Got a secret
Can you keep it?
Swear this one you’ll save
Better lock it in your pocket
Taking this one to the grave
If I show you then I know you
Won’t tell what I said
‘Cause two can keep a secret
If one of them is dead?
— The Pierces
Like most folks I enjoy a good conspiracy theory, and sometimes even a bad one. They allow the listener to suspend disbelief and consider for a moment that the social world in which he or she resides is an illusion. This is somewhat similar to the ability to slip into the fictional world of a story or novel, and bears similarity to the Gnostic assertion that the material realm is an entrapping illusion. In fact many very good films are based on the idea that conspiracies are afoot–the Bourne, Mission Impossible, Star Wars, and of course the Matrix series.
In modern societies, however, there are many conspiracy theories peddled as truth that cause real harm to their believers. One recent example from my country is PizzaGate–a ridiculous story based on the notion that a major Presidential candidate was operating a pederasty ring through a Washington DC pizza place–that led to a believer carrying out an armed raid. Another example is the belief that childhood vaccination is a cause of autism, an assertion founded on falsified data. Failure to vaccinate has led to numerous outbreaks of dangerous epidemic diseases in just the last few years. I should note that the longest running conspiracy theories I know of aren’t theories at all because in the minds of their believers they aren’t disprovable. In that sense they are fictitious certainties,
In his 2016 paper “On the Viability of Conspiratorial Beliefs” Dr David Grimes of Oxford University applied probability models to four popular conspiracy theories–NASA moon landing hoax, climate change fakery, vaccination-autism link, and cancer cure conspiracy. His model simulates the probability of a conspiracy being leaked using a Poisson distribution with conspiracy population parameters based on exponential decay, the Gompertz function, and no change. In order to understand Dr Grimes work a little better, I coded his equations in Octave and simulated his parameters, replicating the generalized results displayed in Figure 1 of the paper.
In examining the work I found one point where I wanted to test a variation. The population parameters and probability distribution used in the paper are continuous–the growth models allow for fractions of a conspirator to leak information. I added an integer rounding function to the population parameters. I also decided on a binomial discrete probability function, where the probability of a leak is treated similarly to the probability of a defective part rolling off an assembly line. I also added a parameter that allows for a leak, presumably to the press, to require verification from additional defectors.
To test this formula I decided to test the fake job numbers conspiracy. The idea is that the US Bureau of Labor Statistics was cooking up job numbers in 2012 to assist the reelection campaign of President Obama. A few problems with this are:
- The BLS employs more than 2500 individuals with a broad range of political preferences
- Job figures are also measured by private firms, such as Gallup and ADP, which show different raw numbers but similar trends
- There really isn’t any evidence that the average American voter pays close attention to economic releases
The down side to the discrete formula is that it uses factorial numbers to calculate the binomial coefficient. Numbers greater than 170! are treated as infinite by Octave. To mitigate this, I used a conspiracy population of 100, with an individual chance of leaking at 0.1% per year, and a press requirement of 2 defectors to publish the leak. While it looks like it would be possible to carry out such a conspiracy one time, the chance of the conspiracy being broken in 80 years is 91.49%. If one assumes a single leaker is sufficient, the chance of discovery rises to more than 99.99%. The chance of carrying it out with the involvement of 2500 must be slim indeed.
The m file code for the discrete distribution is below:
function conspViaDisc(decayType, p, N0, numIt)
x = [1:numIt]; % fill x-vector for x-axis values
alpha = 10^-4;
beta = 0.085; % alpha & beta for gompretzian function
te = 40; % mean age of conspirators
lifeExp = 78; % mean life expectancy of conspirators
lambda = log(2)/(lifeExp-te);
N(1) = N0;
a = 1; % Number of leakers needed to break the conspiracy
if decayType == “G”
for n = 2:numIt
N(n) = int16(N0*exp(0-lambda*n)); % Gompretzian decay
elseif decayType == “E”
for n = 2:numIt
N(n) = int16(N0*exp((alpha/beta)*(1-exp(beta*(te+n))))); % Exponential Decay
for n = 2:numIt
N(n) = N0; % No Decay
L(1) = 0.001; % Estimated probability of discovery in timestep 1
cumL(1) = L(1);
for n = 2:numIt
if N(n) >= a
summ = 0;
for m = 1:a
s = m-1;
binCoef = factorial(N(n))/(factorial(s)*factorial(N(n)-s));
summ += binCoef*(p^s)*((1-p)^(N(1)-s));
L(n) = (1-summ)*(1-cumL(n-1));
cumL(n) = cumL(n-1) + L(n);
L(n) = 0;
cumL(n) = cumL(n-1) + L(n);
“Come and listen to a story ’bout a man named Jed
Poor mountaineer barely kept his family fed
Then one day he was shooting for some food,
And up through the ground come a bubbling crude
(Oil that is, black gold, Texas tea)
Well the first thing you know old Jed’s a millionaire
Kin folk said Jed move away from there
Said California is the place you oughta be…”
—The Ballad of Jed Clampett
Oil and natural gas have been important in the US and globally for nearly 180 years, with that importance spiking following invention and widespread adoption of the internal combustion engine in the 19th and 20th centuries. Beginning in 1973, the US began consuming more crude oil than could be domestically produced. Dependence on global markets made the US economy vulnerable to oil-price shocks in the 1970s, and acted as one of several proximate triggers for the 1990-1991 recession.
In order to better understand production and labor dynamics in US oil and gas production over time, I examined and downloaded production data from the Energy Information Administration (US DoE) and oil and gas employment data from the Bureau of Labor Statistics (US DoL). Graphs of the raw data are below:
A few points stand out when examining the data:
- Natural gas production reached a bottom in the mid-1980s, rising modestly until the mid-1990s, and remained stable until the mid-2000s. Production is currently just off the 2014-2015 peak.
- Crude Oil production in the US peaked in the early 1970s, though oil and gas employment peaked around 1980. Since then, a second production peak occurred in 2015, though production has fallen off slightly since then.
- In spite of recent peaks in production, employment in oil and gas extraction is about 35% off the 1982 peak of 266 thousand workers.
To examine oil and gas combined production and productivity, data was downloaded and opened in MS Excel. Annual barrels of oil and millions of cubic feet of natural gas were converted to British Thermal Units (BTUs). September employment levels were used for each year. The total production series was divided by the employment series to calculate productivity in BTUs per worker.
The productivity data shows a nearly unbroken 20-year increase from 1983 to 2003, a modest decline, and some leveling until 2014, and then a return to productivity growth. As one would expect, productivity increases tend to correspond to falling or stable crude oil prices. Productivity by this measure tends to decline when prices are rising. This makes sense because rising prices can be expected to follow periods of falling production (affecting the numerator), and lead to increases in employment (affecting the denominator).
The following is my opinion only, freely given, worth what you’ve paid for it, isn’t meant to be career or investment advice, and isn’t necessarily the view of my employer or anybody else: The shale revolution has gone global, and demand growth is expected to be slow due to demographics and consumer preferences in the developed world. Because global supply is much less constrained than demand, I expect crude oil prices to stabilize in the 60 to 80 dollar per barrel range for the next several years, assuming no severe shocks to global supply or demand. In order for profitability of US resources to be maintained productivity must continue to improve, implying modest but steady increases in production, and net employment changes occurring very slowly.
The Green Bay Packers and Detroit Lions played their second game of the 2015 season on December 3rd. It ended with a highly unlikely Hail Mary catch by Richard Rodgers to score the winning touchdown. In the days following the game I heard many, many different people expressing excitement and admiration for the play, with one colleague even declaring Aaron Rodgers “The best quarterback in the NFL”.
The league posted a video clip asking “How far exactly did Rodgers’ Hail Mary Travel?”. The official marking of 61 yards is shown at the bottom of the screen. This is the distance from the line of scrimmage at the start of the play to the front of the end zone line. Ben Rohrbach expressed his belief that the total travel distance of the ball was close to 100 yards, including the vertical and horizontal elements.
When a ball is thrown, kicked, or otherwise sent with force into the air, the vertical profile will be shaped like an upside down parabola. This is because gravity acts with constant downward acceleration against the object. The total distance along this segment of a parabola is known as the arc length. It can be calculated by integrating the changing height across the horizontal distance. Wolfram’s discussion is here, and Had to Know’s is here.
With that out of the way, the objective is to calculate the actual arc length of Aaron Rogers December 3rd pass. The same method can be used to calculate golf drives, log throws, rock catapultations, etc…. Note: Air friction is neglected in the following discussion.
The horizontal distance of the pass is calculated via the Pythagorean Theorem using length and width coordinate estimates from the back end zone and visual top of the field. The image below shows how this was done for Aaron Rodgers throw. The throw starts an estimated 39 yards from the back end zone, and ends 103 yards from the same end zone. At the start it’s an estimated 13 yards from the top sideline, and it ends about 18 yards from the same sideline.
The second step is to estimate the total height attained by the ball, minus the height of the passer. Here, I assume that the ball was caught at the same height from which it was thrown. The vertical displacement in yards can be calculated by estimating the hang time, squaring it, and multiplying by 1.34058.
Once the vertical and horizontal displacements are measured, the rest can be calculated in a straightforward manner. Using my estimates of pass start at (13,39), catch at (18,103), and hang time of 5.6 seconds I got the following results:
- Horizontal Displacement: 64.195 yards
- Vertical Displacement: 42.041 yards
- Total Travel Distance: 110.716 yards
- Velocity of Pass: 12.706 yards per second
- Energy Imparted to Football: 21.405 foot pounds
- Power Output of Aaron Rodgers during pass: 0.216 horsepower
This admittedly armchair analysis vindicates Rohrbach, and calculates Aaron Rodgers throwing technique to be as strong as just over a fifth of a horse. The image of my Excel output is below, with the following cell formulas:
- A6 =SQRT(C2^2 + C4^2)
- A11 =SQRT((A6^2/4) + (4*B9^2)) + (A6^2/(8*B9))*ASINH(4*B9/A6)
- A13 =(3*B11)^2*(0.94799/(2*32.174049))
- B9 =1.34058*A9^2
- B11 =SQRT((A6/A9)^2 + 10.72467*(A9/2))
- B13 =(A13/B6)/550
- C2 =ABS(B2-A2)
- C4 =ABS(B4-A4)