Describe how projections play a role in the least squares solution?
In the least squares solution, projections play a crucial role in finding the best-fitting line or plane for a given set of data points.
To understand this, let's start with the concept of a projection. In mathematics, a projection is the process of mapping a vector onto a subspace. In the context of regression analysis, we use projections to map the data points onto a linear subspace spanned by the independent variables.
The objective of the least squares method is to minimize the sum of the squared residuals between the observed data and the predicted values on the fitted line or plane. The residuals are the vertical distances between the observed data points and the fitted line or plane.
To find the least squares solution, we need to find the coefficients or parameters that minimize the sum of the squared residuals. These coefficients represent the values of the regression equation that best fit the data.
In order to do this, we utilize projections. We project each data point onto the linear subspace created by the independent variables. This projection gives us the predicted or expected value of the dependent variable based on the estimated coefficients.
The squared residuals are then calculated as the difference between the observed value and the projected value. These squared residuals indicate how far each data point is from the fitted line or plane.
By minimizing the sum of these squared residuals, we minimize the overall discrepancy between the observed data and the predicted values. This corresponds to finding the best-fitting line or plane. The least squares solution finds the values of the coefficients that achieve this minimum sum of squared residuals.
In summary, projections play a key role in the least squares solution by mapping the data points onto the linear subspace spanned by the independent variables, allowing us to calculate the residuals and find the best-fitting line or plane that minimizes the sum of these residuals.