R help - Mar 2026 - Query on ksmooth(): Theoretical step function vs. plotted diagonal segments

Dear R-help members,

Hello,
I am studying kernel regression and experimenting with the ksmooth()
function in R.

When using a box kernel, the kernel function is an indicator function
(weight = 1 inside the bandwidth and 0 outside). Based on this definition,
I expected the Nadaraya?Watson estimator to produce a step-like function:
the estimate should remain constant while the set of included points is
unchanged, and then jump when a point enters or leaves the bandwidth window.

However, when I run the following code using the cars dataset:

par(mfrow=c(1,1))
with(cars, {
plot(speed, dist)
lines(ksmooth(speed, dist, "normal", bandwidth = 2),
col = "blue", lwd = 3)
lines(ksmooth(speed, dist, "box", bandwidth = 2),
col = "darkorange", lwd = 3)
})
legend("topleft", c("Normal Kernel with h=2", "Box
Kernel with h=2"),
lwd = c(2,2),
col = c("blue","darkorange"), cex = 2)

the curve produced by the box kernel (dark orange) appears to contain
diagonal line segments rather than the step-like shape I expected. I have
attached the resulting plot for reference.

My understanding is that the theoretical estimator should behave like a
step function because the kernel weights are either 0 or 1. Therefore, I
was wondering whether the diagonal segments arise from how ksmooth()
evaluates the estimator on a grid of x values and then connects those
points with straight lines for plotting, or if there is another
implementation detail that explains this behavior.

Could you please clarify whether this is expected behavior?

Thank you very much for your time.

Best regards,
Sangeun Song

-------------- next part --------------
A non-text attachment was scrubbed...
Name: plot_ksmooth_box.png
Type: image/png
Size: 167309 bytes
Desc: not available
URL:
<https://stat.ethz.ch/pipermail/r-help/attachments/20260319/d0cb1687/attachment.png>

You used the lines function to plot the data. Of course it is going to show
straight sloped lines between points.

On March 19, 2026 7:00:00 AM PDT, "???" <helen.song0206 at
gmail.com> wrote:
>Dear R-help members,
>
>Hello,
>I am studying kernel regression and experimenting with the ksmooth()
>function in R.
>
>When using a box kernel, the kernel function is an indicator function
>(weight = 1 inside the bandwidth and 0 outside). Based on this definition,
>I expected the Nadaraya?Watson estimator to produce a step-like function:
>the estimate should remain constant while the set of included points is
>unchanged, and then jump when a point enters or leaves the bandwidth window.
>
>However, when I run the following code using the cars dataset:
>
>par(mfrow=c(1,1))
>with(cars, {
>plot(speed, dist)
>lines(ksmooth(speed, dist, "normal", bandwidth = 2),
>col = "blue", lwd = 3)
>lines(ksmooth(speed, dist, "box", bandwidth = 2),
>col = "darkorange", lwd = 3)
>})
>legend("topleft", c("Normal Kernel with h=2", "Box
Kernel with h=2"),
>lwd = c(2,2),
>col = c("blue","darkorange"), cex = 2)
>
>the curve produced by the box kernel (dark orange) appears to contain
>diagonal line segments rather than the step-like shape I expected. I have
>attached the resulting plot for reference.
>
>My understanding is that the theoretical estimator should behave like a
>step function because the kernel weights are either 0 or 1. Therefore, I
>was wondering whether the diagonal segments arise from how ksmooth()
>evaluates the estimator on a grid of x values and then connects those
>points with straight lines for plotting, or if there is another
>implementation detail that explains this behavior.
>
>Could you please clarify whether this is expected behavior?
>
>Thank you very much for your time.
>
>Best regards,
>Sangeun Song
--
Sent from my phone. Please excuse my brevity.
	[[alternative HTML version deleted]]

?s 14:00 de 19/03/2026, ??? escreveu:
> Dear R-help members,
> 
> Hello,
> I am studying kernel regression and experimenting with the ksmooth()
> function in R.
> 
> When using a box kernel, the kernel function is an indicator function
> (weight = 1 inside the bandwidth and 0 outside). Based on this definition,
> I expected the Nadaraya?Watson estimator to produce a step-like function:
> the estimate should remain constant while the set of included points is
> unchanged, and then jump when a point enters or leaves the bandwidth
window.
> 
> However, when I run the following code using the cars dataset:
> 
> par(mfrow=c(1,1))
> with(cars, {
> plot(speed, dist)
> lines(ksmooth(speed, dist, "normal", bandwidth = 2),
> col = "blue", lwd = 3)
> lines(ksmooth(speed, dist, "box", bandwidth = 2),
> col = "darkorange", lwd = 3)
> })
> legend("topleft", c("Normal Kernel with h=2", "Box
Kernel with h=2"),
> lwd = c(2,2),
> col = c("blue","darkorange"), cex = 2)
> 
> the curve produced by the box kernel (dark orange) appears to contain
> diagonal line segments rather than the step-like shape I expected. I have
> attached the resulting plot for reference.
> 
> My understanding is that the theoretical estimator should behave like a
> step function because the kernel weights are either 0 or 1. Therefore, I
> was wondering whether the diagonal segments arise from how ksmooth()
> evaluates the estimator on a grid of x values and then connects those
> points with straight lines for plotting, or if there is another
> implementation detail that explains this behavior.
> 
> Could you please clarify whether this is expected behavior?
> 
> Thank you very much for your time.
> 
> Best regards,
> Sangeun Song
> 
> 
> ______________________________________________
> R-help at r-project.org mailing list -- To UNSUBSCRIBE and more, see
> https://stat.ethz.ch/mailman/listinfo/r-help
> PLEASE do read the posting guide
https://www.R-project.org/posting-guide.html
> and provide commented, minimal, self-contained, reproducible code.
Hello,

Use type = "s", see help("plot.default") for the possible
values of
argument type.
You have a link to that help page in the documentation of line().

?plot.default
?line

In the code below, I only use it in the 2nd line() call.



# if it's the 1st time you call par(), this does nothing
# (it changes a graphics device from rows*cols == 1*1 to 1*1)
# anyway, save the value so that you can later reset the default
old_par <- par(mfrow = c(1, 1))
# see what mfrow was before
old_par
#> $mfrow
#> [1] 1 1

with(cars, {
   plot(speed, dist)
   lines(ksmooth(speed, dist, "normal", bandwidth = 2),
         col = "blue", lwd = 3)
   lines(ksmooth(speed, dist, "box", bandwidth = 2),
         type = "s",
         col = "darkorange", lwd = 3)
})
legend("topleft", c("Normal Kernel with h=2", "Box
Kernel with h=2"),
        lwd = c(2,2),
        col = c("blue","darkorange"), cex = 2)

# reset the default par values
par(old_par)



Hope this helps,

Rui Barradas

>>>>> ???  
>>>>>     on Thu, 19 Mar 2026 23:00:00 +0900 writes:
    > Dear R-help members,
    > Hello,
    > I am studying kernel regression and experimenting with the ksmooth()
    > function in R.

    > When using a box kernel, the kernel function is an indicator function
    > (weight = 1 inside the bandwidth and 0 outside). Based on this
definition,
    > I expected the Nadaraya?Watson estimator to produce a step-like
function:
    > the estimate should remain constant while the set of included points is
    > unchanged, and then jump when a point enters or leaves the bandwidth
window.

    > However, when I run the following code using the cars dataset:

    > par(mfrow=c(1,1))
    > with(cars, {
    > plot(speed, dist)
    > lines(ksmooth(speed, dist, "normal", bandwidth = 2),
    > col = "blue", lwd = 3)
    > lines(ksmooth(speed, dist, "box", bandwidth = 2),
    > col = "darkorange", lwd = 3)
    > })
    > legend("topleft", c("Normal Kernel with h=2",
"Box Kernel with h=2"),
    > lwd = c(2,2),
    > col = c("blue","darkorange"), cex = 2)

    > the curve produced by the box kernel (dark orange) appears to contain
    > diagonal line segments rather than the step-like shape I expected. I
have
    > attached the resulting plot for reference.

    > My understanding is that the theoretical estimator should behave like a
    > step function because the kernel weights are either 0 or 1. Therefore,
I
    > was wondering whether the diagonal segments arise from how ksmooth()
    > evaluates the estimator on a grid of x values and then connects those
    > points with straight lines for plotting, or if there is another
    > implementation detail that explains this behavior.

    > Could you please clarify whether this is expected behavior?

Definitely, as Jeff and Rui explained.

As solution e.g. for teaching / illustration (otherwise the box
kernel should *never* be used!), I recommend just to evaluate
the resulting function  on a finer grid, e.g.,  at 1000 instead
of by default 100 points:


## MM:  evaluate the curve on a much finer grid, using n.points = 1000  (default
was 100)
with(cars, {
    plot(speed, dist)
    lines(ksmooth(speed, dist, "norm", bandwidth = 2, n.points=1000),
col = "blue", lwd = 3)
    lines(ksmooth(speed, dist, "box",  bandwidth = 2, n.points=1000),
col = "darkorange", lwd = 3)
})
legend("topleft", c("Normal Kernel with h=2", "Box
Kernel with h=2"),
       lwd = 2, col = c("blue","darkorange"),
bty="n")

--
Martin