# Weighted least squares best fit plane

• daviddoria
In summary, the best fit plane through the center of mass can be found by multiplying the scatter matrix with the weighting kernel and then finding the eigenvalues.
daviddoria
I know that the plane through the center of mass whose normal is the eigenvector corresponding to the smallest eigenvalue of the scatter matrix of a set of points is the best fit plane. I now want to do a "weighted least squares" - would I simply multiply the
$\sum(x_j-\overline{x})(x_j-\overline{x})^T$
(from here http://en.wikipedia.org/wiki/Scatter_matrix)

by the weighting kernel

$\sum(x_j-\overline{x})(x_j-\overline{x})^T w(x_j-\overline{x})$

and then proceed to find the eigenvalues as normal?

Thanks,

David

Yes, that is correct. You would multiply the scatter matrix with the weighting kernel and then proceed to find the eigenvalues as normal.

## 1. What is the weighted least squares method for finding the best fit plane?

The weighted least squares method is a mathematical technique used to find the best fitting plane through a set of data points. It takes into account the uncertainties or errors associated with each data point and assigns weights to them accordingly, resulting in a more accurate fit.

## 2. How does the weighted least squares method differ from the ordinary least squares method?

In the ordinary least squares method, all data points are given equal weight, while in the weighted least squares method, the weights are determined based on the uncertainties or errors associated with each data point. This allows the weighted least squares method to account for outliers and produce a more accurate fit.

## 3. What is the significance of the weights in the weighted least squares method?

The weights in the weighted least squares method represent the importance or influence of each data point in the fitting process. Data points with higher weights have a greater impact on the final fit, while data points with lower weights have less influence.

## 4. How is the best fit plane determined using the weighted least squares method?

The best fit plane is determined by minimizing the sum of squared residuals between the actual data points and the plane. This is achieved by adjusting the parameters of the plane (slope and intercept) until the sum of squared residuals is minimized.

## 5. In what situations is the weighted least squares method particularly useful?

The weighted least squares method is particularly useful when there are uncertainties or errors associated with the data points, such as in experimental data or survey data. It can also be used when there are outliers in the data set, as it can account for their influence and still provide an accurate fit.

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