Scatter plot kernel smoothing: ksmooth() does not smooth my data at all

Question

Original question

I want to smooth my explanatory variable, something like Speed data of a vehicle, and then use this smoothed values. I searched a lot, and find nothing that directly is my answer.

I know how to calculate the kernel density estimation (density() or KernSmooth::bkde()) but I don't know then how to calculate the smoothed values of speed.

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Re-edited question

Thanks to @ZheyuanLi, I am able to better explain what I have and what I want to do. So I have re-edited my question as below.

I have some speed measurement of a vehicle during a time, stored as a data frame vehicle:

         t       speed
1        0   0.0000000
2        1   0.0000000
3        2   0.0000000
4        3   0.0000000
5        4   0.0000000
.        .           .
.        .           .
1031  1030   4.8772222
1032  1031   4.4525000
1033  1032   3.2261111
1034  1033   1.8011111
1035  1034   0.2997222
1036  1035   0.2997222

Here is a scatter plot:

I want to smooth speed against t, and I want to use kernel smoothing for this purpose. According to @Zheyuan's advice, I should use ksmooth():

fit <- ksmooth(vehicle$t, vehicle$speed)

However, I found that the smoothed values are exactly the same as my original data:

sum(abs(fit$y - vehicle$speed))  # 0

Why is this happening? Thanks!

Answer 1

Answer to old question

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You need to distinguish "kernel density estimation" and "kernel smoothing".

Density estimation, only works with a single variable. It aims to estimate how spread out this variable is on its physical domain. For example, if we have 1000 normal samples:

x <- rnorm(1000, 0, 1)

We can assess its distribution by kernel density estimator:

k <- density(x)
plot(k); rug(x)

The rugs on the x-axis shows the locations of your x values, while the curve measures the density of those rugs.

Kernel smoother, is actually a regression problem, or scatter plot smoothing problem. You need two variables: one response variable y, and an explanatory variable x. Let's just use the x we have above for the explanatory variable. For response variable y, we generate some toy values from

y <- sin(x) + rnorm(1000, 0, 0.2)

Given the scatter plot between y and x:

we want to find a smooth function to approximate those scattered dots.

The Nadaraya-Watson kernel regression estimate, with R function ksmooth() will help you:

s <- ksmooth(x, y, kernel = "normal")
plot(x,y, main = "kernel smoother")
lines(s, lwd = 2, col = 2)

If you want to interpret everything in terms of prediction:

• kernel density estimation: given x, predict density of x; that is, we have an estimate of the probability P(grid[n] < x < grid[n+1]), where grid is some gird points;
• kernel smoothing: given x, predict y; that is, we have an estimate of the function f(x), which approximates y.

In both cases, you have no smoothed value of explanatory variable x. So your question: "I want to smooth my explanatory variable" makes no sense.

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Do you actually have a time series?

"Speed of a vehicle" sounds like you are monitoring the speed along time t. If so, get a scatter plot between speed and t, and use ksmooth().

Other smoothing approach like loess() and smooth.spline() are not of kernel smoothing class, but you can compare.