# Terrain Ruggedness Index for optimization problem

If I want to study the smoothness of the energy landscape of a cost function, is there any metric similar to Terrain Ruggedness Index used in geology?

For a fixed distance $$i$$, the landscape correlation function is defined as $$$$\rho(i) = \frac{\langle f(s)\cdot f(s^\prime)\rangle_{d(s,s^\prime)=i}-\langle f(s)\rangle^2}{\langle f^2(s)\rangle - \langle f(s)\rangle^2}$$$$ where $$f(s)$$ is the objective function value of a solution $$s$$, $$\langle f(s) \rangle$$ is the average of the solution value for all the solutions $$s$$ in the search space $$S$$, and $$\langle f(s)\cdot f(s^\prime)\rangle_{d(s,s^\prime)=i}$$ is the average of $$f(s)\cdot f(s^\prime)$$ for all the pairs of solutions $$s,s^\prime$$ at distance $$i$$ in $$S$$.
Of course, the formula is not usable in practice, because it requires to analyze the entire search space. One alternative is to perform a random walk on the search space, and compute the empirical autocorrelation function on the sequence of solution values $$f_1, f_2, \dots, f_m$$ observed: $$$$r(i) = \frac{1/(m-i)\cdot \sum_{k=1}^{m-i}(f_k-\bar{f})\cdot(f_{k+i}-\bar{f})}{1/m \cdot \sum_{k=1}^m (f_k-\bar{f})^2}$$$$ where $$m$$ is the length of the random walk, and $$\bar{f}$$ is the average of the sequence of values $$f_1, f_2, \dots, f_m$$ observed in the random walk. This formula gives information about the correlation between solutions at distance $$i$$ in the random walk.