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Update README.md
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mayer79 authored Jul 10, 2023
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Expand Up @@ -315,34 +315,34 @@ $$
with

$$
B_{jkl} = \hat F_{jk}(x_j, x_k) + \hat F_{jl}(x_j, x_l) + \hat F_{kl}(x_k, x_l)
B_{jkl} = F_{jk}(x_j, x_k) + F_{jl}(x_j, x_l) + F_{kl}(x_k, x_l)
$$

and

$$
C_{jkl} = \hat F_j(x_j) + \hat F_k(x_k) + \hat F_l(x_l).
C_{jkl} = F_j(x_j) + F_k(x_k) + F_l(x_l).
$$

The squared and scaled difference between the two sides of the equation leads to the statistic

$$
H_{jkl}^2 = \frac{\frac{1}{n} \sum_{i = 1}^n \big[\hat F_{jkl}(x_{ij}, x_{ik}, x_{il}) - B^{(i)}_{jkl} + C^{(i)}_{jkl}\big]^2}{\frac{1}{n} \sum_{i = 1}^n \hat F_{jkl}(x_{ij}, x_{ik}, x_{il})^2},
H_{jkl}^2 = \frac{\frac{1}{n} \sum_{i = 1}^n \big[\hat F_{jkl}(x_{ij}, x_{ik}, x_{il}) - B^i_{jkl} + C^i_{jkl}\big]^2}{\frac{1}{n} \sum_{i = 1}^n \hat F_{jkl}(x_{ij}, x_{ik}, x_{il})^2},
$$

where

$$
B^{(i)}_{jkl} = F_{jk}(x_{ij}, x_{ik}) + F_{jl}(x_{ij}, x_{il}) + F_{kl}(x_{ik}, x_{il})
B^i_{jkl} = \hat F_{jk}(x_{ij}, x_{ik}) + \hat F_{jl}(x_{ij}, x_{il}) + \hat F_{kl}(x_{ik}, x_{il})
$$

and

$$
C^{(i)}_{jkl} = F_j(x_{ij}) + F_k(x_{ik}) + F_l(x_{il}).
C^i_{jkl} = \hat F_j(x_{ij}) + \hat F_k(x_{ik}) + \hat F_l(x_{il}).
$$

Similar remarks as for $H_{jk}$ apply.
Similar remarks as for $H^2_{jk}$ apply.

#### Total interaction strength of all variables together

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