*3.2. Correlation Analysis*

In this sub-section, we compute correlations coefficients between products at different aggregated levels to make clear their dependencies structure, supporting the subsequent forecasting. At the top level, primary equipment and equipment material are highly correlated. The correlation coefficient of the two families is 0.7784.

As for 15 categories, their correlations are shown in Figure 1. A pink dotted line indicates that the correlation value of two linked nodes is in the interval [0.6, 0.7). Similarly, a blue one corresponds to [0.7, 0.8), while a black line means more than 0.8. Seen from Figure 1, there exists high correlations among 8 categories, including AC transformer, insulator, metal fittings, tower pole, AC disconnector, lightning arrester, high-voltage fuse, and wire & ground wire, especially the first four categories. In addition, the correlation network is clustered by a method proposed by [59] and nodes with same colors in Figure 1 represent that they are clustered in a same group. To be specific, AC transformer, AC disconnector, high-voltage fuse and insulator are in G1, while lightning arrester, tower pole, ground wire and metal fittings are in G2, which are mainly consumed in line laying. According to Table 2, a series of centrality indices of metal fittings are almost the biggest in the correlation network, reflecting that its demand has high correlations with demand of all other categories.

**Figure 1.** Correlation relationships of categories at level 2.


**Table 2.** Centrality indices of categories in the correlation network.

## *3.3. Granger Causality Analysis*

We applied the Granger causal relation test to evaluate the relationship between primary equipment and equipment materials. The result shows that they have no statistically significant causality. In other words, demand of primary equipment does not Granger cause that of equipment materials, with *p* = 0.087. Furthermore, the reverse direction is also not significant, with *p* = 0.201.

In addition, we evaluate the Granger causality relationships among 15 categories at level 2 and visualize the network in Figure 2. Directions of arrows are from Granger causes to Granger results. When the *p*-value of Granger test locates in [0.05, 0.01) and [0.01, 0], the arrow is drawn by a blue dotted line and a black line respectively. Figure 2 shows that the demands of 14 categories have a significant causal influence on each other, except for cable. It is because cables are widely used in power grid construction, not depending on other products. According to the clustering results of causality network, three groups can be found. G1 contains optical cable accessory, optical cable, load switch and switch cabinet, and procurement of first three categories Granger results in that of switch cabinet. G2 consists of cable accessories, AC transformers, insulator, high-voltage fuse, tower pole and AC disconnector, similar to G1 in correlation analysis, existing complex causality relationships. Overall, high-voltage fuse and tower pole locate at core position, which also have causality relationships with out-group products. Lightning arrester, metal fittings, AC circuit breaker and ground wire are clustered in G3, similar to G2 in correlation analysis. The significant direction of causality from the first three categories to wire & ground wire reflects that a growth of their purchase will increase demand for wire & ground wire later.

**Figure 2.** Granger causality relationships of categories at level 2.

To further illustrate Granger causality between categories, we calculate some indices, listed in Table 3. High-voltage fuses and tower poles display the biggest values of in-degree, indicating that their procurement can be greatly explained by lag demand of other categories, while referring to out-degree, high-voltage fuse also ranks first, as well as cable accessory, reflecting that they are strong predictor for follow-up demand of other categories. Besides, switch cabinet, tower pole and high-voltage fuse, with largest centrality indices, are the core nodes in the Granger causality network. what's more, categories in G1 have closer relationship, demonstrated by their higher clustering coefficients.


**Table 3.** Centrality indices of categories in the causality network.

## **4. Empirical Analysis**

## *4.1. Experimental Setup*

In this section, we generate demand forecasting for the above dataset by our new approaches and compare their forecasting performance with four traditional time series models, namely SES, AR, VAR, and DFM, respectively. Except for the enterprise level which only has a single time series, we applied these forecasting approaches on the other four levels of the product hierarchy. Because the number of products varies greatly at the different levels, from several to hundreds, it helps us to investigate whether the new approaches can deal with different data dimensions well. Nevertheless, DFM is not necessary to do forecasting at the family level. VAR ignores the product level and the subcategory level due to a large number of products at the two levels.

An essential step of our new methods is to set up the criteria for constructing proper variable sets, in other words, to define the critical conditions of high correlation and significant Granger causality. If the critical conditions are too strict, most products may not find proper variables, and then their forecasting demand cannot be corrected by association relationships. Conversely, an excessively loose condition will bring too much disturbing information in proper variable sets, even impairing originally forecasting accuracy. Therefore, a rational critical condition is a key point to obtain satisfying prediction results. Considering that in the forecasting approach based on correlation, originally forecasting values of AR are used as explanatory variables, which may further increase uncertainty, a stricter selection criterion is necessary. We set the correlation coe fficient at more than 0.513 as the threshold condition preliminarily. The significance level of the critical value is 0.001 in terms of the size of the training sample. As for the Granger relationship, the standard is that the *p*-value of Granger test is no less than 0.1. Based on the above settings, we ge<sup>t</sup> proper variable sets and then do final demand forecasting. What's more, to further investigate the influence of critical conditions on forecasting performance, we set the critical correlation coe fficient as 0.6, 0.7, 0.8, 0.9 as well as the critical Granger significant level as 0.05 and 0.01 separately to repeat prediction process.

Finally, we evaluate the forecasting performance for these models. To begin with we calculate their absolute errors and relative errors in terms of the equations mentioned before. The approach with a smaller average error is considered to be more accurate. Then we introduce *t*-test to verify whether forecasting errors of our new approaches exist statistically significant di fferences with the baseline model SES. In addition, demand prediction aims to guide subsequent activities including purchase and inventory management. More accurate forecasting may be helpful to avoid high inventory levels or out of stock, reducing the cost loss of enterprises naturally. Therefore, it is common to assess forecasting approaches by simulating the process of inventory management. we also do inventory simulation for 338 products according to forecasting results of di fferent approaches, to compare their performance from the perspective of inventory management.

## *4.2. Results and Analysis*

## 4.2.1. Forecasting Accuracy Analysis

Table 4 presents mean values of absolute errors and relative errors for the proposed approaches, as well as four conventional models. CI and CII refer to the approach based on correlations adopting scheme (I) and scheme (II) to control model size respectively. Similarly, GI and GII refer to the approach based on Granger causality and the Roman numerals represent di fferent schemes. The brackets indicate the critical conditions to construct proper variable sets. The approach based on correlations requires the correlation coe fficient greater than 0.513. As for the approach based on Granger causality, the *p*-value of Granger test should be less than 0.1.


**Table 4.** Forecasting errors of six models at four aggregated levels.

1 SES is the baseline model when calculate relative errors.

We can see from Table 4 that for the approaches based on correlations, evaluation results are consistent whether base on MASE or GMRASE. CII performs best among the six models, having a minimum deviation from real demand. Conversely, forecasting accuracy of CI is low, especially at more disaggregated levels. As the dimension of products increase, forecasting errors of CI grows rapidly, even inferior to the original predictive results (AR). This reflects that the forecasting values of the most correlated time series distort the effect of original predictors, leading to lower accuracy, not aligned with our theoretical expectations. However, the first principal component is equivalent to weighted average of all highly correlated time series, which not only contains more effective information but also offsets errors of different correlated series. Therefore, CII can ge<sup>t</sup> more accurate results. As for the two types of models based on Granger causality, their accuracy is the same basically at all aggregated levels, superior to that of VAR and DFM. When the product dimension is large, Granger II is more advantageous.

We set different critical values to investigate their influence on accuracy of models. For the approach based on correlation, we set the critical values separately as 0.6, 0.7, 0.8 and 0.9. Table 5 shows the forecasting errors in each situation. With the increase of the critical value, CI becomes more precise while CII is absolutely opposite, witnessing an upward tendency in errors. However, even if the critical value is equal to 0.9, CI performs still worse than AR, let alone CII. For the approach based on Granger causality, forecasting results in terms of different critical conditions are displayed in Table 6. We can see that forecasting errors of GII is always lower than that of GI. When the critical value equal to 0.01, GII has the highest accuracy. In conclusion, scheme (II) can help the approach based on association relationships to ge<sup>t</sup> more accurate forecasting results, better than these conventional models. Besides, the critical value should be set in a rationally high level, to ensure that only highly associated time series can be selected to eliminate irrelevant information and the proper variable subset has enough members too to offset errors. In this way, the approaches based on associated relationships can be the most effective.


**Table 5.** Forecasting errors of the approaches based on correlation under different critical conditions.

**Table 6.** Forecasting errors of the approaches based on Granger causality under different critical conditions.


To further validate the accuracy of models, we treat the SES model as the benchmark to do *t*-tests for all other models. A negative test statistic means that average ASE of the model is smaller than that of the baseline model, i.e., more accurate than SES model. The smaller the negative t value is, the higher the significance level is. As shown in Table 7, the *t*-test results for absolute errors and relative errors are not consistent. In terms of absolute errors, at the three levels (production, subcategory and category), forecasting errors of GI and GII are all significantly smaller than SES (*p*-value < 0.01), regardless of the critical values. The models based on correlations outperform SES only when adopt scheme (II) to collect predictors. When it comes to relative errors, CI is still significant superior to the benchmark model only at the category level. GI and GII become insignificant in most cases. In conclusion, significance levels of t-value for relative errors are lower remarkably compared to absolute errors. However, CII still has significantly lower errors, performing best among all models.


**Table 7.** *T*-test of forecasting errors for different approaches at three aggregated levels.

## 4.2.2. Inventory Performance Analysis

In our study, we set five desired service levels and three kinds of cost parameters to do inventory simulations for 338 products at the product level. Total inventory costs are shown in Table 8 and Figure 3 illustrates inventory ratios in different scenarios. When satisfying the same service level, CII enjoys the lowest total costs and the inventory ratio and, as expected, CI spends more costs and maintains higher inventory ratios than the original approach (AR) as well as SES and DFM. As for the two approaches based on Granger causality, the results are inconsistent with that of forecasting errors. GI has fewer stock costs and a lower inventory ratio than GII, although GII's forecast accuracy is higher. In addition, compared with AR and DFM, they need higher inventory levels to realize a certain service level.


**Table 8.** Total inventory costs for different approaches considered satisfying various service levels.

1 Units of total inventory costs: Million.

**Figure 3.** Inventory ratios for different approaches satisfying various service levels.

In addition, we also analyze the inventory performance of the proposed approaches satisfying different threshold values. As shown in Table 9, when the critical value rises, total inventory costs creep up if using CII to forecast demand, and the inventory ratios also reach to the top value (in Figure 4a). CI benefits from high critical values, bringing improvement of inventory performance, but never surpasses that of AR.


**Table 9.** Total inventory costs for the approach based on correlation with different critical values.

1 Units of total inventory costs: Million.

**Figure 4.** Inventory ratios for approaches with different critical values satisfying various service levels. (**a**) is for the approach based on correlation and (**b**) is for the approach based on Granger causality.

For approaches based on Granger causality, their inventory performance is displayed in Table 10 and Figure 4b. We can see that total costs and inventory ratios of GI basically remain the same, regardless of the change of the critical value. GII is more sensitive to the critical value in comparison. A stricter condition to select time series existing Granger causality is helpful to optimize inventory performance for a given service level. As a whole, CII with the critical value equal to 0.513 is the most accurate forecasting model, which can help to reduce inventory costs and improve the service level to the most extent, followed by GII with the critical value equal to 0.01. Therefore, by setting rational thresholds and choosing appropriate methods to mine predictors, the approaches based on associated relationships can achieve higher forecasting accuracy, thereby improving inventory performance. It is helpful to provide effective guidance for actual inventory managemen<sup>t</sup> activities.


**Table 10.** Total inventory costs for the approach based on Granger causality with different critical values.

1 Units of total inventory costs: Million.
