Happenings – 2012 May 26

It’s been another relatively uneventful week.

On the one hand, I have done no further work on color… so I have nothing new from Kang’s “computational color technology”, which was the impetus for the last color post, although it ended up as a very minor part… and I have not continued looking at complementary colors, which depend significantly, of course, on one’s choice of color model (color wheel, if you will).

On the other hand, while working on my next planned regression post, I was reminded of an old open question – and I expect that the answer will constitute this Monday’s post.
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Color: from Spectrum to XYZ and beyond

Intro

Over the last few years, we have learned how to compute the XYZ tristimulus values from a spectrum. In addition, however, over these years, we have learned how to convert XYZ to RGB on my monitor… how to convert RGB to HSV… how to convert XYZ to CIELab… and how to display colors on the artists’ color wheel (which I distinguish from the printers’ color wheel).

Well, let’s have some fun. Some of us don’t have to learn how to do any of these things – we’ve done them all before – and we can focus on putting them all together.

As it happened, I felt a need to once again compute tristimulus values from a spectrum… and I decided to do everything else I could think of once I had XYZ.
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Happenings – 2012 May 19

Well, the past week has been quiet, in more ways than one.

There have been only 2 earthquakes in my immediate vicinity from last Saturday through yesterday… for the month of May, this brings the total to 10.

I made a little progress on the next regression post – the 1st of 2 about the underlying assumptions – but only a little progress. Actually, I’ve probably got all the mathematics done, but the pieces are still disorganized.

As for rings, I’m not sure whether I’m making progress or going backwards. Well, I’ve pretty much got the theory laid out… and I played with the “abstract algebra” package… and now I’m struggling to figure out what examples to add to the theory.

I did take a quick look through my operations research books to check out their chapters on dynamic programming… they have several interesting problems I could look at… but they weren’t interesting enough to distract me from my current overwhelming list of active topics. I also took a look on the Internet, and I found a published excerpt from Richard Bellman’s autobiography, focusing on the beginnings of dynamic programming.


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Discrete Fourier Transform – Trig Parameters in Practice

Preliminaries

This post is a follow-up to the previous one, Discrete Fourier Transform – Trig Parameters in Principle. The titles are deliberately similar – but you will want to distinguish them.

Having shown you how to find trigonometric parameters of a sine wave, I want to show you a real example. I found this in Peter Bloomfield’s “Fourier analysis of time series: an introduction”, 978-0-471-88948-9.

The data, however, I found by searching the Internet. Although the book provides a couple of sources – the data isn’t there any longer.

According to Bloomfield, these numbers are the magnitude of a variable star at midnight on 600 consecutive nights. They have been rescaled.

Before we get started, let me point out that I am a rookie at using the Discrete Fourier Transform. I knew, for example, to expect bin leakage in the previous post… but it hasn’t been all that long that I’ve known about it. Still, what we’re about to do is pretty simple and everything works out fine.

Here are the data…
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Happenings – 2012 May 12

Well, I slept through another earthquake last night. Ho-hum. It was merely a 2.5 instead of the 4.1 I slept through last week.

There were 4 earthquakes of at least magnitude 2.5 in the Bay Area through Friday. If my counting is correct, we are up to 8 so far in May… not counting the early one today.

My alter ego the kid picked up something unusual this past week: “Eye of the hurricane”, the autobiography of the mathematician Richard Bellman. It’s an easy read… but it’s rather discouraging. He was very good, and he worked very hard… I’m not and I don’t. Well, maybe I can be productive today.

To make matters worse, he was either the inventor or the major proponent of “dynamic programming” – and I do not appear to own a book on the subject. Hmm, it’s literally true that I do not own a book… but at least 4 of my operations research books have sections on dynamic programming… and maybe the kid will look at them this morning….

Interestingly, he chose the name for political reasons – that is, for funding: “programming” was hot, and “dynamic” was harmless. And he needed harmless.
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Discrete Fourier Transform – Trig Parameters in Principle

pure sine

I want to look at the Discrete Fourier Transform (henceforth the DFT) of a perfect sine wave. How the heck can I interpret the DFT of some data, if I don’t understand what it tells me in the simplest possible case?

As usual, I am using Mathematica®… in fact I’m using version 7.

Suppose we have the following function: a pure sine wave, with mean 3, amplitude 5, and period 7. (I habitually use prime numbers in examples so that I can see where they end up in any answers.)

Actually, let me do this in stages. First let the mean be zero:

f(x) = 5 \sin \left(\frac{2 \pi  x}{7}\right)\ .

Over the range from 0 to 21, we have the following:

We see that the period is, indeed, 7. Now let me construct the function I really want, with mean 3:
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Happenings – 2012 May 5

According to the USGS, I slept through a magnitude 4.1 earthquake last night. Okay, strictly speaking, all they vouch for is that there was such an earthquake… I’m responsible for the assertion that I slept through it.

The frequency seems to be up recently: there have been 7 earthquakes in the Bay Area since last Saturday, of magnitude at least 2.5. For the record, however, let me show the count is 4 so far in May, through yesterday. I don’t want to count today yet, because it isn’t over.

So much for experimental seismology.

Theoretical seismology, on the other hand, has led me back to continuum mechanics – which includes solids as well as liquids and gases. I finally understand why the combination

\frac{\partial f}{\partial t}+ v\cdot \nabla f

is called “the material derivative” of f – that is, I understand why we say it is computed for a fixed particle. Certainly one can – and I did – do a lot of work treating it as a handy combination to look out for, but I finally understand its interpretation, instead of taking it on faith. I hope to write this up and publish it soon.
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