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Curse of dimensionality in approximation of random fields

100%

Lifshits M. A. , Tulyakova V.

tom Vol. 26, Fasc. 1

97--112

Consider a random field of tensor product-type X(t), t∈[0,1]d, given by [formula] where (λ(i))i>0∈l2(φi)i>0 is an orthonormal system in L2 [0, 1] and (ξk)k∈Nd are non-correlated random variables with zero mean and unit variance. We investigate the quality of approximation (both in the average and in the probabilistic sense) to X by the n-term partial sums Xn minimizing the quadratic error E‖X‒Xn‖2, In the first part of the paper we consider the case of fixed dimension d. In the second part, following the suggestion of H. Woźniakowski, we consider the same problem for d→∞. We show that, for any fixed level of relative error, approximation complexity increases exponentially and we find the ex- plosion coefficient. We also show that the behavior of the probabilistic and average complexity is essentially the same in the large domain of parameters.

Extremes of order statistics of stationary Gaussian processes

100%

Zhao C.

tom Vol. 38, Fasc. 1

61--75

Let {Xi(t), t ≥ 0}, 1 ≤ i ≤ n, be mutually independent and identically distributed centered stationary Gaussian processes. Under some mild assumptions on the covariance function, we derive an asymptotic expansion of P [formula] ]X(r) (t) ≤ u) as u → ∞, where mr(u) = (P([formula] X(r) (t) > u))−1 (1 + o(1)), and {X(r) (t), t ≥ 0} is the rth order statistic process of {Xi(t), t ≥ 0}, 1 ≤ i, r ≤ n. As an application of the derived result, we analyze the asymptotics of supremum of the order statistic process of stationary Gaussian processes over random intervals.

General and regional cross station assessment of machine learning models for estimating reference evapotranspiration

58%

Zouzou Y. , Citakoglu H.

tom Vol. 71, no. 2

927--947

Significant research has been done on estimating reference evapotranspiration (ET0) from limited climatic measurements using machine learning (ML) to facilitate the acquirement of ET0 values in areas with limited access to weather stations. However, the spatial generalizability of ET0 estimating ML models is still questionable, especially in regions with significant climatic variation like Turkey. Aiming at exploring this generalizability, this study compares two ET0 modeling approaches: (1) one general model covering all of Turkey, (2) seven regional models, one model for each of Turkey’s seven regions. In both approaches, ET0 was predicted using 16 input combinations and 3 ML methods: support vector regression (SVR), Gaussian process regression (GPR), and random forest (RF). A cross-station evaluation was used to evaluate the models. Results showed that the use of regional models created using SVR and GPR methods resulted in a reduction in root mean squared error (RMSE) in comparison with the general model approach. Models created using the RF method suffered from overfitting in the regional models’ approach. Furthermore, a randomization test showed that the reduction in RMSE when using these regional models was statistically significant. These results emphasize the importance of defining the spatial extent of ET0 estimating models to maintain their generalizability.