# Decadal Oscillation in the Predictability of Palmer Drought Severity Index in California

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Environmental Setting and Data

#### 2.2. Exponential Smoothing

_{t}is the smoothed PDSI at decadal scale centered on time-year t (Equation (2)); α is the smoothing parameter for the data; I

_{t−p}is the smoothed cycle index at the end of period t, its number being defined by the periods p in the seasonal cycle (Equation (3))

#### 2.3. Transfer Function Models

_{1}(t), X

_{2}(t), …, X

_{k}(t) are the input time series to be considered as explanatory variables contributing to the temporal dynamics of the output series Y(t) and η(t) is a stationary random process. The terms α

_{1}(B), α

_{2}(B), …, α

_{k}(B) are fractional polynomials in the back-shift operator B (such that B

^{S}(X(t) = X(t − s)) of the form:

#### 2.4. Model Validation Methods

## 3. Results and Discussion

#### 3.1. Data Analysis

#### 3.2. Validation Results and PDSI Time Series Predictability

#### 3.3. Simulation Experiment

#### 3.4. Comparison with the Transfer Function Modelling Approach

_{1}presents the sample cross-correlation function (CCF) between El Niño 3.4 series (X) and the PDSI series (Y), and the sample CCF between the pre-whitened X series (residuals after fitting an ARIMA model) and the filtered Y series (after applying the AR(2) filter) in Figure 10b

_{1}. Figure 9b1,b2 show a significant spike at lag = −1 for the CCF between Niño3.4, and PDSI series and PDO and PDSI series. After filtering the input and output time series to discard the autocorrelation effects, Figure 9a1,a2 show the persistent significant leading impact of the Niño3.4 and the PDO series on the PDSI series one year in advance (lag = −1). From the sample CCF functions and according to Box and Jenkins [42], a transfer function model of order $\left({r}_{1},{s}_{1},{d}_{1}\right)=\left(1,1,1\right)$ for input series ${X}_{1}\left(t\right)$ = Niño3.4, and order $\left({r}_{2},{s}_{2},{d}_{2}\right)=\left(1,0,1\right)$ for input series ${X}_{2}\left(t\right)=PDO$, was proposed for this data set. In this case ${\alpha}_{1}\left(B\right)=\left({\delta}_{01}+{\delta}_{11}B\right){B}^{1}/\left(1-{\omega}_{11}{B}^{1}\right)$ and ${\alpha}_{2}\left(B\right)=\left({\delta}_{02}\right){B}^{1}/\left(1-{\omega}_{12}{B}^{1}\right)$.

#### 3.5. Ensemble Forecast with the Transfer Function Model

#### 3.6. Limitations and Perspectives

## 4. Concluding Remarks

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Griffin, D.; Anchukaitis, K.J. How unusual is the 2012–2014 California drought? Geophys. Res. Lett.
**2014**, 41, 9017–9023. [Google Scholar] [CrossRef] - Meko, D.M.; Woodhouse, C.A.; Baisan, C.H.; Knight, T.; Lukas, J.J.; Hughes, M.K.; Salzer, W. Medieval drought in the upper Colorado River basin. Geophys. Res. Lett.
**2007**, 34, L10705. [Google Scholar] [CrossRef] - Raab, L.M.; Larson, D.O. Medieval climatic anomaly and punctuated cultural evolution in coastal Southern California. Am. Antiq.
**1997**, 62, 319–336. [Google Scholar] [CrossRef] - Heusser, L.; Kirby, M.E.; Nichols, J.E. Pollen-based evidence of extreme drought during the last Glacial (32.6–9.0 ka) in coastal southern California. Quat. Sci. Rev.
**2015**, 126, 242–253. [Google Scholar] [CrossRef][Green Version] - Cole, J.E.; Overpeck, J.T.; Cook, E.R. Multiyear La Niña events and persistent drought in the contiguous United States. Geophys. Res. Lett.
**2002**, 29, 25. [Google Scholar] [CrossRef] - California Department of Water Resources. California’s Most Significant Drought: Comparing Historical and Recent Conditions; California Department of Water Resources: Sacramento, CA, USA, 2015. Available online: https://water.ca.gov (accessed on 29 December 2018).
- California’s Sustainable Groundwater Management Act. 2014. Available online: http://groundwater.ucdavis.edu/SGMA (accessed on 29 December 2018).
- Hanak, H.; Lund, J.; Dinar, A.; Gray, B.; Howitt, R.; Mount, J.; Moyle, P.; Thompson, B. Managing California’s Water. From Conflict to Reconciliation; Public Policy Institute of California: San Francisco, CA, USA, 1990; Available online: http://www.ppic.org/content/pubs/report/R_211EHR.pdf (accessed on 29 December 2018).
- Tortajada, C.; Kastner, M.J.; Buurman, J.; Biswas, A.K. The California drought: Coping responses and resilience building. Environ. Sci. Policy
**2017**, 78, 97–113. [Google Scholar] [CrossRef] - Dai, A.; Trenberth, K.E.; Karl, T.R. Global variations in droughts and west spells: 1900–1995. Geophys. Res. Lett.
**1998**, 25, 3367–3370. [Google Scholar] [CrossRef] - Palmer, WC. Meteorological Drought. Research Paper No. 45; Office of Climatology, U.S. Weather Bureau: Washington, DC, USA, 1965.
- Karl, T.R.; Koscielny, A.J. Drought in the United States: 1895-1981. Int. J. Clim.
**1982**, 2, 313–329. [Google Scholar] [CrossRef] - Byun, H.R.; Wilhite, D.A. Objective quantification of drought severity and duration. J. Clim.
**1999**, 12, 2747–2756. [Google Scholar] [CrossRef] - Dai, A.; Trenberth, K.E.; Qian, T. A global dataset of Palmer Drought Severity Index for 1870–2002: Relationship with soil moisture and effects of surface warming. J. Hydrometeorol.
**2004**, 5, 1117–1130. [Google Scholar] [CrossRef] - Shabbar, A.; Skinner, W. Summer drought patterns in Canada and the relationship to global sea surface temperatures. J. Clim.
**2004**, 17, 2866–2880. [Google Scholar] [CrossRef] - Tatli, H. Detecting persistence of meteorological drought via the Hurst exponent. Meteorol. Appl.
**2015**, 22, 763–769. [Google Scholar] [CrossRef][Green Version] - Van der Schrier, G.; Briffa, K.R.; Osborn, T.J.; Cook, E.R. Summer moisture availability across North America. J. Geophys. Res.
**2006**, 111, D11102. [Google Scholar] [CrossRef] - Wells, N.; Goddard, S.; Hayes, M.J. A self-calibrating Palmer Drought Severity Index. J. Clim.
**2004**, 17, 2335–2351. [Google Scholar] [CrossRef] - Flint, L.E.; Flint, A.L.; Mendoza, J.; Kalansky, J.; Ralph, F.M. Characterizing drought in California: New drought indices and scenario-testing in support of resource management. Ecol. Process.
**2018**, 7, 1. [Google Scholar] [CrossRef] - Koster, R.D.; Dirmeyer, P.A.; Guo, Z.; Bonan, G.; Chan, E.; Cox, P.; Gordon, C.T.; Kanae, S.; Kowalczyk, E.; Lawrence, D.; et al. Regions of strong coupling between soil moisture and precipitation. Science
**2004**, 20, 1138–1140. [Google Scholar] [CrossRef] [PubMed] - Cook, E.R.; Seager, R.; Cane, M.A.; Stahle, D.W. North American drought: Reconstructions, causes, and consequences. Earth-Sci. Rev.
**2007**, 81, 93–134. [Google Scholar] [CrossRef] - Han, J.; Kamber, M.; Pie, J. Data mining: Concepts and Techniques; Elsevier: Burlington, MA, USA, 2012. [Google Scholar]
- Huang, F.T.; Mayr, H.G.; Russell, J.M., III; Mlynczak, M.G. Ozone and temperature decadal trends in the stratosphere, mesosphere and lower thermosphere, based on measurements from SABER on TIMED. Ann. Geophys.
**2014**, 32, 935–949. [Google Scholar] [CrossRef][Green Version] - Mossad, A.; Alazba, P. Drought forecasting using stochastic models in a hyper-arid climate. Atmosphere
**2015**, 6, 410–430. [Google Scholar] [CrossRef] - Mishra, A.; Desai, V.R. Drought forecasting using stochastic models. Stoch. Environ. Res. Risk Assess.
**2005**, 19, 326–339. [Google Scholar] [CrossRef] - Cancelliere, A.; Mauro, G.D.; Bonaccorso, B.; Rossi, G. Drought forecasting using the Standardized Precipitation Index. Water Resour. Manag.
**2007**, 21, 801–819. [Google Scholar] [CrossRef] - Fernández, C.; Vega, J.A.; Vega, J.A.; Fonturbel, T.; Jiménez, E. Streamflow drought time series forecasting: A case study in a small watershed in North West Spain. Stoch. Environ. Res. Risk Assess.
**2009**, 23, 1063–1070. [Google Scholar] [CrossRef] - Yoon, J.; Mo, K.; Wood, E.F. Dynamic-model-based seasonal prediction of meteorological drought over the contiguous United States. J. Hydrometeorol.
**2012**, 13, 463–482. [Google Scholar] [CrossRef] - Karavitis, C.A.; Vasilakou, C.G.; Tsesmelis, D.E.; Oikonomou, P.D.; Skondras, N.A.; Stamatakos, D.; Fassouli, V.; Alexandris, S. Short-term drought forecasting combining stochastic and geo-statistical approaches. Eur. Water J.
**2015**, 49, 43–63. [Google Scholar] - Yan, H.; Moradkhani, H. Combined assimilation of streamflow and satellite soil moisture with the particle filter and geostatistical modeling. Adv. Water Resour.
**2016**, 94, 364–378. [Google Scholar] [CrossRef] - Holt, C.C. Forecasting seasonals and trends by exponentially weighted moving averages. Int. J. Forecast.
**2004**, 20, 5–10. [Google Scholar] [CrossRef] - Gardner, E.S., Jr. Exponential smoothing: The state of the art—part II. Int. J. Forecast.
**2006**, 22, 637–666. [Google Scholar] [CrossRef] - Box, G.E.P.; Jenkins, G.M.; Reinsel, G.C. Time Series Analysis: Forecasting and Control, 3rd ed.; Prentice-Hall: Englewood Cliffs, NJ, USA, 1994. [Google Scholar]
- McClain, J.O. Dynamics of exponential smoothing with trend and seasonal terms. Manag. Sci.
**1974**, 20, 1300–1304. [Google Scholar] [CrossRef] - Taylor, J.W. Exponential smoothing with a damped multiplicative trend. Int. J. Forecast.
**2003**, 19, 715–725. [Google Scholar] [CrossRef][Green Version] - Hyndman, R.J.; Koehler, A.B. Another look at measures of forecast accuracy. Int. J. Forecasting
**2006**, 22, 679–688. [Google Scholar] [CrossRef][Green Version] - Armstrong, J.S. Combining forecasts. In Principles of Forecasting: A Handbook for Researchers and Practitioners; Armstrong, J.S., Ed.; Kluwer Academic Publishers: Norwell, MA, USA, 2001. [Google Scholar]
- Diodato, N. Storminess forecast skills in Naples, Southern Italy. In Storminess and Environmental Change; Diodato, N., Bellocchi, G., Eds.; Springer: Dordrecht, The Netherlands, 2014. [Google Scholar]
- Diodato, N.; Bellocchi, G. Long-term winter temperatures in central Mediterranean: Forecast skill of an ensemble statistical model. Appl. Clim.
**2014**, 116, 131–146. [Google Scholar] [CrossRef] - Diodato, N.; Bellocchi, G. Using historical precipitation patterns to forecast daily extremes of rainfall for the coming decades in Naples (Italy). Geosciences
**2018**, 8, 293. [Google Scholar] [CrossRef] - Diodato, N.; Bellocchi, G.; Fiorillo, F.; Ventafridda, G. Case study for investigating groundwater and the future of mountain spring discharges in Southern Italy. J. Mt. Sci.
**2017**, 14, 1791–1800. [Google Scholar] [CrossRef] - Box, G.E.P.; Jenkins, G.M. Time Series Analysis: Forecasting and Control; Holden-Day: San Francisco, CA, USA, 1970. [Google Scholar]
- Shumway, R.H.; Stoffer, D.S. Time Series Analysis and Its Applications; Springer International Publishing: Cham, Switzerland, 2017. [Google Scholar]
- Allen, R.J.; Anderson, R.G. 21st century California drought risk linked to model fidelity of the El Niño teleconnection. npj Clim. Atmos. Sci.
**2018**, 1, 21. [Google Scholar] [CrossRef] - Diffenbaugh, N.S.; Swain, D.L.; Touma, D. Anthropogenic warming has increased drought risk in California. Proc. Natl. Acad. Sci. USA
**2015**, 112, 3931–3936. [Google Scholar] [CrossRef][Green Version] - Box, G.E.P. Understanding exponential smoothing: A simple way to forecast sales and inventory. Qual. Eng.
**1991**, 3, 561–566. [Google Scholar] [CrossRef] - Montgomery, D.C.; Jennings, C.L.; Kulachi, M. Introduction to Time-Series Analysis and Forecasting; Wiley: Hoboken, NJ, USA, 2008. [Google Scholar]
- Wichard, J.D.; Merkwirth, C. Robust long term forecasting of seasonal time series. In Proceedings of the 8th International Work-Conference on Artificial Neural Networks, Barcelona, Spain, 8–10 June 2005. [Google Scholar]
- De Guenni, L.B.; García, M.; Muñoz, Á.G. Predicting monthly precipitation along coastal Ecuador: ENSO and transfer function models. Appl. Clim.
**2017**, 129, 1059–1073. [Google Scholar] [CrossRef] - Hyndman, R.J.; Koehler, A.B.; Ord, J.K.; Snyder, R.D. Forecasting with Exponential Smoothing: The State Space Approach; Springer: Berlin, Germany, 2008. [Google Scholar]
- Franses, P.H. A note on the Mean Absolute Scaled Error. Int. J. Forecast.
**2016**, 32, 20–22. [Google Scholar] [CrossRef][Green Version] - Addiscott, T.M.; Whitmore, A.P. Computer simulation of changes in soil mineral nitrogen and crop nitrogen during autumn, winter and spring. J. Agric. Sci.
**1987**, 109, 141–157. [Google Scholar] [CrossRef] - Wessa, P. Free Statistics Software, Office for Research Development and Education. Version 1.1.23-r7. 2012. Available online: https://www.wessa.net (accessed on 29 December 2018).
- Hurst, H.E. Long-term storage capacity of reservoirs. Trans. Am. Soc. Civ. Eng.
**1951**, 116, 770–808. [Google Scholar] - Karagiannis, T.; Faloutsos, M.; Riedi, R.H. Long-range dependence: Now you see it, now you don’t! In Proceedings of the Global Telecommunications Conference “GLOBECOM ’02”, Taipei, Taiwan, 17–21 November 2002; Volume 3. [Google Scholar]
- Karagiannis, T.; Molle, M.; Faloutsos, M. Long-range dependence: Ten years of internet traffic modeling! IEEE Internet Comput.
**2004**, 8, 57–64. [Google Scholar] [CrossRef] - Belov, I.; Kabašinskas, A.; Sakalauskas, L. A study of stable models of stock markets. Inf. Technol. Control
**2006**, 35, 34–56. [Google Scholar] - Yin, X.-A.; Yang, X.-H.; Yang, F.-Z. Using the R/S method to determine the periodicity of time series. Chaos Solitons Fractals
**2009**, 39, 731–745. [Google Scholar] [CrossRef] - Sheng, H.; Chen, Y.Q. Robustness analysis of the estimators for noisy long-range dependent time series. In Proceedings of the ASME 2009 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference, San Diego, CA, USA, 30 August–2 September 2009. [Google Scholar]
- Chatfield, C. Time-Series Forecasting; Chapman and Hall/CRC: Boca Raton, FL, USA, 2000. [Google Scholar]
- Buishand, T.A. Some methods for testing the homogeneity of rainfall records. J. Hydrol.
**1982**, 58, 11–27. [Google Scholar] [CrossRef] - Pettitt, A.N. A non-parametric approach to the change-point detection. J. R. Stat. Soc. C Appl.
**1979**, 28, 126–135. [Google Scholar] - Daniell, P.J. Discussion following ‘On the theoretical specification and sampling properties of autocorrelated time series’ by M.S. Bartlett. J. R. Stat. Soc.
**1946**, 8, 88–90. [Google Scholar] - R Core Team. R: A Language and Environment for Statistical Computing; R Foundation for Statistical Computing: Vienna, Austria, 2018; Available online: https://www.R-project.org (accessed on 29 December 2018).
- Kim, W.; Cai, W. Second peak in the far eastern Pacific sea surface temperature anomaly following strong El Niño events. Geophys. Res. Lett.
**2013**, 40, 4751–4755. [Google Scholar] [CrossRef][Green Version] - Ramasubramanian, V. Time-Series Analysis, Modelling and Forecasting Using SAS Software; Indian Agricultural Statistics Research Institute: New Delhi, India, 1970; Available online: http://www.iasri.res.in/sscnars/socialsci/5-TS_SAS_lecture.pdf (accessed on 29 December 2018).
- The International Telegraph and Telephone Consultative Committee. International Telephone Service Network Management, Traffic Engineering (Recommendations E.401-E.600), Volume II, Fascicle II.3; The International Telegraph and Telephone Consultative Committee: Geneva, Switzerland, 1985; Available online: http://handle.itu.int/11.1004/020.1000/4.259.43.en.1004 (accessed on 29 December 2018).
- Quian, B.; Rasheed, K. Hurst exponent and financial market predictability. In Proceedings of the 2nd IASTED International Conference on Financial Engineering and Applications, Cambridge, MA, USA, 8–11 November 2004. [Google Scholar]
- Kaklauskas, L.; Sakalauskas, L. Study of on-line measurement of traffic self-similarity. Cent. Eur. J. Oper. Res.
**2013**, 21, 63–84. [Google Scholar] [CrossRef] - Dotov, D.G.; Bardy, B.G.; Dalla Bella, S. The role of environmental constraints in walking: Effects of steering and sharp turns on gait dynamics. Sci. Rep.
**2016**, 6, 28374. [Google Scholar] [CrossRef][Green Version] - Robeson, S.M. Revisiting the recent California drought as an extreme value. Geophys. Res. Lett.
**2015**, 42, 6771–6779. [Google Scholar] [CrossRef][Green Version] - Philander, S.G.H. El Niño, La Niña and the Southern Oscillation; Academic Press: San Diego, CA, USA, 1990. [Google Scholar]
- Trenberth, K.E. The definition of El Niño. Bull. Amer. Meteorol. Soc.
**1997**, 78, 2771–2777. [Google Scholar] [CrossRef] - Wang, S.; Huang, J.; He, Y.; Guan, Y. Combined effects of the Pacific Decadal Oscillation and El Niño-Southern Oscillation on global land dry-wet changes. Sci. Rep.
**2014**, 4, 6651. [Google Scholar] [CrossRef] [PubMed] - Benson, L.; Linsley, B.; Smoot, J.; Mensing, S.; Lund, S. Influence of the Pacific Decadal Oscillation on the climate of the Sierra Nevada, California and Nevada. Quat. Res.
**2003**, 59, 151–159. [Google Scholar] [CrossRef][Green Version] - Ropelewski, C.F.; Halpert, M.S. Precipitation patterns associated with the high index phase of the Southern Oscillation. J. Clim.
**1986**, 2, 268–284. [Google Scholar] [CrossRef] - Shukla, S.; Steinemann, A.; Iacobellis, S.F.; Cayan, D.R. Annual drought in California: Association with monthly precipitation and climate phases. J. Clim.
**2015**, 54, 2273–2281. [Google Scholar] [CrossRef] - Spliid, H. Marima: Multivariate ARIMA and ARIMA-X Analysis. R Package Version 2.2. 2017. Available online: https://cran.r-project.org/web/packages/marima (accessed on 29 December 2018).
- Collopy, F.; Armstrong, J.S. Rule-based forecasting: Development and validation of an expert systems approach to combining time series extrapolations. Manag. Sci.
**1992**, 38, 1394–1414. [Google Scholar] [CrossRef] - Chu, P.-S.; Chen, Y.R.; Schroeder, A. Changes in precipitation extremes in the Hawaiian Islands in a warming climate. J. Clim.
**2010**, 23, 4881–4900. [Google Scholar] [CrossRef] - Sheffield, J.; Wood, E.F. Global trends and variability in soil moisture and drought characteristics, 1950–2000, from observation-driven simulations of the terrestrial hydrologic cycle. J. Clim.
**2008**, 21, 432–458. [Google Scholar] [CrossRef] - Esfahani, A.A.; Friedel, M.J. Forecasting conditional climate-change using a hybrid approach. Environ. Model. Softw.
**2014**, 52, 83–97. [Google Scholar] [CrossRef] - Hazeleger, W.; van den Hurk, B.J.J.M.; Min, E.; van Oldenborgh, G.J.; Petersen, A.C.; Stainforth, D.A.; Vasileiadou, E.; Smith, L.A. Tales of future weather. Nat. Clim. Chang.
**2015**, 5, 107–113. [Google Scholar] [CrossRef][Green Version] - Ingram, W. Extreme precipitation: Increases all round. Nat. Clim. Chang.
**2016**, 6, 443–444. [Google Scholar] [CrossRef]

**Figure 1.**Mapped patterns of reconstructed PDSI for some intervals in 19th century across the U.S. (modified from Cole et al. [5]).

**Figure 2.**(

**a**) Rainfall monthly regime with relative bioclimatic patterns for California; (

**b**) mean annual smoothed 20-km spatial precipitation over the period 1961–1990; (

**c**) the corresponding annual reference evapotranspiration (arranged via LocClim FAO software, http://www.fao.org/land-water/land/land-governance/land-resources-planning-toolbox/category/details/en/c/1032167).

**Figure 3.**(

**a**) Observed Palmer Drought Severity Index time-series (blue curve 1801–2014) with training and validation periods; (

**b**) for the validation period, the simulated series (plume, light grey) with both the ensemble mean (red curve) and the observed Gaussian Filter with 11-year smoothing (bold grey curve).

**Figure 4.**Smoothed periodogram of the PDSI time series (bandwidth is a measure of the width of the frequency interval used in the smoothing procedure).

**Figure 5.**(

**a**,

**a**) Residual histogram and normal Q-Q plot between PDSI forecasted and observed in validation period for the official run; (

_{1}**b**,

**b**) Residual histogram and normal Q-Q plot between PDSI forecasted and observed in validation period for the ensemble mean.

_{1}**Figure 6.**Evolution of observed annual PDSI (black curve) with its smoothed 11-year Gaussian filter in bold blue curve (1942–2014), and exponential smoothing forecasts (2015–2054) with plume prediction (light gray) and the ensemble mean value (bold black line). PDSI classes are also reported.

**Figure 7.**Time series of PDO, ENSO indices (source: https://www.esrl.noaa.gov/psd/data/climateindices/list) and PDSI. MEI-1871 and MEI-1950 are the Multivariate ENSO Index series starting in 1871 and 1950 simultaneously; indices Niño1+2, Niño3, Niño34 and Niño4 are the mean Sea Surface Temperature anomalies in the Pacific Ocean regions: 0–10 S, 90 W–80 W; 5 N–5 S, 150 W–90 W; 5 N–5 S, 170–120 W; 5 N–5 S, 160 E–150 W, respectively; ONI is the Oceanic Niño Index; PDO is the Pacific Decadal Oscillation; SOI is the Southern Oscillation Index.

**Figure 8.**Sample cross-correlation functions between the PDSI series and all ENSO indices shown in Figure 7. Numbers indicate the associated lag at the peak value.

**Figure 9.**(

**a1**) Sample cross-correlation function (CCF) between the pre-whitened Pacific Decadal Oscillation (PDO) series denoted as X and the filtered PDSI series denoted as Y for the training period; (

**a2**) the sample CCF between the PDO series and the PDSI series; (

**b1**) CCF pre-whitened Niño3.4 and the filtered PDSI; (

**b2**) the sample CCF between the Niño3.4 series and the PDSI series.

**Figure 10.**(

**a**) Cross-correlation and (

**b**) partial autocorrelation functions (ACF and PACF, respectively) of the estimated residuals (${\widehat{\eta}}_{t}$) for the fitted model.

**Figure 11.**Observed PDSI time series (black line) with the training dataset to build the model (blue line) for the period 1900–1953, including the filtered observed values (navy blue). Also comparison between the observed values (black line) and predicted values with the TF model for the validation period (1954–2014) (red line), including the corresponding 95% confidence intervals (red dash).

**Figure 12.**Residual histogram and normal Q-Q plot between the PDSI predicted values and observed PDSI time series for the validation period (1954–2014).

**Figure 13.**Observed PDSI time series (black line) with the simulation plume (grey lines) for the period 2015–2054, including the filtered observed series (navy-blue line), the median of the simulated values (thick grey line) and the 2.5% (bottom dashed line) and 97.5% quantile (top dashed line) of the simulated values.

**Figure 14.**Comparison between estimates from the exponential smoothing model (ESM) and the Transfer Function model (TFM) for both the validation period (1954–2014) and the forecast period (2015–2054).

**Table 1.**Estimated values of the Hurst (H) exponent (with two methods) for the PDSI annual series as a whole and for a reduced number of years.

Hurst (H) Exponent/Estimation Method | Whole Series (1801–2014) | Reduced Series (1901–2014) |
---|---|---|

Rescaled range (R/S) | 0.611 | 0.743 |

Ratio variance of residuals | 0.611 | 0.550 |

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**MDPI and ACS Style**

Diodato, N.; De Guenni, L.B.; Garcia, M.; Bellocchi, G. Decadal Oscillation in the Predictability of Palmer Drought Severity Index in California. *Climate* **2019**, *7*, 6.
https://doi.org/10.3390/cli7010006

**AMA Style**

Diodato N, De Guenni LB, Garcia M, Bellocchi G. Decadal Oscillation in the Predictability of Palmer Drought Severity Index in California. *Climate*. 2019; 7(1):6.
https://doi.org/10.3390/cli7010006

**Chicago/Turabian Style**

Diodato, Nazzareno, Lelys Bravo De Guenni, Mariangel Garcia, and Gianni Bellocchi. 2019. "Decadal Oscillation in the Predictability of Palmer Drought Severity Index in California" *Climate* 7, no. 1: 6.
https://doi.org/10.3390/cli7010006