azay said:Given a homogeneous linear least squares problem:
[tex]
A^{T}y=0
[/tex]
What is the difference between minimizing
[tex]
y^{T}AA^{T}y
[/tex] (the least square error)
and:
[tex]
y^{T}AA^{+}y=y^{T}A(A^{T}A)^{-1}A^{T}y
[/tex]
?
Thanks.
Homogeneous least squares is a mathematical method used to find the best fitting line or curve for a set of data points. It is used to minimize the sum of the squared differences between the observed data and the predicted values.
Homogeneous least squares is a special case of ordinary least squares, where the intercept term is forced to be zero. This means that the resulting line or curve will pass through the origin, while ordinary least squares allows for a non-zero intercept term.
The main assumption of homogeneous least squares is that the errors in the data are normally distributed. Additionally, the data should be linearly related and the errors should be independent and have equal variances.
The best fitting line or curve is determined by finding the coefficients that minimize the sum of the squared differences between the observed data and the predicted values. This is typically done using a mathematical technique called gradient descent.
Homogeneous least squares is commonly used in many fields, including statistics, machine learning, econometrics, and engineering. It is a fundamental tool for data analysis and is often used to model relationships between variables and make predictions.