从GAM光滑对象确定导数 [英] Determining derivatives from GAM smooth object
问题描述
我有一个非常简单的时间序列数据集，其中包含一个单变量("AVERAGE")的年平均值.我希望研究时间序列趋势"分量的变化率(一阶导数)和加速度(二阶导数)以及相关的标准误差.我已经使用MGCV的GAM和PREDICT功能获得了趋势"，如下所示:
A <- gam(AVERAGE ~ s(YEAR), data=DF, na.action=na.omit)
B <- predict(A, type="response", se.fit=TRUE)
我已经通过两种独立的方法确定了导数，分别应用高自由度三次方平滑样条和通过第一和第二次差异(轻度平滑)并进行自举以近似误差，两者均产生可比较的结果.
我注意到"gam.fit3"函数有助于确定最多二阶导数，但并未直接调用.我还注意到，使用类型为"lpmatrix"的"predict.gam"有助于平滑的派生.我想直接使用"GAM"函数来计算一阶和二阶导数，但是对计算或提取这些导数的技能不够熟练.我试图在"Predict.gam"帮助页面末尾为一个变量重新配置Wood的示例，但是没有成功.任何帮助我朝正确方向前进的帮助都将是非常棒的.谢谢菲尔.
predict.gam中的示例使用有限差异以近似平滑项的导数
这里是一个针对单个预测器模型执行此操作的示例.这比帮助中的示例更直接.
A <- gam(AVERAGE ~ s(YEAR), data=DF, na.action=na.omit)
# new data for prediction
newDF <- with(DF, data.frame(YEAR = unique(YEAR)))
# prediction of smoothed estimates at each unique year value
# with standard error
B <- predict(A, newDF, type="response", se.fit=TRUE)
# finite difference approach to derivatives following
# example from ?predict.gam
eps <- 1e-7
X0 <- predict(A, newDF, type = 'lpmatrix')
newDFeps_p <- newDF + eps
X1 <- predict(A, newDFeps_p, type = 'lpmatrix')
# finite difference approximation of first derivative
# the design matrix
Xp <- (X0 - X1) / eps
# first derivative
fd_d1 <- Xp %*% coef(A)
# second derivative
newDFeps_m <- newDF - eps
X_1 <- predict(A, newDFeps_m, type = 'lpmatrix')
# design matrix for second derivative
Xpp <- (X1 + X_1 - 2*X0) / eps^2
# second derivative
fd_d2 <- Xpp %*% coef(A)
如果您使用引导捆绑获取置信区间，则应该能够获得这些近似值的置信区间.
I have a quite simple time series data set consisting of annual averages of a singe variable ("AVERAGE"). I wish to investigate the rate of change (1st derivative) and acceleration (2nd derivative) and associated standard errors of the "trend" component of the time series. I have obtained the "trend" using the GAM and PREDICT functions of MGCV simply as follows:
A <- gam(AVERAGE ~ s(YEAR), data=DF, na.action=na.omit)
B <- predict(A, type="response", se.fit=TRUE)
I have determined derivatives through 2 separate methods, applying a high DoF cubic smooth spline and via first and second differences (lightly smoothed) and bootstrapping to approximate errors with both producing comparable results.
I note that the "gam.fit3" function facilitates determining upto 2nd order derivatives but is not called directly. I also note that using "predict.gam" with the type "lpmatrix" facilitates derivatives of smooths. I would like to use the "GAM" function directly to calculate the 1st and 2nd derivatives but am not sufficiently skilled to calculate or extract these derivatives. I tried to reconfigure Wood's example at the end of the "Predict.gam" help page for one variable but with no success. Any help to get me headed in the right direction would be terrific. Thanks Phil.
The example from predict.gam uses finite differences to approximate the derivatives of the smoothed terms
Here is an example to do this for a single predictor model. This is more straightforward that the example from the help.
A <- gam(AVERAGE ~ s(YEAR), data=DF, na.action=na.omit)
# new data for prediction
newDF <- with(DF, data.frame(YEAR = unique(YEAR)))
# prediction of smoothed estimates at each unique year value
# with standard error
B <- predict(A, newDF, type="response", se.fit=TRUE)
# finite difference approach to derivatives following
# example from ?predict.gam
eps <- 1e-7
X0 <- predict(A, newDF, type = 'lpmatrix')
newDFeps_p <- newDF + eps
X1 <- predict(A, newDFeps_p, type = 'lpmatrix')
# finite difference approximation of first derivative
# the design matrix
Xp <- (X0 - X1) / eps
# first derivative
fd_d1 <- Xp %*% coef(A)
# second derivative
newDFeps_m <- newDF - eps
X_1 <- predict(A, newDFeps_m, type = 'lpmatrix')
# design matrix for second derivative
Xpp <- (X1 + X_1 - 2*X0) / eps^2
# second derivative
fd_d2 <- Xpp %*% coef(A)
If you are using boot strapping to get the confidence intervals, you should be able to get confidence intervals on these approximations.
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