where E[f(st)] and var(f(st**)) are the expected value and the variance of** f(st). Estimates r(k) of autocorrelation coefficients ρ(k**) can be calculated**
with a time series (s1, s2, . . . , sL**) of length** L :

$$r(k)={\frac{\sum_{j=1}^{L-k}(f(s_{j})-{\bar{f}})(f(s_{j+k})-{\bar{f}})}{\sum_{j=1}^{L}(f(s_{j})-{\bar{f}})^{2}}}$$

where ¯f =1L
PL
j=1 f(sj ), and L >> **0. A random walk is representative of**
the entire landscape when the landscape is statistically isotropic. In **this case,**
whatever the starting point of random walks and the selected neighbors during the walks, estimates of r(n**) must be nearly the same. Estimation error**
diminishes with the walk length. The correlation length τ **measures how the autocorrelation function decreases** and it summarizes the ruggedness of the landscape: the larger the **correlation**
length, the smoother is the landscape. Weinberger's definition τ = −1 ln(ρ(1))
makes the assumption that the autocorrelation function decreases exponentially. Here we will use another definition that comes from a more general analysis of time series, the Box-Jenkins approach [32], introduced in **the field**
of fitness landscapes by Hordijk [33]. The time series of fitness values will be approached by an autoregressive moving-average (ARMA) model. In ARMA(**p, q**)
model, the current value depends linearly on the p **previous values and the** q previous white noises.

$$f(s_{t})=c+\sum_{i=1}^{p}\alpha_{i}f(s_{t-i})+\epsilon_{t}+\sum_{i=1}^{q}\beta_{i}\epsilon_{t-i}{\mathrm{~where~}}\epsilon_{t}{\mathrm{~are~white~noises.}}$$

The approach consists in the iteration of three stages [32]. The **identification**
stage consists in determining the value of p and q **using the autocorrelation**
function (acf) and the partial autocorrelation function (pacf) of the time series.

The estimation stage consists in determining the values c, αi and βi **using the**
pacf. The significance of this values is tested by a t-test. The value **is not** significant if t-test is below 2. The diagnostic checking stage **is composed of**
two parts. The first one checks the adequation between data and **estimated**
data. We use the square correlation R2 **between observed data of the time**
series and estimated data produced by the model and the **Akaide information** criterion AIC:

$$A I C(p,q)=l o g(\hat{\sigma}^{2})+2(p+q)/L\mathrm{~where~}\hat{\sigma}^{2}=L^{-1}\sum_{j=1}^{L}(y_{j}-\hat{y}_{j})^{2}$$

The second one checks the white noise of residuals which is the difference between observed data value and estimated values. For this, the autocorrelation of residuals and the p-value of Ljung-Box test are computed.