1. Gaussian correlation function The Gaussian correlation function R(xi x j|) introduced in Sect. 20.3, p....

1. Gaussian correlation function The Gaussian correlation function R(xi − x j|ξ) introduced in Sect. 20.3, p. 768, quantifies the correlation between outputs at two points xi and x j based on the distance between them. For parts

(a) and

(b) below, consider the case of d = 1 input variable.

(a) To investigate the effect of the value of parameter θ on the correlation between outputs at two points, calculate the twenty five correlations R(xi − x j|θ) for θ ∈ {0.5, 2, 5, 20, 100} and |xi − x j|∈{0.1, 0.2, 0.4, 0.6, 0.7}, where |xi − x j| is the absolute value of the distance between xi and x j .

(b) Construct a plot with |xi −x j| on the x-axis and R(xi −x j|θ) on the y-axis, plot R(xi −x j|θ)for each value of θ on the same plot, and comment on the relationship between θ and R(xi −x j|θ).