Functional single-index $k$-nearest neighbor relative error regression under weak dependence

We study estimation of the relative error regression function in a functional single-index regression model under quasi-associated dependence. We introduce a $k$-nearest neighbors ( $k$-NN) estimator whose smoothing adapts to the local concentration of the covariate trajectories. Almost complete convergence rates are established under mild small-ball and dependence conditions. A simulation study compares the proposed $k$-NN estimator with its kernel counterpart, and a real data application to air quality forecasting (NOx $\to$ ozone) illustrates it's practical performance.