Suppose the x -coordinates of the data (x1, y1), …,(xn, yn) are in mean deviation form, so that

step 1 Let's write the matrix of the coefficients, big x, has the first column is all ones, and the sec

Suppose the x -coordinates of the data (x1, y1), …,(xn, yn)...

Suppose the $x$ -coordinates of the data $\left(x_{1}, y_{1}\right), \ldots,\left(x_{n}, y_{n}\right)$ are in mean deviation form, so that $\sum x_{i}=0 .$ Show that if $X$ is the design matrix for the least-squares line in this case, then $X^{T} X$ is a diagonal matrix.

Key Concepts

Diagonal Matrix

A diagonal matrix is a matrix in which the off-diagonal entries are all zero. In the context of the normal equations for linear regression, if the cross-product matrix (X?X) is diagonal, it implies that the estimation of one parameter does not affect the estimation of the other. This decoupling of parameters simplifies both the theoretical understanding and the computational aspects of parameter estimation.

Centering of Data (Mean Deviation Form)

Centering the data, or using mean deviation form, involves adjusting the predictor variables so that their sum (or mean) is zero. This transformation simplifies the regression analysis as it often eliminates the covariance between the intercept term and the predictor variables, which can lead to simplifications such as a diagonal structure in the cross-product matrix.

Design Matrix

The design matrix is a structured matrix that organizes the data in a way that facilitates the computation of the regression parameters. It typically includes a column of ones to account for the intercept and additional columns for each predictor variable, allowing the formulation of the normal equations in matrix form.

Least Squares Regression

Least squares regression is a method used to fit a model, typically a line, to a set of data by minimizing the sum of the squares of the differences between the observed values and the model’s predictions. This approach leads to solving a set of normal equations that determine the optimal parameters of the model.

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