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I am working with simulation software to generate training data. Specifically, I want to create a regression model predicting the weight of plastic parts from an injection molding process.

Besides an accurate prediction, I am interested in quantifying a prediction interval, which I am computing as:

$$\hat\theta x_{new}\pm t_{\frac{\alpha}{2},n-p}s\sqrt{1+x_{new}(X^\top X)^{-1}x_{new}^\top}$$

Since I am obtaining the training data from simulation, I would like to specify the design matrix $X$ in such a way, that I tend to minimize the average prediction interval (in a relevant interval for future predictions $[L,U]$). In the simple two-dimensional case, I can derive the following statements: Since $$SE(y)=MSE*\left[ 1+\frac{1}{n}+\frac{(x_{new}-\bar X)^2}{\sum_{\forall i} (x_i-\bar X)^2} \right]$$

it makes sense to center the data collection efforts on the interval where future predictions are to be made to minimize $x_{new}-\bar X$, , while the observations $x_i$ should spread as widely from the center as possible, to maximize the term $\sum_{\forall i} (x_i-\bar X)^2$. Intuitively, I understand that a straight line fit is 'fixed' more by distant points than by close points.

I would like to generalize this to the multivariate regression setting, to specify how to simulate data (given the objective of minimizing the uncertainty intervals). From this post:

Multicollinearity: Why does highly correlated columns in the design matrix lead to high variance of the regression coefficient?

I understand that it would make sense to generate the data s.t. the columns of $X$ are not linearly dependent. Intuitively, this sounds to me like a similar version to the above statement. I am left with the question: What is a mathematically sound approach of specifying the Design matrix to minimize the average prediction interval in the multivariate regression case?

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You always can generate a "degenerate case" where the number of $\beta$s equals the number of observations (without complete colineariaty). In this case you will have a $0$ variance, i.e., $\hat{Y} = Y$.

For a more realistic case, when you have $n>>p$ and you cannot control the noise term $\varepsilon$, then $\mathrm{X}$ with (piarwise) orthogonal columns will give you the most "stable" $\mathrm{(X'X)}^{-1}$ matrix.

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  • $\begingroup$ Thank you very much for the input! Am I right in the assumption that I should thus try to find columns s.t. the dot products of pairwise column vectors get as close to zero as possible? $\endgroup$
    – Sean
    Jul 19, 2017 at 19:00
  • $\begingroup$ Yes, it is right. $\endgroup$
    – V. Vancak
    Jul 19, 2017 at 19:01
  • $\begingroup$ Again, thanks alot! It seems to me that this is a nonlinear optimization problem, where the objective is to minimize $\sum_{\forall i,j; i\not=j}|v_i \cdot v_j|$ with the variables $x_i,x_j$ representing the vector entries for vector $i$ and $j$ respectively. Do you see any heuristical approach to intelligently, not necessarily optimally, selecting the vectors, i.e. the $x_is$ & $x_js$? $\endgroup$
    – Sean
    Jul 19, 2017 at 20:08
  • $\begingroup$ Vectors like $(x_1, x_2, x_3, 0,0,0)$ and $(0,0,0,y_1, y_2, y_3)$ are trivial examples. Maybe following this logic will help. Anyway, you're welcome! (BTW, you can up-vote/accept the answer if it helps you. $\endgroup$
    – V. Vancak
    Jul 20, 2017 at 10:03

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