Forecasting of meteorological drought using ARIMA model

Transcription

1 Indian J. Agric. Res., 51 (2) 2017 : Print ISSN: / Online ISSN: X AGRICULTURAL RESEARCH COMMUNICATION CENTRE Forecasting of meteorological drought using ARIMA model M. Karthika*, Krishnaveni 1 and V.Thirunavukkarasu 2 Centre For Water Resources, Anna University, Chennai, , Tamil Nadu, India. Recieved: Accepted: DOI: /ijare.v0iOF.7631 ABSTRACT Drought is a global phenomenon that occurs virtually in all landscapes causing significant damage. Due to the random nature of contributing factors, occurrence and severity of droughts can be treated as stochastic in nature. Early indication of possible drought can help to set out drought mitigation strategies and measures in advance. Therefore drought forecasting plays an important role in the planning and management of water resource systems. The principal objective of the study is to carryout short-term annual forecasting of meteorological drought using Auto Regressive Integrated Moving Average (ARIMA) model in Lower Thirumanimuthar Sub-basin located in semi-arid region of Tamilnadu, India is chosen as the study area which is predominantly affected by drought over few decades. Suitable linear stochastic model, non seasonal autoregressive integrated moving average (ARIMA) was developed to predict drought. The best model was selected based on minimum Akaike Information Criterion (AIC) and Schwarz Bayesian Criterion (SBC). Parameter estimation step indicates that the estimated model parameters are significantly different from zero. The predicted data using the best ARIMA model were compared to the observed data for model validation purpose in which the predicted data show reasonably good agreement with the actual data. Hence the models were applied to forecast drought in the Lower Thirumanimuthar subbasin region up to 2 years in advance with good accuracy. Key words: Arima, Drought, Forecasting,Stochastic model. INTRODUCTION A drought is an extended period of months or years when region notes a deficiency in its water supply, whether surface or underground water. Generally this occurs when a region receives consistently below average precipitation. Meteorological drought is brought about when there is a prolonged period with less than average precipitation. Meteorological drought usually precedes the other kinds of drought. Hydrological drought refers to deficiencies in surface and subsurface water supplies. It is measured as stream flow as lake, reservoir, and groundwater levels. Agricultural drought various characteristics of meteorological or hydrological drought to agricultural impacts, focusing on precipitation shortages, differences between actual potential evapo-transpiration, soil, soil water deficits and reduced ground water or reservoir levels. Socioeconomic drought is associated with the demand and supply aspect of economic goods together with elements of all other kinds of drought. Literature survey:van Rooy, (1965) developed Rainfall Anomaly Index (RAI) based on ratios of rainfall departure from normal to departure of threshold value from normal. It is completely dependent on long term meteorological rainfall observations. RAI shows the relation between a regional humidity index and the actual evaluation of dry periods during the rainy seasons. It is still used in certain drought studies due to its simplicity in estimation and easiness in usage. The demerit of this index is that it uses the observations from only rainfall. Alley, (1984) stated, the Palmer Drought Severity Index (PDSI) was developed for Meteorological drought assessment using precipitation, evapotranspiration and soil moisture conditions as the key inputs. The PDSI is efficient in addressing two of the most significant properties of drought and they are the intensity of drought and its onset and offset time. However, PDSI is very complicated to compute and requires a long term observations of multiple parameters which makes it usable at only limited regions. It has some other limitations too, due to which, the conventional time series models may not be able to capture the stochastic properties of PDSI. Tsakiris and Vangelis, (2005) used the Reconnaissance Drought Index (RDI) for the assessment of drought severity. Here PET (Potential Evapotranspiration) is calculated using Thronthwaithe formula. Rainfall and temperature data are obtained for monthly time steps. Since PET is used, a realistic dtermination of water deficit is obtained. This paper finally *Corresponding author s karthikamozhi@gmail.com 1 Centre For Water Resources, Anna University, Guindy, Chennai, Tamil Nadu. 2 Sri Jayaram Institute of Engineering and Technology, Gummidipondi, Tamil Nadu.

2 104 INDIAN JOURNAL OF AGRICULTURAL RESEARCH Suitable linear stochastic model, viz. seasonal and non seasonal autoregressive integrated moving average (ARIMA) developed to predict drought at different time scale. The best model was selected based on minimum Akaike Information Criterion (AIC) and Schwarz Bayesian Criterion (SBC). Statistical analysis revealed that nonseasonal ARIMA model was appropriate for 3-month SPI series while seasonal ARIMA models have been found promising for SPI series at 6, 9, 12 and 24-month time scale. The predicted data using the best ARIMA model were compared to the observed data for model validation purpose in which the predicted data show reasonably good agreement with the actual data. Study area: The study area, Lower Thirumanimuttar River Sub Basin of Cauvery River, South India lies within the geographic co-ordinate of the Namakkal District in the interior of Tamil Nadu between the Latitudes 11 N and N and longitudes E and E is shown in Figure1.The Thirumanimuthar rises in the southern flank of Shervaroy hills and Manjavadi Ghat and flows towards the North East of Namakkal.After it is flowing through Namakkal districts nearly 102 km it confluences with the river cauvery at Koodutharai in Paramathi Taluk of Namakkal district. The major source for recharge of water in this area is rainfall, during monsoon season. The normal average rainfall of the sub-basin is 640mm to 800mm. About 35% of total rainfall is received during South west monsoon and 45% during North east monsoon. MATERIALS AND METHODS For this study, meteorological data pertaining to monthly precipitation data were collected from Department of Statistics and Economics, Teynampet, Chennai,Tamilnadu for the period of 35 years ranging from for eight rain gauge stations namely Rasipuram, Elachipalayam, Mohanoor, Paramathy, Sendamangalam, Mallasamudram, Namakkal and Tiruchengode are shown in Figure 2. By using monthly precipitation data, annual rainfall was calculated for assessing annual drought severity using IMD(India Meteorological Department) as one of the drought indices. At last annual meteorological drought was forecasted using ARIMA model. Auto Regressive Integrated Moving Average (ARIMA) model: The autoregressive integrated moving average (ARIMA) process generates non-stationary series that are integrated of order d, denoted I (d). A non-stationary I (d) process is one that can be made stationary by taking D differences. Such processes are often called differencestationary or unit root processes. A series that you can model as a stationary ARMA (p, q) process after being differenced D times is denoted by ARIMA (p, d, q). Model identification: This step consists of identifying the possible ARIMA model that represents the behavior of the time series. The series behavior was investigated by the autocorrelation function (ACF) and partial autocorrelation function (PACF). The ACF and PACF were used to assist in determining the order of the model. The information provided TAMILNADU NAMAKKAL DISTRICT LOWER THIRUMANIMUTHAR SUBBASIN Figure 1:Location map of study area. Figure 2: Rain Gauge Stations in study area.

3 by ACF and PACF is useful to suggest the type of models that can be built. The final model was then selected using the penalty function statistics through the Akaike information criterion (AIC) and Schwarz-Bayesian criterion (SBC). These criteria help to rank models (models having the lowest value of criterion being the best). The AIC and SBC take the mathematical form as shown below: AIC= -2 log (L) + 2k SBC= -2 log (L) + k ln (n) where k is number of parameters in the model, (p + q + P + Q); L is the likelihood function of the ARIMA model; and n is the number of observations. Parameter estimation: After identifying the appropriate model as an essential step, the estimation of model parameters was achieved. The model estimate values for the AR and MA parts were calculated using Yule-Walker equation. The AR and MA parameters were tested to make sure that they are statistically significant or not. The associated parameters, such as standard error of estimates and their related t-values, are also calculated. Diagnostic checking: Diagnosing the ARIMA model is a crucial part and the last step of the model development. It involves in checking the adequacy of selected model. Several diagnostic statistics and plots of residuals are investigated to see if the residuals are correlated white noise or not. In this study the residual ACF function(racf) and Pormantateau lack-of-fit test are the diagnostic checks adopted. The residual ACF function(racf) should be obtained to determine whether residuals are white noise. There are two useful applications related to RACF for the independence of residuals. The first is the correlogram drawn by plotting correlation against lag. If some of the RACFs are significantly different from zero, this may indicate that the present model is inadequate. Pormantateau lack-of-fit test is used in this study to test the adequacy of the model. The test is computed as follows: Here n is the number of observations in series and Q(k) is distributed as a Chi-square with (K-p-q-P-Q) degrees of freedom. These Q(r) stat values are compared with χ 2 distribution with respective degrees of freedom at a 5% significant level. The calculated value should be less than the actual χ 2 value, so that the model will be adequate. If these assumptions are not satisfied, we need to fit a more appropriate model. That is, we go back to the model identification step and try to develop a better model. Once these Diagnostic checks are satisfied, we can forecast the best fit model. Model forecasting: One of the most important tests of any model is how well it forecasts. When assessing how well a model forecasts, we need to compare it to the actual data, this then produces a forecast value, an actual value and a Volume 51 Issue 2 (2017) 105 forecast error (difference between forecast and actual values) for each individual observation used for the forecast. Then the accuracy of the forecast needs to be measured, this can be done by: i. Plot of forecast values against actual values ii. Use of a statistic such as the Mean or variance of actual against Forecasted. RESULTS AND DISCUSSION DROUGHT FORECASTING USING ARIMA MODEL Autocorrelation and partial autocorrelation plot: Autocorrelation plot is a plot of the sample autocorrelations versus their respective lags. From the shape of the auto correlation plot we can identify the model. Alternating positive and negative, decaying to zero mean AR model. Decay, starting after a few lags means mixed AR and MA model. Auto correlation plot for different rain gauge station is shown in Table 1. Partial Autocorrelation function gives partial correlation of a time series with its own lagged values. Partial Autocorrelation plot is shown in Table 2. Order of the model: The model which gives minimum Akaike Information Criteria (AIC) and Schwarz-Bayesian Information Criteria (SBC) is the best fit model. Table 3 shows the time series selection results. In Elachipalayam rain gauge station model 2,1,1 gives minimum AIC and SBC. Parameter estimation: After the identification of model using the AIC and SBC criteria, estimation of parameters is done. Model estimates were calculated simultaneously for AR and MA Parameters. The value of parameters, associated standard errors, t-ratio and p-values are listed in Table 4. Diagnostic check: After the estimation of model parameters, diagnostic checking was performed to verify the adequacy of the model. The residuals are studied to see if any pattern remains unaccounted. All calibration tests are summarized briefly below. ACF of residuals: The residual ACF function (RACF) should be obtained to determine whether residuals are white noise. There are two useful applications related to RACF for the independence of residuals. If some of the RACFs are significantly different from zero, this may indicate that the present model is inadequate. The ACF of Residuals for Rasipuram is shown in figure 3. Since the most of the values does not satisfies the condition, Significant if Correlation > This indicates that these autocorrelations are. Nonsignificant, so the residuals are white noise. This condition is satisfied by all other Rain gauge stations Pormantateau lack-of-fit test: The Portmanteau Test (sometimes called the Box-Pierce-Ljung statistic) is used in this study to test the adequacy of the model. The test is computed as follows:

4 106 INDIAN JOURNAL OF AGRICULTURAL RESEARCH Table 1: Autocorrelation plot Station Autocorrelation plot Paramathy Rasipuram Sendamangalam Tiruchengode

5 Volume 51 Issue 2 (2017) 107 Table 2: Partial Autocorrelation plot Station Partial Autocorrelation plot Mohanoor Rasipuram Paramathy Elachipalayam

6 108 INDIAN JOURNAL OF AGRICULTURAL RESEARCH Table 3: Order of the Model Station P Q d AIC SBC Elachipalayam Mallasamudram Rasipuram Paramathy Mohanoor Namakkal Sendamangalam Table 4: Model Parameters Station Model Parameters Parameter Estimates Standard Error T-Value Probability level Elachipalayam Φ θ Mallasamudram θ Mohanoor θ Namakkal Φ θ Paramathy Φ θ Rasipuram θ θ Sendamangalam θ θ Tiruchengode

7 Volume 51 Issue 2 (2017) 109 Figure 3: ACF plot of Residuals TABLE 5: Q(r) stat Calculation of Residuals Station Model Q(r) stat Degrees of freedom χ 2 Decision(0.05) Elachipalayam ARIMA(211) Adequate Mohanoor ARIMA(011) Adequate Mallasamudram ARIMA(011) Adequate Namakkal ARIMA(211) Adequate Paramathy ARIMA(211) Adequate Rasipuram ARIMA(112) Adequate Sendamangalam ARIMA(112) Adequate Tiruchengode ARIMA(110) Adequate Figure 4: Comparison of Actual data and Forecasted data using best ARIMA models For Mohanoor Station Figure 5: Comparison of Actual data and Forecasted data using best ARIMA models For Paramathy Station

8 110 INDIAN JOURNAL OF AGRICULTURAL RESEARCH Table 6: Validation plot Station Validation Plot Namakkal Rasipuram Paramathy

9 Table 7: Comparison of statistic properties of the Actual and Forecasted data Volume 51 Issue 2 (2017) 111 Station Mean Mean Decision Variance Variance Decision Observed Forecasted Z cal <1.96 observed Forecasted F cal <1.462 Elachipalyam < <1.462 Mallasamudram < <1.462 Mohanoor < <1.462 Namakkal < <1.462 Paramathy < <1.462 Rasipuram < <1.462 Sendamangalam < <1.462 Tiruchengode < <1.462 Here n is the number of observations in series and Q(k) is distributed as a Chi-square with (K-p-q-P-Q) degrees of freedom. The first 17 ACF of the residual from the model are taken for calculation of Q(r) stat is shown in table 5.These Q(r) stat values are compared with 2 distribution with respective degrees of freedom at a 5% significant level. It is observed that the calculated value is less than the actual 2 value, which signified that the present models are adequate on the available data. Drought forecasting from selected models: After selecting the best time series model, the model was validated using the D i series for the period 1995 to 2014 was used. The Validation plot is shown in table 6. Once the data was validated, the forecast was done for 2-years lead time using the best selected models. The plot between observed data and predicted data using the selected best model for all Di series is shown in Figure 4 to 5. It is observed that the predicted data follows the observed data very closely. Basic statistical properties are compared between observed and forecasted data for one month lead time, using Z-test for the means and F-test for standard deviation, shown in Table 7. Since Zcal values related to means were between Z-critical table values (±1.96 for two tailed at a 5% significance level), the data shows that there is no significant difference between the mean values of observed and predicted data. Similarly, the Fcal values of standard deviation were smaller than the F-critical values at a 5% significance level. Thus, the results show that predicted data preserves the basic statistical properties of the observed series. The forecast is done with 1-year to 3-year lead-time. For example 1-year ahead forecast means that during year 2000 the forecast for the year 2001 is done. Therefore the selected best models from ARIMA building approach using a time series data of D i series can be used for the drought forecasting. CONCLUSION Based on the study and results from forecasting using ARIMA model, the following conclusions were drawn. i.time series analysis using ARIMA model was carried out for annual rainfall data. ARIMA(211), ARIMA(011), ARIMA(112) are the best fit model for Elachipalayam, Mohanoor, Namakkal, Rasipuram stations respectively. ii.the ARIMA models developed in Lower Thirumanimuthar sub- basin can be used for the development of a drought preparedness plan in the region so as to ensure sustainable water resource planning within the sub-basin. REFERENCES Alley.W.M. (1984), The Palmer Drought Severity Index: Limitations & assumptions, Journal of Climate and Applied Meteorology, 23, Tsakiris, G and Vangelis, H. (2005), Establishing a drought index incorporating evapotranspiration,european Water Resources Association, European Water,3-11. Van Rooy, M. P. (1965), A rainfall anomaly index independent of time and space,notos, 14: