Comment from Mike Woodhouse
Time: Thursday, October 9, 2008, 5:51 am

I ran the data through Curvexpert which got the best fit to be a Gaussian model:

y = a*exp(-(x-b)^2/(2*c^2)) ( I think)

where

a = 407.68954
b = 4.8840398
c = 3.2081591

It fits slightly better than the quadratic (r = 0.9286) but I wouldn’t give either much credit, especially when we subsequently read that we don’t have the whole data set.

And a sixth-order polynomial is over-fitting to an appallingly dangerous degree unless you know exactly what you’re doing (in which case the question wouldn’t have been asked in the first place!)

Comment from Rob
Time: Monday, October 13, 2008, 8:52 am

a good example of the 6th order overkill is Runge’s phenomenon

Comment from Will Dwinnell
Time: Sunday, March 8, 2009, 8:43 pm

It is possible to fit splines sets of connected simple curves to data (typically collections of cubic polynomials). Your pair of linear fits would be a linear spline. Almost always, spline fitting is performed as going exactly through the given data points, but this is not necessary. A trend for your 80-point series could probably be reasonably fit using 6 or 7 spline “knots” (places where the simple curves are connected.

Comment from DaleW
Time: Thursday, October 22, 2009, 10:16 am

TableCurve finds that a Lorentzian peak is a slightly better fit than a Gaussian peak, but I agree with Mike that your starting data, at first glance and with no context, looks more like a peak than a process that is in a state of statistical control.

SPC is not very powerful with only 10 points, and we might be better off using direct hypothesis testing if we can’t see the larger data set. At the end of your ten points, you’ve got 5 points in a row steadily decreasing. Assuming no ties, the odds that six random points in a row are sequentially ordered from any given point is two in (1*2*3*4*5*6) or 1 in 360. One more point would typically be considered an SPC out of control rule violation, and your data set is so small that standard SPC rules, which are certainly a great tool for larger datasets, are too forgiving here.

A look at the larger data set might show just randomness as you suggested. Or it might show intermittent peaks, or other out of control drift of the mean. From these ten points alone, not knowing if they were cherry-picked or typical of the larger distribution, shouldn’t we tentatively affirm that this limited evidence favors the existence of a local peak?

Comment from DaleW
Time: Thursday, October 22, 2009, 8:35 pm

Jon,

I’d hate to see you fall into absolute skepticism whenever there are only 10 data points.

Based solely on these 10 points, a rather compelling case can be made that we do NOT here have a random sample from a process that is in a state of statistical control. That conclusion seems meaningful to me, and suggests SPC charting of a fuller dataset would very likely find it to be out of control. (Or perhaps that you were presented with a worst case example from a much larger dataset.)

As our friendly default hypothesis, let’s assume we have your hypothesized stable process where each measurement has only independent random error; this is the hypothesis against which an SPC chart is intended to detect exceptions.

The alternate hypothesis on the table is a particular nonrandom pattern in our data, some type of peak which spans more than a single data point (not just an isolated outlier). Let’s adopt this alternative only if the default hypothesis fails to explain our data, say at p<0.01 (p<0.05 is typical, but let's be more conservative here since there are several different patterns that would each cause us to reject the default hypothesis).

What test do we use to detect peak-ness? We may need to improvise one. The t-test is a good starting point, since our default hypothesis assumes we have a constant but unknown population variance.

Any distribution with unique values has a largest value. Under our default hypothesis, the points closest to it in time ("Near") should not have any higher value that the points ("Far") further from it in time. We might define our Near category as the half of the points closer in time to our maximum, and our Far category as the half of points further in time from it, excluding the maximum point from either set to keep the test fair, with an expected difference of zero. Then we can use a one-sided pooled t-test to determine if Near > Far by a significant amount. =TTEST({400,325,360,305},{190,250,255,191,123),1,2) evaluates to 0.001526 for a p-value, or 1 chance in 655 for a metric that extreme for a process that is in a state of statistical control. Unlikely!

Alternatively, we might define “near” as absolute difference and try to fit an inverse v-peak to our data. Then a linear fit of Y v. ABS(X-Xmax) for our remaining nine points not including Xmax has a t-score of 6.07 for the slope, which has a two-tailed probability of 0.0005 by regression using the Data Analysis add-in. That’s one chance in 1984 times for our default hypothesis — really half that likely, since only one tail supports a peak claim. When only a 1 in 4000 tail of our default model fits the data, it’s probably time to look at a different model, wouldn’t you say?

This isn’t at all likely to be random data — not in the sense that we define random for an SPC chart.