Explaining hardness modeling with XAI of C45 steel spur-gear induction hardening

Abstract

This work presents an interpretability study with XAI tools to explain an XGBoost model for hardness prediction in the simultaneous double-frequency induction hardening. Experiments were carried out on C45 steel spur-gear. In order to explain the model, firstly, the built-in tool of the XGBoost library was used to interpret the feature importance. Then, a more advanced approach with the SHAP library was employed to highlight local and global explanations. Finally, the implementation of an interpretable surrogate model allowed to illustrate rules for prediction, making the explanation, although approximate, clear. This study proposes a relevant approach of AI to explain the results obtained by black box models which is currently a major element for the industry allowing to justify the quality of the results in a clear way. It is concluded that the model is consistent with physical principles.

Appendix A: Example : SHAP value calculation

This appendix present a short example of SHAP values calculation. The case presented in Eq. 1 shows a space of variables X which is too large for explanation. Indeed,

$$\begin{aligned} X = \{MF, HF, P_{MF}, P_{HF}, T, d, t\} \end{aligned}$$

(8)

The number of elements of X is denoted by card(X) i.e. $card(X) = 6$. F being the powerset of X i.e. the space of all possible combinations of the elements in X, $card(F) = 2^6 = 64$ which is too many elements to graphicly present the concepts. Hence, a subset will be used for this example as the space of variables is now defined as:

$$\begin{aligned} X_{shap} = \{d, T, t\} \end{aligned}$$

(9)

As mentioned, the powerset F is the space containing all the different combinations of variable. Here it can be illustrated like in Fig. 21.

It is then possible to defined F as:

$$\begin{aligned} F = \{\{\varnothing \}, \{d\}, \{T\}, \{t\}, \{d, T\}, \{d, t\}, \{T, t\} , X_{shap}\} \end{aligned}$$

(10)

Hence, it comes $card(F) = 2^3 = 8$. Each node of the powerset represents a model trained with the set of variables of the node. Hence each node has its own predicted value of hardness H. For a single value $x_0$ to predict, every hardness predictions $\hat{y_0}$ in each node are recorded in Fig. 22.

Fig. 21

The powerset F of the variables of $X_{shap}$

Fig. 22

The powerset and the predicted values $\hat{y}_0$ of hardness H of each node

Each edge is the impact of a variable added on a previous model to the new one. Therefore, each edge represents the marginal contribution of a variable, it is calculated with the difference of the predicted values $\hat{y_0}$ between 2 nodes, the first one should have not the evaluated variable it, while the second one should. In this example, the marginal contribution of the temperature T between the first two nodes is defined as:

$$\begin{aligned} mc_{T,\{T\}}(x_0) = \hat{y}_\varnothing (x_0) - \hat{y}_T(x_0) \end{aligned}$$

(11)

Moreover, this marginal contribution is weighted. The weighted are defined as follow:

• the sum of the weights of a row must be equal to the sum of a any other row.

• all weights involved in the marginal contribution must be equal to 1.

• all weights of the same row are equal to each other.

The edges concerned with the marginal contribution calculation of the temperature T are illustrated in Fig. 23.

Fig. 23

The powerset and the weighted of the edges for marginal contribution (green) of T

Therefore, in this example, $w_1 = w_2 + w_3 = w_4$ with $w_2 = w_3$. It comes $w_1 = w_4 = \frac{1}{3}$ and $w_2 = w_3 = \frac{1}{6}$.

Moreover, with the Eq. 4 presented in the SHAP value section, the weights are defined as:

$$\begin{aligned} \frac{|S|! (|X| - |S| - 1)!|}{|X|!} \end{aligned}$$

(12)

The weight values can be verified, here the calculation of $w_2$ as an example:

$$\begin{aligned} {\begin{matrix} w_2 &{} = \frac{|S|! (|X| - |S| - 1)!|}{|X|!} \\ &{} = \frac{1! \times (3 - 1 - 1)!}{3!} \\ &{} = \frac{1}{3!} = \frac{1}{6}\\ \end{matrix}} \end{aligned}$$

(13)

Now it is possible to have the SHAP value of T for the whole powerset, it is defined as :

$$\begin{aligned} {\begin{matrix} SHAP_T(x_0) &{} = w_1 \cdot mc_{T,\{T\}}(x_0) + w_2 \cdot mc_{T,\{d,T\}}(x_0) \\ &{} + w_3 \cdot mc_{T,\{T, t\}}(x_0) + w_4 \cdot mc_{T,\{d,T,t\}}(x_0) \\ &{} = \frac{1}{3} \cdot (500 - 700) + \frac{1}{6} \cdot (600 - 650) \\ &{} +\frac{1}{6} \cdot (550 - 650) + \frac{1}{3} \cdot (650 - 625) \\ &{} = {\textbf {- 83.2}} \end{matrix}} \end{aligned}$$

(14)

The shap value is calculated and can be used for several plotting functions to better understand how the model actually predict.