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ThesisItem Open Access Fitting linear mixed effects models for unbalanced longitudinal data(CCSHAU, Hisar, 2020-08) Ravita; Verma, UrmilThe classical linear regression model is an important statistical tool but its use is limited because of its standard assumptions. Regression models using time series data occur quite oftenly and the assumption of uncorrelated or independent errors is often not appropriate. Moreover, many time series having complex structure calls for the addition of fixed and random effects accounting for the observational design. Such effects are straightforward to add in a mixed model environment (accommodating unbalanced data). The fixed-effects parameters can be either qualitative (as in the traditional analysis of variance) or quantitative (as in standard linear regression). BLUP is a standard method for estimating random effects of a mixed model. The mixed procedure uses the REML method, also known as residual maximum likelihood. It is here that the Gaussian assumptions are exploited. One such class is varying coefficient models, where the response variable is allowed to depend linearly on some regressors, with coefficients as smooth functions of some other predictor variables, called the effect modifiers. Varying coefficient models, where the effect modifier variable is calendar time, leads to time-varying coefficient models. The statistical modelling approaches viz., multiple linear regression and linear mixed effects were applied to develop mustard yield forecasts models on agro-climatic zone basis in Haryana. The mustard yield data for the period 1980-81 to 2016-17 of Hisar, Bhiwani, Sirsa, Mahendragarh and Gurugram, 1989-90 to 2016-17 of Rewari and 1997-98 to 2016-17 of Jhajjar and Fatehabad districts alongwith fortnightly weather data were used for the purpose. The zonal yield forecast models have been developed on the basis of time-trend and weather data from 1980-81 to 2009-10 while the data from 2010-11 to 2016-17 were used for validity checking of the developed models. Trend yield/time variable was included to take care of variation between districts within zone as the weather data are not available for all the districts, though the zonal model utilized the same weather information in adjoining districts under the zone. The linear mixed effects models with time both as fixed and random effects and weather as random effects with covariance structures; VC, AR(1) and Toeplitz have been fitted. The post-sample predictive performance(s) of alternative LMMs and regression based weather-yield models were observed in terms of percent relative deviations from real-time yield(s) and root mean square error(s), and that differed markedly among the alternative models. LMMs with weather as random effect(s) consistently showed the superiority over regression based weather-yield models in capturing lower percent relative deviations. The LMMs with weather as random effects performed well with lower error metrics as compared to the alternative mixed effects/regression models in most of the post-sample time regimes. Sevensteps ahead (i.e. 2010-11 to 2016-17) predicted values favour the use of LMMs. A critical in-depth of the results indicates the preference of using varying coefficients models in comparison to conventional, i.e., constant/fixed coefficients models developed under this empirical study. The linear mixed effects models with Toeplitz type structure substantially improved the predictive accuracy and produced what can be considered as satisfactory district-level mustard yield prediction in Haryana.ThesisItem Open Access Time series intervention modeling and simulation for mustard yield forecasting in Haryana(CCSHAU,HiSAR, 2020-10) Ajay Kumar; Verma, UrmilModeling and Simulation is a discipline for developing a level of understanding of the interaction of the parts of a system, and of the system as a whole. A model is a simplified representation of a system at some particular point in time or space intended to promote understanding of the real system. Simulation permits the evaluation of operating performance prior to the implementation of a system. The study compares the efficacy of time series Intervention models and simulation in quantifying the pre-harvest mustard yield in Hisar, Bhiwani, Sirsa, Fatehabad, Mahendragarh, Rewari, Jhajjar and Gurugram districts of Haryana. The objective of this study was to assess the forecast accuracy of the contending models for district-level mustard yield forecasts in Haryana. The fortnightly weather data on rainfall, minimum temperature and maximum temperature over the crop growth period (September-October to February-March) have been utilized from 1980-81 to 2010-11 for the models‟ building. The weather-yield data from 2011-12 to 2015-16 have been used to check the post-sample validity of the fitted models for mustard yield forecasts in comparison to those obtained from State Department of Agriculture crop yield(s) estimates. The statistical modeling approaches viz., multiple linear regression, ARIMA, regression with ARIMA errors (RegARIMA) and ARIMA-Intervention were applied for the purpose. First of all, weather-yield models based on multiple linear regression were developed to relate mustard yield to fortnightly weather input alongwith linear time-trend yield/crop condition term as an indicator variable.Alternatively, ARIMA, RegARIMA, and ARIMA-Intervention models were fitted as per targeted objectives. Additionally, Student‟s t-copula in SAS is applied as a simulation tool and compared the output to the time series forecasts. The forecasts are compared to determine if there is either a consistent or significant difference between the two output. The forecast performance(s) of the alternative models were observed in terms of percent relative deviations of mustard yield forecasts from observed yield(s) and root mean square error(s). RegARIMA models performed well with lower error metrics as compared to the alternative models in most of the time regimes. Five-steps ahead forecast figures i.e. 2011-12 to 2015-16 favour the use of RegARIMA models to obtain pre-harvest mustard yield forecasts in the districts under study. The forecasts generated by RegARIMA are remarkably close to the forecasts obtained through the simulation process. Empirical evidence from this study confirms that the RegARIMA model can produce reliable forecasts and would therefore provide a more robust approach of forecasting with limited data sets.using the developed forecast models, the district-level mustard yield estimates could be computed successfully well in advance of the actual harvest. On the other hand, the State Department of Agriculture crop yield estimates are obtained quite late after the actual crop harvest.ThesisItem Open Access State Space Modelling with Weather as Exogenous Input for Sugarcane Yield Prediction in Haryana(CCSHAU, Hisar, 2020-05) Hooda, Ekta; Verma, UrmilParameter constancy is a fundamental issue for empirical models to be useful for forecasting, analyzing or testing any theory. This work addresses the concept of parameter constancy and the implications of predictive failure. Predictive failure is uniquely a post-sample problem. Unlike classical regression analysis, the state space models are time varying parameters models as they allow for known changes in the structure of the system over time and provide a flexible class of dynamic and structural time series models. The study has been performed in two parts i.e. the development of state space models in two forms (the state space and unobserved component approach), and the state space models with weather as an exogenous input for sugarcane yield prediction in Ambala, Karnal, Kurukshetra, Panipat and Yamunanagar districts of Haryana. The time series sugarcane yield data for the period 1966-67 to 2009-10 of Ambala and Karnal districts, 1972-73 to 2009-10 of Kurukshetra and 1970-71 to 2009-10 of Panipat and Yamunanagar districts were used for the development of different models. The validity of fitted models have been checked for the subsequent years i.e., 2010-11 to 2016-17, not included in the development of the yield forecast models. The selection of autoregressive orders, i.e., five, three, two, four and five looked reasonable for Ambala, Karnal, Kurukshetra, Panipat and Yamunanagar districts respectively helped in determining the amount of past information to be used in the canonical correlation analysis and further leading to the selection of state vector. Information from the canonical correlation and preliminary autoregression analyses were used to form preliminary estimate of the parameters of state space models and that provided the sugarcane yield estimates using Kalman filtering technique. The UCMs with level, trend and irregular components were fitted to study the trend of sugarcane yield. For all the five districts, the irregular component was found to be highly significant while both level and trend component variances were observed non-significant. Lastly, the state space models with weather as exogenous input using different types of growth trends viz., polynomial splines; PS(1), PS(2) and PS(3) were developed. The weather variables used for each district were selected on the basis of stepwise regression method and PS(2) with weather input was selected as the best suited model for all districts. The post-sample sugarcane yield estimates were obtained on the basis of fitted SS, UCM and SSM with exogenous input. The predictive performance(s) of the contending models were observed in terms of percent relative deviations and RMSEs of sugarcane yield forecasts in relation to observed yield(s). The SSMs with weather input consistently showed the superiority over SS and UCM models in capturing lower percent relative deviations. Thus, it is inferred that the state space models may be effectively used pertaining to Indian agriculture data, as it takes into account the time dependency of the underlying parameters which may further enhance the predictive accuracy of time-series models with parameter constancy.
