Modified Functional Principal Component Regression Open Access

Abstract

Using simulated temperature data as a proof of concept, we present a novel approach to prediction via the development and application of Modified Functional Principal Component Regression (MFPCR). By employing bivariate splines over triangulations in a functional linear regression model, this study offers a proof of concept for the modified approach that include additional data to improve the Function Principal Component Regression (FPCR). The mock temperature prediction problem has few complexities and clear influencing factors. Our comparative analysis of MFPCR and FPCR reveals insights into predictive accuracy, uncertainty quantification, and the spatial distribution of functional data, setting the stage for more precise neighborhood-level forecasting in future investigations. 

Table of Contents

1 Abstract 2

2 Introduction and Background 2

3 Modified Functional Principal Component Regression 5

3.1 Autoregressive Process........................ 6

3.2 Computational Method Outline: .................. 8

4 Simulated Temperature Data 9

4.1 Constructing a Temperature Data Set ............... 10

4.2 Topological Features ......................... 10

4.3 North-South Effect.......................... 10

4.4 Hourly Variation ........................... 10

4.5 Seasonality .............................. 11

4.6 Adding Noise to the Signal ..................... 12

4.7 Combining the Features to Generate the Data. . . . . . . . . . . 12

5 Simulated Temperature Experiments 13

5.1 Fixed and Adjusted Attributes of the Simulations . . . . . . . . . 13

5.2 Principal Component Analysis ................... 14

5.3 Assessing the Prediction Function through MSE Comparison Amongst Adjusted Attributes ......................... 17

5.3.1 Base Case: Mountain Valley................. 18

5.3.2 Altered Input Function ................... 19

5.3.3 Weight Adjustments ..................... 19

5.3.4 Noise ............................. 19

5.3.5 Number of Eigenvalues.................... 20

5.3.6 Overall MSE Comparison .................. 20

6 Comparing the Predictions 21

6.1 MFPCR vs. FPCR.......................... 22

6.2 Comparing Prediction Functions with Updated Station Data . . 22

7 Conclusion and Extension to the Ground-Level Ozone Application 25 

About this Honors Thesis

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School	Emory College
Department	Mathematics
Degree	B.S.
Submission	Honors Thesis
Language	English
Research Field	Applied Mathematics
Keyword	Ground-Level Ozone Functional Data Analysis Principal Component Analysis
Committee Chair / Thesis Advisor	Bree Ettinger, Emory University
Committee Members	Shomu Banerjee, Emory University Talea Mayo, Emory University Julianne Chung, Emory University

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