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   A Horten Ho 229—the only surviving one.  Although in parts as it is being restored so one can not tell, this aircraft looks more like the 1990s B-2 bomber than a 1940s German plane.  An impressive bit of engineering for the time, the craft was fast, long-range, and even had a reduced radar cross-section (i.e. stealth).  I will have to return to see it again when it is completed.  It has only been at the Smithsonian National Air and Space Museum since the summer.  I watched a bit of video about the restoration center, and these people are serious about what they do.  So their efforts should be impressive when complete.
   A MIG 21 on display at the Smithsonian National Air and Space Museum.  What I find interesting about this aircraft is a small blurb about the plane:
This MiG-21F-13 was displayed in a Soviet military hardware exhibit at Bolling Air Force Base, Maryland, as part of a "Soviet Awareness" training program. Its service history remains unknown.
   I read "service history remains unknown" as "we can't tell you."
   The Smithsonian National Air and Space Museum had the original mothership model from Close Encounters Of The Third Kind.  Apparently it had several jokes worked into it, including a Volkswagen bus, a submarine, the R2-D2 android, a U.S. mailbox, an aircraft, and a small cemetery plot.  The only one I found was R2-D2.  I enjoyed the movie Close Encounters (as I do many of Steven Spielberg's movies), and I think part of this has to do with the portrayal of benign aliens.  Spielberg makes them a mystery and even a bit scary with how powerful they are, but in the end no one is hurt.  We're not exactly sure what took place or what happens next, but the aliens don't seem bent on genocide and takeover.  Sadly, this is not the norm when aliens are encountered in movies.

March 08, 2015

Mathematics of Weighted Polynomial Regression

Weighted polynomial regression is a method by which some terms are considered more strongly than others. There are a number of reasons this may be desired. If the regression coefficients are being used to extrapolate additional data it may make for a more accurate prediction if the newest data is considered more strongly than older data. A set of data that is based on measurements might have a margin of error that is unique for each data point. The data points with lower margins of error can be considered more strongly than those of higher error. A set of data points might also represent an average over a different amount of data. In this case, the amount of data each point represents can be used as the weight. There are a number of reasons to weight data points individually.

We will begin with an example of a weighted average. A classic example has to do with considering an average of averages, where each average has a different number of data points. Consider two classes that have taken the same test. One class has 20 students, the other 30.

First class:

71 71 68 68 67 67 70 73 73 65 72 67 68 70 69 74 72 67 74 73                              

Second class:

94 85 86 89 91 94 86 92 89 93 91 89 89 88 94 92 86 91 85 92 91 92 89 85 91 94 94 91 91 86

The average for the first class is 70.0, and for the second class 90.0. The average of all the grades together is 82.0. However, the average of the two averages is 80.0. This is because the average of averages considers both sets to be equal. This can be corrected by considering the number of students.

Here, a weighting value has been added to each term and reproduces the same value as the average of all data points. Let us look at this in a general form:

To prove this, let's plug in the values:

w0 20
w1 30
y0 70
y1 90

To see how we came to the general form, let us consider how to introduce a weighting term into the average. The mean average is actually just polynomial regression of degree 0. So let's write out the average as polynomial regression in expanded matrix form:

This doesn't look like the mean average, but it can be show to be so if we reduce. Here a is the average. A 1x1 matrix is the same as if it wasn't in a matrix, so we can remove the matrices.

Normally one does not see that averages have x values associated with y values, but they do—kind of. The reason they are not seen is because the x values are raised to the 0th power, which means every x value is just 1. And the summation of a sequence of 1 is just the count of that sequence (if you add up n ones, you get n). So the series can be reduced:

Now solve for a by moving n to the opposite side:

And this is how one is use to seeing the average represented. Thus our matrix representation is equivalent. Let's return to the matrix form and introduce a weighting term, W. Consider what happens when both sides are multiplied by this constant:

Algebraically W does nothing—it simply cancels out. We can move W inside the matrices:

And move W inside the summations:

Thus far we have assume W is a constant, but that was just to maneuver it into place. Now that W is in where it should be we can stop making that assumption and allow W to become a weighting term. Let W = wi so that there is a sequence of weights that can be applied to every y value. Our equation becomes:

We can drop the matrices, and get rid of the x0 because that is just 1.

Solve for a:

This produces a weighted average with value yi weighted by wi. This matches our earlier representation of a weighted average. It is also equivalent to the mean average if the weighting term is constant, and we can quickly show this:

We can apply the same weighting method to linear regression. We will use linear regression because the equation is not nearly as large as polynomial regression. First, the expanded representation of linear regression:

Now introduce a constant weighting term:

Move this term into the matrix:

And then into the summations:

Once there we no longer make the assumption that the weight value is constant, but is instead a sequence:

Reduce the powers:

And we now have the weighted form of linear regression. What was done here should be pretty straightforward—wi was just placed in all the summations. We take what we did with linear regression and can quickly apply it to polynomial regression. First start with the expanded form of polynomial regression:

Here, m is the number of coefficients for the polynomial, and n is the number of x/y data points. Now add wi was just placed in all the summations.

Clean this up a little to simplify:

This is the general form of weighted polynomial regression. Again, if the weighting term is constant for each index it will simply reduce out and one will be left with unweighted polynomial regression.

Just as with using polynomial regression, how weighting is actually used is not so easy to generalize. One must carefully consider what phenomena is being modeled before adding a weighting term to the data. However once this is understood a weighting term can be very useful.

This article has been added to the polynomial regression site and is mostly a recap of a previous article.

March 07, 2015

Increasing Sunlight

   I have noticed we are getting a lot more light on the roof as measured by ππ

    This graph shows all the light data we have so far in Wh/m2·day.  The trend line is a second degree polynomial and emphasizes the recent light recovery.  Adding to this data has been the clear skies and I think the sun has now crept above the tree line during noon hours. 

The other day I got a negative comment on the online polynomial regression page.  It reads:

this website is stupid and u have no life

My first thought was “You came to a mathematics site and took the time to complain the author had no life. How ironic.” Then I started to wonder: how did this person get to my site? The online polynomial regression page has become increasingly popular, to the point it is now the most active site on my server. The other day I saw hits coming from A little searching around I found a link to the site being used as part of a math course. And looking through my log file I saw a huge number of hits from This domain belongs to the Las Virgenes Unified School District. I speculate that students at this school were required to use the online polynomial regression page as part of a class assignment, and the person who commented is a disgruntled teen who neither enjoys the assignment, or my website.  Leave it to primary school to ruin the intrigue of any subject.  I likely would have felt the same.  I now interpret the message as "This class is stupid, I don't want to use your website, and I would rather be doing something else with my life."  So the message now stands as a testament to discontent.

2 comments have been made.

From Roger

Portland, OR

March 12, 2015 at 12:06 PM

I understand your logic, but keep in mind that many of your visitors (like me) go straight to your polynomial regression page.  The negative comment stood out and, in my opinion, tarnished your useful website.  If I were you I would remove, or at modify the comment with your explanation above.  I see no value in letting a disgruntled student mar something you put a lot of work in creating.

From Andrew Que (

Middleton, WI

March 14, 2015 at 5:10 AM

Thank you for your comment Rodger of Portland, OR.  I thought for awhile how best to address the comment on my polynomial page, but in the end I think it is better to take the bad with the good.  To invite comments and censor those you don't agree with seems like lying to yourself and the world.  As long as the site doesn't end up spamed with comments I can't justify removing them.  If anything the negative feedback shows educators their students have the same freedoms as the rest of the world when using the site--like they are being treated as real people.  If they require their students to use the site, the students could make comments that reflect the student's views--views that may not coincide with the those of the educator.  Maybe I could put a heading with a message to that effect on the site.

CDC 3800

CDC 3800

   The Control Data 3800 computer.  This one was on display at the Smithsonian National Air and Space Museum and used from the 1960s though to the 1990s to operate Air Force satellites.  A 30 year run for a computer is impressive.  This computer was also designed by Seymour Cray, the father of supercomputers.  In the large cabinet was 128 KB of memory and a CPU that could run at least 1 MIPS which would qualify it as a supercomputer for its day.  By comparison cellphones of the day can run over 2,000 DMIPS and contain hundreds of megabytes of memory.  Still this computer represents the cutting edge of its day and a stepping stone for computer technology.
   This billboard says that "hell is real" found on the roadside heading south between Gary and Indianapolis, Indiana.  My initial reaction was "prove it."  But I wonder what kind of message this message billboard is trying to convey.  A billboards purpose is to advertise, typically for marketing.  So what is being marketed?  I'm pretty sure this isn't an advertisement to encourage people to go to hell, so in all likelihood it's a warning.  This is the "believe or else" school of religious thought designed to keep people afraid and thereby secure adherence.  It doesn't say "Heaven is real."
   The indented audience is primarily people of faith.  Non-religious people would no more be convince by this billboard than we would be if it claimed unicorns were real.  A factitious statement is no more true when printed in large white letters on a black background.  So this billboard is unlikely to effect the non-religious.  However, those who do have faith might get a feeling of guilt when looking at this billboard.  They might think about things they are doing that are considered sinful, and consider what they should do about it.  An equivalent would be to print "big brother is watching" on a billboard by the roadside.  It's essentially the same message: better watch yourself because you are going to be punished if you falter.  Now my question has become: is this the intended function of this message?  Is it is simple as threatening people to keep them conforming?

1 comment has been made.

From Rob

from Rockton

September 07, 2015 at 2:54 AM

I've always assumed that this meant 'Hell is Indiana".

March 02, 2015

Road trip day 9--The Return Home

East Chicago

East Chicago

   Turns out my accident yesterday bent a strut and there was no way to correct for the alignment.  I decided to wrap the trip up and take Eve home where I could take her to a shop I trust to do the repair work.  She handled fine, and despite having a pretty good ding in the front driver side tire rim I had no vibrations.  In fact she handled just fine the entire drive home.
   It was a nice drive through the mountains of southern New York/Northern Pennsylvanian.  I pulled into the drive way around 2:00 am.  Having marked the odometer before and after the trip I logged exactly 3067 miles. 
   Although cut short it was a good trip and I enjoyed each of my stops.  I picked up a new hobby, got to see a few things I had wanted to see for a long time, and visited several people I had not seen in a very long time.

1 comment has been made.

From colleen


March 05, 2015 at 12:19 AM

Was so good seeing you! :)