Desertification
Luigi Caputi, 02/02/2023
| DESERTIFICATION |
BASIC STATISTICSThe supplied dataset (“Desertification”) consists of 1,1600 observations from 29 localities. Data are organized by monthly values of Wind, Temperature, Rainfall, and Destert storm occurrences per year (from 2003 to 2012). Herein, I will first reorder the data to have them more handle, and then make four different data frames, one for each of the four variables.
For each of the variable (Wind, Sandstorms, Rainfall and temperature) a barplot (with stabdard deviations) and a boxplot are produced. Moreover, by stations trends are visualized.
WINDMean Annual Wind per station -
Boxplots showing the Median, the range and the outliers of Wind per station -
- Wind annual trends with 95% C.I. per station -
SANDSTORMS
Mean Annual Sandstorm per station -
Boxplots showing the Median, the range and the outliers of Sandstorm per station -
Sandstorms annual trends with 95% C.I. per station -
TEMPERATURE
Mean Annual Temperature Values per station -
Boxplots showing the Median, the range and the outliers of Temperature per station -
Temperature annual trends with 95% C.I. per station -
RAINFALL
Mean Annual Temperature Values per station -
Boxplots showing the Median, the range and the outliers of RAINFALL per station -
Rainfall annual trends with 95% C.I. per station -
FORECASTING FUTURE SANDSTORMSModelling future sand storms using a generalized mixed model -
A Generalized Linear Model (GLM) was chosen to model the addittive role of Temperature, Wind and Rainfall in determining and predicting Sandstorms. GLM is a generalization of ordinary linear regression that allows for response variables that have error distribution models other than a normal distribution like Gaussian distribution.
## # A tibble: 29 × 9
## station Meansandstorms sd.x MeanWind sd.y Meantemperature sd.x.x
## <chr> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
## 1 Abha 0.0818 0.335 6.27 1.22 28.3 3.76
## 2 Al-Ahsa 1.65 2.84 6.31 1.70 39.9 7.69
## 3 Al-Baha 0.118 0.631 6.19 1.66 32.9 6.45
## 4 Al-Jouf 0.7 1.40 7.55 1.24 34.6 7.99
## 5 Arar 0.327 0.743 8.05 1.41 35.4 8.87
## 6 Bisha 0.264 0.725 4.3 0.934 37.1 4.38
## 7 Dhahran 0.218 0.612 8.15 1.25 38.6 7.91
## 8 Dmmam 0.636 1.22 8.48 1.29 38.6 7.87
## 9 Guriat 0.373 0.876 8.43 2.20 33.6 8.08
## 10 Hafr Elbatten 1.06 1.88 7.67 1.61 37.5 8.25
## # ℹ 19 more rows
## # ℹ 2 more variables: Meanrainfall <dbl>, sd.y.y <dbl>##
## Call:
## glm(formula = Meansandstorms ~ Meantemperature + MeanWind + Meanrainfall,
## family = "poisson", data = Means)
##
## Deviance Residuals:
## Min 1Q Median 3Q Max
## -0.7797 -0.3492 -0.1001 0.1803 1.1082
##
## Coefficients:
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) -8.80887 5.99329 -1.470 0.142
## Meantemperature 0.16489 0.13217 1.248 0.212
## MeanWind 0.25520 0.24219 1.054 0.292
## Meanrainfall 0.02699 0.24476 0.110 0.912
##
## (Dispersion parameter for poisson family taken to be 1)
##
## Null deviance: 8.6125 on 27 degrees of freedom
## Residual deviance: 6.1563 on 24 degrees of freedom
## (1 osservazione eliminata a causa di un valore mancante)
## AIC: Inf
##
## Number of Fisher Scoring iterations: 5To interpret the above results, keep in mind that the coefficient estimate in the output indicate the average change in the log odds of the response variable associated with a one unit increase in each predictor variable.
Now, let’s use the model to make a prediction
## 1 2 3 4 5 6 7
## 0.09733685 0.60025727 0.18987974 0.32544857 0.42013593 0.22021218 0.77628705
## 8 9 10 11 12 13 14
## 0.86454739 0.35550305 0.57210792 0.32319406 0.54070079 0.29301858 0.12853908
## 15 16 17 18 19 20 21
## 0.42714928 0.28626118 0.20534505 0.46640977 0.46036550 0.42740638 0.39790099
## 23 24 25 26 27 28 29
## 0.47377305 0.64299266 0.18614660 0.26771322 0.24547088 0.34611663 0.63250762Results are not too clear, indicating that this is maybe not the best way to predict the response variable. The results are not clear. However, it is possible to affirm the odds of sandstorms have some positively affected by increases in wind and temperature, and negatively affected by increase in rainfall
- Modelling future sand storms using an ARIMA model -
The Autoregressive Integrated Moving Average (ARIMA) model is used to to forecast upcoming series points. ARIMA models are applied in some cases where data show evidence of non-stationarity data.
To use the ARIMA model, we first have to check for the non-stationarity of data. We will do this only for a single station (Al-Ahsa), but results will be valid for all he stations.
As first step, let’s transform the data in a format accepted by R as time series, then let’s check for non-stationarity.
##
## #######################
## # KPSS Unit Root Test #
## #######################
##
## Test is of type: mu with 4 lags.
##
## Value of test-statistic is: 0.6065
##
## Critical value for a significance level of:
## 10pct 5pct 2.5pct 1pct
## critical values 0.347 0.463 0.574 0.739##
## ###############################################
## # Augmented Dickey-Fuller Test Unit Root Test #
## ###############################################
##
## Test regression none
##
##
## Call:
## lm(formula = z.diff ~ z.lag.1 - 1 + z.diff.lag)
##
## Residuals:
## Min 1Q Median 3Q Max
## -5.8656 -0.0533 0.0000 1.4332 13.3945
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## z.lag.1 -0.464824 0.094194 -4.935 3.03e-06 ***
## z.diff.lag -0.007108 0.097479 -0.073 0.942
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.83 on 105 degrees of freedom
## Multiple R-squared: 0.2347, Adjusted R-squared: 0.2202
## F-statistic: 16.1 on 2 and 105 DF, p-value: 7.939e-07
##
##
## Value of test-statistic is: -4.9348
##
## Critical values for test statistics:
## 1pct 5pct 10pct
## tau1 -2.58 -1.95 -1.62Remember that the two test have different null hypothesis. KPSS Test: Data are stationary D-F Test: Data are non-stationary Data are thus clearly non-stationary. Since all data have been sampled in the same way, we assume that all data are non-stationary.
Now, we will perform ARIMA analysis for the five stations with the highest sandstorms occurrences. At the end of each per-station analysis, a data table and a plot with the predicted future sandstorm for the next 24 months are given. NOTE THAT: only Al-Ahsa predictions makes sense. This is due to the relatively short time span of the analysis (9 years) and to the relatively few datapoints in all bu Al-Ahsa stations
- Al-Ahsa -##
## ARIMA(2,1,2)(1,0,1)[12] with drift : Inf
## ARIMA(0,1,0) with drift : 561.1516
## ARIMA(1,1,0)(1,0,0)[12] with drift : 554.0325
## ARIMA(0,1,1)(0,0,1)[12] with drift : 534.7235
## ARIMA(0,1,0) : 559.1516
## ARIMA(0,1,1) with drift : 535.7758
## ARIMA(0,1,1)(1,0,1)[12] with drift : 536.7672
## ARIMA(0,1,1)(0,0,2)[12] with drift : 536.7203
## ARIMA(0,1,1)(1,0,0)[12] with drift : 534.7544
## ARIMA(0,1,1)(1,0,2)[12] with drift : Inf
## ARIMA(0,1,0)(0,0,1)[12] with drift : 562.533
## ARIMA(1,1,1)(0,0,1)[12] with drift : Inf
## ARIMA(0,1,2)(0,0,1)[12] with drift : Inf
## ARIMA(1,1,0)(0,0,1)[12] with drift : 555.0102
## ARIMA(1,1,2)(0,0,1)[12] with drift : Inf
## ARIMA(0,1,1)(0,0,1)[12] : 532.9767
## ARIMA(0,1,1) : 533.9156
## ARIMA(0,1,1)(1,0,1)[12] : 534.8942
## ARIMA(0,1,1)(0,0,2)[12] : 534.9761
## ARIMA(0,1,1)(1,0,0)[12] : 532.9619
## ARIMA(0,1,1)(2,0,0)[12] : 534.9602
## ARIMA(0,1,1)(2,0,1)[12] : Inf
## ARIMA(0,1,0)(1,0,0)[12] : 560.2934
## ARIMA(1,1,1)(1,0,0)[12] : 525.8295
## ARIMA(1,1,1) : 525.1277
## ARIMA(1,1,1)(0,0,1)[12] : 526.0592
## ARIMA(1,1,1)(1,0,1)[12] : 526.7145
## ARIMA(1,1,0) : 554.841
## ARIMA(2,1,1) : 524.7834
## ARIMA(2,1,1)(1,0,0)[12] : 526.6345
## ARIMA(2,1,1)(0,0,1)[12] : 526.6719
## ARIMA(2,1,1)(1,0,1)[12] : 527.6806
## ARIMA(2,1,0) : 552.7808
## ARIMA(3,1,1) : 522.1566
## ARIMA(3,1,1)(1,0,0)[12] : 523.6693
## ARIMA(3,1,1)(0,0,1)[12] : 523.7121
## ARIMA(3,1,1)(1,0,1)[12] : 525.4025
## ARIMA(3,1,0) : 530.9177
## ARIMA(4,1,1) : 520.9456
## ARIMA(4,1,1)(1,0,0)[12] : 522.7825
## ARIMA(4,1,1)(0,0,1)[12] : 522.788
## ARIMA(4,1,1)(1,0,1)[12] : 524.7285
## ARIMA(4,1,0) : 532.3472
## ARIMA(5,1,1) : 520.9891
## ARIMA(4,1,2) : Inf
## ARIMA(3,1,2) : Inf
## ARIMA(5,1,0) : 531.8403
## ARIMA(5,1,2) : Inf
## ARIMA(4,1,1) with drift : Inf
##
## Best model: ARIMA(4,1,1)## Series: AA_train[, 5]
## ARIMA(4,1,1)
##
## Coefficients:
## ar1 ar2 ar3 ar4 ma1
## 0.3667 -0.0702 -0.2438 0.1908 -0.9362
## s.e. 0.1109 0.1000 0.0996 0.1051 0.0556
##
## sigma^2 = 6.711: log likelihood = -254.47
## AIC=520.95 AICc=521.78 BIC=537.04## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## Feb 2012 0.8582904 -2.461645 4.178226 -4.219112 5.935693
## Mar 2012 0.7360849 -2.878429 4.350599 -4.791837 6.264007
## Apr 2012 0.4574103 -3.191901 4.106722 -5.123730 6.038550
## May 2012 0.5891722 -3.096108 4.274452 -5.046978 6.225322
## Jun 2012 0.6598181 -3.035576 4.355213 -4.991800 6.311436
## Jul 2012 0.7210951 -3.007740 4.449931 -4.981667 6.423857
## Aug 2012 0.6533017 -3.122951 4.429555 -5.121979 6.428582
## Sep 2012 0.6320557 -3.151255 4.415367 -5.154019 6.418131
## Oct 2012 0.6275668 -3.160280 4.415413 -5.165445 6.420578
## Nov 2012 0.6556348 -3.138208 4.449478 -5.146548 6.457817
## Dec 2012 0.6584866 -3.150078 4.467051 -5.166211 6.483184
## Jan 2013 0.6546012 -3.167255 4.476457 -5.190423 6.499625
## Feb 2013 0.6452761 -3.187454 4.478006 -5.216379 6.506931
## Mar 2013 0.6467901 -3.194375 4.487955 -5.227764 6.521344
## Apr 2013 0.6494917 -3.201226 4.500210 -5.239673 6.538657
## May 2013 0.6519083 -3.209255 4.513072 -5.253232 6.557048
## Jun 2013 0.6504561 -3.221577 4.522489 -5.271307 6.572219
## Jul 2013 0.6493841 -3.232809 4.531577 -5.287918 6.586686
## Aug 2013 0.6490193 -3.243040 4.541079 -5.303372 6.601411
## Sep 2013 0.6497760 -3.252141 4.551693 -5.317691 6.617243
## Oct 2013 0.6500634 -3.261954 4.562080 -5.332850 6.632977
## Nov 2013 0.6500000 -3.272128 4.572128 -5.348377 6.648377
## Dec 2013 0.6497025 -3.282450 4.581855 -5.364006 6.663411
## Jan 2014 0.6496721 -3.292393 4.591738 -5.379197 6.678541
- Hafr Elbatten -##
## ARIMA(2,1,2)(1,0,1)[12] with drift : 425.1989
## ARIMA(0,1,0) with drift : 462.1922
## ARIMA(1,1,0)(1,0,0)[12] with drift : 441.1444
## ARIMA(0,1,1)(0,0,1)[12] with drift : 420.0431
## ARIMA(0,1,0) : 460.1922
## ARIMA(0,1,1) with drift : 419.7021
## ARIMA(0,1,1)(1,0,0)[12] with drift : 419.9446
## ARIMA(0,1,1)(1,0,1)[12] with drift : 421.718
## ARIMA(1,1,1) with drift : 420.7987
## ARIMA(0,1,2) with drift : 420.739
## ARIMA(1,1,0) with drift : 443.6226
## ARIMA(1,1,2) with drift : 420.6268
## ARIMA(0,1,1) : 417.7059
## ARIMA(0,1,1)(1,0,0)[12] : 417.9546
## ARIMA(0,1,1)(0,0,1)[12] : 418.0552
## ARIMA(0,1,1)(1,0,1)[12] : 419.7239
## ARIMA(1,1,1) : 418.8088
## ARIMA(0,1,2) : 418.7486
## ARIMA(1,1,0) : 441.6226
## ARIMA(1,1,2) : 418.6339
##
## Best model: ARIMA(0,1,1)## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## Feb 2012 0.06869195 -2.036534 2.173918 -3.150973 3.288357
## Mar 2012 0.06869195 -2.079517 2.216901 -3.216710 3.354094
## Apr 2012 0.06869195 -2.121657 2.259041 -3.281158 3.418542
## May 2012 0.06869195 -2.163001 2.300385 -3.344388 3.481772
## Jun 2012 0.06869195 -2.203593 2.340977 -3.406469 3.543853
## Jul 2012 0.06869195 -2.243473 2.380857 -3.467459 3.604843
## Aug 2012 0.06869195 -2.282676 2.420060 -3.527415 3.664799
## Sep 2012 0.06869195 -2.321237 2.458621 -3.586388 3.723772
## Oct 2012 0.06869195 -2.359185 2.496569 -3.644425 3.781809
## Nov 2012 0.06869195 -2.396549 2.533932 -3.701568 3.838952
## Dec 2012 0.06869195 -2.433355 2.570738 -3.757858 3.895242
## Jan 2013 0.06869195 -2.469627 2.607011 -3.813332 3.950716
## Feb 2013 0.06869195 -2.505388 2.642772 -3.868024 4.005408
## Mar 2013 0.06869195 -2.540659 2.678043 -3.921967 4.059351
## Apr 2013 0.06869195 -2.575460 2.712844 -3.975190 4.112574
## May 2013 0.06869195 -2.609809 2.747193 -4.027722 4.165106
## Jun 2013 0.06869195 -2.643723 2.781107 -4.079588 4.216972
## Jul 2013 0.06869195 -2.677218 2.814601 -4.130814 4.268198
## Aug 2013 0.06869195 -2.710309 2.847693 -4.181423 4.318807
## Sep 2013 0.06869195 -2.743011 2.880394 -4.231436 4.368820
## Oct 2013 0.06869195 -2.775336 2.912720 -4.280874 4.418258
## Nov 2013 0.06869195 -2.807299 2.944683 -4.329756 4.467140
## Dec 2013 0.06869195 -2.838910 2.976294 -4.378101 4.515485
## Jan 2014 0.06869195 -2.870181 3.007565 -4.425927 4.563310
- Qaisumah -##
## ARIMA(2,1,2)(1,0,1)[12] with drift : Inf
## ARIMA(0,1,0) with drift : 501.982
## ARIMA(1,1,0)(1,0,0)[12] with drift : 476.4037
## ARIMA(0,1,1)(0,0,1)[12] with drift : 439.867
## ARIMA(0,1,0) : 499.982
## ARIMA(0,1,1) with drift : 437.888
## ARIMA(0,1,1)(1,0,0)[12] with drift : 439.8678
## ARIMA(0,1,1)(1,0,1)[12] with drift : 441.6647
## ARIMA(1,1,1) with drift : 439.5038
## ARIMA(0,1,2) with drift : 439.3593
## ARIMA(1,1,0) with drift : 476.4333
## ARIMA(1,1,2) with drift : 438.2166
## ARIMA(0,1,1) : 435.8901
## ARIMA(0,1,1)(1,0,0)[12] : 437.8704
## ARIMA(0,1,1)(0,0,1)[12] : 437.8696
## ARIMA(0,1,1)(1,0,1)[12] : 439.6726
## ARIMA(1,1,1) : 437.504
## ARIMA(0,1,2) : 437.3593
## ARIMA(1,1,0) : 474.4333
## ARIMA(1,1,2) : 436.2948
##
## Best model: ARIMA(0,1,1)## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## Feb 2012 0.310662 -1.974959 2.596283 -3.184894 3.806218
## Mar 2012 0.310662 -1.993411 2.614735 -3.213114 3.834438
## Apr 2012 0.310662 -2.011716 2.633040 -3.241109 3.862433
## May 2012 0.310662 -2.029878 2.651202 -3.268885 3.890209
## Jun 2012 0.310662 -2.047900 2.669224 -3.296447 3.917771
## Jul 2012 0.310662 -2.065785 2.687109 -3.323800 3.945124
## Aug 2012 0.310662 -2.083537 2.704861 -3.350949 3.972273
## Sep 2012 0.310662 -2.101158 2.722482 -3.377898 3.999222
## Oct 2012 0.310662 -2.118651 2.739975 -3.404652 4.025976
## Nov 2012 0.310662 -2.136019 2.757343 -3.431214 4.052538
## Dec 2012 0.310662 -2.153265 2.774589 -3.457589 4.078913
## Jan 2013 0.310662 -2.170391 2.791715 -3.483781 4.105105
## Feb 2013 0.310662 -2.187399 2.808723 -3.509793 4.131117
## Mar 2013 0.310662 -2.204293 2.825617 -3.535629 4.156953
## Apr 2013 0.310662 -2.221073 2.842397 -3.561293 4.182617
## May 2013 0.310662 -2.237744 2.859068 -3.586788 4.208112
## Jun 2013 0.310662 -2.254305 2.875629 -3.612117 4.233441
## Jul 2013 0.310662 -2.270761 2.892085 -3.637284 4.258608
## Aug 2013 0.310662 -2.287112 2.908436 -3.662291 4.283615
## Sep 2013 0.310662 -2.303361 2.924686 -3.687142 4.308466
## Oct 2013 0.310662 -2.319510 2.940834 -3.711839 4.333163
## Nov 2013 0.310662 -2.335560 2.956884 -3.736386 4.357710
## Dec 2013 0.310662 -2.351514 2.972838 -3.760785 4.382109
## Jan 2014 0.310662 -2.367372 2.988696 -3.785038 4.406362
- Rafha -##
## ARIMA(2,1,2)(1,0,1)[12] with drift : 383.1335
## ARIMA(0,1,0) with drift : 424.3704
## ARIMA(1,1,0)(1,0,0)[12] with drift : 409.9084
## ARIMA(0,1,1)(0,0,1)[12] with drift : 382.6348
## ARIMA(0,1,0) : 422.3704
## ARIMA(0,1,1) with drift : 380.9847
## ARIMA(0,1,1)(1,0,0)[12] with drift : 382.6789
## ARIMA(0,1,1)(1,0,1)[12] with drift : 384.477
## ARIMA(1,1,1) with drift : 381.9588
## ARIMA(0,1,2) with drift : 381.7438
## ARIMA(1,1,0) with drift : 408.9989
## ARIMA(1,1,2) with drift : 383.4506
## ARIMA(0,1,1) : 378.9848
## ARIMA(0,1,1)(1,0,0)[12] : 380.679
## ARIMA(0,1,1)(0,0,1)[12] : 380.6349
## ARIMA(0,1,1)(1,0,1)[12] : 382.4775
## ARIMA(1,1,1) : 379.9591
## ARIMA(0,1,2) : 379.7441
## ARIMA(1,1,0) : 406.9989
## ARIMA(1,1,2) : 381.4507
##
## Best model: ARIMA(0,1,1)## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## Feb 2012 0.2099093 -1.549121 1.968940 -2.480296 2.900114
## Mar 2012 0.2099093 -1.578831 1.998650 -2.525734 2.945552
## Apr 2012 0.2099093 -1.608056 2.027875 -2.570429 2.990247
## May 2012 0.2099093 -1.636818 2.056637 -2.614417 3.034235
## Jun 2012 0.2099093 -1.665139 2.084958 -2.657730 3.077549
## Jul 2012 0.2099093 -1.693039 2.112857 -2.700399 3.120217
## Aug 2012 0.2099093 -1.720535 2.140354 -2.742451 3.162269
## Sep 2012 0.2099093 -1.747646 2.167464 -2.783912 3.203731
## Oct 2012 0.2099093 -1.774385 2.194204 -2.824808 3.244626
## Nov 2012 0.2099093 -1.800770 2.220588 -2.865159 3.284977
## Dec 2012 0.2099093 -1.826812 2.246631 -2.904988 3.324806
## Jan 2013 0.2099093 -1.852526 2.272345 -2.944313 3.364132
## Feb 2013 0.2099093 -1.877923 2.297742 -2.983155 3.402973
## Mar 2013 0.2099093 -1.903015 2.322833 -3.021529 3.441348
## Apr 2013 0.2099093 -1.927812 2.347631 -3.059454 3.479272
## May 2013 0.2099093 -1.952325 2.372144 -3.096943 3.516761
## Jun 2013 0.2099093 -1.976563 2.396382 -3.134012 3.553830
## Jul 2013 0.2099093 -2.000536 2.420354 -3.170674 3.590493
## Aug 2013 0.2099093 -2.024251 2.444069 -3.206944 3.626762
## Sep 2013 0.2099093 -2.047717 2.467535 -3.242832 3.662651
## Oct 2013 0.2099093 -2.070942 2.490760 -3.278351 3.698170
## Nov 2013 0.2099093 -2.093932 2.513751 -3.313512 3.733331
## Dec 2013 0.2099093 -2.116695 2.536514 -3.348326 3.768144
## Jan 2014 0.2099093 -2.139238 2.559057 -3.382802 3.802620
- Sulayel -##
## ARIMA(2,1,2)(1,0,1)[12] with drift : 383.1335
## ARIMA(0,1,0) with drift : 424.3704
## ARIMA(1,1,0)(1,0,0)[12] with drift : 409.9084
## ARIMA(0,1,1)(0,0,1)[12] with drift : 382.6348
## ARIMA(0,1,0) : 422.3704
## ARIMA(0,1,1) with drift : 380.9847
## ARIMA(0,1,1)(1,0,0)[12] with drift : 382.6789
## ARIMA(0,1,1)(1,0,1)[12] with drift : 384.477
## ARIMA(1,1,1) with drift : 381.9588
## ARIMA(0,1,2) with drift : 381.7438
## ARIMA(1,1,0) with drift : 408.9989
## ARIMA(1,1,2) with drift : 383.4506
## ARIMA(0,1,1) : 378.9848
## ARIMA(0,1,1)(1,0,0)[12] : 380.679
## ARIMA(0,1,1)(0,0,1)[12] : 380.6349
## ARIMA(0,1,1)(1,0,1)[12] : 382.4775
## ARIMA(1,1,1) : 379.9591
## ARIMA(0,1,2) : 379.7441
## ARIMA(1,1,0) : 406.9989
## ARIMA(1,1,2) : 381.4507
##
## Best model: ARIMA(0,1,1)## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## Feb 2012 0.2099093 -1.549121 1.968940 -2.480296 2.900114
## Mar 2012 0.2099093 -1.578831 1.998650 -2.525734 2.945552
## Apr 2012 0.2099093 -1.608056 2.027875 -2.570429 2.990247
## May 2012 0.2099093 -1.636818 2.056637 -2.614417 3.034235
## Jun 2012 0.2099093 -1.665139 2.084958 -2.657730 3.077549
## Jul 2012 0.2099093 -1.693039 2.112857 -2.700399 3.120217
## Aug 2012 0.2099093 -1.720535 2.140354 -2.742451 3.162269
## Sep 2012 0.2099093 -1.747646 2.167464 -2.783912 3.203731
## Oct 2012 0.2099093 -1.774385 2.194204 -2.824808 3.244626
## Nov 2012 0.2099093 -1.800770 2.220588 -2.865159 3.284977
## Dec 2012 0.2099093 -1.826812 2.246631 -2.904988 3.324806
## Jan 2013 0.2099093 -1.852526 2.272345 -2.944313 3.364132
## Feb 2013 0.2099093 -1.877923 2.297742 -2.983155 3.402973
## Mar 2013 0.2099093 -1.903015 2.322833 -3.021529 3.441348
## Apr 2013 0.2099093 -1.927812 2.347631 -3.059454 3.479272
## May 2013 0.2099093 -1.952325 2.372144 -3.096943 3.516761
## Jun 2013 0.2099093 -1.976563 2.396382 -3.134012 3.553830
## Jul 2013 0.2099093 -2.000536 2.420354 -3.170674 3.590493
## Aug 2013 0.2099093 -2.024251 2.444069 -3.206944 3.626762
## Sep 2013 0.2099093 -2.047717 2.467535 -3.242832 3.662651
## Oct 2013 0.2099093 -2.070942 2.490760 -3.278351 3.698170
## Nov 2013 0.2099093 -2.093932 2.513751 -3.313512 3.733331
## Dec 2013 0.2099093 -2.116695 2.536514 -3.348326 3.768144
## Jan 2014 0.2099093 -2.139238 2.559057 -3.382802 3.802620