Final Answer:
In linear regression, θ₀ and θ₁ are chosen by iteratively minimizing the cost function, J(θ), using techniques like gradient descent.
Step-by-step explanation:
In linear regression, determining the values of θ₀ (intercept) and θ₁ (slope) involves an optimization process aimed at minimizing the cost function, J(θ). This cost function quantifies the squared differences between the predicted and actual values in the dataset. Through an iterative approach, gradient descent is employed to update θ₀ and θ₁ incrementally.
The update rules are governed by the partial derivatives of the cost function with respect to each parameter. Specifically, θ₀ and θ₁ are adjusted in the direction that reduces the overall cost. The learning rate, denoted as α, plays a crucial role in controlling the step size during optimization. Too large a step may lead to overshooting the minimum, while too small a step can result in slow convergence.
The process continues iteratively until convergence, where the parameters achieve values that minimize the cost function, signifying the optimal configuration for the linear regression model. In essence, the choice of θ₀ and θ₁ is a dynamic process driven by the interplay of the cost function, gradient descent, and the learning rate, ensuring that the resulting linear model best captures the underlying patterns within the given dataset.