*Covariance Functions*

The covariance function *<sup>k</sup><sup>x</sup>p*, *<sup>x</sup>q* determines the measure of similarity between outputs *yp* and *yq* based on the input points *<sup>x</sup>p*, *<sup>x</sup>q* locations. For stationary kernels (analyzed herein), only the distance between the point is used in the calculation, a radial basis function [18]. The smaller the distance between points, the higher the similarity between output values expected with a higher value of covariance. Each predictor included in the input vector *x* can be scaled by its length-scale parameter. If the value of the length-scale parameter (found by the maximum marginal likelihood optimization) approaches infinity, the corresponding dimension (predictor) can be ignored. This type of kernel is called an automatic relevance determination (ARD) covariance function. Such covariance functions were implemented to determine the crucial fatigue damage quantities for the tested CuZn37 brass. The distance *r* between points *<sup>x</sup>p*, *<sup>x</sup>q* is defined as

$$r = \sqrt{\left(\mathbf{x}\_p - \mathbf{x}\_q\right)^T \mathbf{M} \left(\mathbf{x}\_p - \mathbf{x}\_q\right)}\tag{A10}$$

where *M* = diag(*li*)−<sup>2</sup> is a diagonal matrix of length-scale parameters *li* = *l*1, ... , *ld* (each is assigned to a particular predictor). Four popular covariance functions were analyzed for fatigue life prediction, as presented below:

• Exponential (EX)

$$k(\mathbf{x}\_{p\prime}\mathbf{x}\_q) = \sigma\_k^2 \exp(-r) \tag{A11}$$

• Matern 3/2 (M3/2)

$$k(\mathbf{x}\_{p}, \mathbf{x}\_{q}) = \sigma\_{k}^{2} \left( 1 + \sqrt{3}r \right) \exp\left(-\sqrt{3}r \right) \tag{A12}$$

• Matern 5/2 (M5/2)

$$k(\mathbf{x}\_{\mathcal{P}}, \mathbf{x}\_{\emptyset}) = \sigma\_k^2 \left( 1 + \sqrt{5}r + \frac{5}{3}r^2 \right) \exp\left( -\sqrt{5}r \right) \tag{A13}$$

• Rational quadratic (RQ)

$$k\left(\mathbf{x}\_{\mathcal{P}}, \mathbf{x}\_{\emptyset}\right) = \sigma\_k^2 \left(1 + \frac{r^2}{2\alpha}\right)^{-\alpha} \tag{A14}$$

• Squared exponential (SE)

$$k\left(\mathbf{x}\_{\mathcal{V}}, \mathbf{x}\_{\eta}\right) = \sigma\_k^2 \exp\left(-\frac{1}{2}r^2\right). \tag{A15}$$
