2. Power exponential correlation function The Gaussian correlation function (20.3.3) is a special case of a Power exponential correlation function. The latter is given by R(xi − x j|ξ) =  d k=1 exp(−θk |xik − x jk | pk ) where ξ represents the parameters (θ1,..., θd , p1,..., pd ), with all θk ≥ 0, and 0 < pk ≤ 2.

(a) Suppose there is d = 1 input variable. To investigate the effect of θ on the correlation between outputs at two points xi and x j for the Power exponential correlation function, calculate the nine correlations R(xi − x j|θ) for θ ∈ {0.5, 5, 100} and |xi − x j|∈{0.1, 0.4, 0.7}, for each value of p ∈ {0.5, 1, 1.5, 2}.

(b) Construct four plots, one for each value of p, 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) for each value of p.