# Model-Based and Model-Free Point Prediction Algorithms for Locally Stationary Random Fields
## 7 Conclusions and Future Work  
In this paper, we investigate the problem of one-sided prediction over random fields that are stationary only across a limited part of their entire region of definition. For such locally stationary random fields, we develop frameworks for point prediction using both a Model-Based approach, which includes a coordinate changing trend and/or variance and also by using the Model-Free principle proposed by [21; 22]. We apply our algorithms to both synthetic data as well as a real-life dataset consisting of images from the CIFAR-10 dataset. In the latter case, we obtain the best performance by using the Model-Free approach and thereby demonstrate the superiority of this technique versus the Model-Based case where an additive model is assumed arbitrarily for purposes of prediction. In future work, we plan to investigate both Model-Based and Model-Free prediction using random fields with nonuniform spacing of data as well as consider extending our algorithms for estimating prediction intervals.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Prediction Method** & **Residual Type** & **MSE** \\ \hline MB-LC & P & \(1.98\times 10^{-1}\) \\ \hline  & F & \(2.20\times 10^{-1}\) \\ \hline MB-LL & P & \(1.79\times 10^{-1}\) \\ \hline  & F & \(1.95\times 10^{-1}\) \\ \hline MF-LC & P & \(1.79\times 10^{-1}\) \\ \hline  & F & \(2.12\times 10^{-1}\) \\ \hline MF-LLH & P & \(1.60\times 10^{-1}\) \\ \hline  & F & \(1.89\times 10^{-1}\) \\ \hline MF-LLM & P & \(1.64\times 10^{-1}\) \\ \hline  & F & \(1.70\times 10^{-1}\) \\ \hline LL & Not Applicable & \(2.12\times 10^{-1}\) \\ \hline LLH & Not Applicable & \(2.38\times 10^{-1}\) \\ \hline LLM & Not Applicable & \(2.14\times 10^{-1}\) \\ \hline \hline \end{tabular}
\end{table}
Table 2: Point Prediction performance for CIFAR-10 dataset.  
Figure 4: Sample images from the CIFAR-10 dataset with label dog (Note: Here, full images are shown although only part of it is used for prediction).  
**Author Contributions:** Software and experiments--S.D. and Y.Z.; conceptualization and writing--S.D. and D.N.P. All authors have read and agreed to the published version of the manuscript.  
**Funding:** D.N.P. was partially supported by NSF grant DMS 19-14556. S.D. and Y.Z. did not receive any external funding.  
**Institutional Review Board Statement:** Not applicable.  
**Informed Consent Statement:** Not applicable.  
**Data Availability Statement:** The CIFAR-10 publicly available dataset was analyzed in this study. This data can be found in [32]. R code for the algorithms discussed in this paper can be found at [[https://github.com/srinjoyd/randomfields_pp](https://github.com/srinjoyd/randomfields_pp)]([https://github.com/srinjoyd/randomfields_pp](https://github.com/srinjoyd/randomfields_pp)) (accessed on 24 July 2023).  
**Acknowledgments:** This research was partially supported by NSF grant DMS 19-14556. The authors would like to acknowledge the Pacific Research Platform, NSF Project ACI-1541349 and [PERSON] (PI, Calit2 at UCSD) for providing the computing infrastructure used in this project.  
**Conflicts of Interest:** The authors declare no conflict of interest.