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## Section: New Results

### Calibration of TRANUS Using Maximum Likelihood Estimation

Calibration of the TRANUS land use module typically involves determination of key parameters which dictate land use assignments and prices. As mentioned earlier, It is a difficult task to calibrate a LUTI model as the number of parameters involved are large and are uncertain. Traditionally, these models are calibrated manually by experts, who try to estimate the parameters using their prior experience. However, such a method is difficult as well as time consuming, especially when the parameter space is large and uncertain. Hence, an algorithmic procedure to estimate parameters from mathematical model is desired.

We have proposed an algorithm to calibrate the land use module of TRANUS using maximum likelihood estimation (MLE). The observed outputs of the land use module is modeled to follow a Gaussian process. The covariance matrix is represented as a function of inputs of the land use module and hyperparameters. A MLE optimization problem is then formulated to estimate the parameters of the land use module and the hyperparameters of the Gaussian covariance kernel. The resulting nonlinear programming (NLP) problem is then solved using NLP solvers based on sequential quadratic programming.

The proposed calibration algorithm has been successfully applied to the model of Grenoble, France ; and the performance of the proposed calibration methodology, has been compared to traditional calibration techniques. The metric to judge performance is assumed to be the ${ℒ}_{2}$ norm of the difference between observed and calculated land use assignments obtained using the calibrated model.

Before this calibration task is performed, a sensitivity analysis has been carried out. Hence, sensitivity analysis of the parameters on the output is important as is helps us identify major sources of uncertainty in terms of their contribution towards output space variability. Here, the total effect of the land use parameters on a quantity of interest or QoI is assessed. The QoI is assumed to be the ${ℒ}_{2}$ norm of the difference between observed and calculated land use assignments. For this Grenoble model, the number of uncertain parameters involved are 100, and finally is is observed that only 3 amongst them contribute towards $99.2%$ of QoI variability. [14] , [13]