2.6.1. Binary Logistic Regression Classification (BLRC)

Logistic regression is a generalized linear regression model that can explain the relationships between variables. This model has been extensively applied to classification and regression problems. Because lodging recognition is a binary classification problem in this study, we adopted a binary logistic regression model as a classifier [40].

In this binary logistic regression model, the maize sample values extracted from di fferent image features form the independent variables. The predicted dependent variable is a function of the probability that maize lodging will occur. Moreover, there is a basic assumption about the dependent variable: the probability of the dependent variable takes the value of 1 (positive response). According to the logistic curve, the value of the dependent variable can be calculated using the following formula [41]:

$$P(y=1|X) = \frac{\exp(\sum \text{CX})}{1 + \exp(\sum \text{CX})} \tag{6}$$

where *P* represents the probability of maize lodging (the dependent variable); *X* indicates the independent variables; and *C* represents the estimated parameters.

The logit transformation was performed to linearize the above model and eliminate the 0/1 boundaries for the original dependent variable, which is the probability. The new dependent variable is boundless and continuous within the range from 0 to 1. After logit transformation, the above model is expressed by the following equation:

$$\ln\left(\frac{P}{(1-P)}\right) = \text{Co} + \text{C}\_1\text{X}\_1 + \text{C}\_2\text{X}\_2 + \text{C}\_3\text{X}\_3 + \dots + \text{C}\_k\text{X}\_k\tag{7}$$

where *P* represents the probability of maize lodging (the dependent variable), *X1, X2, X3,* ... *, Xk* are the independent variables, and *C0, C1, C2,* ... *, Ck* represent the estimated parameters.

The binary logistic regression model for predicting maize lodging probability encoded by R programming language was trained using R Studio software. The encoded program automatically determined the coefficients of all independent variables according to the relationship between independent and dependent variables. The fitted model was inputted into the "Band Math" module of ENVI software to obtain the lodging probability map of the study area. Finally, a uniform and optimal probability threshold was determined for all pixels to judge whether they represented lodging.
