Statistical Distances and Their Applications to Biophysical Parameter Estimation: Information Measures, M-Estimates, and Minimum Contrast Methods

Abstract

Radiative transfer models predicting the bidirectional reflectance factor (BRF) of leaf canopies are powerful tools that relate biophysical parameters such as leaf area index (LAI), fractional vegetation cover f V and the fraction of photosynthetically active radiation absorbed by the green parts of the vegetation canopy (f APAR ) to remotely sensed reflectance data. One of the most successful approaches to biophysical parameter estimation is the inversion of detailed radiative transfer models through the construction of Look-Up Tables (LUTs). The solution of the inverse problem requires additional information on canopy structure, soil background and leaf properties, and the relationships between these parameters and the measured reflectance data are often nonlinear. The commonly used approach for optimization of a solution is based on minimization of the least squares estimate between model and observations (referred to as cost function or distance; here we will also use the terms "statistical distance" or "divergence" or "metric", which are common in the statistical literature). This paper investigates how least-squares minimization and alternative distances affect the solution to the inverse problem. The paper provides a comprehensive list of different cost functions from the statistical literature, which can be divided into three major classes: information measures, M-estimates and minimum contrast methods. We found that, for the conditions investigated, Least Square Estimation (LSE) is not an optimal statistical distance for the estimation of biophysical parameters. Our results indicate that other statistical distances, such as the two power measures, Hellinger, Pearson chi-squared measure, Arimoto and Koenker-Basset distances result in better estimates of biophysical parameters than LSE; in some cases the parameter estimation was improved by 15%.Remote Sens. 2013, 5 1356

Multiple Cost Functions and Regularization Options for Improved Retrieval of Leaf Chlorophyll Content and LAI through Inversion of the PROSAIL Model

“…However, in case of outliers and nonlinearity, the residuals are distorted and therefore the key assumption for using RMSE (maximum likelihood estimation with the Gaussian noise) is violated [36]. The latter authors suggested that alternative CFs may provide a more robust way to estimate biophysical parameters since they allow retrievals for cases where errors are not normally distributed and allow dealing with nonlinear high-parametric problems.…”

Retrieval of Seasonal Leaf Area Index from Simulated EnMAP Data through Optimized LUT-Based Inversion of the PROSAIL Model

“…Some studies focused the influence of an appropriate band selection for the retrieval [25,34,35], others analyzed the potential of adding artificial noise to the data in an additive [32], multiplicative [31,36,37] or a combined form [33]. Besides the very well approved RMSE [23,26,31], alternative cost functions have been subject of various studies [36][37][38]. To counteract the influence of the ill-posed problem, the use of a multiple number of best fits is widely accepted [15,26,27,37], yet investigations of the averaging method are still rare, like the one from Darvishzadeh et al [25], who analyzed the potential of mean and median.…”

“…Recently, several studies have investigated the potential of alternative cost functions for the retrieval of best fits between measured and simulated data (e.g., [36][37][38]). One of the most common measures in this context is the root mean square error (RMSE), which has been applied in several studies (e.g., [23,26,31]).…”

Hyperspectral Classification Based on Texture Feature Enhancement and Deep Belief Networks

“…Hyperspectral classification is a key technique employed in aforementioned applications. A majority of classification methods have been promoted in the last several decades to distinguish physical objects and classify each pixel into a unique land-cover label, such as maximum likelihood [5], minimum distance [6], K-nearest neighbors [7,8], random forests [9], Bayesian models [10,11], neural networks, etc., and their improvements [12][13][14][15]. Among these supervised classifiers, one of the most important classifiers is kernel-based support vector machine (SVM), which can also be considered as a kind of neural network.…”