In 1960, R.E. Kalman published his famous paper describing a recursive solution, the Kalman filter, to the discrete-data linear filtering problem. In the following decades, thanks to the continuous progress of numerical computing, as well as the increasing demand for weather prediction, target tracking, and many other problems, the Kalman filter has gradually become one of the most important tools in science and engineering. With the continuous improvement of its theory, the Kalman filter and its derivative algorithms have become one of the core algorithms in optimal estimation. This paper attempts to systematically collect and sort out the basic principles of the Kalman filter and some of its important derivative algorithms (mainly including the Extended Kalman filter (EKF), the Unscented Kalman filter (UKF), the Ensemble Kalman filter (EnKF)), as well as the scope of their application, and also to compare their advantages and limitations. In addition, because there are a large number of applications based on the Kalman filter in data assimilation, this paper also provides examples and classifies the applications of both the Kalman filter and its derivative algorithms in the field of data assimilation.

Camellia oleifera is an oilseed crop that holds significant economic, ecological, and social value. In the realm of Camellia oleifera cultivation, utilizing hyperspectral analysis techniques to estimate chlorophyll content can enhance our understanding of its physiological parameters and response characteristics. However, hyperspectral datasets contain information from many wavelengths, resulting in high-dimensional data. Therefore, selecting effective wavelengths is crucial for processing hyperspectral data and modeling in retrieval studies. In this study, by using hyperspectral data and chlorophyll content from Camellia oleifera samples, three different dimensionality reduction methods (Taylor-CC, NCC, and PCC) are used in the first round of dimensionality reduction. Combined with these methods, various thresholds and dimensionality reduction methods (with/without further dimensionality reduction) are used in the second round of dimensionality reduction; different sets of core wavelengths with equal size are identified respectively. Using hyperspectral reflectance data at different sets of core wavelengths, multiple machine learning models (Lasso, ANN, and RF) are constructed to predict the chlorophyll content of Camellia oleifera. The purpose of this study is to compare the performance of various dimensionality reduction methods in conjunction with machine learning models for predicting the chlorophyll content of Camellia oleifera. Results show that (1) the Taylor-CC method can effectively select core wavelengths with high sensitivity to chlorophyll variation; (2) the two-stage hybrid dimensionality reduction methods demonstrate superiority in three models; (3) the Taylor-CC + NCC method combined with an ANN achieves the best predictive performance of chlorophyll content. The new two-stage dimensionality reduction method proposed in this study not only improves both the efficiency of hyperspectral data processing and the predictive accuracy of models, but can serve as a complement to the study of Camellia oleifera properties using the Taylor-CC method.

The interference of background noise leads to the extremely high spatial complexity of hyperspectral data. Sensitive band selecting is an important task to minimize or eliminate the influence of non-target elements. In this study, Taylor expansion is innovatively used to identify core wavelengths/bands of hyperspectral data. Unlike other traditional methods, this proposed Taylor-CC method considers more local and global information of spectral function to estimate the linear/nonlinear correlation between two wavelengths. Using samples of hyperspectral data with a wavelength range of 350–2500 nm and SPAD for Camellia oleifera, this Taylor-CC method is compared with the traditional PCC method derived from the Pearson correlation coefficient. Using the 240 samples with their different 57 core wavelengths identified by the Taylor-CC method and PCC method, three machine models (i.e., random forest-RF, linear regression-LR, and artificial neural network-ANN) are trained to compare their performances. Their results show that the correlation matrix from the Taylor-CC method represents a clear diagonal pattern with near zero values at most locations away from the diagonal, and all three models confirm that the Taylor-CC method is superior to the PCC method. Moreover, the SPAD spectral response relationship based on machine learning algorithms is constructed, and ANN is the best prediction performance among the three models when using the core wavelengths identified by the Taylor-CC method. The Taylor-CC method proposed in this study not only lays a mathematical foundation for the next analysis of the response mechanism between spectral characteristics and nutrient content of Camellia leaf, but also provides a new idea for the correlation analysis of adjacent spectral bands for hyperspectral signals in many applications.