3.1.1. ARIMA Model
TCE data are a non-stationary series, also known as weak stationarity, characterized by dependence, i.e., the value of a specific time in the future depends on its past information. The ARIMA model is a time series and prediction method [
30]. Its basic principle is first to use the d-order difference to stabilize the non-stationary time series and then use Autoregressive AR(p), Moving Average MA(q), Autocorrelation Function (ACF), and Partial Correlation Coefficient (PCF) to identify the model for the stabilized time series. This model is often used for time series analysis.
First, the primary variables involved in the formula are described.
is a time series,
represents the
t-th point in the time series (
t is an integer from 1 to N), and N represents the length of the series. In this model, relevant variables are shown in
Table 1.
Secondly, the stationarity of the time series is tested. Generally, ACF and PACF functions are used to judge the type. For ACF, the calculation formula is as follows.
The correlation coefficient ACF (unbiased) is as follows:
The correlation coefficient ACF (biased) is as follows:
For PACF, the calculation process is more complex, and the following assumptions are generally made first:
In this formula,
(
) is the linear correlation coefficient,
is noise, i.e., we assume that the point
is linearly related to the first k points, as follows
,
,
,
. PACF represents the correlation between
and
. Therefore, the PACF formula of the sequence is as follows:
The solution process of is omitted here, which can be determined by programming.
If the time series
fails to pass the stationarity test, the original data must be stabilized and transformed into a weakly stationary series by difference. In practical application, d is usually equal to 1 or 2, and the determination method is that the data pass the stationarity test after d-order difference. ARMA(p, q) model has many identification methods, but it is generally identified by autocorrelation coefficient (ACF) and partial correlation coefficient (PCF). If the d-order difference of
is a stable ARIMA process, it is called the autoregressive moving average summation model. The solution formula of d is as follows:
If follows model, is said to be an process.
ARIMA includes three components: autoregressive, differential, and moving average. p, d, and q represent autoregressive order (Lags of time series data used in the prediction model, also called AR/Auto Recursive term.), difference number (How many orders of real-time data need to be differentiated to obtain stable data, also called Integrated term.), and moving average order (Lags of prediction error used in the prediction model, also called MA/Moving Average.), respectively, and the Bayesian Information Criterion (BIC) can be used to calculate the BIC value to select p value and q value.
Bayesian decision theory is a part of BIC. It means that under incomplete reporting, some unknown states are estimated with subjective probability, and then the occurrence probability is modified with the Bayesian formula. Finally, the expected value and the modified possibility are used to make the optimal decision, with the formula as follows:
where
h is the number of model parameters, and
h = 5 is taken in this paper.
L is the likelihood function and
is the penalty term. When the dimension is too large, and the training sample data are small, dimension disaster can be effectively avoided. The order of the optimal
model is the
p-value, and
q-value that minimizes the BIC value.
For the time series that have passed the stationarity test, the stationary process
can be used to replace the position of the unstable
in the ARIMA model, namely:
Represented by the lag operator:
where, {
} is a white noise process,
model after the d-order difference change is called
model. Equation (15) is equivalent to the following equation:
Finally, the that has been established is used to predict the changes of subsequent index values, and the final prediction results are obtained. ARIMA model is suitable for short-term prediction. In this paper, an adaptive method is proposed to predict carbon emissions in the field of transportation by using JupyterLab 3.0 software.
3.1.2. Fuzzy Comprehensive Evaluation Method
The concept of fuzzy set theory was put forward by the American automatic control expert Zadeh in 1965 to express the uncertainty of things, which is an important part of fuzzy mathematics and the theoretical source of fuzzy comprehensive evaluation and analysis [
31].
The fuzzy comprehensive evaluation method blurs all aspects and factors of the evaluation object and then gets the final evaluation result through the fuzzy comprehensive operation. The basic steps of the fuzzy comprehensive evaluation method include establishing a factor set for a comprehensive evaluation, establishing an evaluation set for a comprehensive evaluation, determining the fuzzy comprehensive evaluation matrix, determining the weights of each factor, and calculating the comprehensive evaluation index.
- Step 1:
Establish a comprehensive evaluation factor set.
A factor set is a general set, usually represented by U, composed of various factors that affect the evaluation object, and these factors have varying degrees of ambiguity. Establishing a comprehensive evaluation factor set is the foundation of fuzzy comprehensive evaluation. Due to the different degrees of correlation between different factors and evaluation objects, the selection of indicators will also affect the final evaluation results.
where
represents the factors that affect the evaluation object, and m is the number of evaluation indicators.
- Step 2:
Establish an evaluation set of a comprehensive evaluation.
In the factor set, each factor influences the evaluation results differently. To this end, give the weighing
for each factor
, and the fuzzy set of the weight collection of each factor, which is represented by
.
where a represents the elements of assessment;
represents the number of evaluation concentration elements, which is determined by the nature of the evaluation object and the evaluation process. The specific evaluation level is determined by the appropriate language the evaluation object uses, such as “strong, medium, weak” language.
- Step 3:
Determine the fuzzy comprehensive appraisal matrix.
If the membership grade of the first element in the factors u in the evaluation set A is R11, the results of the bullies of the first element single factor evaluation are represented as
. The matrix
is composed of m single-factor evaluation sets
, which is called a fuzzy comprehensive evaluation matrix.
- Step 4:
Determine the weight of each factor.
In the evaluation process, the importance of various factors will be different. Therefore, give the factors
a weight
, and the weight collection of each factor is represented by
E:
The weight has an important impact on the results of the final model. Therefore, the determination of weight directly affects the rationality of the evaluation model. Different weights will lead to different research results, so the weight-determining method is significant. There are many ways to determine weights, such as the Delphi (expert investigation method), the weighted average method, the analytic hierarchy process (AHP), and the evaluation method. When data are available, the entropy method is usually used to calculate the weight.
- Step 5:
Calculate the comprehensive evaluation index.
Perform the matrix synthesis operation to get matrix C.
Finally, compare and sort the evaluation results of multiple evaluation objects and calculate the comprehensive evaluation index of each indicator.