mgcv中基于张量积的高斯过程更平滑

Question

我正在尝试运行 GAM 模型，其中 X 和 Y 之间的交互使用高斯过程建模。下面的代码在 mgcv 中使用默认平滑 (s()) 时工作​​正常，但我想用张量积 (te()) 对我的数据进行建模，因为我了解 te 产品专门解决各向异性交互。但是，当我使用te() 时，我似乎无法将所需的参数传递给模型。下面的例子。

library(mgcv)
set.seed(540)
df <- gamSim(2, n = 300, scale = 0.15)[[1]]
df$x <- scales::rescale(df$x, to = c(-180,180))
df$y <- scales::rescale(df$y, to = c(-90,90))
head(df)
# works fine using s()
m1 <- gam(z ~ s(x, y, bs = "gp", m = c(2, 5, 2), k = 30), data = df)
summary(m1)
Family: gaussian 
Link function: identity 

Formula:
z ~ s(x, y, bs = "gp", m = c(2, 5, 2), k = 30)

Parametric coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)  0.46143    0.01581   29.19   <2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Approximate significance of smooth terms:
         edf Ref.df     F p-value
s(x,y) 8.055  12.73 0.333   0.983

R-sq.(adj) =  0.0248   Deviance explained = 5.11%
GCV = 0.077292  Scale est. = 0.074959  n = 300

# Fails
# pass list as per example in ?te for multivariate marginal
mp <- list(c(2, 5, 2))
m2 <- gam(z ~ te(x, y, bs = "gp", m = mp, k = 30), data = df)
Error in gpE(knt, knt, object$p.order) : 
  incorrect arguments to GP smoother
In addition: Warning message:
In mgcv::te(x, y, bs = "gp", m = mp, k = 30) : m wrong length and ignored.

# try passing directly
m2 <- gam(z ~ te(x, y, bs = "gp", m = c(2,5,2), k = 30), data = df) # FAILS
Error in gpE(knt, knt, object$p.order) : 
  incorrect arguments to GP smoother
In addition: Warning message:
In mgcv::te(x, y, bs = "gp", m = c(2, 5, 2), k = 30) :
  m wrong length and ignored.

# don't pass m
m2 <- gam(z ~ te(x, y, bs = "gp", k = 10), data = df) # works
summary(m2)
Family: gaussian 
Link function: identity 

Formula:
z ~ te(x, y, bs = "gp", k = 10)

Parametric coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)  0.46143    0.01299   35.52   <2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Approximate significance of smooth terms:
          edf Ref.df     F p-value    
te(x,y) 14.69  20.14 7.633  <2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

R-sq.(adj) =  0.341   Deviance explained = 37.4%
GCV = 0.05341  Scale est. = 0.050618  n = 300

gam.check(m2) # k has been ignored!?
Method: GCV   Optimizer: magic
Smoothing parameter selection converged after 15 iterations.
The RMS GCV score gradient at convergence was 6.499192e-07 .
The Hessian was positive definite.
Model rank =  100 / 100 

Basis dimension (k) checking results. Low p-value (k-index<1) may
indicate that k is too low, especially if edf is close to k'.

          k'  edf k-index p-value
te(x,y) 99.0 14.7    1.08    0.94

Answer 1

在意识到d 必须在te() 函数中显式设置后，我解决了这个问题。下面的代码有效。

library(mgcv)
set.seed(540)
df <- gamSim(2, n = 300, scale = 0.15)[[1]]
df$x <- scales::rescale(df$x, to = c(-180,180))
df$y <- scales::rescale(df$y, to = c(-90,90))
head(df)

m1 <- gam(z ~ s(x, y, bs = "gp", m = c(2, 5, 2), k = 30), data = df, method = "ML")

mp <- list(c(2, 5, 2))
m2 <- gam(z ~ te(x, y, bs = "gp", m = mp, d = 2, k = 30), data = df, method = "ML")

Answer 2

您想要的似乎不存在（还没有？），即高斯过程 (GP) 平滑本质上是各向同性的。您可以从 MGCV 包中可用于 GP 平滑的可能自相关函数中看到它们是各向同性的（参见 Wood 2017 的第 242 页，或者在加载库后在 R 中简单地键入“?gp.smooth”）：所有这些都依赖于 仅在数据点之间的距离 d 上。如果有类似各向异性 GP 平滑的东西，您可以为每个轴的自相关标度定义不同的值，但似乎并非如此。

如果您的 x 和 y 数据具有不同的单位，并且您确实需要 GP 平滑，据我所知，您唯一能做的就是将它们缩放为单位（“就像在 LOESS 平滑中经常做的那样”，见第 227 页，Wood 2017）。

Wood 2017：“广义加法模型。R. 第二版简介”。