Is my color process going awry?

This blog is a first in a series of blog posts giving some concrete examples of how the newly-invented technique of ColorSPC and ellipsification can be used to answer real-world questions being asked by real-world people about real-world problems for color manufacturers.

So, picture this scenario. I am running a machine that puts color onto (or into) a product. Maybe it's some kind of printing press; maybe it mixes pigment into plastic; maybe this is about dyeing textiles or maybe it's about painting cars. The same principles apply.

John the Math Guy really Lays color SPC on the line

Today's question: I got this fancy-pants spectrophotometer that spits out color measurements of my product. How can I use it to alert me when the color is starting to wander outside of its normal operating zone?

An important distinction

There are two main reasons to measure parts coming off an assembly line:

     1. Is the product meeting customer tolerances?

     2. Is my machine behaving normally?

Conformance and SPC (statistical process control). These are intertwined. Generally, one implies the other. But consider two scenarios where the two answers are different. 

It could be that the product is meeting tolerances, but the machine is a bit wonky. Not wonky enough to be spitting out red parts instead of green, but there is definitely something different than yesterday. Should we do anything about this? Maybe, maybe not. It's certainly not a reason to run out of the building with our hair on fire. But it could be your machine's way of asking for a little TLC in the form of preventative maintenance.

Or it could be that your machine is operating within its normal range, and is producing product that is outside the customer tolerances. This the case you need to worry about. Futzing with the usual control knobs ain't gonna bring things in line. You need to change something about your process.

Use of DE for SPC

The color difference formulas, such as DE₀₀, were designed specifically to be industrial tolerances for color. While DE₀₀ may well be the second ugliest formula ever developed by a sentient being in this universe, it does a fair job of correlating with our own perception of whether two colors are an acceptable match. 

But is it a good way to assess whether the machine is operating in a stable manner? I mean, you just track DE over time, and if it blips, you know something is going on. Right? Let's try it out on a set of real data.

The plot below is a runtime chart of just over 1,000 measurements of pink spot color that I received from Company B. These are all measurements from a single run. I don't know for sure what the customer tolerance was, but I took a guess at 3.0 DE₀₀, and added that as an orange dashed line.

It sure looks like a lot of measurements were out of tolerance!

Uh-oh. It looks like we got a problem. There are a whole lot of measurements that are well above that tolerance... maybe one out of three are out of tolerance?

But maybe it's not as bad as it looks. The determination lies in how one interprets tolerance. Here is one interpretation from a technical report from the Committee for Graphic Arts Technologies Standards (CGATS TR 016, Graphic technology — Printing Tolerance and Conformity Assessment):

"The printing run should be sampled randomly over the length of the run and a minimum of 20 samples collected. The metric for production variation is the 70th percentile of the distribution of the color difference between production samples and the substrate-corrected process control aims."

TR 016 defines a number of conformance levels. (For a description of what those values mean, check out my blog on How Big is a DE₀₀) It says that 3.0 DE₀₀  is "Level II conformance", so the orange dashed line is a quite reasonable acceptance criteria for a press run. But a runtime chart is not at all useful for identifying those "Danger Will Robinson" moments. I mean, how do you decide if a single measurement is outside of a tolerance that requires 20 measurements? 

If we want to do SPC, then we must set the upper control limit differently.

Use of DE for SPC, take 2

The basic approach from statistical process control -- the whole six sigma shtick -- is to set the upper control limit based on what the data tells us about the process, and not based on customer tolerances. It is traditional to use the average plus three times the standard deviation as the upper limit. For our test data set, this works out to 5.28 DE₀₀.

The process looks in control now!

This new chart looks a lot more like a chart that we can use to identify goobers. In fact, I did just that with the two red arrows. Gosh darn it, everything looks pretty good.

But I think we need a bit closer look at what the upper limit DE means. The following pair of plots give us a perspective of this data in CIELAB. The plot on the left is looking down from the top at the a*b* values. The plot on the right is looking at the data points from the side with chroma on the horizontal axis and L* on the vertical.

The green dots are each of the measurements. The red diamond is the target color, and the ovoids are the upper limit tolerances of 5.28 DE₀₀. (Note: in DE₀₀, the tolerance regions are not truly ellipses, but are properly called ovoids. One should ovoid calling them ellipses, and also ovoid making really bad puns.)

Those are some big eggs!

The next image is  closeup of the C*L* plot, showing (with red arrows) the small set of wonky points that were identified with the DE runtime chart. I would say that these are pretty likely to be outliers. But look at the smattering of points that are well outside the cluster of data points, but are still within the ovoid that serves as the upper limit for DE. These should have stuck out in the runtime chart, if it were doing its job), but are deemed OK.

Wonkyville

Now, listen carefully... If you are using a runtime plot of "DE₀₀ from the target color", you are in effect saying that everything within the ovoids represents normal behavior for your process. So long as measurements are within those ovoids, you will conclude that nothing has changed in your process. That's just silly talk!

Here is my summary of DE runtime charts: JUST SAY NO! Well... unless your are looking at conformance, and your customer tolerance is an absolute, as in, "don't you never go above 4 DE₀₀!"

Use of Zc for a SPC

I know this was a long time ago, but remember the Z statistic from Stats 101? You compute the average and standard deviation of your data, and then normalize your data points to give you a parameter called Z. If a data point had a Z value that was much smaller than -3, or much larger than +3, then it was suspicious. This is mathematically equivalent to what's going on with the upper limit in a runtime chart.

I have extended this idea to three-dimensional data (such as color data). I call the statistic Zc. This is the keystone of ColorSPC.

Now, remember back when I showed the CIELAB plots of the data along with a DE₀₀ ovoid? Didn't you just want to grab a red pencil and draw in some ellipses that represented the data better? That's what I did, only I used my slide rule instead of a pencil. There is a mathematical algorithm that I call ellipsification that adjusts the axes lengths and orientation of a three-dimensional ellipsoid to "fit" the data. Ellipsification is the keystone of ColorSPC.

Ellipsification charts in CIELAB

The concentric ellipses in the drawings above are the points where Zc = 1, 2, 3, and 4. That is to say, all points on the innermost ellipse have Zc of 1. All points between the innermost and the next ellipse have Zc between 1 and 2.

Zc is a much better way to do SPC on color data. Here is a runtime plot of Zc for this production run. The red dashed line is set to 3.75. That number is the 3D equivalent of the Z = 3 upper limit used in traditional SPC.

Finally, a runtime chart we can believe!

As can easily be seen (if you click on the image, and then get out a magnifying glass) this view of the data provides us with a much better indication of data points which are outside of the typical variation of the process. Nine outliers are identified, and many of them stick out like sore thumbs. Kinda what we would expect from the CIELAB plots.

But wait!

In the previous DE analysis, we computed DE from the target value. In a paper by Brian Gamm (The Analysis Of Inline Color Measurements For Package And Labels Printing Using Statistical Process Monitoring Techniques, TAGA 2017), he pointed out this problem with DE runtinme charts, and advocated the use of the DE, but with DE measured from the average L*a*b* value, rather than the target. The graphs below show the result of this analysis on our favorite data set.

DE00 ovoids based on computing color difference from average

Addendum Feb 22, 2018: 

I would like to update the previous paragraph based on conversations with Brian.

First, he wanted to reiterate something that I have said before, and which bears re-reiterating. Looking at a runtime chart of DE is the correct thing to do when you are doing QA -- if your question is "did my product meet the conformance criteria from my customer?" But his paper (and this blog post) show that DE is not the proper tool for finding aberrant data. Both are necessary and useful.

Second, he advocated something a bit different than what I said. Subtle, but important difference. I said "... but with DE measured from the average L*a*b* value". Brian advocated "... but with DE measured from the initial L*a*b* value". Brian is looking at the drift during a production run. The assumption is made that color was dialed in pretty decent at the start, but may be gradually changing over time.

Thanks, Brian!

It is interesting to note that the DE₀₀ ovoid in a*b* (on the left) is similar to the to the ovoid produced by ellipsifcation. Larger, and not quite as eccentric, but similar in orientation. This is a good thing, and will often be the case. This will not be the case for any pigments that have a hook, which is to say, those that change in hue as strength is changed. This includes cyan and magenta printing inks.

However, it can be seen that the orientation of the DE₀₀ ovoid in C*L* (on the right) does not orient with the data in orientation. This is soooo typical of C*L* ovoids!

So, DE₀₀  from the average is a much better metric than DE₀₀ from target color. If you have nothing else to use, this is preferred. If you are reading this shortly after this blog was posted, and you aren't using my computer, then you don't have nothing else to use, since these wonderful algorithms have not migrated beyond my computer as I write this. I hope to change that soon.

Conclusion

For the purpose of conformance testing, there is no question that DE is the choice. DE₀₀ is preferred to ΔE_ab(or even DE_CMC  or DE₉₄  or DIN 99).

For the purpose of SPC -- characterizing your color process to outliers -- the DE from target metric is lousy. The use of DE from average is preferable, but the best metric is Zc, which is based on Color SPC and fitting ellipses to your data.

Statistics of multi-dimensional data, example

In the previous blog post, Statistics of multi-dimensional data, theory, I introduced a generalization of the standard deviation to three-dimensional data. I called it ellipsification. In this blog post I am going to apply this ellipsification thing to real data to demonstrate the application to statistical process control of color.

I posted this cuz there just aren't enough trolls on the internet

Is the data normal?

In traditional SPC, the assumption is almost always made that the underlying variation is normally distributed. (This assumption is rarely challenged, so we blithely use the hammers that are conveniently in our toolbox -- standard SPC tools -- to drive in screws. But that's another rant.)

The question of normalcy is worth addressing. First off, since I am at least pretending to be a math guy, I should at least pay lip service to stuff that has to do with math. Second, we are venturing into uncharted territory, so it pays to be cautious. Third, we already have a warning that deltaE color difference is not normal. Ok, maybe a bunch of warnings. Mostly from me.

I demonstrate in the next section that my selected data set can be transformed into another data set with components that are uncorrelated, have zero mean and standard deviation of 1.0, and which give every indication of being normal. So, one could us this transform on the color data and apply traditional SPC techniques on the individual components, but you will see that I take this one step further.

    Original data

I use the solid magenta data from the data set that I describe below in the section below called "Provenance of the data". I picked magenta because it is well known that it has a "hook". In other words, as you increase pigment level or ink film thickness, it changes hue. The thicker the magenta ink, the redder it goes. Note that this can be seen in the far left graph as a tilt to the ellipsoid.

I show three views of the data below. The black ellipses are slices through the middle of the ellipsification in the a*b* plane, the L*a* plane, and the L*b* plane, respectively.

View from above

View from the b* axis

View from the a* axis

    Standardized data

Recall for the moment when you were in Stats 201. I know that probably brings up memories of that cute guy or girl that sat in the third row, but that's not what I am talking about. I am talking about standardizing the data to create a Z score. You subtracted the mean and then divided by the standard deviation so that the standardized data set has zero mean, and standard deviation of 1.0.

I will do the same standardization, but generalized to multiple dimensions. One change, though. I need an extra step to rotate the axes of the ellipsoid so that all the axes are aligned with the coordinate axes. The cool thing is that the new scores (call them Z1, Z2, and Z3, if you like) are now all uncorrelated.

Geometrically, the operations are as follows: subtract the mean, rotate the ellipsoid, and then squish or expand the individual axes to make the standard deviations all equal to 1.0. The plot below show three views of the data after standardization. (Don't ask me which axes are L*, a*, and b*, by the way. These are not L*, a*, or b*.)

Standardized version of the L*, a*, and b* variation charts

Not much to look at -- some circular blobs with perhaps a tighter pattern nearer the origin. That's what I would hope to see. 

Here are the stats on this data:

	Mean	Stdev	Skew	Kurtosis
Z1	0.000	1.000	-0.282	-0.064
Z2	0.000	1.000	0.291	0.163
Z3	0.000	1.000	-0.092	-0.658

The mean and standard deviation are exactly 0.000 and 1.000. This is reassuring, but not a surprise. It just means that I did the arithmetic correctly. I designed the technique to do this! Another thing that happened by design is that the correlations between Z1 and Z2, and between Z1 and Z3 are both exactly 0.000. Again, not a surprise. Driving those correlations to zero was the whole point of rotating the ellipsoid, which I don't mind saying was no easy feat.

The skew and kurtosis are more interesting. For an ideal normal distribution, these two values will be zero. Are they close enough to zero? None of these numbers are big enough to raise a red flag. (In the section below entitled "Range for skew and kurtosis", I give some numbers to go by to scale our expectation of skew and kurtosis.)

In the typical doublespeak of a statistician, I can say that there is no evidence that the standardized  color variation is not normal. Of course, that's not to say that the standardized color variation actually is normal, but a statement like that would be asking too much from a statistician. Suffice it to say that it walks like normally distributed data and quacks like normally distributed data.

Dr. Bunsen Honeydew lectures on proper statistical grammar

This is an important finding. At least for this one data set, we know that the standardized scores Z1, Z2, and Z3 can be treated independently as normally distributed variables. Or, as we shall see in the next section, we can combine them into one number that has a known distribution.

Can we expect that all color variation data behaves this nicely when it is standardized by ellipsification? Certainly not. If the data is slowly drifting, the standardization might yield something more like a uniform distribution. If the color is bouncing back and forth between two different colors, then we expect the standardized distributions to be bi-modal. But I intend to look at a lot of color to try to see if 3D normal distribution is the norm for processes that are in control.

In the words of every great research paper every written, "clearly more research is called for".

The Zc statistic

I propose a statistic for SPC of color, which I call Zc. This is a generalization of the Z statistic that we all know and love. This new statistic could be applied to any multi-dimensional data that we like, but I am reserving the name to apply to three-dimensional data, in particular, to color data. (The c stands for "color". If you have trouble remembering that, then note that c is the first letter of my middle name.)

Zc is determined by first ellispifying the data set. The data set is then standardized, and then each data point is reduced to a single number (a scalar), as described in the plot below. The red points are a standardization of the data set we have been working with.the data set we have been working with. I have added circles at Zc of 1, 2, 3, 4. Any data points on one of these circles will have a Zc score of the corresponding circle. Points in between will have intermediate values, which are the distance from the origin. Algebraically, Zc is the sum in quadrature of the individual three components, that is to say, the square root of the sum of the squares of the three individual components.

A two-dimensional view of the Z scores

Now that we have standardized our data into three uncorrelated random variables that are (presumably) Gaussian with zero mean and unit standard deviation, we can build on some established statistics. The sum of the squares of our standardized variable will follow a chi-squared distribution, and the square root of the sums of the squares will follow a chi distribution. Note that this quantity is the distance from the data point to the origin.

Chi is the Greek version of our letter X. It is pronounced with the hard K sound, although I have heard neophytes embarrass themselves by pronouncing it with the ch sound. To make things even more confusing, there is a Hebrew letter chai which is pronounced kinda like hi, only with that rasping thing in the back of your throat. Even more confusing is the fact that the Hebrew chai looks a lot like the Greek letter pi, which is the mathematical symbol for all things circular like pie and cups for chai tea. But the Greek letter chi has nothing to do with either chai tea, or its Spoonerism tai chi.

Whew. Glad I got that outa the way.

Why is it important that we can put a name on the distribution? This gives us a yardstick from which to gauge the probability that any given data point belongs to the set of typical data. The table below gives some probabilities for the Zc distribution. Here is an example that will explain the table a bit. The fifth row of the table says that 97% of the data points that represent typical behavior will have Zc scores of less than 3.0. Thus the chance that a given data point will have a Zc score larger than that is 1 in 34.

Levels of significance of Zc

Zc	P(Zc)	Chance
1.0	0.19875	1
1.5	0.47783	2
2.0	0.73854	4
2.5	0.89994	10
3.0	0.97071	34
3.5	0.99343	152
4.0	0.99887	882
4.5	0.99985	6623
5.0	0.99999	66667

The graph below is a run time chart of the Zc scores for the 204 data points that we have been dealing with. The largest score is about 3.5. We would be hard pressed at calling this an aberrant point, since the table above says that there is a 1 in 152 chance of such data happening at random. By the way, we had close to 152 data points, so we should expect 1 data point above 3.5. A further test: I count eight data points where the Zc score is above 3.0. Based on the table, I expect about 6.

My conclusion is that there is nothing funky about this data.

Runtime chart for Zc of the solid magenta patches

Where do we draw the line between common cause and special cause variation? In traditional SPC, we use Z > 3 as the test for individual points. Note that for a normal distribution, the probability of Z < 3 is 0.99865, or one chance in 741 of Z < 3.0. This is pretty close to the probability of Zc < 4 for a chi distribution. In other words, if you are using Z > 3 as a threshold for QC with normally distributed data, then you should use Zc > 4 when using my proposed Zc statistic for color data. Four is the new three.

Provenance for this data

In 2006, the SNAP committee (Specifications for Newspaper Advertising Production) took on a large project to come to some consensus about what color you get when you mix specific quantities of CMYK ink on newsprint. A total of 102 newspapers printed a test form on its presses. The test form had 928 color patches. All of the test forms were measured by one very busy spectrophotometer. The data was averaged by patch type, and it became known as CGATS TR 002.

Some of the patches were duplicated on the sheet for quality control. In particular all of the solids were duplicated. Thus, in the blog post, I was dealing with 204 measurements of a magenta solid patch from 102 different newspaper printing presses.

Range for skew and kurtosis

How do we decide when a value of skew or kurtosis is indicative of a non-normal distribution? Skew should be 0.0 for normal variation, but can it be 0.01 and still be normal? Or 0.1? Where is the cutoff?

Consider this: the values for skew and kurtosis that we compute from a data set are just estimates of some metaphysical skew and kurtosis. If we asked all the same printers to submit another data set the following day, we would likely have a somewhat different value of all the statistics. If we had the leisure of collecting a Gillian or a Brilliant or even a vermillion measurements, we would have a more accurate estimate of these statistical measures. 

Luckily some math guy figgered out a simple formula that allows us to put a reliability on the estimates of skew and kurtosis that we compute.

Our estimate of skew has a standard deviation of sqrt (6 / N). For N = 204 (as in our case) this works out to 0.171. So, an estimate of skew that is outside of the range from -0.342 to 0.342 is suspect, and outside the range of -0.513 to 0.513 is very suspect.

For kurtosis, the standard deviation of the estimate is sqrt (24/N), which gives us a range of +/- 0.686 for suspicious and +/- 1.029 for very suspicious.

Statistics of multi-dimensional data, theory

This blog post is the culmination of a long series of blog posts on the statistics of color difference data. Most of them just basically said "yeah, normal stats don't work". Lotta help that is, eh? Several blog posts alluded to the fact that I did indeed have a solution. The most recent of which alluded to a method that works in the very title of the blog post: Statistical process control of color, approaching a method that works.

Now it's time to unveil the method.

Generalization of the standard deviation

One way of describing the technique is to call it a generalization of the standard deviation to multiple dimensions -- three dimensions if we are dealing with color data. That's a rather abstract concept, so I will explain.

     One dimensional standard deviation

We can think of our good friends, the standard deviation and mean, as describing a line segment on the number line, as illustrated below. If the data is normally distributed (also called Gaussian, or bell curve), then you would expect that about 68% of the data will fall on the line segment within one standard deviation unit (one sigma) of the mean, 95.45% of the data will fall within two sigma of the mean, and 99.73% of the data will be within three sigma of the mean.

As an aside, note that not all data is normally distributed. This holds true for color difference data, which is the issue that got me started down this path!

So, a one-dimensional standard deviation can be thought of as a line segment that is 2 sigma long, and centered on the mean of the data. It is a one-dimensional summary of all the underlying data.

     Two-dimensional standard deviation

Naturally, a two-dimensional standard deviation is a two-dimensional summary of the underlying two-dimensional data. But instead of a (one-dimensional) line segment, we get an ellipse in two dimensions.

In the simplest case, the two-dimensional standard deviation is a circle (shown in orange below) which is centered on the average of the data points. The circle has a radius of one sigma. If you want to get all mathematical about this, the circle represents a portion of a two-dimensional Gaussian distribution with 39% of the data falling inside the circle, and 61% falling outside.

Two dimensional histogram of a simple set of two dimensional data
The orange line encompasses 39% of the volume.

I slipped a number into that last paragraph that deserves to be underlined: 39%. Back when we were dealing with one-dimensional data, +/- one sigma would encompass 68% of normally distributed data. The number for two-dimensional data is 39%. Toto, I have a feeling we're not in one-dimensional-normal-distribution-ville anymore.

Of course, not all two-dimensional standard deviations are circular like the one in the drawing above. More generally, they will be ellipses. The the length of the semi-major and semi-minor axes of the ellipse are the major and minor standard deviation.

--- Taking a break for a moment

I better stop to review some pesky vocabulary terms. A circle has a radius, which is the distance from the center of the circle to any point on the circle. A circle also has a diameter, which is the distance between opposite points on the circle. The diameter is twice the radius.

When we talk about ellipses, we generally refer to the two axes of the ellipse. The major axis is the longest line segment that goes through the center of the ellipse. The minor axis is the shortest line segment that goes through the center of the ellipse. The lengths of the major and minor axes are essentially the extremes of the diameters of the ellipse. They run perpendicular to each other.

An ellipse, showing off the most gorgeous set of axes I've ever seen

There is no convenient word for the two "radii" of an ellipse. All we have is the inconvenient phrases semi-major axis and semi-major axis. These are half the length of the major and minor axes, respectively.

--- Break over, time to get back to work

The axes of the ellipses won't necessarily be straight up and down and left-to-right on a graph. So, the full description of the two-dimensional standard deviation must include information to identify the orientation of these axes.

The image below shows a set of hypothetical two-dimensional data that has been ellipsified. The red dots are random data that was generated using Mathematica. I asked it to give me 200 normally distributed x data points with a standard deviation of 3, and 200 normally distributed y data points  with a standard deviation of 1. These original data points (the x and y values) were uncorrelated.

This collection of data points were then rotated by 15 degrees so that the new x values had a bit of y in them, and the new y values had a bit of x in them. In other words, there was some correlation (r = 0.6) between the new x and y. I then added 6 to the new x values and 3 to the new y values to move the center of the ellipse. So, the red data points are designed to represent some arbitrary data set that could just happen in real life.

I performed an ellipsification, and have plotted the one, two, and three sigma ellipses (in pink). The major and minor axes of the one sigma ellipse are shown in blue.

Gosh darn! that's purdy!

The result of ellipsifying this data is all the parameters pertaining to the innermost of the ellipses in the image above. This is an ellipse that is centered on {6.11, 3.08}, with major axis of 3.19 and minor axis of 1.00. The ellipse is oriented at 15.8 degrees. These are all rather close to the original parameters that I started with, so I musta done sumthin right.

I also counted the number of data points within the three ellipses. I counted 38.5% in the 1 sigma ellipse, 88.5% in the 2 sigma ellipse, and 99% in the 3 sigma ellipse. (Of course when I say I did this, I really mean that Mathematica gave me a little help.) If the data follows a two-dimensional normal distribution, then the ellipses will encompass 39%, 86.5%, and 98.9% of the data. This is one indication that this condition is met.

The following pieces of information are determined in the ellipsification process of two-dimensional data:

     a) The average of the data which is the center of the ellipse (two numbers, for the horizontal and vertical values)
     b) The orientation of the ellipse (which could be a single number, such as the rotation angle)
     c) The lengths of the semi-major and semi-minor axes of the ellipse (two numbers)

The ellipsification can be described in other ways, but these five numbers will tell me everything about the ellipse. The ellipse is the statistical proxy for the whole data set.

     Three-dimensional standard deviation

The extension to three-dimensional standard deviation is "obvious". (Obvious is a mathematician's way of saying "I don't have the patience to explain this to mere mortals.")

The result of ellipsifying three-dimensional data is the following nine pieces of information that are necessary to describe an arbitrary (three-dimensional) ellipsoid:

    a) The average of the data (three numbers, for the average of the x, y, and z values)
    b) The orientation of the ellipsoid (three numbers defining the direction that the axes point)
    c) The lengths of the semi-major, semi-medial, and semi-minor axes of the ellipse (three numbers)

The image below is an ellipsification of real color data. The data is the color of a solid patch as produced by 102 different newspaper printing presses. There were two samples of this patch from each press, so the number of dots is 204.

The 204 dots were used to compute the three-dimensional standard deviation, which is represented by the three lines. The longest line, the red line, is the major axis of the ellipse, and has a length of 5.6 CIELAB units. The green and blue lines are the medial and minor axes, respectively. They 2.2 and 2.1 CIELAB units long. All three of the axes meet at the mean of all the data points, and all three are two-sigma long (+/-1 sigma from the mean). Depending on the angle you are looking, it may not appear that the axes are all perpendicular to each other, but they are.

Ellipsification of some real data, as shown with the axes of the ellipsoid

Trouble viewing the image above? The image is a .gif image, so you should see it rotating. If it doesn't, then try a different browser, or download it to your computer and view it directly.

What can we do with the ellipsification?

The ellipsification of color data is a three-dimensional version of the standard deviation, so in theory, we can use it for anything that we would use the standard deviation for. The most common use (in the realm of statistical process control) is to decide whether a given data point is an example of typical process variation, or if there is some nefarious agent at work. (Deming would call that a special cause.)

We will see an example of this on real data in the next blog post on the topic: Statistics of multi-dimensional data, example.