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Local linear estimation for spatial random processes with stochastic trend and stationary noise.

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This study introduces a new model for estimating stochastic trends in spatial data, finding that treating trends as deterministic or stochastic yields similar estimation accuracy. The convergence rate depends on spatial correlation, not just trend complexity.

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Area of Science:

  • Spatial statistics
  • Time series analysis
  • Nonparametric regression

Background:

  • Spatial random process models often include deterministic or stochastic trends.
  • Existing methods may not fully capture the complexities of stochastic spatial trends.
  • Generalizing time series structural models offers a new approach for spatial trend analysis.

Purpose of the Study:

  • To propose and analyze a novel model for stochastic trends in spatial random processes.
  • To develop a nonparametric estimation method for stochastic trends.
  • To investigate the impact of stochasticity on trend estimation accuracy and convergence rates.

Main Methods:

  • Proposed a generalized structural model for stochastic spatial trends.
  • Employed local linear regression for nonparametric trend estimation.
  • Derived asymptotic mean squared error and analyzed convergence rates.
  • Developed a data-dependent bandwidth selection procedure (Mallows' C_p generalization).

Main Results:

  • The asymptotic mean squared error for stochastic trends is comparable to deterministic trends of similar complexity.
  • The estimation convergence rate is influenced by the decay of the stationary process's correlation function.
  • The proposed bandwidth selection method is effective.

Conclusions:

  • For estimation purposes under stationary noise, the distinction between deterministic and stochastic trends in spatial processes is often negligible.
  • The spatial correlation structure significantly impacts the efficiency of trend estimation.
  • The methodology is validated through simulations and real-world surface temperature anomaly data.