spei: Calculation of the Standardized...

Description Usage Arguments Details Value Note Author(s) References See Also Examples

View source: R/spei.R View source: R/spei.R

Description

Given a time series of the climatic water balance (precipitation minus potential evapotranspiration), gives a time series of the Standardized Precipitation-Evapotranspiration Index (SPEI).

Usage

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spei(data, scale, kernel = list(type = 'rectangular', shift = 0),
	distribution = 'log-Logistic', fit = 'ub-pwm', na.rm = FALSE,
	ref.start=NULL, ref.end=NULL, x=FALSE, params=NULL, ...)

spi(data, scale, kernel = list(type = 'rectangular', shift = 0),
	distribution = 'Gamma', fit = 'ub-pwm', na.rm = FALSE,
	ref.start=NULL, ref.end=NULL, x=FALSE, params=NULL, ...)

Arguments

data

a vector, matrix or data frame with time ordered values of precipitation (for the SPI) or of the climatic balance precipitation minus potential evapotranspiration (for the SPEI).

scale

an integer, representing the time scale at which the SPEI / SPI will be computed.

kernel

optional, a list defining the type of kernel used for computing the SPEI / SPI at scales higher than one. Defaults to unshifted rectangular kernel.

distribution

optional, name of the distribution function to be used for computing the SPEI / SPI (one of 'log-Logistic', 'Gamma' and 'PearsonIII'). Defaults to 'log-Logistic' for spei, and to 'Gamma' for spi.

fit

optional, name of the method used for computing the distribution function parameters (one of 'ub-pwm', 'pp-pwm' and 'max-lik'). Defaults to 'ub-pwm'.

na.rm

optional, a logical value indicating whether NA values should be stripped from the computations. Defaults to FALSE, i.e. no NA are allowed in the data.

ref.start

optional, starting point of the reference period used for computing the index. Defaults to NULL, indicating that the first value in data will be used as starting point.

ref.end

optional, ending point of the reference period used for computing the index. Defaults to NULL, indicating that the last value in data will be used as ending point.

x

optional, a logical value indicating wether the data used for fitting the model should be kept. Defaults to FALSE.

params

optional, an array of parameters for computing the spei. This option overrides computation of fitting parameters.

...

other possible parameters.

Details

The spei and spi functions allow computing the SPEI and the SPI indices. These are climatic proxies widely used for drought quantification and monitoring. Both functions are identical (in fact, spi is just a wrapper for spei), but they are kept separated for clarity. Basically, the functions standardize a variable following a log-Logistic (or Gamma, or PearsonIII) distribution function (i.e., they transform it to a standard Gaussian variate with zero mean and standard deviation of one).

Input data

The input variable is a time ordered series of precipitation values for spi, or a series of the climatic water balance (precipitation minus potential evapotranspiration) for spei. When used with the default options, it would yield values of both indices exactly as defined in the references given below.

The SPEI and the SPI were defined for monthly data. Since the PDFs of the data are not homogenous from month to month, the data is split into twelve series (one for each month) and independent PDFs are fit to each series. If data is a vector or a matrix it will be treated as a sequence of monthly values starting in January. If it is a (univariate or multivariate) time series then the function cycle will be used to determine the position of each observation within the year (month), allowing the data to start in a month other than January.

Time scales

An important advantage of the SPEI and the SPI is that they can be computed at different time scales. This way it is possible to incorporate the influence of the past values of the variable in the computation enabling the index to adapt to the memory of the system under study. The magnitude of this memory is controlled by parameter scale. For example, a value of six would imply that data from the current month and of the past five months will be used for computing the SPEI or SPI value for a given month. By default all past data will have the same weight in computing the index, as it was originally proposed in the references below. Other kernels, however, are available through parameter kernel. The parameter kernel is a list defining the shape of the kernel and a time shift. These parameters are then passed to the function kern.

Probability distributions

Following the original definitions spei uses a log-Logistic distribution by default, and spi uses a Gamma distribution. This behaviour can be modified, however, through parameter distribution.

Fitting methods

The default method for parameter fitting is based on unbiased Probability Weighted Moments ('ub-pwm'), but other methods can be used through parameter fit. A valid alternative is the plotting-position PWM ('pp-pwm') method. For the log-Logistic distribution, also the maximum likelihood method ('max-lik') is available.

User-provided parameters

An option exists to override parameter fitting and provide user default parameters. This is activated with the parameter params. The exact values provided tothis parameter depend on the distribution function being used. For log-Logistic and PearsonII it should be a three-dimensional array with dimensions (3,number of series in data,12), containing twelve parameter triads (xi, alpha, kappa) for each data series, one for each month. For Gamma, a three-dimensional array with dimensions (2,number of series in data,12), containing twelve parameter pairs (alpha, beta). It is a good idea to look at the coefficients slot of a previously fit spei spei object in order to understand the structure of the parameter array. The parameter distribution is still used under this option in order to know what distribution function should be used.

Reference period

The default behaviour of the functions is using all the values provided in data for parameter fitting. However, this can be modified with help of parameters ref.start and ref.end. These parameters allow defining a subset of values that will be used for parameter fitting, i.e. a reference period. The functions, however, will compute the values of the indices for the whole data set. For these options to work it is necessary that data will be a time series object. The starting and ending points of the reference period will then be defined as pairs of year and month values, e.g. c(1900,1).

Processing large datasets

It is possible to use the spei and spi functions for processing multivariate datasets at once. If a matrix or data frame is supplied as data, with time series of precipitation or precipitation minus potential evapotranspiration arranged in columns, the result would be a matrix (data frame) of spi or spei series. This makes processing large datasets extremely easy, since no loops need to be used.

Value

Functions spei and spi return an object of class spei. The generic functions print and summary can be used to obtain summaries of the results. The generic accessor functions coefficients and fitted extract useful features of the object.

An object of class spei is a list containing at least the following components:

call: the call to spei or spi used to generate the object.

fitted: time series with the values of the Standardized Precipitation-Evapotranspiration Index (SPEI) or the Standardized Precipitation Index (SPI). If data consists of several columns the function will treat each column as independent data, and the result will be a multivariate time series. The names of the columns in data will be used as column names in fitted.

coefficients: an array with the values of the coefficients of the distribution function fitted to the data. The first dimension of the array contains the three (or two) coefficients, the second dimension will typically consist of twelve values corresponding to each month, and the third dimension will be equal to the number of columns (series) in data. If a time scale greater than one has been used then the first elements will have NA value since the kernel can not be applied. The first element with valid data will be the one corresponding to the time scale chosen.

scale: the scale parameter used to generate the object.

kernel: the parameters and values of the kernel used to generate the object.

distribution: the distribution function used to generate the object.

fit: the fitting method used to generate the object.

na.action: the value of the na.action parameter used.

data: if requested, the input data used.

Note

Dependencies: the spei function depends on the library lmomco.

Author(s)

Santiago Beguer<c3><ad>a and Sergio M. Vicente-Serrano. Maintainer: Santiago Beguer<c3><ad>a.

References

S.M. Vicente-Serrano, S. Beguer<c3><ad>a, J.I. L<c3><b3>pez-Moreno. 2010. A Multi-scalar drought index sensitive to global warming: The Standardized Precipitation Evapotranspiration Index <e2><80><93> SPEI. Journal of Climate 23: 1696, DOI: 10.1175/2009JCLI2909.1.

Beguer<c3><ad>a S, Vicente-Serrano SM, Reig F, Latorre B. 2014. Standardized precipitation evapotranspiration index (SPEI) revisited: parameter fitting, evapotranspiration models, tools, datasets and drought monitoring. International Journal of Climatology 34(10): 3001-3023.

http://sac.csic.es/spei/

See Also

kern for different kernel functions available. thornthwaite, hargreaves and penman for ways of calculating potential evapotranspiration. summary.spei and print.spei for summaries of spei objects. plot.spei for plotting spei objects.

Examples

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# Load data
data(wichita)

# Compute potential evapotranspiration (PET) and climatic water balance (BAL)
wichita$PET <- thornthwaite(wichita$TMED, 37.6475)
wichita$BAL <- wichita$PRCP-wichita$PET

# Convert to a ts (time series) object for convenience
wichita <- ts(wichita[,-c(1,2)], end=c(2011,10), frequency=12)
plot(wichita)

# One and tvelwe-months SPEI
spei1 <- spei(wichita[,'BAL'], 1)
spei12 <- spei(wichita[,'BAL'], 12)
class(spei1)

# Extract information from spei object: summary, call function, fitted values, and coefficients
summary(spei1)
names(spei1)
spei1$call
spei1$fitted
spei1$coefficients

# Plot spei object
par(mfrow=c(2,1))
plot(spei1, main='Wichita, SPEI-1')
plot(spei12, main='Wichita, SPEI-12')

# One and tvelwe-months SPI
spi_1 <- spi(wichita[,'PRCP'], 1)
spi_12 <- spi(wichita[,'PRCP'], 12)

par(mfrow=c(2,1))
plot(spi_1, 'Wichita, SPI-1')
plot(spi_12, 'Wichita, SPI-12')

# Time series not starting in January
par(mfrow=c(1,1))
plot(spei(ts(wichita[,'BAL'], freq=12, start=c(1980,6)), 12))

# Using a particular reference period (1980-2000) for computing the parameters
plot(spei(ts(wichita[,'BAL'], freq=12, start=c(1980,6)), 12,
	ref.start=c(1980,1), ref.end=c(2000,1)))

# Using different kernels
spei24 <- spei(wichita[,'BAL'],24)
spei24_gau <- spei(wichita[,'BAL'], 24, kernel=list(type='gaussian', shift=0))
par(mfrow=c(2,1))
plot(spei24, main='SPEI-24 with rectangular kernel')
plot(spei24_gau, main='SPEI-24 with gaussian kernel')

# Computing several time series at a time
# Dataset balance contains time series of the climatic water balance at 12 locations
data(balance)
head(balance)
bal_spei12 <- spei(balance, 12)
plot(bal_spei12)

# Using custom (user provided) parameters
coe <- spei1$coefficients
dim(coe)
spei(wichita[,'BAL'], 1, params=coe)

Example output

Loading required package: lmomco
Loading required package: parallel
Loading required package: ggplot2
# Package SPEI (1.7) loaded [try SPEINews()].
[1] "spei"
Call:
spei(data = wichita[, "BAL"], scale = 1)

Coefficients:
	Series 1:
           xi     alpha       kappa
1   17.157119  9.992808 -0.18785891
2   21.290746 14.157734 -0.16806381
3   40.360700 23.047271 -0.15865070
4    2.496139 26.248322 -0.23076885
5    3.177373 35.876511 -0.22675938
6  -13.337079 42.629452  0.02487904
7  -93.748211 33.595773  0.03340238
8  -74.317275 38.218627 -0.14222759
9  -47.693803 30.827470 -0.37631795
10   5.102042 28.872061 -0.23503990
11  12.537650 16.103267 -0.24361084
12  22.149226 12.515723 -0.30940103

Fitted:
              Jan          Feb          Mar          Apr          May
1980  1.346769500 -0.026239224  1.107899056 -0.505369023 -0.373317709
1981 -0.796748602 -1.102047011 -0.257822701 -2.137032848  1.042500041
1982  1.228401555 -0.080094170 -0.229186296 -0.683825486  1.287999193
1983  1.210734854  0.337645651  1.172012249  1.189827631  0.450489785
1984 -0.831319847  0.058106444  2.218027241  1.021085144 -1.210439627
1985 -0.713625297  1.112618111 -0.757894854 -0.182279618 -1.121917546
1986 -1.576886016  0.283067019 -1.325463408 -0.396232192 -0.495605177
1987  0.945247374  1.630360662  0.967710881 -1.154019305  1.155798383
1988 -0.270545411 -0.843131641  0.446556896  1.194009625 -0.950066613
1989 -0.132790396 -0.534732551 -0.057058873 -1.722671756  0.436681892
1990  1.031991430  1.054035606  0.154678165 -0.700732664 -1.133075898
1991 -0.157582124 -1.758163958 -1.544216759 -0.098202497 -0.260668903
1992 -0.372611983 -1.161600730  0.542474712 -0.468149443  0.008603568
1993  0.638126554  1.237600638 -0.275392741  0.278627194  1.686657578
1994 -1.148045093 -0.714325334 -1.965250149  0.923871422 -1.623849571
1995 -0.403409123 -0.896299487  0.086132260  0.885909233  1.054560530
1996 -1.011028101 -1.133454072 -0.095310165 -0.300087884 -0.040249643
1997 -0.540702819  1.089685824 -1.570946104  1.087671092  0.561283497
1998  0.477163418 -1.056530706  1.233560038  0.776915571 -1.585002800
1999  0.892449667 -1.300874265 -0.261106820  1.554091737  1.140605612
2000  0.336419629  1.296022838  1.639267763 -0.481921474 -0.686973746
2001  1.025347707  2.042528470  0.093471608 -0.817668791 -0.492343928
2002  0.507474622 -0.201154791 -1.235294347  0.184361952  0.814516115
2003 -0.989997946  1.011027923  0.733225622  0.859346215 -0.302582498
2004  1.362688581  0.465065095  0.505270127  0.655347159 -0.339311085
2005  2.169826173  0.681841861 -0.426459567 -0.784543460 -1.174517375
2006 -1.930903001 -1.168294305 -0.261611957 -0.699868881  0.912837097
2007  0.738298319 -0.479832579  1.256913243  0.537228466 -0.068771026
2008 -0.556357747  0.684831704  0.476875679  0.017830374  2.102014053
2009 -1.057908753 -0.786503816 -0.346063935  2.268975978 -0.416300736
2010 -0.428125340  0.290443031 -0.482207902 -1.211849911  0.939351517
2011 -0.576343431  0.425161042 -1.198689908 -0.629803873 -0.742842303
              Jun          Jul          Aug          Sep          Oct
1980 -1.495912191 -1.850564599 -0.254028648 -1.557542920 -0.595880042
1981 -0.451497102 -1.043483757 -0.099625465  0.014735455  1.144941164
1982  1.492864728 -1.137557844 -0.959189890 -0.703993999 -1.118570575
1983  1.142089017  0.301865628 -1.318266542  0.091539448  0.391998156
1984 -1.085179304 -1.248547421 -1.396506961  0.107275388  0.319405895
1985  0.030155658  0.352392585  0.042736238  1.269597122  1.343855368
1986 -0.094122892 -0.048576778  1.195777989  0.566958429  0.711065516
1987 -0.253330064 -0.352525344  1.474838888  0.065958834 -0.598873504
1988 -1.276602771 -0.952698468 -1.250665845 -1.171503681 -0.639941186
1989  1.300657257  0.665465608  1.060427114  1.590970977 -1.399215871
1990 -1.446599591 -0.691123547 -0.618372590 -0.380683186 -0.983640699
1991 -1.458257193 -0.487798477  1.363449513  0.119858595 -0.823380503
1992  1.115061545  0.991611153 -0.050390988  0.393625623  0.272046907
1993 -0.131591055  1.349198420 -0.967080667  0.133009473 -0.087285560
1994 -1.058316456  1.573383183 -0.327636754 -0.536276139  0.494917742
1995  1.570668303  0.663179773  0.764583911  0.153045519 -1.317674797
1996 -1.129552547  0.612786035  0.897752475  0.852742080 -0.167328616
1997 -0.122191788  1.435419245  0.662545785  0.511302257  0.544408316
1998 -1.657763983  0.891518871 -0.893887504 -0.036070402  2.010091514
1999  1.095736702  0.054545776 -1.011478447  1.929699917 -1.421396807
2000  0.907658584  0.345223036 -2.054651937 -1.083635246  0.971493576
2001 -0.240978083 -1.286840500 -0.852867546  0.518959845 -0.754378140
2002  0.064181170 -0.580152425  0.602142929 -1.168984425  1.984849314
2003 -0.130521989 -1.397643881  1.069580984  1.060450986  0.503638638
2004  1.222086380  1.966848274 -0.076890678 -1.539012479  0.457458461
2005  0.702603066  0.704558853  2.220188940 -1.477911600 -0.124313908
2006  0.415989811 -0.685802528  0.844286388 -0.518498350 -0.733524494
2007  1.365826462  0.601419339 -0.113392351 -1.300372413  0.765204354
2008  0.801000801  0.430842181  0.073609759  2.102105501  0.863638707
2009 -0.440741085  0.497103936  0.542144594  1.150154924  1.029604261
2010 -0.274184906 -0.320166142  0.113111520  0.245722265 -1.452876725
2011 -0.549278812 -1.460900906 -0.357119408 -0.486948915 -1.009854898
              Nov          Dec
1980 -0.746120768  1.025436150
1981  1.221130499 -0.901377820
1982 -0.207862785  0.586555881
1983  0.985484125  0.302731949
1984  0.306811081  2.052979856
1985  0.723015386 -0.355678901
1986 -0.180731489  0.311393649
1987  0.219129837  1.156730424
1988 -0.519161418 -0.853122703
1989 -1.321112267 -0.621507058
1990  0.465619165 -0.115799189
1991  1.183189058  0.930598795
1992  2.043472623  0.318346838
1993 -0.268958681 -1.177616585
1994  1.302139618  0.048724814
1995 -1.052653904 -0.273495143
1996  1.654350308 -1.460805032
1997  0.383460176  1.370789642
1998  1.254712548  0.132081403
1999 -0.310796506  1.868129509
2000  0.197788080 -0.761387181
2001 -1.016523245 -1.775819884
2002 -0.953849811  0.317926614
2003 -1.188879716 -0.157359834
2004  1.014415856 -1.176058457
2005 -1.545667838 -0.400246679
2006 -0.977575658  0.633560000
2007 -1.187010293  1.382166928
2008  0.202414951  0.430120620
2009 -1.073134839 -0.700600560
2010  0.060880781 -1.287666788
2011                          
[1] "call"         "fitted"       "coefficients" "scale"        "kernel"      
[6] "distribution" "fit"          "na.action"   
spei(data = wichita[, "BAL"], scale = 1)
              Jan          Feb          Mar          Apr          May
1980  1.346769500 -0.026239224  1.107899056 -0.505369023 -0.373317709
1981 -0.796748602 -1.102047011 -0.257822701 -2.137032848  1.042500041
1982  1.228401555 -0.080094170 -0.229186296 -0.683825486  1.287999193
1983  1.210734854  0.337645651  1.172012249  1.189827631  0.450489785
1984 -0.831319847  0.058106444  2.218027241  1.021085144 -1.210439627
1985 -0.713625297  1.112618111 -0.757894854 -0.182279618 -1.121917546
1986 -1.576886016  0.283067019 -1.325463408 -0.396232192 -0.495605177
1987  0.945247374  1.630360662  0.967710881 -1.154019305  1.155798383
1988 -0.270545411 -0.843131641  0.446556896  1.194009625 -0.950066613
1989 -0.132790396 -0.534732551 -0.057058873 -1.722671756  0.436681892
1990  1.031991430  1.054035606  0.154678165 -0.700732664 -1.133075898
1991 -0.157582124 -1.758163958 -1.544216759 -0.098202497 -0.260668903
1992 -0.372611983 -1.161600730  0.542474712 -0.468149443  0.008603568
1993  0.638126554  1.237600638 -0.275392741  0.278627194  1.686657578
1994 -1.148045093 -0.714325334 -1.965250149  0.923871422 -1.623849571
1995 -0.403409123 -0.896299487  0.086132260  0.885909233  1.054560530
1996 -1.011028101 -1.133454072 -0.095310165 -0.300087884 -0.040249643
1997 -0.540702819  1.089685824 -1.570946104  1.087671092  0.561283497
1998  0.477163418 -1.056530706  1.233560038  0.776915571 -1.585002800
1999  0.892449667 -1.300874265 -0.261106820  1.554091737  1.140605612
2000  0.336419629  1.296022838  1.639267763 -0.481921474 -0.686973746
2001  1.025347707  2.042528470  0.093471608 -0.817668791 -0.492343928
2002  0.507474622 -0.201154791 -1.235294347  0.184361952  0.814516115
2003 -0.989997946  1.011027923  0.733225622  0.859346215 -0.302582498
2004  1.362688581  0.465065095  0.505270127  0.655347159 -0.339311085
2005  2.169826173  0.681841861 -0.426459567 -0.784543460 -1.174517375
2006 -1.930903001 -1.168294305 -0.261611957 -0.699868881  0.912837097
2007  0.738298319 -0.479832579  1.256913243  0.537228466 -0.068771026
2008 -0.556357747  0.684831704  0.476875679  0.017830374  2.102014053
2009 -1.057908753 -0.786503816 -0.346063935  2.268975978 -0.416300736
2010 -0.428125340  0.290443031 -0.482207902 -1.211849911  0.939351517
2011 -0.576343431  0.425161042 -1.198689908 -0.629803873 -0.742842303
              Jun          Jul          Aug          Sep          Oct
1980 -1.495912191 -1.850564599 -0.254028648 -1.557542920 -0.595880042
1981 -0.451497102 -1.043483757 -0.099625465  0.014735455  1.144941164
1982  1.492864728 -1.137557844 -0.959189890 -0.703993999 -1.118570575
1983  1.142089017  0.301865628 -1.318266542  0.091539448  0.391998156
1984 -1.085179304 -1.248547421 -1.396506961  0.107275388  0.319405895
1985  0.030155658  0.352392585  0.042736238  1.269597122  1.343855368
1986 -0.094122892 -0.048576778  1.195777989  0.566958429  0.711065516
1987 -0.253330064 -0.352525344  1.474838888  0.065958834 -0.598873504
1988 -1.276602771 -0.952698468 -1.250665845 -1.171503681 -0.639941186
1989  1.300657257  0.665465608  1.060427114  1.590970977 -1.399215871
1990 -1.446599591 -0.691123547 -0.618372590 -0.380683186 -0.983640699
1991 -1.458257193 -0.487798477  1.363449513  0.119858595 -0.823380503
1992  1.115061545  0.991611153 -0.050390988  0.393625623  0.272046907
1993 -0.131591055  1.349198420 -0.967080667  0.133009473 -0.087285560
1994 -1.058316456  1.573383183 -0.327636754 -0.536276139  0.494917742
1995  1.570668303  0.663179773  0.764583911  0.153045519 -1.317674797
1996 -1.129552547  0.612786035  0.897752475  0.852742080 -0.167328616
1997 -0.122191788  1.435419245  0.662545785  0.511302257  0.544408316
1998 -1.657763983  0.891518871 -0.893887504 -0.036070402  2.010091514
1999  1.095736702  0.054545776 -1.011478447  1.929699917 -1.421396807
2000  0.907658584  0.345223036 -2.054651937 -1.083635246  0.971493576
2001 -0.240978083 -1.286840500 -0.852867546  0.518959845 -0.754378140
2002  0.064181170 -0.580152425  0.602142929 -1.168984425  1.984849314
2003 -0.130521989 -1.397643881  1.069580984  1.060450986  0.503638638
2004  1.222086380  1.966848274 -0.076890678 -1.539012479  0.457458461
2005  0.702603066  0.704558853  2.220188940 -1.477911600 -0.124313908
2006  0.415989811 -0.685802528  0.844286388 -0.518498350 -0.733524494
2007  1.365826462  0.601419339 -0.113392351 -1.300372413  0.765204354
2008  0.801000801  0.430842181  0.073609759  2.102105501  0.863638707
2009 -0.440741085  0.497103936  0.542144594  1.150154924  1.029604261
2010 -0.274184906 -0.320166142  0.113111520  0.245722265 -1.452876725
2011 -0.549278812 -1.460900906 -0.357119408 -0.486948915 -1.009854898
              Nov          Dec
1980 -0.746120768  1.025436150
1981  1.221130499 -0.901377820
1982 -0.207862785  0.586555881
1983  0.985484125  0.302731949
1984  0.306811081  2.052979856
1985  0.723015386 -0.355678901
1986 -0.180731489  0.311393649
1987  0.219129837  1.156730424
1988 -0.519161418 -0.853122703
1989 -1.321112267 -0.621507058
1990  0.465619165 -0.115799189
1991  1.183189058  0.930598795
1992  2.043472623  0.318346838
1993 -0.268958681 -1.177616585
1994  1.302139618  0.048724814
1995 -1.052653904 -0.273495143
1996  1.654350308 -1.460805032
1997  0.383460176  1.370789642
1998  1.254712548  0.132081403
1999 -0.310796506  1.868129509
2000  0.197788080 -0.761387181
2001 -1.016523245 -1.775819884
2002 -0.953849811  0.317926614
2003 -1.188879716 -0.157359834
2004  1.014415856 -1.176058457
2005 -1.545667838 -0.400246679
2006 -0.977575658  0.633560000
2007 -1.187010293  1.382166928
2008  0.202414951  0.430120620
2009 -1.073134839 -0.700600560
2010  0.060880781 -1.287666788
2011                          
, , 1

       
par       Series 1
  xi    17.1571193
  alpha  9.9928078
  kappa -0.1878589

, , 2

       
par       Series 1
  xi    21.2907465
  alpha 14.1577338
  kappa -0.1680638

, , 3

       
par       Series 1
  xi    40.3607005
  alpha 23.0472715
  kappa -0.1586507

, , 4

       
par       Series 1
  xi     2.4961392
  alpha 26.2483224
  kappa -0.2307689

, , 5

       
par       Series 1
  xi     3.1773725
  alpha 35.8765107
  kappa -0.2267594

, , 6

       
par         Series 1
  xi    -13.33707886
  alpha  42.62945170
  kappa   0.02487904

, , 7

       
par         Series 1
  xi    -93.74821061
  alpha  33.59577275
  kappa   0.03340238

, , 8

       
par        Series 1
  xi    -74.3172747
  alpha  38.2186268
  kappa  -0.1422276

, , 9

       
par       Series 1
  xi    -47.693803
  alpha  30.827470
  kappa  -0.376318

, , 10

       
par       Series 1
  xi     5.1020424
  alpha 28.8720610
  kappa -0.2350399

, , 11

       
par       Series 1
  xi    12.5376503
  alpha 16.1032665
  kappa -0.2436108

, , 12

       
par      Series 1
  xi    22.149226
  alpha 12.515723
  kappa -0.309401

   indore kimberley albuquerque valencia  viena abashiri  tampa sao_paulo
1  -25.73    -84.70       11.97   -11.78 127.34    26.10  90.04    104.04
2  -52.71    -80.69        2.12    -8.08  26.36    19.40  79.78     73.40
3 -119.63     16.37      -20.23    24.74 119.28    17.70 146.62     76.11
4 -199.76    -21.87      -21.63   -16.15  49.37    11.40  28.46     43.76
5 -249.20    -46.18      -50.39   -30.37  18.63    30.05 -40.48     93.72
6 -210.47     -5.66     -102.92   -47.62   5.20    54.85  77.86     -1.62
   lahore punta_arenas helsinki
1    3.61       -33.61    56.00
2   -9.28        -6.71    82.00
3  -71.45       -50.41    45.00
4  -98.52       -15.00    44.08
5 -230.38        -8.47     3.08
6 -354.94        -2.27   -19.67
[1]  3  1 12
              Jan          Feb          Mar          Apr          May
1980  1.346769500 -0.026239224  1.107899056 -0.505369023 -0.373317709
1981 -0.796748602 -1.102047011 -0.257822701 -2.137032848  1.042500041
1982  1.228401555 -0.080094170 -0.229186296 -0.683825486  1.287999193
1983  1.210734854  0.337645651  1.172012249  1.189827631  0.450489785
1984 -0.831319847  0.058106444  2.218027241  1.021085144 -1.210439627
1985 -0.713625297  1.112618111 -0.757894854 -0.182279618 -1.121917546
1986 -1.576886016  0.283067019 -1.325463408 -0.396232192 -0.495605177
1987  0.945247374  1.630360662  0.967710881 -1.154019305  1.155798383
1988 -0.270545411 -0.843131641  0.446556896  1.194009625 -0.950066613
1989 -0.132790396 -0.534732551 -0.057058873 -1.722671756  0.436681892
1990  1.031991430  1.054035606  0.154678165 -0.700732664 -1.133075898
1991 -0.157582124 -1.758163958 -1.544216759 -0.098202497 -0.260668903
1992 -0.372611983 -1.161600730  0.542474712 -0.468149443  0.008603568
1993  0.638126554  1.237600638 -0.275392741  0.278627194  1.686657578
1994 -1.148045093 -0.714325334 -1.965250149  0.923871422 -1.623849571
1995 -0.403409123 -0.896299487  0.086132260  0.885909233  1.054560530
1996 -1.011028101 -1.133454072 -0.095310165 -0.300087884 -0.040249643
1997 -0.540702819  1.089685824 -1.570946104  1.087671092  0.561283497
1998  0.477163418 -1.056530706  1.233560038  0.776915571 -1.585002800
1999  0.892449667 -1.300874265 -0.261106820  1.554091737  1.140605612
2000  0.336419629  1.296022838  1.639267763 -0.481921474 -0.686973746
2001  1.025347707  2.042528470  0.093471608 -0.817668791 -0.492343928
2002  0.507474622 -0.201154791 -1.235294347  0.184361952  0.814516115
2003 -0.989997946  1.011027923  0.733225622  0.859346215 -0.302582498
2004  1.362688581  0.465065095  0.505270127  0.655347159 -0.339311085
2005  2.169826173  0.681841861 -0.426459567 -0.784543460 -1.174517375
2006 -1.930903001 -1.168294305 -0.261611957 -0.699868881  0.912837097
2007  0.738298319 -0.479832579  1.256913243  0.537228466 -0.068771026
2008 -0.556357747  0.684831704  0.476875679  0.017830374  2.102014053
2009 -1.057908753 -0.786503816 -0.346063935  2.268975978 -0.416300736
2010 -0.428125340  0.290443031 -0.482207902 -1.211849911  0.939351517
2011 -0.576343431  0.425161042 -1.198689908 -0.629803873 -0.742842303
              Jun          Jul          Aug          Sep          Oct
1980 -1.495912191 -1.850564599 -0.254028648 -1.557542920 -0.595880042
1981 -0.451497102 -1.043483757 -0.099625465  0.014735455  1.144941164
1982  1.492864728 -1.137557844 -0.959189890 -0.703993999 -1.118570575
1983  1.142089017  0.301865628 -1.318266542  0.091539448  0.391998156
1984 -1.085179304 -1.248547421 -1.396506961  0.107275388  0.319405895
1985  0.030155658  0.352392585  0.042736238  1.269597122  1.343855368
1986 -0.094122892 -0.048576778  1.195777989  0.566958429  0.711065516
1987 -0.253330064 -0.352525344  1.474838888  0.065958834 -0.598873504
1988 -1.276602771 -0.952698468 -1.250665845 -1.171503681 -0.639941186
1989  1.300657257  0.665465608  1.060427114  1.590970977 -1.399215871
1990 -1.446599591 -0.691123547 -0.618372590 -0.380683186 -0.983640699
1991 -1.458257193 -0.487798477  1.363449513  0.119858595 -0.823380503
1992  1.115061545  0.991611153 -0.050390988  0.393625623  0.272046907
1993 -0.131591055  1.349198420 -0.967080667  0.133009473 -0.087285560
1994 -1.058316456  1.573383183 -0.327636754 -0.536276139  0.494917742
1995  1.570668303  0.663179773  0.764583911  0.153045519 -1.317674797
1996 -1.129552547  0.612786035  0.897752475  0.852742080 -0.167328616
1997 -0.122191788  1.435419245  0.662545785  0.511302257  0.544408316
1998 -1.657763983  0.891518871 -0.893887504 -0.036070402  2.010091514
1999  1.095736702  0.054545776 -1.011478447  1.929699917 -1.421396807
2000  0.907658584  0.345223036 -2.054651937 -1.083635246  0.971493576
2001 -0.240978083 -1.286840500 -0.852867546  0.518959845 -0.754378140
2002  0.064181170 -0.580152425  0.602142929 -1.168984425  1.984849314
2003 -0.130521989 -1.397643881  1.069580984  1.060450986  0.503638638
2004  1.222086380  1.966848274 -0.076890678 -1.539012479  0.457458461
2005  0.702603066  0.704558853  2.220188940 -1.477911600 -0.124313908
2006  0.415989811 -0.685802528  0.844286388 -0.518498350 -0.733524494
2007  1.365826462  0.601419339 -0.113392351 -1.300372413  0.765204354
2008  0.801000801  0.430842181  0.073609759  2.102105501  0.863638707
2009 -0.440741085  0.497103936  0.542144594  1.150154924  1.029604261
2010 -0.274184906 -0.320166142  0.113111520  0.245722265 -1.452876725
2011 -0.549278812 -1.460900906 -0.357119408 -0.486948915 -1.009854898
              Nov          Dec
1980 -0.746120768  1.025436150
1981  1.221130499 -0.901377820
1982 -0.207862785  0.586555881
1983  0.985484125  0.302731949
1984  0.306811081  2.052979856
1985  0.723015386 -0.355678901
1986 -0.180731489  0.311393649
1987  0.219129837  1.156730424
1988 -0.519161418 -0.853122703
1989 -1.321112267 -0.621507058
1990  0.465619165 -0.115799189
1991  1.183189058  0.930598795
1992  2.043472623  0.318346838
1993 -0.268958681 -1.177616585
1994  1.302139618  0.048724814
1995 -1.052653904 -0.273495143
1996  1.654350308 -1.460805032
1997  0.383460176  1.370789642
1998  1.254712548  0.132081403
1999 -0.310796506  1.868129509
2000  0.197788080 -0.761387181
2001 -1.016523245 -1.775819884
2002 -0.953849811  0.317926614
2003 -1.188879716 -0.157359834
2004  1.014415856 -1.176058457
2005 -1.545667838 -0.400246679
2006 -0.977575658  0.633560000
2007 -1.187010293  1.382166928
2008  0.202414951  0.430120620
2009 -1.073134839 -0.700600560
2010  0.060880781 -1.287666788
2011                          

SPEI documentation built on May 2, 2019, 11:05 a.m.