diff --git a/doc/src/Regression/Regression.do.txt b/doc/src/Regression/Regression.do.txt index c5bfc053a..43990852c 100644 --- a/doc/src/Regression/Regression.do.txt +++ b/doc/src/Regression/Regression.do.txt @@ -1901,13 +1901,13 @@ $X_j$, ($i\neq j$): !split ===== Covariance example ===== -Suppose we have defined three vectors $\hat{x}, \hat{y}, \hat{z}$ with +Suppose we have defined three vectors $\bm{x}, \bm{y}, \bm{z}$ with $n$ elements each. The covariance matrix is defined as !bt \[ -\hat{\Sigma} = \begin{bmatrix} \sigma_{xx} & \sigma_{xy} & \sigma_{xz} \\ +\bm{\Sigma} = \begin{bmatrix} \sigma_{xx} & \sigma_{xy} & \sigma_{xz} \\ \sigma_{yx} & \sigma_{yy} & \sigma_{yz} \\ \sigma_{zx} & \sigma_{zy} & \sigma_{zz} \end{bmatrix}, @@ -1926,11 +1926,11 @@ the exact mean valu\ es. The following simple function uses the _np.vstack_ function which takes each vector of dimension $1\times n$ and produces a $3\times n$ -matrix $\hat{W}$ +matrix $\bm{W}$ !bt \[ -\hat{W} = \begin{bmatrix} x_0 & y_0 & z_0 \\ +\bm{W} = \begin{bmatrix} x_0 & y_0 & z_0 \\ x_1 & y_1 & z_1 \\ x_2 & y_2 & z_2 \\ \dots & \dots & \dots \\ @@ -1941,8 +1941,8 @@ matrix $\hat{W}$ !et which in turn is converted into into the $3\times 3$ covariance matrix -$\hat{\Sigma}$ via the Numpy function _np.cov()_. We note that we can -also calculate the mean value of each set of samples $\hat{x}$ etc +$\bm{\Sigma}$ via the Numpy function _np.cov()_. We note that we can +also calculate the mean value of each set of samples $\bm{x}$ etc using the Numpy function _np.mean(x)_. We can also extract the eigenvalues of the covariance matrix through the _np.linalg.eig()_ function. @@ -2550,7 +2550,7 @@ where we have used that $\mathbb{E} (\mathbf{Y} \mathbf{Y}^{T}) = \sigma^2 \, \mathbf{I}_{nn}$. From $\mbox{Var}(\bm{\beta}) = \sigma^2 \, (\mathbf{X}^{T} \mathbf{X})^{-1}$, one obtains an estimate of the variance of the estimate of the $j$-th regression coefficient: -$\hat{\sigma}^2 (\hat{\beta}_j ) = \hat{\sigma}^2 \sqrt{ +$\bm{\sigma}^2 (\bm{\beta}_j ) = \bm{\sigma}^2 \sqrt{ [(\mathbf{X}^{T} \mathbf{X})^{-1}]_{jj} }$. This may be used to construct a confidence interval for the estimates.