黄土以新的x值预测 [英] loess predict with new x values

问题描述

我试图理解predict.loess函数如何能够在原始数据中不存在的点x处计算新的预测值(y_hat).例如(这是一个简单的例子，我知道这种例子显然不需要黄土，但它说明了这一点):

I am attempting to understand how the predict.loess function is able to compute new predicted values (y_hat) at points x that do not exist in the original data. For example (this is a simple example and I realize loess is obviously not needed for an example of this sort but it illustrates the point):

x <- 1:10
y <- x^2
mdl <- loess(y ~ x)
predict(mdl, 1.5)
[1] 2.25

loess回归是通过在每个x处使用多项式来进行的，因此在每个y处都创建了预测的y_hat.但是，由于没有存储系数，因此在这种情况下，模型"只是用于预测每个y_hat(例如，span或degree)的详细信息.当我执行predict(mdl, 1.5)时，predict如何在这个新的x处产生一个值?是否在两个最接近的现有x值及其关联的y_hat之间进行插值?如果是这样，它如何执行此操作的详细信息是什么?

loess regression works by using polynomials at each x and thus it creates a predicted y_hat at each y. However, because there are no coefficients being stored, the "model" in this case is simply the details of what was used to predict each y_hat, for example, the span or degree. When I do predict(mdl, 1.5), how is predict able to produce a value at this new x? Is it interpolating between two nearest existing x values and their associated y_hat? If so, what are the details behind how it is doing this?

我已经在线阅读了cloess文档，但是找不到在何处进行讨论.

I have read the cloess documentation online but am unable to find where it discusses this.

推荐答案

但是，由于没有存储系数，因此在这种情况下，模型"只是用于预测每个y_hat的细节

However, because there are no coefficients being stored, the "model" in this case is simply the details of what was used to predict each y_hat

也许您已经使用过print(mdl)命令，或者只是使用了mdl来查看模型mdl包含的内容，但事实并非如此.该模型非常复杂，并且存储了大量参数.

Maybe you have used print(mdl) command or simply mdl to see what the model mdl contains, but this is not the case. The model is really complicated and stores a big number of parameters.

要了解其中的内容，可以使用unlist(mdl)并查看其中的大量参数.

To have an idea what's inside, you may use unlist(mdl) and see the big list of parameters in it.

这是命令手册的一部分，描述了命令的实际工作方式:

This is a part of the manual of the command describing how it really works:

安装是在本地完成的.也就是说，对于点x处的拟合，使用x邻域中的点进行拟合，并按它们与x的距离加权(计算距离时会忽略参数"变量中的差异).邻域的大小由α控制(由跨度或enp.target设置).对于α＜在图1中，邻域包含点的比例α，并且它们具有三次三次加权(与(1--(dist/maxdist)^ 3)^ 3成比例).对于α> 1，将使用所有点，其中最大距离"假定为p个解释变量的实际最大距离的α^(1/p)倍.

Fitting is done locally. That is, for the fit at point x, the fit is made using points in a neighbourhood of x, weighted by their distance from x (with differences in ‘parametric’ variables being ignored when computing the distance). The size of the neighbourhood is controlled by α (set by span or enp.target). For α < 1, the neighbourhood includes proportion α of the points, and these have tricubic weighting (proportional to (1 - (dist/maxdist)^3)^3). For α > 1, all points are used, with the ‘maximum distance’ assumed to be α^(1/p) times the actual maximum distance for p explanatory variables.

对于默认族，拟合度是(加权)最小二乘.为了 family ="symmetric"的M估计过程的一些迭代 使用Tukey的biweight.请注意，因为初始值是 最小二乘拟合，这不必是非常可靠的拟合.

For the default family, fitting is by (weighted) least squares. For family="symmetric" a few iterations of an M-estimation procedure with Tukey's biweight are used. Be aware that as the initial value is the least-squares fit, this need not be a very resistant fit.

我相信它试图在每个点的附近拟合一个多项式模型(不仅仅是整个集合的一个多项式).但是，邻域并不仅仅意味着一个点之前和之后的一个点，如果我正在实现这样的功能，我会给与点x最接近的点赋予较大的权重，而给远端点赋予较轻的权重，并尝试拟合一个适合最高的总重量.

What I believe is that it tries to fit a polynomial model in the neighborhood of every point (not just a single polynomial for the whole set). But the neighborhood does not mean only one point before and one point after, if I was implementing such a function I put a big weight on the nearest points to the point x, and lower weights to distal points, and tried to fit a polynomial that fits the highest total weight.

然后，如果要为其预测高度的给定x'最接近点x，我尝试使用拟合在点x的邻域上的多项式-说P(x)-并将其应用于x'-说P(x')-那将是预测.

Then if the given x' for which height should be predicted is closest to point x, I tried to use the polynomial fitted on the neighborhoods of the point x - say P(x) - and applied it over x' - say P(x') - and that would be the prediction.

如果您正在寻找特殊的东西，请告诉我.

Let me know if you are looking for anything special.

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