.369 0.892
0.077 2.703 0.835
VI
(0.036)
0.5479
(0.017)
(0.036)
(0.019)
-0.1629
(0.019)
- 0.001
- 0.1439
0.5377
(0.017)
0.016
(0.019)
-0.1508
(0.019)
-0.1509
(0.019)
-0.1715
(0.019)
-0.1416
(0.019)
-0.2199
a2
2) + asy(t
(0.036)
0.5645
(0.017)
0.022
(0.036)
0.5569
(0.017)
(0.036)
(0.036)
- 0.021
0.5671
(0.017)
- 0.036
-
-
(0.019)
0.2786
(0.019)
0.2777
(0.019)
0.2561
(0.019)
0.2654
(0.019)
0.2481
(0.019)
0.2665
(0.019)
0.2199
a3
0.2651
(0.019)
-0.1621
(0.019)
- 0.1774
(0.019)
-0.1350
(0.019)
-0.1665
(0.019)
-0.1600
(0.019)
--0.1682
(0.019)
0.1704 (0.017)
(0.018)
-0.1289
(0.018)
- 0.0909
(0.018)
- 0.0945
(0.018)
-0.1353
(0.018)
- 0.1452
(0.018)
-0.1165
-
-
with Model: 5) + %jy(t
a5
index
4) + (&jy(t -
014
-
for Kansas drought
3) + (Y&t
terms in AR models
AR(5) given in the first row is a valid model, p,, = 5.373. Standard error of the parameters are given within parentheses.
18
17
16
15
14
0.5730
(0.017)
- 0.007
(0.017)
0.5834
a1
1) + &y(t
13
a0
-
of non-contiguous
@e + (Y,y(t
- 0.002 (0.036)
=
-
i
y(t)
Investigation
TABLE
0.0766 (0.017)
(0.017)
0.065
(0.017)
0.1266
0.1438 (0.017)
(0.017)
-0.0108
-0.0933 (0.017)
ff6
i)
3.520
3.480
3.468
3.535
3.504
3.514
3.535
P
0.42
1.55
1.90
0.00
0.88
0.59
-
from AR( 5) model
% reduction of residual variance
221
Fig. 12. One-step-ahead forecasts of monthly drought index series of (A) Kansas and (B) Iowa (FCl forecasts).
228 TABLE VII Mean square error of one-step-ahead forecasts Drought index series
Kansas (m) Iowa (m) Kansas (y) Iowa (y)
MSS
MSE FCl
FC2
0.9274 0.9704 10.464
7.36 6.31 3.1000 3.384
5.373 4.072
m = monthly; y = yearly.
accurately forecast the drought indices 1 month ahead. This is not surprising as the monthly PDI is strongly persistent. However, FCl forecasts of annual Kansas drought index has a very high mean square error compared to the mean square value of data that is being forecast. This may be because the parameters are estimated by using a small number of observations. FC2 forecasts of annual PDI series computed by using recursive parameter estimates are much better. However, the annual PDI values cannot be forecast as accurately as monthly values. This is evident from the high mean square error values of annual models. On the whole, monthly models have better forecasting capability. By the nature of monthly models. monthly PDI values may be forecast accurately several months ahead also. CONCLUSIONS
The analysis of the Palmer’s drought index series data indicate that longterm oscillations and high persistence structure are important characteristics of drought indices and hence of droughts. The periodic behavior in drought index series is apparent in the conventional power spectra and also in the spectra computed by using high-resolution spectral methods. Resealed range-lag characteristics also support these conclusions. Nevertheless, the PDI series can be modeled by simple stochastic models which preserve the important statistical characteristics of the original data as demonstrated in this study. The simple models of PDI series developed can be used for generating synthetic PDI data and for forecasting PDI values which are used in agricultural planning and optimal operation of irrigation systems, respectively.
REFERENCES Akaike, H., 1974. A new look at statistical model identification. Control, AC-19: 716-722.
I.E.E.E. Trans. Autom.
229
Ausuabel, J. and Biswas, A.K., 1980. Climate Constraints and Human Activity. Pergamon, New York, N.Y. Hecht, A.D., 1981. The challenge of climate to man. EOS (Trans. Am. Geophys. Union), 62(51): 1193-1197. Jenkins, G.M. and Watts, D.G., 1968. Spectral Analysis and Its Applications. Holden-Day, San Francisco, Calif. Kashyap, R.L., 1977. A Bayesian comparison of different classes of dynamic models using empirical data. I.E.E.E. Trans. Autom. Control, AC-22: 1284-1292. Kashyap, R.L., 1980. Inconsistency of the AIC rule for estimating the order of autoregressive models. I.E.E.E. Trans. Autom. Control, AC-25: 996-998. Kashyap, R.L. and Rao, A.R., 1976. Dynamic Stochastic Models from Empirical Data. Academic Press, New York, N.Y. King, J.W., Hurst, E., Slater, A.J., Smith, P.A. and Tamkin, B., 1974. Agriculture and Sunspots. Nature (London), 252 : 2-3. Mandelbrot, B.B and Wallis, J.R., 1969. Some long run properties of geophysical records. Water Resour. Res., 5(2): 321-340. Mitchell, J.M., Stockton, C.W. and Meko, D.M., 1979. Evidence of a 22-year rhythm of drought in the western U.S. related to the hale solar cycle since the 17th century. In: B.M. McCormace and T.A. Sehga (Editors), Solar-Terrestrial Influences on Weather and Climate. D. Reidel, Dordrecht. Padmanabhan, G. and Rao, A.R.,.1980. Stochastic analysis and modeling of hydrologic and climatologic time series II: Analysis of causal connections between solar activity and climatologic time series and of Palmer’s drought index series. School Civ. Eng., Purdue Univ., West Lafayette, Ind., Tech. Rep. CE-HSE-80-2. Palmer, W.C., 1965. Meteorlogicaldrought. Weather Bur., U.S. Dep.Commer., Washington, D.C., Res. Pap. No. 45, pp. l-58. Rao, K.N., George, M., Morey, C.J., and Mehta, N.K., 1973. Spectral analysis of drought index (Palmer) for India. Ind. J. Meteorol. Geophys., 24(3) 257-270. Rosenberg, N.J. (Editor), 1978. North American Drought. Westview, Boulder, Cola. Sen, A., 1982. Poverty and Famines: An Essay on Entitlement and Deprivation. Oxford University Press, Oxford. Stockton, C.W. and Boggess, W.R., 1979. Geohydrological implications of climate change on water resources development. Rep. U.S. Army Coast. Eng. Res. Cent., Fort Belvoir, Va., 206 pp. Thomas, H.E., 1962. The meteorological phenomena of drought in the southwest, U.S. Geol. Surv., Prof. Pap. 342A. Vines, R.G., 1982. Rainfall patterns in the western United States. J. Geophys. Res., 87(No. C9): 7303-7311.
Journal
211
ANALYSIS AND MODELING OF PALMER’s DROUGHT INDEX SERIES
A. RAMACHANDRA
RAO and G. PADMANABHAN
School of Civil Engineering, Purdue University, Department of Civil Engineering, North Dakota
West Lafayette, IN 47907 (U.S.A.) State University, Fargo, ND 58105
(U.S.A.)
(Revised and accepted for publication
November 2, 1982)
ABSTRACT Rao, A.R. and Padmanabhan, G., 1984. Analysis and modeling of Palmer’s drought index series. In: G.E. Stout and G.H. Davis (Editors), Global Water: Science and Engineering -The Ven Te Chow Memorial Volume. J. Hydrol., 68: 211-229.
The objective of the present study is to investigate the stochastic nature of yearly and monthly Palmer’s drought index (PDI) series and to characterize them via valid stochastic models which may be used to forecast and to simulate the PDI series. The monthly and annual PDI series for Iowa (1930-1962) and Kansas (1887-1962) are analyzed in the present study. The period of data includes the severe droughts experienced in U.S.A. in the 1930’s. Valid autoregressive models are fitted to these time series and the final models are selected on the basis of class selection rules. Statistical characteristics of the observed data and those generated by the selected models are compared. The selected models are also tested for their forecasting ability. Results of these analyses and the modeling effort demonstrate that PDI series can be forecast with reasonable accuracy, one to several months ahead. The yearly forecasts are much less accurate. The models can also be used to generate synthetic data which preserve the important statistical characteristics such as the long-term oscillations of the original data.
INTRODUCTION
Because droughts cause serious economic as well as social impacts, it has been investigated by hydrologists (Stockton and Boggess, 1979) and others (Rosenberg, 1978; Ausuabel and Biswas, 1980; Hecht, 1981; Sen, 1982). Due to the random nature of the contributing factors, occurrence and severity of droughts are stochastic in nature. An important and often discussed characteristic of droughts and their indices is their apparent periodic behavior. Thomas (1962) has pointed out the 20-22 yr. periodicity in drought in the Great Plains region. Recently, Mitchell et al. (1979) have shown by an analysis of data reconstructed from tree rings, that the spatial extent of drought also follows an approximate 22-yr. cycle. Vines (1982) has analyzed rainfall data west of Mississippi River and has concluded that the rainfall has
0022-1694/84/$03.00
o 1984 Elsevier Science Publishers B.V.
212
both a 22-yr. and a 19-yr. (lunar) cycle. The evidence of a 22-yr. cycle in droughts gives rise to speculations about solar influence on droughts because the annual sunspot number series coded for polarity also has a period of 22yr. In spite of comments about the periodicity in the occurrence of droughts, the stochastic structure of time series of drought indicators does not appear to have been analyzed to establish and understand the nature of these series. Consequently, the basic objective of the present paper is to investigate the stochastic characteristics of Palmer’s drought index (PDI) series. One of the basic deficiencies in mitigating the effects of droughts is the inability to predict drought conditions accurately for months or years in advance (King et al., 1974). Accurate drought forecasts would enable optimal operation of irrigation systems. The data generated from valid models of drought indicator time series are useful in agricultural planning. Therefore, there is a clear need to examine the stochastic characteristics of drought indicators with special reference to long-term oscillations and to develop simple and effective models of these indicators which may be used for prediction of droughts and to generate synthetic drought index time series. There are many definitions and indices of drought, some of which are based only on rainfall, others on both rainfall and temperature, and still others which are based on related factors. Although the main cause of droughts is the lack of rainfall, the manifestations of droughts are varied. These include low streamflows, lower ground water-table levels, low reservoir fills and prolonged periods of soil-moisture deficiency with their attendant effect on agriculture. One of the best known indices of droughts is that proposed by Palmer (1965) and this index is therefore analyzed in this study. Palmer’s drought index is an indicator of drought severity which is being extensively and routinely used in the U.S.A. and many other countries of the world for agricultural planning, forecasting crop yields and other related purposes. The index is based on a definition of drought as a prolonged period of abnormal moisture deficiency, and that drought severity is a function of moisture demand as well as supply. The scale of severity of droughts in terms of Palmer’s drought index is given in Table I. The PDI is computed by using observed precipitation and temperature data and a moisture accounting procedure (Palmer, 1965). The PDI of various areas in the U.S.A. are routinely published in the Week@ Weather and Crop Bulletin. The main objective of the present study is to investigate the stochastic nature of yearly and monthly time PDI series and to characterize them via valid stochastic models which may be used to forecast and simulate the PDI series. The monthly and annual PDI series for Iowa (1930-1962) and Kansas (1887-1962), computed by Palmer (1965), are used in the present study, and include the severe droughts experienced in the U.S.A. in the 1930’s. The preliminary data analyses discussed herein give an insight to the stochastic nature of the data series. Valid autoregressive models are fitted
213 TABLE I Drought classification by Palmer’s drought index Palmer’s index
Degree of drought
PDSI < - 4.0 < PDSI < - 3.0 < PDSI < -2.O
- 4.0 - 3.0 - 2.0
extremely dry severely dry moderately dry mildly dry near normal mildly wet moderately wet severely wet extremely wet
+ 2.0 + 3.0 + 4.0
to these time series and final models are selected on the basis of class selection rules. Statistical characteristics of the original and synthetic data generated by the selected models are compared next. The selected models are next tested for their forecasting ability. Results of these analyses and modeling effort demonstrate that the PDI series can be forecast with reasonable accuracy, one to several months ahead. These models can also be used to generate synthetic data which preserve the important statistical characteristics, including the long-term oscillations, of the original data.
CHARACTERISTICS
OF THE DATA USED IN THE STUDY
The monthly PDI series are shown in Fig. 1. No obvious periodicity is evident in the plot of monthly PDI series of either Kansas or Iowa. In fact the data appear to be random. The elementary statistics of the data are given
,OqD
-10
IOWA
DROUGHT
oo/ 00
1000
MOO MONTHS
@9 10001
INDEX
300.0
4000
00
KANSAS
2000
DROUGHT
4000
INDEX
6000
a000
MONTHS
Fig. 1. Monthly Palmer’s drought index series for (A) Iowa and (B) Kansas.
700
214 TABLE
II
Elementary
statistics
of the data series
No.
Series
Mean
Standard deviation
Coefficient of variation
Skewness coefficient
Kurtosis
Hurst’s H
1 2 3 4
KAN(Y) IOW(Y) KAN(M) IOW(M)
- 0.004 0.353 - 0.004 0.353
2.32 2.02 2.71 2.51
579.5 5.7 678.0 7.11
-0.133 - 0.424 0.159 - 0.465
2.286 2.788 2.466 2.369
0.67 0.75 0.79 0.89
KAN(Y), IOW(Y) are yearly data of Kansas and Iowa, respectively; monthly data of Kansas and Iowa, respectively.
KAN(M),
IOW(M) are
in Table II. The mean values of monthly and annual PDI are close to zero for Kansas data and they are significantly greater than zero for Iowa data. The standard deviations of both Kansas and Iowa PDI series are large, which indicate a rather wide variability in drought conditions. The data are not normally distributed according to the skewness coefficient and kurtosis values given in Table II. Histograms of the data shown in Fig. 2 also support these observations. The monthly means, standard deviations and skewness coefficients of the data series (Fig. 3) indicate some periodicity in the monthly means of Kansas but not in Iowa data. The monthly standard deviations and skewness coefficients do not have any periodicity and show rather wide variations from one month to the next. The correlograms and power spectral density functions of the data estimated by using standard methods (Jenkins and Watts, 1968) and the Hamming window are shown in Figs. 4 and 5. Also shown in Figs. 4 and 5 are the correlograms and spectral densities of simulated data generated by using valid models and these are discussed later. Although long-period oscillations can be seen in the correlograms they are not very pronounced. The two-standard error (2~) limits are also shown on the correlograms. The power spectra obtained by using standard methods for the yearly and monthly series shown in Figs. 4 and 5 clearly indicate the presence of longterm oscillations. The spectra of these series are also estimated by using the high-resolution maximum-entropy spectral method (Padmanabhan and Rao, 1980) and these are shown in Fig. 6. These maximum-entropy spectra confirm the presence of long-term oscillations in the PDI series of approximate periods ranging from 13 to 20 yr. in the data. Resealed range-lag characteristics of data provide an important statistic which may be used in investigating long-term oscillations (Mandelbrot and Wallis, 1969; Kashyap and Rao, 1976). The statistic H, derived from log-log plots of resealed range [R(s)/o(s)] against lag s, indicates the nonstationarity and persistence in the series. If there is a predominant periodic component in the data series the log [R(s)/o(s)] vs. log (s) plot of the data will deviate from
215 @
04
TNSAS
SIMULATED
~:
SERIES,
02 5
ORIGINAL
DATA
,
02
01
ifI 2 i$
rSAS
:, 0’
5
00 -583
3
00
-5 17
u.
-524
4 81
u-
s
IOWA SIMULATED
9 ~
SERIES
o4iOWA
ORIGINAL
DATA
DROUGHT
INDEX
,
02
2
04 00 m -2 6
4 26 DROUGHT
@.mj~~s~~
INDEX
SIMULATED
KANSAS
SERIES,
ORIGINAL
DATA
.3oy1 20
>
s Y 8
E w 2
IOWA
SIMULATED
.I0
0 -7.05
e w .30
SERIES
3O
IOWA
926 ORIGINAL
DATA
? J K .I0
IO t
200 m -6.65
1.72 DROUGHT
INDEX
2 .200 t;-_l -6.70
7.72 DROUGHT
INDEX
Fig. 2. Histograms of original (right) and simulated (left) data of (A) yearly and (B) monthly drought index series of Iowa and Kansas.
being a straight line around lags corresponding to this periodicity. The resealed ranges of PDI series are plotted against the corresponding lags on log-log paper and are shown in Fig. 7. These plots again indicate the presence of long-term oscillations in the data around 13-20 yr. Hurst’s Hvalues given in Table II for the data indicate high persistence in the index and hence in droughts. The results of data analysis discussed above clearly point to the fact that the drought index series are highly persistent and have strong long-term oscillations. Evidence of nearly significant to significant long-term oscillations in PDI series has been reported in other countries also (Rao et al., 1973). Therefore, models proposed for drought index data should be capable of preserving these long-term oscillations.
216
‘U
0.: 03 w y 02
06
9 $01 *
0
JFMAMJJASOND MONTHS
0 1’11-11 JFMAMJ KANSAS
JASOND MONTHS
IOWA
Fig. 3. Monthly mean, standard deviation series of Kansas (left) and Iowa (right). SELECTION
OF CANDIDATE
and
skewness
coefficient
of drought
index
MODELS
Based on the data analysis discussed in the previous section, it is quite evident that the PDI series do not have the annual and semi-annual periods which are usually present in hydrologic time series. Long-term periodicities, however, are present in these data. Therefore, models belonging to the autoregressive moving average (ARMA) class are selected to model the PDI series. AR models of different orders m, such as those in the following equation were fitted to the PDI values which are denoted by y(t): y(t)
= ol,+a,y(t-l)+a,
+...+o,y(t-m)+W(t)
(1)
In many of these models of monthly data the value of o1 are found to be close to unity which indicates that the data are highly correlated. The reduction in variance of residuals w(t) from AR(l) to AR(6) models of monthly data is also very small. However, for models of annual series, residual variance decreases considerably with increasing model orders. In order to select the best models from these candidate models, the statistics Ui(W) and hi(W) given in eqs. 2 and 3, respectively, are used. These statistics developed by Akaike (1974) (eq. 2) and Kashyap (1977) (eq. 3) are given by: Ui(W) = n In@,) + 2pi hi(W) =
-(n
-mi)
In pi + [(p -Pi)/n]
(2) ‘-pi In(n) -mi
In(p) (3)
IOWR SIMLILRTION 1.33. IBWR DROUGHT
-Am ‘.w.m
INDEX
LRG IN YERRS
IOWR DROUGHT
INDEX
IOWR SIMIJLRTION B.on
""1
~
.ccol
q,,
;*‘""
DROUGHT
j
INDEX
.20x
.Bmo
.Yrml
FREQUENCY IN CYCLES PER YERR KANS
.a1303
SIHULFiTIBN
"m/
I
I
.w
,o.m
KRNSAS
DROUGHT
2o.m
LAG IN YEARS
INDEX
30.m
I
ro.m
-.m
( .m
KRNS
lOLo
2o.m
LRG IN YERRS
mm
i
w.03
SIMULRTI3N
Fig. 4. Correlograms and power spectra of original (left) drought index series of (A) Iowa and (B) Kansas.
and simulated
(right) yearly
i! b
RUTOCCIRRELRTION
$
RUTOCORRELRTION
8
J
SPECTRAL
SPECTRRL
DENSITY
DENSITY
RUTOCORRELAT
RUlCIC(1RRELFITION
rClN
@
KAN
ANNU%
DF: INDEX
I51
e.goo 1
KRNSRS
fiONTHLY OR INDEX
e0.m
Fig. 6. Maximum-entropy and (B) spectra of monthly
(601
0
61
spectra of the data series.
data
series:
(A)
spectra
of yearly
data
series;
In eqs. 2 and 3, n is the number of observations used in fitting the model; pi is the number of parameters; mi the maximum lag in the ith model; pi the residual variance of the ith model; and p the variance of the PDI series y(t). The model which gives the minimum ai or the maximum hi(W) is accepted as the best model for the series. The decision rule based on hi(W) is called the posterior probability rule and that based on ai( the AIC rule. These statistics ai and hi(w) are given in Tables III and IV. Akaike’s criterion ai indicates that the AR(3) and AR(5) models are the best models for monthly PDI series of Kansas and Iowa, respectively. It must be emphasized that Akaike’s criterion is not consistent (Kashyap, 1980). The posterior probability criterion based on the statistic hi(W), in which the principle of parsimony of parameters is emphasized more, indicates that the AR(l) model is the best model for both the monthly Kansas and Iowa series. However, as AR(l) models do not account for the long-term periodicities present in the data, the AR(3) and AR(5) models are selected for monthly PDI series. For annual PDI series both criteria indicate the AR(5) and AR(4) models for Kansas and Iowa data, respectively. However, for Iowa yearly data, the posterior probability rule decisively picks up AR(4),
40 60 s (YEARS)
LAG
LAG
100
s (YEARS)
30 a0 R(s) 2. IO
. . . OBSERVED
3 2 I
IO
DATA
* * * GENERATED
DATA
” CONFIDENCE
BAND
20
100
50 LAG
s (MONTHS)
200
1 400
IO
I
:fyf$2yf
20 LAG
50
100
s( MONTHS
200
‘
1
IOWA
KANSAS
Fig. 7. R(s)/o(s) characteristics of original drought index series of Kansas and Iowa.
and simulated
(A) yearly
and (B) monthly
whereas the AIC rule shows only a local minimum at AR(4) and later continues to decrease with increasing order of models (Table IV). One of the advantages in using the posterior probability rule (Kashyap, 1977) is that the probability of a model in correctly representing a system can be estimated by using it. Examples of these probabilities are shown in Fig. 8, for annual PDI series. The posterior probability rule clearly shows the superiority of AR(5) and AR(4) models for the Kansas and Iowa data, respectively. The AIC, on the other hand, does not give these probabilities.
VALIDATION
OF MODELS
AND SIMULATION
OF DROUGHT
INDEX
SERIES
The models selected by methods discussed in the previous section are validated by testing their residuals for whiteness and presence of periodicities. The residuals from the selected models of yearly data, their
221 TABLE III Selection of monthly models i
1 2 3 4 5 6
ai(w
hi(w)
Y
Kansas
Iowa
Kansas
Iowa
-45.16 -- 48.20 -51.63* -50.70 --49.20 - 50.81
-9.41 -7.94 -11.00 -11.64 -14.53* -12.74
28.34* 24.87 22.88 15.57 8.18 - 28.30
-4.91* -11.23 -13.12 -17.42 - 19.54 - 26.20
* Selected model. TABLE IV Selection of yearly models i
1 2 3 4 5 6
ai(w
)
hi(w
1
Kansas
Iowa
Kansas
Iowa
112.45 113.64 113.35 110.49 107.97* 109.73
38.96 39.02 35.41 22.25* 22.45 22.14
-115.20 - 116.55 -116.56 -114.42 -112.80* -114.90
- 39.10 -40.59 - 37.93 - 23.63* - 30.86 - 32.31
* Selected model.
correlograms, power spectra, and cumulative periodograms are shown in Fig. 9. The results of tests on residuals from monthly models are shown in Fig. 10. These results in Figs. 9 and 10 show that the residuals are white and have no periodicity. Residual histograms are also tested to determine whether they are normally distributed by using the Kolmogorov-Smirnov test. The results of this test indicated that they may be approximated by normal distribution. The normal distributions fitted to the histograms of residuals are shown in Fig. 11. The mean and standard deviation of the residuals are used to simulate y(t)-values. Normally distributed random numbers with their respective mean and variance values are generated and used with the AR models to synthetically generate drought index values. The statistical characteristics of the synthetic data are compared with those of observed data. Histograms of original and simulated data are given in Fig. 3, the correlograms and the power spectra in Figs. 4 and 5, and the resealed range-lag characteristics in Fig. 7. The elementary statistics and Hurst coefficients of the original and simulated data are given in Table V.
KANSAS
d----L--
Order of the AR Model
Fig. 8. Probability and (B) Iowa.
that the selected orders are correct
2
3
4
5
6
7
Order of the AR Model
for yearly models:
(A) Kansas
These results indicate that the characteristics of generated data correspond very well to those of original data. The models thus “preserve” the correlation, resealed range and probability distribution of the original data. Because a cycle of - 13-l 8 yr. was evident from the spectral characteristics of Kansas data, another experiment was undertaken to investigate noncontiguous AR models for Kansas yearly data. The noncontiguous terms of the lag close to the period of the cycle were introduced along with AR terms up to order 5, which were selected by using the selection criteria described in the previous section. Terms with lags varying from 13.---I8 (i = 12,14, . . . ,18) were used in: y(t)
= 010 + o!,y(t - 1) + ac,y(t - 2) + cX,y(t - 3) + a,y(t - 4) i = 13,14.. . . ,18 + a,y(t - 5) + cr,y(t - i),
(4)
The models and corresponding residual variances are given in Table VI. Of all the noncontiguous terms, the one with a lag of 16yr. appears to bring about the maximum reduction of - 2% in residual variance from that of AR(5) model already selected. This is not very significant considering that the percentage reduction is variance brought about by the AR(5) model is - 30%. Consequently, the noncontiguous models are not investigated further. FORECASTING
EXPERIMENTS
The forecasting ability of the AR models fitted to the monthly and annual data are tested by using split samples. The available observations are split in half, the first half of which is used to estimate the model parameters and the second half to check the forecasting accuracy. This one-step-ahead forecast is referred to herein as FCl. However, conducting such a forecasting experiment is not possible for the yearly Iowa data since
224
KRNSAS RESIDUALS
KANSRS RESIDURLS ““7
CENTRRL IGWR
IOWR AESIDL~ALS
RESlOURLS
IW s 1.m
,’
E ,’ 2
.m
z k 5.m F&I
,’
,,,,y:,.,. I
,’ lzl
’ IiT%LES
,emmi
Fig. 10. Residual tests for monthly models for: (A) Kansas and (B) Iowa drought index series.
225 KANSAS
YEARLY
MODEL
AR(5)
KANSAS
0
3.84
RESIDUALS If
0.6
w
IOWA
YEARLY
MODEL
IOWA
AR(3)
MODEL
@
AR (51 MODEL
AR(4)
0.2 0i -6.38
6.52 RESIDUALS
-3.47
3.44
RESIDUALS
Fig. 11. Histograms of residuals of selected models for Kansas and Iowa: (A) yearly and (B) monthly drought index series and the fitted normal distributions.
only 33 observations are available in this case. Therefore, the residual variance is used as an estimate of the forecast accuracy of the model of annual Iowa data. Finally, the parameters in the models are recursively estimated and the updated parameters are used to forecast PDI values one step ahead and these are designated FC2. Results of forecast analyses are presented in Table VII. The one-step-ahead forecasts FCl of the monthly models for Iowa [AR(5)] and Kansas [AR(3)] are also shown in Fig. 12. Mean square error of forecasts (FCl) for monthly models in Table VII are much smaller than the mean square PDI values (shown as MSS in Table VII). Consequently, these models fitted to monthly data may be used to TABLE V Statistics of original
and simulated
Yearly
data Monthly
data
Kansas
Iowa
original
simulated
original
Mean Variance
-0.003 5.373
-0.009 3.806
0.353 4.072
Skewness coefficient Kurtosis Hurst’s H
-0.134 2.286 0.672
0.042 3.023 0.651
-0.424 2.788 0.751
simulated
data
_ Iowa
Kansas
original
_
original
simulated
simulated
0.334 2.610
-0.004 7.359
-0.005 6.037
0.353 6.316
0.408 4.330
0.076 2.444 0.518
0.159 2.466 0.790
~ 0.044 2.704 0.807
-0.465 2.369 0.892
0.077 2.703 0.835
VI
(0.036)
0.5479
(0.017)
(0.036)
(0.019)
-0.1629
(0.019)
- 0.001
- 0.1439
0.5377
(0.017)
0.016
(0.019)
-0.1508
(0.019)
-0.1509
(0.019)
-0.1715
(0.019)
-0.1416
(0.019)
-0.2199
a2
2) + asy(t
(0.036)
0.5645
(0.017)
0.022
(0.036)
0.5569
(0.017)
(0.036)
(0.036)
- 0.021
0.5671
(0.017)
- 0.036
-
-
(0.019)
0.2786
(0.019)
0.2777
(0.019)
0.2561
(0.019)
0.2654
(0.019)
0.2481
(0.019)
0.2665
(0.019)
0.2199
a3
0.2651
(0.019)
-0.1621
(0.019)
- 0.1774
(0.019)
-0.1350
(0.019)
-0.1665
(0.019)
-0.1600
(0.019)
--0.1682
(0.019)
0.1704 (0.017)
(0.018)
-0.1289
(0.018)
- 0.0909
(0.018)
- 0.0945
(0.018)
-0.1353
(0.018)
- 0.1452
(0.018)
-0.1165
-
-
with Model: 5) + %jy(t
a5
index
4) + (&jy(t -
014
-
for Kansas drought
3) + (Y&t
terms in AR models
AR(5) given in the first row is a valid model, p,, = 5.373. Standard error of the parameters are given within parentheses.
18
17
16
15
14
0.5730
(0.017)
- 0.007
(0.017)
0.5834
a1
1) + &y(t
13
a0
-
of non-contiguous
@e + (Y,y(t
- 0.002 (0.036)
=
-
i
y(t)
Investigation
TABLE
0.0766 (0.017)
(0.017)
0.065
(0.017)
0.1266
0.1438 (0.017)
(0.017)
-0.0108
-0.0933 (0.017)
ff6
i)
3.520
3.480
3.468
3.535
3.504
3.514
3.535
P
0.42
1.55
1.90
0.00
0.88
0.59
-
from AR( 5) model
% reduction of residual variance
221
Fig. 12. One-step-ahead forecasts of monthly drought index series of (A) Kansas and (B) Iowa (FCl forecasts).
228 TABLE VII Mean square error of one-step-ahead forecasts Drought index series
Kansas (m) Iowa (m) Kansas (y) Iowa (y)
MSS
MSE FCl
FC2
0.9274 0.9704 10.464
7.36 6.31 3.1000 3.384
5.373 4.072
m = monthly; y = yearly.
accurately forecast the drought indices 1 month ahead. This is not surprising as the monthly PDI is strongly persistent. However, FCl forecasts of annual Kansas drought index has a very high mean square error compared to the mean square value of data that is being forecast. This may be because the parameters are estimated by using a small number of observations. FC2 forecasts of annual PDI series computed by using recursive parameter estimates are much better. However, the annual PDI values cannot be forecast as accurately as monthly values. This is evident from the high mean square error values of annual models. On the whole, monthly models have better forecasting capability. By the nature of monthly models. monthly PDI values may be forecast accurately several months ahead also. CONCLUSIONS
The analysis of the Palmer’s drought index series data indicate that longterm oscillations and high persistence structure are important characteristics of drought indices and hence of droughts. The periodic behavior in drought index series is apparent in the conventional power spectra and also in the spectra computed by using high-resolution spectral methods. Resealed range-lag characteristics also support these conclusions. Nevertheless, the PDI series can be modeled by simple stochastic models which preserve the important statistical characteristics of the original data as demonstrated in this study. The simple models of PDI series developed can be used for generating synthetic PDI data and for forecasting PDI values which are used in agricultural planning and optimal operation of irrigation systems, respectively.
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