stylistic:style_transfer
From random forest to KeRF

Thus random forest estimates satisfy, for all x ∈ d {\displaystyle \mathbf {x} \in ^{d}}, m M, n (x, Θ 1, …, Θ M) = 1 M ∑ j = 1 M (∑ i = 1 n Y i 1 X i ∈ A n (x, Θ j) N n (x, Θ j)) {\displaystyle m_{M,n}(\mathbf {x},\Theta _{1},\ldots,\Theta _{M})={\frac {1}{M}}\sum _{j=1}^{M}\left(\sum _{i=1}^{n}{\frac {Y_{i}\mathbf {1} _{\mathbf {X} _{i}\in A_{n}(\mathbf {x},\Theta _{j})}}{N_{n}(\mathbf {x},\Theta _{j})}}\right)}. Random regression forest has two levels of averaging, first over the samples in the target cell of a tree, then over all trees. Thus the contributions of observations that are in cells with a high density of data points are smaller than that of observations which belong to less populated cells. In order to improve the random forest methods and compensate the misestimation, Scornet defined KeRF by m ~ M, n (x, Θ 1, …, Θ M) = 1 ∑ j = 1 M N n (x, Θ j) ∑ j = 1 M ∑ i = 1 n Y i 1 X i ∈ A n (x, Θ j), {\displaystyle {\tilde {m}}_{M,n}(\mathbf {x},\Theta _{1},\ldots,\Theta _{M})={\frac {1}{\sum _{j=1}^{M}N_{n}(\mathbf {x},\Theta _{j})}}\sum _{j=1}^{M}\sum _{i=1}^{n}Y_{i}\mathbf {1} _{\mathbf {X} _{i}\in A_{n}(\mathbf {x},\Theta _{j})},} which is equal to the mean of the Y i {\displaystyle Y_{i}} 's falling in the cells containing x {\displaystyle \mathbf {x} } in the forest. If we define the connection function of the M {\displaystyle M} finite forest as K M, n (x, z) = 1 M ∑ j = 1 M 1 z ∈ A n (x, Θ j) {\displaystyle K_{M,n}(\mathbf {x},\mathbf {z})={\frac {1}{M}}\sum _{j=1}^{M}\mathbf {1} _{\mathbf {z} \in A_{n}(\mathbf {x},\Theta _{j})}}, i.e. the proportion of cells shared between x {\displaystyle \mathbf {x} } and z {\displaystyle \mathbf {z} }, then almost surely we have m ~ M, n (x, Θ 1, …, Θ M) = ∑ i = 1 n Y i K M, n (x, x i) ∑ ℓ = 1 n K M, n (x, x ℓ) {\displaystyle {\tilde {m}}_{M,n}(\mathbf {x},\Theta _{1},\ldots,\Theta _{M})={\frac {\sum _{i=1}^{n}Y_{i}K_{M,n}(\mathbf {x},\mathbf {x} _{i})}{\sum _{\ell =1}^{n}K_{M,n}(\mathbf {x},\mathbf {x} _{\ell })}}}, which defines the KeRF.