# Predictive Analysis of Vehicular Lane Changes: An Integrated LSTM Approach
## 2 Methodologies
### Algorithms and Program
#### 2.2.1 Maneuver Classification  
Regression  
The regression model consists of multiple methods that can predict the target variable (y) based on one or multiple features (x). Mathematically, we can define the regression model as a function of the label \(y\) and feature vector \(x\). In this experiment, we use four kinds of regression methods--logistic regression [15], stochastic gradient descent (SGD) regression [16], ridge regression [17], and lasso regression [18]--to predict the lane change maneuver based on the input features. The Sk-learn package will be used to build the models. Logistic regression measures the relationship between the dependent variable (the label to be predicted) and one or more independent variables (features) by using its inherent logistic function (sigmoid function) to estimate the probability. The linear regression algorithm aims to create a polynomial function to predict the label value of a new sample. In the normal gradient descent process, the cost of all samples in the data set is calculated for each iteration, and the value of the parameter \(\theta\) is adjusted accordingly to make the cost function \(J(\theta)\) gradually approach the most merit (lowest point). The stochastic gradient descent (SGD) method randomly selects a sample each time to adjust the parameters. Compared with the default gradient descent (using all samples to calculate the parameters), stochastic gradient descent solves the computation demand problem caused by the excessive number of samples. Ridge regression is a biased estimation regression method dedicated to collinearity data analysis and is essentially an improved least squares estimation method. Ridge regression abandons the least square method's unbiased nature and obtains more realistic and reliable regression coefficients at the cost of losing some information and reducing accuracy. Its fitting to ill-conditioned data is stronger than the least squares method. Like ridge regression, the least absolute shrinkage and selection operator also adds a penalty to the regression coefficient's absolute value. Aside from that, it can reduce bias and improve the accuracy of linear regression models. Lasso regression uses absolute values instead of square values in the penalty part. This leads to a penalty (that is, it constrains the sum of the estimates' absolute values) to make some parameter estimates equal to zero. Having explored regression techniques, another powerful method comes to light which specializes in binary classification tasks. The support vector machine (SVM) [19] is a perfect algorithm for binary classification problems. The data can be mapped to higher dimensions and be separated by a hyperplane. Since the input data might be nonlinearly separable, both linear support vector classification models and support vector classification models using nonlinear kernels will be used. To design a better SVM model, the performance of the models using different kernel functions will be compared. The kernel functions are as follows:  
* Polynomial: \((x^{T}y+1)p\);
* Radial basis function: \(\exp(-\frac{1}{2\sigma}||x-y||^{2})\);* Sigmoid: \(\tanh(\beta_{0}x^{T}y+1)\). While SVMs provide a distinct approach to binary classification, neural networks present another versatile avenue for maneuver classification, especially for complex, nonlinear data. Multilayer Perceptron (MLP)  
The multilayer perceptron [20] is a class of feedforward artificial neural networks. An MLP utilizes a supervised learning technique of backpropagation for training. Its multiple layers and nonlinear activation function can distinguish data that are not linearly separable. An MLP in the experiment is built with one input layer with the dimensions of the input training data, 2 hidden layers with 100 neurons and an ReLU activation function, and 1 output layer with a tanh activation function.