I want to place a Multiple Regression model into a production system and use the Prediction Interval as a threshold for anomalies.
I've seen how I can calculate the Prediction Interval two ways:
$$ \hat{y} \pm 1.96 \hat{\sigma} \sqrt{1 + \mathbf{X}^* (\mathbf{X}'\mathbf{X})^{-1} (\mathbf{X}^*)'}. $$ Referenced here
and
$$ \hat{y}_h \pm t_{(\alpha/2, n-p)} \times \sqrt{MSE + [\textrm{se}(\hat{y}_{h})]^2} $$ Referenced here
With the first it seems like the prediction interval value changes based on new observations ($\mathbf{X}^*$) and the second appears to be a fixed prediction interval based off of one initial calculation after creating the regression model.
I would personally like to use a fixed prediction interval for the case I'm considering but I'm not certain if I'm thinking about this all wrong.
Are these two prediction interval calculations different from each other?