College Team Preformance

I watched Corey's Lecture today and was curious about the end where he plotted what he hoped to be the future of the program.  He was envisioning Yuk making it to Nationals in a few years or so - a great goal, but I wonder if it is attainable.  I downloaded the data / copied it from Score Reporter, and posted it as a google doc.  Some caution should be taken for the years 2008 and 2009, there was a lot of unknown teams in the top college spots.  The data was also weird in that it seemed to be done in late December / Early January, not at the end of the Spring Season. 

What I saw was not a lot of variation in the performance of the selected teams; the good stayed good and the medium stayed medium.  But CMU Ultimate is designed to be a program, so I wondered if a B-teams performance is an indicator of an successful A-team; thinking that B-team's performance provides an indicator the the programs performance as players matriculate from a successful B-Team to A-Team.  I don't think that the data is there to reach such a conclusion - partly due to changes in the  RRI / Ranking calculation, partly due to the wild fluctuations in 100 and below teams.

Kinda weird how Stanford A and B follow a trend, as does Pitt's A and B though.

Shenandoah National Park (Camping with the Brats + David)

I saw three more bears last weekend; an adult and two cubs!  They disappeared into the woods before I could wrassle them (it would have been real cute), but Eli still saw them so I will count it as a win.  I found the trip very enjoyable.  We hiked in on Thursday and camped beside what sounded like the Niagara Falls - the entire night I woke up at various points in the night thinking it was raining.  On Friday morning I got to cross off a  life goal of bathing in a mountain stream.  By a small miscalculation of time we ended up at our night camp at 1 p.m.. We spent the afternoon damning up a small stream, watching David flick the frying pan at Isaac off a stick, playing cards, and David and I taking pot shots at each other with a homemade bow and arrow.  Saturday we hiked the Devil's Stairs - it was just a trail up a gorge that crossed the stream multiple times - and a couple of miles of the AT.

Knoxville Summer League

Green and Yellow Hat got ups!
From Regionals against Georgia

The first games of Knoxville Summer League were tonight.  First ultimate I played since regionals.  Came out flat and slow but warmed up towards the end once I found my legs.  We ended up beating our opponents 17-7 or so.   Highlights included handling, caching and throwing multiple hammers, throwing  a terrible scobber, throwing a huck, and shredding a zone.  Lowlights included being  beat on multiple in cuts because of laziness, fouling some poor dude (I thought he was going to make a play on it but he just decided to stand still - I told him I fouled him and gave him the disc, but then I followed it up by point blocking him), a terrible break shot, and getting a calf cramp at the end.  I don't know why I am getting them - I got them every time I had a hard practice  / track workout in the spring.

Going off on a tangent, if you don't have the throws / don't know how to handle don't do it!  We had a lot of people pick up the disk off a turn and then just throw it away, or ignore their dump.   I know that they are their to learn / have fun, but I groaned inwardly.  Also, what is with people just running around willy nily?  It seems to be a lot more work for me to guard a cutter who doesn't know what they are doing compared to a cutter who does.

Gaussian Distribution (Probability Function Approximations)

I took the derivative of a Gaussian distribution, $$\frac{\mathrm{d} }{\mathrm{d} x}\frac{1}{\sqrt{2\pi \sigma^2}}e^{-(x-\mu)^2/{2\sigma^2}}=-\frac{e^{-\frac{(x-\mu )^2}{2 \sigma ^2}} (x-\mu )}{\sqrt{2 \pi } \sigma ^2 \sqrt{\sigma ^2}} $$, in order to find the type of terms I am looking - looking for something like $$x e^{x^2}$$ I expect $$\mu$$ to be zero because I am centering the distribution around zero, and the width of the distribution ($$\sigma$$) to depend on the linearly on n, the depth.

The first approximation I tried was using a simple a simple exponential; $$x!\approx e^{x}$$.  The approximation was not that good, but it was simple.  This did not pan as terms canceled in the integral. The next approximation I tried was $$x!\approx c^{x^2}$$.   The approximation still rose too quickly, and then integrating yielded the error function - not good.

The final approximation that I am still working on is the Sterling Approximation.  Now I am just trying a power expansion, because I cannot integrate $$\frac{n^{\frac{1}{2}+n} p^{-1+n} \int (n-r)^{-\frac{1}{2}-n+r} r^{-\frac{1}{2}-r} \, dr}{\sqrt{2 \pi }}$$.

PSD

Tentative PSD Algorithm

I have gotten caught up in doing PSD (Pulse Spectrum Discrimination) on our films.  Results seem promising!

I worked a little more on the mathematics of the Falling Balls / Gaussian Distribution.  The idea is to integrate all of the probabilities across the rows
$$ Ae^{-(x-\mu)^2/2\sigma^2} ?=\int_{0}^{n-1} \frac{r!}{n!(r-n!)}p^{n-1} $$
- I think that will provide the probability density function (PDF). Turns out the continuous analog of the factorial  is the gamma function, (for positive values, which we have).    Turns out that nobody has ever analytically integrated the gamma function before, either.  So now I am looking for a nice, integratable approximation for the factorial / gamma function (trying to do a Taylor series expansion of the terms I get from the Sterling Approximation).

Gaussian Distribution

After being stupid (trying to be clever with the indices, but being too clever) I developed a dynamic programming solution to the peg network problem.  The figure on the left shows the what happens if we fit the results to a Gaussian distribution; we get excellent agreement, with small residuals.  However, the residuals are not randomly distributed about zero, which usually provides information of a systematic error.  I don't have an explanation for why those features showed up.

function [pdf grid data ]= DPSol(n)
% Making sure the input is odd; if it is not making it the next odd
if mod(n,2) == 0
    n = n +1;
end

% Setting up the the grid to be two larger than the probabilities
grid = gridSetup(n+2);

p = 0.5;
pdf = zeros(n,2*n+1);
mid = ceil(length(pdf(1,:))/2);

% Itterating down the "pegs"
pdf(1,mid) = 1.0;
for j = 2:n
    % Setting the pegs on that row
    for i=(mid-j+1):(mid+j-1)
        if(mod(j,2)==0)
            if( mod(i,2) == 1)
                pdf(j,i) = (pdf(j-1,i-1)+pdf(j-1,i+1))*rand;
            end
        else
            if( mod(i,2) == 0)
                pdf(j,i) = (pdf(j-1,i-1)+pdf(j-1,i+1))*rand;
            end
        end
    end
end

In the version I used to derive the graph I used a constant probability of 0.5, the following figure was produced using a random probability at each intersection.  You can see the initial spike at the first node of 1.0, and after that it all seems to die out as the ball progresses its way down the network.

Isotropic Scattering (Derivation of Gaussian)

Where we last left off I had shown through the conservation of momentum that if a ball coming from the left hit the left side of a peg, it would stay on the left side of the peg.  This undermines the assumption I made that there was a 50/50 probability that the ball would go left or right when hitting the pin - though that assumption was that the ball was being dropped straight down unto the peg so it would have neither left nor right leanings (in fact, conservation of momentum would have the ball be at rest atop the peg).

I want to look into this more now to see if this problem can be resolved.  On the left is a schematic of the problem (negating gravity).

Let's let b (impact parameter) represent the distance between the marble and the peg. If $$b < r_1+r_2$$ than scattering occurs, if $$b>R_1+R_2$$ than scattering does not.  We can then look at how a small change in impact parameter effects the scatter angle $$\psi$$, $$2 \pi b db = -\sigma(\psi) 2 \pi sin(\psi) d\psi$$.  From geometry we find that $$ b = (R_1 + R_2) cos(\psi/2) $$, and so we can take the derivative of that to get $$ db = (R_1 + R_2) (-1/2) sin (\psi/2) d\psi $$.  Plugging $$ \frac{db}{d\psi} $$ back into the small change in impact parameter allows us to solve for the differential scattering cross section, $$ \sigma(\psi)=1/4 (R_1 + R_2) $$, which doesn't depend on the impact parameter; i.e. it is isotropic, but all that means that the scattering cross section is the same for all impact parameters.

I wanted to show results of the DP, but I can't seem to get the DP to make sense.  I got DP to work on Pascal's Triangle, but I think Pascal's Triangle would only work with constant probability, and I want to randomly pick the probability as the ball falls down.

Goverment Budget Silliness

 As a final project for one of my courses I am supposed to devise the energy policy of the United States (or the world, but lets be realistic here).  I was wondering what sort of budget I was dealing with, and I found a neat little semi-interactive chart (left) that breaks down how the U.S. Budget.  It doesn't explain how each of the smaller components utilizes its funds (I sent them an email asking for this feature to be released in the next update). 0.84% of the budget is spent on Technology Programs, and it seems that the DOE labs as "Other Science and Research Laboratories" (anything not NASA) getting 0.39% of the national budget.  I think that might make it worse than it seems; there is probably lab science money buried in the Defense.

Speaking of defense, check out the following two charts (source data).

Source Excel

I only plotted certain items that I found interesting; that is why the total doesn't look like the sum of the lines underneath it.  I was then curious to see if the defense budget dropped after the end of WW II, or if just continued to rise.  As shown below, it did drop a little.

Source Excel

 I will end my trend of badly scaled buget plots by one more, which shows how certain "functions" fared.  I have no idea what a function is other than the name that was provided in the source data.






Finally, do you know where the energy in your house goes? 

Stolen from Dr. Steve Koohin

There is a reason why I don't get any of my work completed.

Gaussian Distribution

This ones for you, Brando.

At the ARI Conference in DC I visited Camilo where we discussed  a science fair project he did where he derived a Gaussian distribution by dropping marbles down a grid network of evenly spaced nails among other things (I am terrible at estimating girls weights.  113.5 lbs, really?).

The symmetry of the problem is apparent, and then the model can be simplified by only worrying about the positive x values.

Symmetric Model

In the above model if you assume that the probability p of the marble going left or right it becomes possible to derive the probability that the marble will be a specific location.



A couple of things to notice:

• A check for the symmetry condition can be completed by simply summing the probabilities over a j - they should equal 1/2.  (They do if you remember to divide the probability of i=0 nodes by 2, since they are halfway on the boundary)
• The coefficients look familiar - Can you guess it?
• It is simple to write a recursive expression for the probability of a given node: $$P(i,j)=[P(i-\frac{1}{3},j-1)+P(i+\frac{1}{2},j-1)]p$$  This can be expanded into a general function, as below:

function p = P(i,j)
% Setting the intial probability
p_init = 0.5;

if( i < 0)
    i = abs(i);
end

% Base Case, ball is on the zero level (First Peg)
if(j == 0 )
    if (i == 1/2)
        p = p_init;
    else
        p = 0;
    end
    %  Recursion bit
else
    p = (P(i-1/2, j-1) + P(i+1/2,j-1))*p_init;
end
end

• Hello dynamic programming!

How Realistic is the 50/50 probability assumption?
If we make the following assumptions;

• that the collision is elastic (which is okay, since marbles and pegs are hard)
• no mass is transfered from the marble to the peg and vice versa
• the peg does not experience a change in momentum (always at rest)

than the conservation of momentum simplifies from $$m_{1,i}\mathbf{v}_{1,i} + m_{2,i}\mathbf{v}_{2,i} = m_{1,f}\mathbf{v}_{1,f} + m_{2,f}\mathbf{v}_{2,f}$$ to $$\mathbf{v}_{1,i} = \mathbf{v}_{1,f}$$.  From here we can see the components must balance out.  Ignoring the y component since we are only concerned with horizontal motion we see that if the ball is initially on the left of the peg than it should stay on the left, and if it is on the right, than it should stay on the right.  It the collision is head one (neither starting on the left or right), than it is a 50/50 probability.

More work to come later; right now I have a presentation to create!

Regionals Afterthoughts (Emo post)

It is weird, but now I find myself being less satisfied with the teams performance.  I think back to the Florida game and the bid they got on me, thinking if I would have just noticed I could have out bid him and we could have made the score one point closer - maybe that would have been the fire we needed.  In the Georgia game what if I was just a bit faster and my straight up mark had more of a force to it, what would have given us the energy to beat them.  I feel confident that we could have beaten Georgia; but the questions are how I could have given team members the energy and confidence they needed to succeed.

There is a part about making goals (or maybe being human) where the current state is not good enough.  Always room for improvement.  Always the desire for a perfect game.  I think that this makes sense - this is what drives us (me?) to improve ourselves.

I received an invitation to try out for Chain this weekend.  I don't want to pass up this opportunity, but at the same time I don't know if another year of competitive Frisbee is right for me, and I don't want to do something half-assed.

I kinda feel like Zip Tip's Last Post: "just remember, we're better than those guys".

South East Regionals

Highlights

• Sunday at Regionals
• Shutting down Mr. Sage and Mr. Sullivan on Florida (with Trey) as they tried to do a weave against Trey and I.  It was great forcing them back field.
• I had a really nice bid against Georgia.
• Phil's Callahan
• Beating Georgia Southern (we previously lost to them twice at CCC)

Lowlights

• Having three turns (though only one was my fault - the rest were poorly thrown that I got my hands on after bidding)
• Losing to Florida
• The Georgia Game

 I am content with how the season turned out (series record is 10-2, total record is 26-12).  It would have been nice to end my college career making nationals, but I can settle for 3rd at regionals (technically tied for 3rd, but . . . ).  I don't think that I will play club this fall; I want to learn kung fu or something like that.

Finally, below is a somewhat embellished write up by Richey.
"Our first game Saturday morning was against Auburn, who we took down 13-5 with great contributions from some of our younger players, notably freshman Jordan Eddy.
In our next game of the day we faced a Mississippi State team that took us to overtime the last time we had played them.  We came out excited to play and dominated them 13-2.  The highlight of the game was an insane layout defensive score (CALLAHAN) by Captain Phil Brock.
Our third game of the day we played the #2 in the nation ranked Florida Gators.  Again, Agent Orange started the game ready to fight hard and took an early 3-0 lead.  As fatigue began to set in Florida's speed and intensity became too much for us as we lost 8-10.
Luckily we had time to rest our legs in a BYE round before playing our last game on Saturday against the Home team Florida State.  Excellent work by both our offensive and defensive lines helped us to defeat an accomplished Florida State team 13-5.
Sunday morning we arrived at the fields knowing our chances of making Nationals for the first time ever would require us to win all 4 of our upcoming games in which we were to play some of the best teams in the nation.
Our first game was against cross state bitter rival Vanderbilt.  The game was close at first but we put them away late 10-5.
Our next game was against a Georgia Southern team who was leading the tournament after going undefeated Saturday, beating Florida on universe point.  Georgia Southern took an early lead on us, which we fought hard to regain and eventually win the match-up 11-8.  This victory put us into the Semifinals for a Sectional finals rematch with Georgia.
Our thirteen players warmed up for the game facing Georgia's army of at least thirty players knowing how badly both sides wanted the win.  Every player on the small Tennessee squad gave everything we had and more.  We were all sunburned, tired, and barely able to stand but we ran our hardest and fought for every point.  However, we came up short 9-13 and will not get to make the trip to Nationals.

As I know all of our players are heartbroken over this loss, I would like to point out the unbelievable amount of heart, desire, and commitment we have all shown this season.  Taking third at Regionals is the best any Tennessee team has done in at least the past 10 years, and we did so in the face of insurmountable odds.  This has been a year to remember and I will treasure being a part of this team forever.  As we look forward to a promising season next year I would like to thank Jake Altemus for coaching us through the series for providing us with knowledge and motivation that moved us to the next level, as well as our Seniors leaving us John Kerrigan, Matthew Urpher, and our Captain and MVP Phil Brock.

- Richey Ward"

And that folks, is why you never use a nickname that is spelled slightly different than your last name.