Parameter Estimation

Mathematical Modeling · Axiom Academy

LESSON Parameter Estimation - Mathematical Modeling Unit 7: Data-Driven Modeling - Mathematical Modeling Parameter estimation is the process of using observed data to determine the values of unknown parameters in a mathematical model. It bridges the gap between theoretical models and real-world observations. Model + Data --> Parameter Values Given a model structure where represents unknown parameters, parameter estimation finds the values of that best explain the observed data. Consider a general model with observations: y_i are the observed outputs (measurements) x_i are the observed inputs (independent variables) represents measurement error or noise Example: Population Growth Model Given population measurements over time, estimate growth rate r and carrying capacity K in: Here are the parameters to estimate from observed population data. The most common approach minimizes the sum of squared residuals between model predictions and observed data. Find parameter values that minimize: where is the residual for observation i . Positive and negative errors don't cancel Larger errors are penalized more heavily Leads to smooth, differentiable objective function Optimal under assumption of normally distributed errors Computationally tractable for linear models Linear Case: Analytical Solution For a linear model (in matrix form), the least squares solution is: Take derivative and set to zero: Maximum Likelihood Estimation (MLE)

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