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.
To be specific, I had read that we should look at residuals e as a function of predicted values yhat, because these two are not correlated… while the residuals and the data values y are correlated. In fact, we know even more. If we fit a line for e as a function of yhat, we will find a slope of 0; if, instead, we fit a line for e as a function of y, we will find a slope of 1 – R^2, where R^2 is computed for the original regression – that is, the regression from which we got e and yhat in the first place.
I want to show you that: not just that there’s always a slope for e of y, but that it’s always 1-R^2.
For my ongoing record of earthquakes in my vicinity, there was only one in the past week… a 2.8 about 75 miles north of me. That brings the total to 11 so far in May.
In other earthquake news, after a long period of silence about that manslaughter trial of 6 seismologists and a civil servant in Italy, there are 2 new reports. One of them says that a California seismologist is testifying for the prosecution. The other says that the civil servant’s boss specified the media report that would come out of the meeting… before the meeting was held.
Now I want to chatter for a little while.
Before I started working on regression this time around, I had had a fair bit of experience running regressions. I was reasonably comfortable using dummy variables and low order polynomials, and forward selection. On the other hand, I was not widely read in the subject… and I’ve learned a lot in the past couple of years: backward selection… selection criteria… multicollinearity… single deletion statistics… variance inflation factors.
(For the record, I have a lot more to learn.)
My experience with regression is in marked contrast to my experience with control theory – in which I am rather widely read, and wretchedly inexperienced.
My experience with color was intermediate between these two: I had neither experience – except that I see in color! – nor was I widely read. I just jumped in, doing whatever mathematics presented itself. (Interestingly, my dictation software had written “nor was I wisely read” – which is true, too.)
Anyway, as a consequence of being so widely read in control theory, I look at an example in a book and think, “But what about this instead… or that instead… and this other alternative… and that one…” and so on.
Well, I’ve decided 2 things.
One, almost all of what little work I’ve done so far has been without Wolfram’s “control system professional” package. Now that they’ve moved most of its capabilities into Mathematica version 8, I really should learn to use it for my studying. This means I get to follow along doing the elementary control theory, while I learn exactly how to use the sophisticated tools available to me.
If you will, I’ve simply found a good excuse to work elementary problems as presented, without searching for alternatives… not to begin with anyway.
Two, it seems that far too many of the examples I’ve looked at end up at: unfortunately, our standard design techniques have left us with the following problem (which we’ve never mentioned before), and the solution is this (which we’ve never mentioned before).
To be specific, I’m thinking of an example in Carstens’ “automatic control systems and components” – which is a really fine book, by the way, and it’s on my bibliography page. On pages 270-271, he says
Theoretically, we wouldn’t need to perform any response curves for our system, since we can calculate virtually every feature of these curves. However, there are certain things we wish to avoid that are mathematically difficult to predict…. Figure 9–7 is the open loop Bode plot of our steering control system up to this point. Everything looks normal; the phase margin is right where we would like it to be. But notice where the zero gain crossover point occurs – right at the corner frequency or bend in our gain curve. This is coincidental, and not too desirable….
… Let’s consider what would happen if we were to add a rate generator to our system.
Never mind that, having not really registered his discussion of rate generators 50 pages before, I didn’t know what a rate generator was. I was annoyed because it came out of nowhere.
On top of that, even after I went back and read about rate generators, his considering the use of one to fix this issue still came out of left field.
Now, I figure I don’t understand something when the solution surprises the hell out of me. And yet, I’m not trying to become an expert in control theory… I want to understand a fair bit, but I do not ever propose to design a real world control system.
Well, I found another way to think of it. I can imagine myself as a non-control engineer who has been asked, “How do you think the subcontractor will control this?” I personally want to be able to do the straightforward standard design… to take the problem to the point where I could confidently return it to my boss and say, “This is as far as I can go… and while I do not see any problems, I wouldn’t be surprised if I’m missing something.”
First, then, I’m going to start at the beginning and hone my mathematical skills for control theory. 2nd, while I will try to collect tricks that seem to come from nowhere, I’m going to focus on becoming an apprentice, if you will – not even a journeyman – learning the easy stuff well.
We see how it goes.