# How to Open a Black Box Classifier for Tabular Data
## 2 Materials and Methods
### Data sets used  
#### 2.2.1 Synthetic Data  
(a) 2D circle. We implemented a sample size of n = 10,000 with unbalanced classes, which is the case for all synthetic data sets in this paper. In this case, the _logit_ has two separate univariate components,  
\[logit(P(C|(x_{1},x_{2}))=10\times\left[\left(x_{1}-\frac{1}{2}\right)^{2}+ \left(x_{2}-\frac{1}{2}\right)^{2}-0.08\right] \tag{31}\]  
This data set is similar to that used in [15] but instead of generating clean data by allocating different classes on either side of the boundary, we use noisy data by generating binary targets with a Bernoulli distribution, which is also common for all the synthetic data sets that we report,  
\[Y\sim Bin(n,P(C|(x_{1},x_{2}))). \tag{32}\]  
The factor of 10 in Equation (31) is to reduce the amount of noise and so ensure a reasonable value for the AUC. The values of \((x_{1},x_{2})\) are generated using \(x_{i}\) = 0.5 \(\times\)\((u_{i}+w)\), where both \(u_{i}\) and \(w\) are uniform distributions in the range [0,1], to demonstrate the prediction accuracy when the two input variables are correlated. There are only two univariate main effects and no interaction term. (b) XOR function. The purest bivariate interaction is the XOR, represented in the multilinear form appropriate for continuous Boolean algebra [28],  
\[P(C|(x_{3},x_{4}))=x_{3}+x_{4}-2x_{3}x_{4},x_{i}\in]0,1[ \tag{33}\]  
Each variable will be generated by a uniform distribution in [0,1]. This density function has the property that  
\[logit(P(C|(x_{3},x_{4})))=log\left(\frac{x_{3}+x_{4}-2x_{3}x_{4}}{1-x_{3}-x_{4} +2x_{3}x_{4}}\right) \tag{34}\]Therefore, \(logit\left(P\left(C\middle|\left(x_{3},\frac{1}{2}\right)\right)=0\) making it a pure interaction for the ANOVA decomposition with the _Dirac measure_ anchored at (1/2,1/2). Similarly, for the _Lebesgue measure_, it is readily shown that  
\[logit\left(P(C|\left(x_{3},x_{4}\right)\right)=-logit\left(P(C|\left(1-x_{3},x_{ 4}\right)\right)\right. \tag{35}\]  
which is similar to the other dimension; therefore, the integrals corresponding to the univariate terms vanish, once again leaving the pure interaction term. (c) Logical AND function. A combination of univariate and bivariate terms by generating data according to the logical AND function, which in continuous Boolean algebra is represented by the following atomic term:  
\[P(C|\left(x_{5},x_{6}\right))=x_{5}x_{6},x_{i}\in]0,1[ \tag{36}\]  
This time, the ANOVA expansion with the _Dirac measure_ anchored at (1/2,1/2) is  
\[logit\left(P(C|\left(x_{5},x_{6}\right))\right) \tag{37}\] \[=-\log(3)+\sum\limits_{i}\left[\log\left(\frac{x_{i}}{2-x_{i}} \right)+\log(3)\right]\] \[+\left[\log\left(\frac{(2-x_{5})(2-x_{6})}{1-x_{5}x_{6}}\right) -\log(3)\right)\right]\]  
The _Lebesgue measure_ yields univariate terms given by  
\[\varphi_{i}^{Lebesgue}=log\left(x_{i}/(1-x_{i})\right)+\frac{log(1-x_{i})}{x_ {i}}+Li_{2}(1) \tag{38}\]  
where \(Li_{2}(1)\) is the polylogarithm function of second order evaluated at 1. The bivariate term is given by the explicit ANOVA decomposition in Equation (5) and does not reduce to a simpler algebraic form. (d) Three-way interaction. The final synthetic data set comprises a data set that cannot be modelled with univariate and bivariate terms only. The purpose is now to see how well PRiSM models work to model a high-order effect. \[Y\sim Bin(n,P(C|(x_{1},x_{2}))). \tag{32}\]0  
Note that the complete set of input variables for the synthetic data set is calculated only once. In this way, the same sample of 9-dimensional input data will be used for all classifiers. Only the target classes differ, thus generating four separate binary classification tasks. In each case, two or three variables will carry signal and the others comprise noise. A minimum requirement of all classifiers is to identify the relevant input dimensions and discard the rest.