Question

Consider the problem of a single input variable x and output variable y. Assume we want to fit a quadratic model (with no linear term) of the form y = θ₁x² +θ₀, and accordingly find the best value of θ₁ and θ₀. Assume we have N measurements of the input and output where the n-th measurement is given by xₙ and yₙ with corresponding estimate yₙ = θ₁xₙ² +θ₀. To find the best model, we want to minimize the mean square error (MSE), MSE EN=1/nΣₙᴺ yn – yn)²

Find the optimum values of θ₁ and θ₀, that minimizes the MSE.

Answer 1

To minimize the MSE for the quadratic model y = θ1x² + θ0, we use calculus to differentiate the MSE with respect to θ1 and θ0, set those derivatives to zero, and solve the resulting equations to find the values of θ1 and θ0 that correspond to the least-squares regression.

Step-by-step explanation:

To find the optimum values of θ1 (θ1) and θ0 (θ0) that minimize the Mean Square Error (MSE) for a quadratic model of the form y = θ1x² + θ0, we must differentiate the MSE with respect to both θ1 and θ0 and set the results to zero to find the minimum value. This process is a part of least-squares regression used to fit the line of best fit.

The MSE is defined as MSE = (1/N) ∑ from n=1 to N (yn - θ1x² - θ0)², where yn are the observed values and θ1x² + θ0 are the estimated values based on the model. Differentiating the MSE with respect to θ1 and θ0 and setting both derivatives equal to zero yields two normal equations. Solving these equations simultaneously will give us the estimators θ1 and θ0 that minimize the MSE.

This procedure resembles the way we find the slope (b) and intercept (a) for a linear regression line, which is similar but with a different model (y = ax + b). However, in this specific problem, the model is quadratic and there is no linear term (hence, no 'x' term with θ1).