Universal kriging for spatio‐temporal data

Abstract

In this article we have used wide applicable classes of spatio‐temporal nonseparable and separable covariance models. One of the objectives of this paper is to furnish a possibility how to avoid the usage of complicated covariance functions. Assuming regression model for mean function the analytical expressions for the optimal linear prediction (universal kriging) and mean squared prediction error (MSPE) was obtained. Parameterized spatio‐temporal covariance functions were fitted for the real data. Prediction values and MSPE were presented. For visualization of results on graphics are used free available software Gstat.

Straipsnyje lemos pavidalu pateiktos analitines išraiškos UK (universalaus krigingo) ir MSPE (vidutines kvadratines prognozes klaidos), kai erdves‐laiko kovariacine funkcija yra atskiriama, naudojant sandaugos modeli. Taip pat gautos kovariaciniu modeliu išraiškos, eliminavus laiko itaka stebejimams bei kovariaciniai modeliai atskiriems sezonams, kurie svorinio vidurkio pagalba gali būti apjungti i sezonini vidurkio modeli.

Pateiktu formuliu pagalba, realiems duomenis (Klaipedos jūru tyrimo centro duomenys apie druskingumo kieki devyniose Baltijos jūros stotyse), ivertinti erdves ir laiko kovariaciniu modeliu parametrai ir atlikta optimali prognoze žinomame taške (prieš tai ji eliminavus iš duomenu).

Semivariogramu modeliu grafikai gauti programinio paketo Gstat pagalba. Lyginant gautus rezultatus, galima teigti, kad šiuos duomenis geriausiai aprašo nepriklausomu laike stebejimu kovariacinis modelis.

Key words:

• Spatio‐temporal random process
• nonseparable and separable covariance functions
• universal kriging
• temporal independence
• seasonal average model