Overview#
Big Ideas#
Suupose a variable \(y\) depends on be independent variables \(x_1,\dots,x_p\). Use the notation \(\mathbf{x} = (x_1,\dots,x_p) \in \mathbb{R}^p\) to denote the vector of independent variables. The construction of a data-driven model consists of the following steps:
A data-driven model is a function \(f(x,\beta)\) and relates an independent variable \(x\) to an dependent variable $
which depends on a parameter \(\beta\) where \(\varepsilon\) is the error.
When the dependent variable is continuous then we call
Observe samples \(y_0,\dots,y_N\) of the dependent variable and corresponding values \(\mathbf{x}_0,\dots,\mathbf{x}_N\) of the dependent variables
Choose a regression function \(f(\mathbf{x}; \boldsymbol{\beta})\) which depends on some vector of parameters \(\boldsymbol{\beta} = (\beta_0,\beta_1,\dots)\)
Choose a cost function \(C\) which measures the goodness-of-fit of the function \(f\) relative to the observed data
Compute the parameters \(\boldsymbol{\beta}\) which minimize the cost \(C\)
Analyze the regression model \(y = f(\mathbf{x} ; \boldsymbol{\beta}) + \varepsilon\) where \(\varepsilon\) is the random error representing the discrepancy in the approximation
Learning Goals#
Under construction