问题
This is a follow up question to How can I fit a smooth hysteresis in R?. A straightforward application of smooth.spline fits my actual poorly, although it proved a useful generic idea for this type of problem and had worked well on my simulated toy dataset.
I uploaded an example of my dataset here.
Following image is created by the code at the end.
Thysteresis is not fully closed - but that's not an issue.
Applying smooth.spline produces a way too messy output (red). Using argument spar in smooth.spline I was able to get a sufficiently smooth approximation (blue) - though it is only poorly following the curve of my data points on the upper right.
However what really disturbs me is the "knot" around the points in the lower left - even for the smooth curve (blue) it is there. It also has an artificial "loop" when it moves from the lower left to the upper right, although there are not data points near there.
library(dplyr)
library(ggplot2)
library(tidyr)
exdata <- readRDS("data/example.Rdata")
exdata %<>%
mutate(y_smooth = smofun(y),
x_smooth = smofun(x),
y_smooth2 = smofun(y, spar = 0.7),
x_smooth2= smofun(x, spar = 0.7)
)
ggplot(exdata, mapping = aes(x=x, y=y)) +
geom_point(alpha = 0.3) +
geom_path(mapping = aes(x=x_smooth, y=y_smooth), colour = "red") +
geom_path(mapping = aes(x=x_smooth2, y=y_smooth2), colour = "blue") +
theme_light()
回答1:
As I said in our previous thread: How can I fit a smooth hysteresis in R?, we want to smooth each coordinate separately.
## you have 713 data
plot(1:713, exdata$x); points(1:713, exdata$y)
Usually I would use smooth.spline for each. It worked good on your original toy dataset, but not that good on your real dataset sketched above. Directly using smooth.spline will lead us to an awkward situation as here: R smooth.spline(): smoothing spline is not smooth but overfitting my data. Obviously there are two change points in your data, and you require different amount of smoothness / degree of freedom on different segments. In the linked case study I was able to come up with a transformation to overcome the problem, but here I don't have a clue on what suitable transform I could do. Instead, I suggest that we place knots of the spline manually in order to have a direct control on the smoothness.
smooth.spline does not allow us to place knots manually. You can tell it how many knots you want (by using argument nknots) and it internally places knots by quantile. In fact, we don't have to rely on smooth.spline. This is just a regression problem. We can do it with any kind of regression method, as long as it satisfies you. In the following I use regression B-splines provided by splines package (one comes with R so no need to install) and fit a model using ordinary least squares (via lm).
library(splines)
test <- function (b1, b2, n1, n2, n3, degree = 1, exdata) {
x <- exdata$x; y <- exdata$y; t <- 1:length(x)
## place knots:
## n1 knots on [1, b1)
## n2 knots on [b1, b2]
## n3 knots on (b3, 713]
t1 <- seq(1, b1, length = n1 + 1)
t2 <- seq(b1, b2, length = n2)
t3 <- seq(b2, 713, length = n3 + 1)
kt <- c(t1[1:n1], t2, t3[-1])
kt_inner <- kt[2:(length(kt) - 1)]
fitx <- lm( x ~ bs(t, degree = degree, knots = kt_inner) )
fity <- lm( y ~ bs(t, degree = degree, knots = kt_inner) )
xh <- fitted(fitx); yh <- fitted(fity)
par(mfrow = c(1, 2))
plot(t, x, col = 8); points(t, y, col = 8)
abline(v = kt, lty = 2, col = 8)
lines(t, xh, lwd = 2); lines(t, yh, lwd = 2)
plot(x, y, col = 8)
lines(xh, yh, lwd = 2)
z <- list(x = xh, y = yh)
}
The test function allows you to specify two break points b1, b2. Then, on [1, b1) you want n1 knots, on [b1, b2] you want n2 knots, and on (b3, 713] you want n3 knots. Setting degree allows us to try linear spline, quadratic spline and cubic spline. The test function would produce fitted splines, and return invisiblely the smoothed x, y values.
Upon a few try, I found the following is probably the best:
z <- test(b1 = 90, b2 = 130, n1 = 4, n2 = 5, n3 = 10, degree = 3, exdata)
No penalization is imposed here. But penalized regression is possible, by using mgcv::gam in place of lm + splines::bs. Package mgcv allows us to specify knots (see: mgcv: How to set number and / or locations of knots for splines).
If we don't want to bother specifying knots, we may use "adaptive spline" smoother in mgcv, which allows smoothing parameter to vary locally (it essentially produces a set of smoothing parameters). This really overcomes the limit in smooth.spline where you only have a single smooth parameter that controls global penalization.
Note that in spline approach, choosing break points is hidden in the choice of knots, as break points are among the set of knots.
There are also none spline based methods. Package segmented does piece-wise linear regression by automatically choosing change / break points (but you have to tell it ahead how many break points there are by using argument psi).
There is plenty of room for comparison and discussion of different methods. As a backup for possible future re-investigation, I attached your dataset here (in case the dropbox link breaks).
exdata <- structure(list(x = c(129.5695, 131.7681, 111.0764, 104.072, 117.4685, 124.6366, 123.3217, 127.4646, 128.0695, 110.6105, 105.3012, 98.3239, 94.7057, 103.349, 101.5618, 97.0452, 99.2581, 96.869, 87.8457, 89.5727, 101.3341, 103.7006, 100.9025, 101.2322, 107.0066, 99.4793, 101.446, 110.8227, 115.8369, 113.8645, 119.1815, 115.382, 106.6097, 100.0037, 99.4583, 93.7219, 93.5242, 94.6776, 98.66, 92.5156, 99.0487, 99.2332, 97.9979, 98.5008, 99.4881, 96.2035, 99.3862, 108.1677, 107.2952, 107.5039, 109.8523, 107.0338, 98.331, 106.3463, 105.3598, 100.5509, 108.4587, 111.785, 102.911, 104.0221, 103.0966, 96.0376, 95.0978, 92.9253, 94.8009, 96.7117, 101.732, 107.1491, 107.0559, 107.6543, 108.8192, 101.6353, 94.3223, 94.3889, 93.4375, 99.0691, 101.1942, 102.7105, 105.8936, 104.8027, 93.6381, 99.6715, 98.8952, 100.2534, 104.2578, 105.9263, 106.0875, 118.8101, 121.278, 117.5301, 121.667, 428.0926, 731.1513, 775.6144, 795.4499, 817.3595, 535.1731, 276.7803, 330.6044, 439.8171, 541.2278, 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来源：https://stackoverflow.com/questions/52300914/how-to-fit-a-smooth-hysteresis-in-a-poorly-distributed-data-set