Modeling Uncertainty Quantification of NDVI of Agricultural Fields through Bayesian Linear Regression in Time Series Prediction

Abstract

The current research discusses the applications of Bayesian linear regression to predict the uncertainty of remote sensing data. To predict the uncertainty, the study considered the SENTINEL-2 satellite data of agricultural fields of Uttar Pradesh state of India. Using the stratified sampling method in Google Earth Engine, the random points generated are mapped to agricultural fields. Data was collected in the form of maximum Normalized Difference Vegetation Index (NDVI) values of each agricultural field. The dynamics of the time series predictions were explored with Bayesian linear regression, a probabilistic deep learning method. The model uncertainty defined as epistemic uncertainty is evaluated with the prior and posterior probability parameters of Bayesian statistics in linear regression. The number of regression lines predicted for the same data shows evidence of uncertainty. The Bayesian linear regression models show evidence of high uncertainty for the predicted NDVI values. The variation in model uncertainty is measured by dividing the dataset into samples and it is observed that with increase in data the uncertainty is reduced. Also, with the increase in data, the posterior density becomes sharper which corresponds to a decrease in variance. Further, the study extended the concept of regression analysis with Gaussian basis functions to determine the effect of model uncertainty with an increase in data. The analysis has shown the same result in knowing the effect of uncertainty with the increase in data. Further, a nonlinear polynomial regression model with a Gaussian distribution as a basis function was developed to evaluate the marginal probabilities of the evidence function in capturing the uncertainty with varying degrees of freedom. The polynomial regression with a Gaussian distribution using Bayesian statistics has captured the uncertainty and confirmed that the uncertainty is captured at lower degrees of freedom.

Keywords: SENTINEL-2, epistemic uncertainty, normalized difference vegetation index, bayesian regression, gaussian basis function