Power Customer Complaint Prediction Model Based on Time Series Analysis

2.1 Theoretical bases of the three models

2.1.1 ARIMA time series algorithm

Time series change with the elapse of time. The value of a time series at a moment depends on various factors. However, it is impossible to consider the impact of every factor in customer complaint prediction, but to determine the size and weight the impact of each factor. Therefore, the relevant factors were divided into four categories for time series analysis: long-term change T_l, seasonal change S_c, cyclic change C_c, and irregular change I_c.

The time series is usually combined through addition or multiplication. The additive and multiplicative time series can be respectively expressed as:

$y_{c}=T_{l}+S_{c}+C_{c}+I_{c}$      (1)

$y_{c}=T_{l} \times S_{c} \times C_{c} \times I_{c}$     (2)

Time series analysis aims to decipher the change law of the target system with time, and redesign the system based on the analysis results. The change law can be inferred from the features of the time series of the system, and used to build an accurate model of the dependence between elements in the series. The established model could predict the future development of the system. Therefore, the time series adopted in prediction model must meet one of the following requirements: (1) the time series is stationary; (2) the elements of the time series are correlated with each other.

The common algorithms for time series analysis include autoregressive algorithm AR(p), moving average algorithm MA(q), autoregressive moving average algorithm ARMA(p, q) , and ARIMA (p, d, q) algorithm.

The AR(p) algorithm can be defined as:

$y_{c}=\varepsilon_{1} y_{c-1}+\varepsilon_{2} y_{c-2}+\cdots+\varepsilon_{p} y_{c-p}+U_{c}$      (3)

where, U_c is the linear function of residual term.

Using the backward operator, formula (3) can be rewritten as:

$y_{c}=\varepsilon_{1} L y_{c}+\varepsilon_{2} L^{2} y_{c}+\cdots+\varepsilon_{p} L^{p} y_{c}+U_{c}$       (4)

Thus, AR(p) algorithm can be expressed as:

$\varphi(L) y_{c}=U_{c}$       (5)

The MA(q) algorithm can be defined as:

$y_{c}=U_{c}-\theta_{1} U_{c-1}-\theta_{2} U_{c-2}-\cdots-\theta_{q} U_{c-q}$     (6)

Using the backward operator, formula (6) can be rewritten as:

$y_{c}=\left(1-\theta_{1} L_{1}-\theta_{2} L_{2}-\cdots-\theta_{q} L_{q}\right) U_{c}$      (7)

Thus, MA(q) algorithm can be expressed as:

$y_{c}=\Phi(L) U_{c}$

The ARMA (p, q) algorithm, which couples AR(p) algorithm and MA(q) algorithm, can be defined as:

$y_{c}=\alpha_{0}+\alpha_{1} y_{c-1}+\alpha_{2} y_{c-2}+\cdots+\alpha_{p} y_{c-p}+\sigma_{c}$$+\beta_{1} \sigma_{c-1}+\beta_{2} \sigma_{c-2}+\cdots+\beta_{q} \sigma_{c-q}$     (8)

Using the backward operator, formula (8) can be rewritten as:

$\begin{aligned}\left(1-\alpha_{1} L-\alpha_{2} L^{2}\right.&\left.-\cdots-\alpha_{p} L^{p}\right) y_{c} \\ &=c+\left(1+\beta_{1} L+\beta_{2} L^{2}+\cdots\right.\\ &\left.+\beta_{q} L^{q}\right) \sigma_{c} \end{aligned}$    (9)

Thus, ARMA (p, q) algorithm can be expressed as:

$y=c /\left(1-\beta_{1}-\beta_{2}-\cdots-\beta_{p}\right)$    (10)

The ARIMA (p, d, q) algorithm, which couples AR(p) algorithm, MA(q) algorithm, and difference method DX=diff(y, i), can be defined as follows:

$\vartheta(L)\left(\Delta^{d} y_{c}\right)=\theta_{0}+\Phi(L) U_{c}$      (11)

where, $\vartheta(L)$ is the AR algorithm of order $p ; \Phi(L)$ is the MA algorithm of order $q ; \Delta^{d} y_{c}$ is the MA algorithm, in which the predicted value $y$ is divided by $d$ times.

2.1.2 MLR algorithm

The MLR algorithm unveils the relationship between a dependent variable and multiple independent variables:

y=\beta_{0}+\beta_{1} x_{1}+\cdots+\beta_{p} x_{p}+\tau      (12)

where, y is the dependent variable with a randomly observed value; β₀ is a constant; β_i is the partial regression coefficient.

Suppose there are p independent variables, whose vectors are x₁, x₂,…,x_p, and n sets of observed data, y₁, y₂, …, y_n. Then, it can be assumed that the dependent variable is linearly correlated with the independent variables:

$y_{i}=\hat{y}+\varepsilon_{i}=b_{0}+b_{1} x_{i 1}+\cdots+b_{p} x_{i p}+\varepsilon_{i}$     (13)

where, ε_i is a normally distributed evaluation value.

In the MLR, the independent variables are often selected by stepwise regression. The basic idea of stepwise regression is to consider adding or subtracting a variable from the set of independent variables based on preset criterion in each step. The prediction target y and all the candidate independent variables x_p can be established by linear pairwise regression equations:

$\left\{\begin{array}{c}y=\beta_{0}+\beta_{1} x_{1}+u \\ \cdots \\ y=\beta_{0}+\beta_{1} x_{m}+u\end{array}\right.$      (14)

First, x_n with the largest value is selected as the first filtered argument. Then, x_n and the remaining m-1 independent variables are combined into an m-1 linear pairwise regression equation. Next, x_i with the largest value is selected as the second filtered argument. By analogy, the independent variables are screened gradually until the marginal contribution of the next new independent variable is too small or the algorithm meets the demand.

2.1.3 BPNN algorithm

The BPNN algorithm consists of the forward propagation of data (the forward calculating of data) and the reverse propagation of error signal (the reverse calculation of error). The typical structure of the BPNN has three layers: an input layer, a hidden layer, and an output layer. The information is imported to the input layer, transmitted to the hidden layer, and eventually to the output layer. Each layer has one or multiple nodes. The training results are saved by connecting the stored weights and thresholds between nodes. Figure 1 is the sketch map of node j in the BPNN.

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Figure 1. The sketch map of a node in the BPNN

Note: $x_{i}$ is the input of node $i ; w_{i j}$ is connection weight between nodes $i$ and $j ; b_{i}$ is the threshold of node $j ; f(\cdot)$ is the transfer function; $y_{j}$ is the output of node $j ; S_{j}$ is the input of node $j$

The input of node j S_j can be defined as:

$S_{j}=\sum_{i=1}^{n} w_{i j} * x_{i}+b_{j}=W_{j} X+b_{j}$      (15)

where, $X=\left[x_{1}, x_{2}, \ldots, x_{i}, \ldots, x_{n}\right]^{T}$;

$w_{j}=\left[w_{j 1}, w_{j 2}, \ldots w_{j i}, \ldots, w_{j n}\right]$.

2.2 Preprocessing of customer complaint data

This paper takes the total number of work orders in response to future customer complaints in a power enterprise as the prediction target. According to business regulations of the power enterprise, 25 secondary work orders, denoted as x₁,x₂,…,x₂₅ , were adopted as the initial independent variables of the prediction model, and combined into the experimental data (real number of customer complaints) (as shown in Table 1).

Table 1. The experimental data

Considering the different types of customer complaints and the features of ARIMA time series algorithm, the sample period was set as October 28, 2016 - September 30, 2019. For MLP and BPNN algorithms, the total number of complaints in a week was taken as the dependent variable, and the sample period was set s July 2018 – September 2019.

Before modeling, the independent variables of 25 dimensions were preprocessed through principal component analysis (PCA) [14] (Figure 2) to reduce the dimensionality of the experimental data (independent variables). The preprocessing was designed to overcome the following defects of the BPNN and MLP algorithms.

For the BPNN algorithm, the network training will be disrupted by the strong correlations between the multiple dimensions of independent variables, because the NN boasts strong non-mapping ability. For the MLR algorithm, it is too complex and unstable to select independent variables for the prediction model, if the 25 dimensions are directly analyzed through the MLR. Besides, the algorithm will face a high error due to the multicollinearity among variables.

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Figure 2. The workflow of data preprocessing

To determine the data preprocessing rules, the time difference index of the secondary work orders was calculated by:

$\left|\hat{y}_{c}(l)-y_{c}\right|=\min \left\{\left|f_{1}\left(x_{c}\right)-y_{c}\right|, \mid f_{2}\left(x_{c}\right)\right.$$\left.-y_{c} \mid, \ldots,\right\}, l \in[L, c-L]$      (16)

where, $t$ is the current time; $n$ is the time difference between the dependent variable and independent variables; $\hat{y}_{c}(n)$ is the predicted value of the dependent variable with time difference of $n ; f_{n}\left(x_{c}\right)$ is the prediction function with time difference $n$ $L$ is the best time difference between the dependent variable and independent variables.

The PCA function [15] was adopted to quantify the amount of information contained in the experimental data, and derive the proportion of retained features. Through the PCA, the original 25 dimensions of experimental data were reduced, producing 25 principal components z_i. The contribution rate of each principal component is given in Table 2.

As shown in Table 2, the cumulative contribution rate of the top 10 principal components was greater than the standard of 95%. Therefore, the top 10 principal components were selected for modeling. The 10 independent variables composed of the original variables are denoted as $z_{1}, z_{2}, z_{3}, \ldots, z_{10}$. The post-PCA data are real numbers (Table 3).

Table 2. The contribution rate of each principal component

Table 3. The post-PCA data

2.3 Three prediction models

The ARIMA time series prediction model was constructed by the steps in Figure 3.

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Figure 3. The construction steps of ARIMA time series prediction model

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Figure 4. The comparison between the predicted results and the actual data of ARIMA time series prediction model

The time series of each Monday was chosen as experimental data to verify the accuracy of the ARIMA (p, d, q) customer complaint prediction model in predicting the number of complaints in the coming week. The parameters of the model were configured as p=2, d=1, and q=2. Figure 4 compares the predicted results with the actual data.

As shown in Figure 4, the time series fluctuated through the sample period, and the curve of predicted value agreed well with the curve of the real value. The two curves exhibited very similar volatility. Although the prediction results are reasonable, the ARIMA time series prediction model only applies to short-term prediction, failing to achieve a good fitting effect in the long run.

The MLR aims to solve the MLR equation based on the relationship between a dependent variable and multiple independent variables. The ARIMA time series prediction model was constructed by the steps in Figure 5.

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Figure 5. The construction steps of MLR prediction model

The establishment of the MLR equation is essentially to estimate the coefficient of the MLR model and look for a suitable expression. In the MLR prediction model, the parameter vectors were estimated by the least squares (LS) method. The basic idea of solving the MLR equation is to calculate the partial regression coefficient in the MLR model by the LS principle, so as to minimize the sum of squares of the residual errors between the observed values and the regression values. After the MLR model was established, the MLR fitting equation was obtained as:

$y=0.133 x_{1}+0.72 x_{2}+0.201 x_{3}-1.652 x_{4}$$\quad+0.602 x_{5}-3.468 x_{6}-0.93 x_{7}$$-13.419 x_{8}-1.17 x_{9}+0.94 x_{10}$-2.31      (17)

Figure 6 compares the predicted results with the actual data.

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Figure 6. The comparison between the predicted results and the actual data of MLR prediction model

As shown in Figure 6, the actual value and predicted value had basically the same trend, but with a large error between them. Thus, the MLR prediction model has a poor predictive effect: the model cannot effectively regress and summarize the complex relationship between independent variables and the dependent variable.

The BPNN prediction model was constructed by the steps in Figure 7.

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Figure 7. The construction steps of BPNN prediction model

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Figure 8. The comparison between the predicted results and the actual data of BPNN prediction model

In this experiment, the principal components of the original data after dimensionality reduction were taken as the independent variables (inputs) of the prediction, and the total number of customer complaints in the coming week was taken as the dependent variable (output). Since 10 principal components were selected through the PCA, the BPNN has a total of 10 input layer nodes.

Even if the parameters remain constant, the BPNN algorithm might output different weights to the model, and make different predictions in each run. To solve the problem, the BPNN algorithm was ran 200 times. The ten minimum and ten maximum results were removed. Then, the remaining 180 results were averaged as the final prediction. Figure 8 compares the predicted results with the actual data.

As shown in Figure 8, the predicted value was almost identical with the actual value, an evidence of the ultrahigh accuracy of the BPNN algorithm.

Table 4 compares the relative errors of the three prediction models.

Table 4. The relative errors of the three prediction models

It can be seen from Table 4 that the BPNN prediction model achieved much higher accuracy than the other two models. Judging by prediction accuracy and fitness between dependent and independent variables, the BPNN algorithm is clearly the best algorithm to build the customer complaint prediction model.