Rationale for the use of Regressogram (Bin-Smooth)

I am taking a class in data mining and we have recently been introduced to bin-smoothing in regression analysis but i cannot seem to understand the usefulness of this method nor how the method works or why it works. Basically, an illustration was given of a data set which cannot be fit using a linear model and the bin smooth was mentioned as a better model. See this lecture slide for the data set. This is not from my class but the bin smooth on the slide is identical. Could one explain the rational behind using averages as is done in bin-smooth?

• Although I believe most readers would have some idea of what "bin-smoothing" might be, it's a broad technique and conceivably could be applied in several distinct ways. Could you please tell us more precisely what you mean by it in the context of regression analysis? Could you supply a link to a reference? – whuber Mar 10 '13 at 15:00
• @whuber bin smoothing in the context of piecewise constant regression. – cobie Mar 10 '13 at 15:10
• Yes, but exactly how? Please either describe this method in sufficient detail so that at least an expert (or preferably anybody) can know what you are referring to, or else provide an easily accessible reference. – whuber Mar 10 '13 at 15:12
• cobie, could you give some more context perhaps? What exactly was said about it in your class? Do you have a text which describes it? Your question, as it stands, risks closure. It would be a pity, because I think there's a very good question trying to come out here and I'd like to give a good answer to it. Please, try to identify more clearly what precisely you're talking about and if possible where your confusions lie. You MUST have been given more detail than just the name of a technique. – Glen_b Mar 10 '13 at 21:53
• basically, an illustration was just given of a data set which cannot be fit using a linear model and the bin smooth was mentioned as a better model. See this lecture slide for the data set amath.colorado.edu/courses/7400/2010Spr/lecture38.pdf. This is not from my class but the bin smooth on the slide is identical.cheers – cobie Mar 10 '13 at 23:12

So we're saying "Y is related to X", but we don't know the form. We want to estimate $E(Y|X=x)$. But at any observation, we have noise, and we need to be able to estimate it between observations.

One (fairly naive) way we can approximate it is to assume that while the average changes, that it doesn't change too rapidly (i.e. that it's 'slowly varying' in a particular sense). Consequently, we could slice the domain up into sections (bins), and calculate an estimate of $E(Y)$ - here just the sample mean of the $y$'s - for all the $x$-values in each section (bin).

That is to say, the averages of the $y$'s in a narrow strip of $x$'s will typically be closer to those $x$'s than averaging more widely (because of variation in the mean of y over the range of x - that is, lower bias), but much smoother than just taking say the nearest x-value (because you're typically averaging a bunch of x's - all the ones in the bin).

The graph then is just a horizontal line drawn within the bin, at the mean of the observations in that bin. Those horizontal bars will 'follow' the data simply because they're local means - within each bin they're the average of the observations in the bin. Obviously as the relationship moves up and down the local bin-means will also.

This is fairly simple and can cope with complicated relationships, if they don't change too rapidly.

But it's not smooth! There are discontinuities at the bin boundaries.

There's a tradeoff between bias (in the sense that wider bins mean that we move further from accuracy in approximating the mean at a particular $x$ as we get near the ends of the bin) and variance (in the sense that narrower bins mean we have less data, so the noise dominates more).

A slightly more sophisticated version of this kind of general idea would be Nadaraya-Watson kernel smoothing, in the same way that kernel density estimation is related to the histogram.

http://en.wikipedia.org/wiki/Kernel_regression

and then we can work from there up to say local linear or local polynomial smoothing - fitting local lines or curves rather than local means.

Edit: Here's an outline of the basic steps in the example data you pointed to.

Here's the first few observations in the motorcycle data (which is in mcycle in the MASS package in R):

times accel
1    2.4   0.0
2    2.6  -1.3
3    3.2  -2.7
4    3.6   0.0
5    4.0  -2.7
6    6.2  -2.7
7    6.6  -2.7
8    6.8  -1.3
9    7.8  -2.7
10   8.2  -2.7

Härdle (who I said hello to just a few days ago when he visited) does a regressogram of this data in "Applied Nonparametric Regression". He says he uses a bin width of 4, and that his bin origin is 0. This is plainly false since he gets a lot more than 5 bins between 0 and 20. But lets take his suggestion of a bin width of 4 and on origin of 0. I'm going to run my bins to exclude the left limit and include the right one (simply because that's the way the R cut function does it; otherwise I'd be inclined to run the other way).

So that means our first bin runs from time 0 to time 4 ($(0,4]$). What are the accelerations in that? The first 5 times (circled in red) are in the first bin. The next 4 times (in blue) are in the second. We average the accelerations in those time periods. So $\hat{m}(x)$ is -1.34 for $x$ in $(0,4]$ and then it's -2.35 for $x$ in $(4,8]$ and then it's -2.844 ...

In R:

with(mcycle,print(tapply(accel,cut(times,seq(0,60,4)),mean),3))
(0,4]    (4,8]   (8,12]  (12,16]  (16,20]  (20,24]  (24,28]  (28,32]  (32,36]
-1.340   -2.350   -2.844  -22.365  -78.176 -119.167  -45.159   27.990   25.040
(36,40]  (40,44]  (44,48]  (48,52]  (52,56]  (56,60]
3.643    4.862   -4.020   -0.867   -2.350   10.700

So if we plot those values across those ranges: Incidentally, Härdle's actual binwidth appears to be 2.1 - if I use that binwidth with bin origin 0, I reproduce his bin smooth: • so its pretty much about getting average values for estimators in a range? – cobie Mar 10 '13 at 15:49
• What? You're trying to estimate $E(Y|X=x)$ at each possible $x$ in the range. There are obvious reasons why you need to blend information across different $x$'s (already addressed). One way to do that - not particularly great, but doable - is this way. – Glen_b Mar 10 '13 at 15:51
• It sounds like both the questioner and the answerer are guessing about what is being discussed--not a good recipe for a solid post, is it? – whuber Mar 10 '13 at 15:53
• @whuber I think it's at least broadly clear enough to hazard a first attempt at an answer which might be improved. I answered in the hopes that it would help the questioner identify their question more clearly, or so that they could explain that I had misunderstood what they wanted. In this case, at least, I'd rather give some response to what I believe is a genuine question than leave the OP with nothing on which to try to improve the question. I have an idea for improving my answer, which I hope to get to today. – Glen_b Mar 10 '13 at 21:51
• Beautiful reply--thorough, well-illustrated, accurate, helpful. (+1) – whuber Mar 11 '13 at 16:53