is built, where is a parameter of the algorithm. Uniform forest Uniform forest is another simplified model for Breiman's original random forest, which uniformly selects a feature among all features and performs splits at a point uniformly drawn on the side of the cell, along the preselected feature. From random forest to KeRF Given a training sample of -valued independent random variables distributed as the independent prototype pair , where . We aim at predicting the response , associated with the random variable , by estimating the regression function . A random regression forest is an ensemble of randomized regression trees. Denote the predicted value at point by the -th tree, where are independent random variables, distributed as a generic random variable , independent of the sample . This random variable can be used to describe the randomness induced by node splitting and the sampling procedure for tree construction. The trees are combined to form the finite forest estimate . For regression trees, we have , where is the cell containing , designed with randomness and dataset , and . Thus random forest estimates satisfy, for all , . 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 which is equal to the mean of the 's falling in the cells containing in the forest. If we define the connection function of the finite forest as , i.e. the proportion of cells shared between and , then almost surely we have , which defines the KeRF. Centered KeRF The construction of Centered KeRF of level is the same as for centered forest, except that predictions are made by , the corresponding kernel function, or connection function is Uniform KeRF Uniform KeRF is built in the same way as uniform forest, except that predictions are made by , the corresponding kernel function, or connection function is Properties Relation between KeRF and random forest Predictions given by KeRF and random forests are close if the number of points in each cell is controlled: Assume that there exist sequences such that, almost surely, Then almost