Social Security

 The Name Game

Come on everybody
I say now let's play a game
I betcha I can make a rhyme
Out of anybody's name
The first letter of the name
I treat it like it wasn't there
But a "B" or an "F"
Or an "M" will appear

The NAME is Social Security INSURANCE

Social Security (full disclosure, I am collecting Social Security. Hey, there have to be some perks to being 71) is an INSURANCE program.  An insurance program is NOT a Ponzi scheme. An insurance program is the OPPOSITE of risky behavior.  Payments into an insurance program cease being my money as soon as I make a payment. The fact that everyone with a wage pays into this insurance program does not make it any less of an insurance program. A payment from an insurance program is NOT an entitlement. The fact that the administrators of the program are the government does not make it any less of an insurance program.

Claim: Social Security is a Ponzi scheme.

A Ponzi scheme is one that collects money and makes payments, allegedly as interest, from the principal, in order to trick someone into believing that they are receiving interest. The payments into Social Security are made before you reach retirement age and the payments from Social Security are made after you reach retirement age. This is no different than any insurance. You make payments to an insurance company before a claim and receive payments from the insurance company after a claim. The fact that the payments into insurance happened before the claim, in the past, and the payments from insurance happened after the claim, in the future, does not make it a Ponzi scheme.

Finding: False.

Claim: The stock market has a better rate of return than Social Security.

The mean rate of return is not relevant here. The rate of return on an individual’s investment is what is relevant The average return on investment for the stock market might be 5%, but that might be one investor having a -5% rate of return and another investor having a 15% rate of return. In social security everyone has the same rate of return, so this is not a valid comparison. We don’t live in Lake Wobegon where all of the children are above average.

Finding: True, but irrelevant.

Claim: Money paid into Social security is your money.

Tell that to MetLife! Once you made a payment to Social Security Insurance, it ceased being YOUR money. If you die before retirement age, the payments are NOT part of your estate. The fact that the payments from Social Security are based on the payments you made into Social Security does not mean that while in the Social Security Trust Fund, it was your money. And besides, are the payments by employers into Social Security based on your wages, your money, or the employer’s money?

Finding: False

Claim: Social Security is a government program.

I love the sign “Government keep you hands off my Social Security." Uh…the government collects the payments, and the government pays the benefits. Is YOUR ”Social Security” the payments you made or the benefits that you receive.  Both are government actions, hands.

Social Security is a government operated INSURANCE program. The fact that is operated by the government does not make it NOT any less an INSURANCE program, I do not get Social Security. I get payments from Social Security Insurance. I might call it AMICA, but that is only its name, not what it is. AMICA never pretends that is not an insurance company.

Finding: True but it is a government INSURANCE program, NOT an entitlement.

The rINOs objecting to Social Security are “republicans In Name Only. They object to any government, including the constitutionally mandated Republic that is the United States. They seem to have forgotten the pledge we all make.  “I pledge allegiance to the flag of the United States of America and to the Republic for which it stands, one nation under God, indivisible, with liberty and justice for all."

Decisions

 

Cocaine Blues

Into the courtroom my trial began
Where I was judged by twelve honest men
Yes as the jury started walkin' out
I saw that little judge commence to look about

Why twelve honest men on the jury?

A jury is charged with making a finding of the truth. But we live in a random universe. So how certain can we be certain that the jury’s finding is indeed the truth? The scientific standard is 3 Sigma, 3σ. If the odds are 50%, True and False, then if the jury consisted of only 1 person, then a finding of Truth would only have a 50% chance of being certain, which is no better than the odds. If the jury consists of two persons, then the chances that both persons made a finding of truth is one out of four outcomes, which means that two people have a 75% percent certainty of being correct. To reach a scientific standard of certainty, it requires that 12 members make a finding. If all twelve members make a finding that the evidence is true, then there is a 99.976% certainty, while 3 Sigma is 99.97% certainty, that they are correct.

Might the jury still be certain but wrong? Remember the wisdom of Abraham Lincoln that “you can fool all of the people some of the time, and some of the people all of the time.” The evidence might have fooled the jurors such that at least one of their findings of truth should have been false. In statistics this would be called a false positive. It is also possible that any juror could have a bias such that a finding NOT in evidence affected their decision. For that reason even a 99.976% certainty does not mean that the jury might still be wrong. That is why findings of trial by jury should always be reversible.

Similarly, the scientific standard can be used to determine when a decision might have passed a lower standard, such a 1 Sigma, or the mean plus one Standard Deviation.  Decisions by a group are no more certain than the odds if no Standard Deviation is used. Most decisions do NOT require a decision by the whole group and thus a decision that is greater than the mean will be more timely. The group decision passes the 1 Sigma test for certainty only if it is by 68% percent of the group. It takes a group of at least three  to approximate such certainty. If the group has, for example, 435 members, but that group has only two parties, and decisions are on a party line, the certainty is only 50%, which is no better than the odds. Which is why the most important decisions affecting the group, such as declarations of war, require 2/3, approximately a certainty of 1 Sigma. It also means that decisions of say a Supreme Court of nine members should not be considered certain unless they are 6-3 decisions. Judges are bound by the same rules of math as are juries. What is good for the goose, is good for the gander.

Correlation is not Causation

 Taking a Chance on Love

Here I go again

I hear those trumpets blow again

All aglow again

Taking a chance on love

Are you considering chances, randomness, when you are doing regressions?

Statistician George Box famously said that all models are wrong, but some are useful. However if they are very wrong, but their regression has a good correlation, they may also be misleading.

Models are typically developed from regressions of observed data. That regression is generally linear but can also be non-linear. However regression, is only the process of developing coefficients that are validated against an assumption of the pattern in the data. A fundamental question which is often embedded in that regression is an assumption that the data is of a deterministic event which is being observed. This can lead to a regression of the data that is completely wrong although it appears highly correlated.

For example, a random event will produce a normal distribution of data. One such normal random distribution is the logistics distribution, also known as the hyperbolic secant squared distribution. If its range is 0.5, which it should be in a on/off, yes/no, heads/tails, distribution, then it should have average odds of 50%, i.e. 0.5.  This means that, no matter what the value is of the mean, the range, s, should be 0.5. This requires that the average value of the distribution be 0.5 and its Cumulative Distribution Function should vary between zero and 1 without repeating. The chart of this distribution, with a mean of zero, is shown in Figure 1.

Figure 1

As a random event, it does not repeat and is a plot of ½*sech²x. However, if the data was erroneously thought not to be random, and will repeat, the substitution of a traditional trigonometric cosine, for the hyperbolic secant, i.e. ½*cos²x, has almost the same shape as the logistics distribution function around a mean of 0, although it does repeat as shown in Figure 2..

Figure 2

If the data from the logistics distribution between -4.0 and 4.0, which is equivalent to random nonzero x data with a mean of 4, was used to fit to the equation a*cos²(x*b) using a non-linear regression, the amplitude of the cosine would be a= 0.33491 and the inverse of the period, b/2π, of the cosine would be, b=0.479546 as shown in Figure 3. This is a smaller amplitude and a longer period than the theoretical value of the repeating event. However the regression with the random data would be quite good, with a coefficient of determination of 0.784252 and a correlation coefficient of 0.88558. 

Figure 3

However, like the trigonometric deterministic function, the regression repeats, while the observed random data does not repeat. Care should be taken to examine the original premise of the data being random or deterministic. If the Cumulative Distribution Function, CDF, of the regression were shown, as it is in Figure 4, it would erroneously show that its value increases as the observation increases. It would also erroneously assume that at the mean, in this case zero, the CDF at the mean is 0%, not 50% as it should be. The regression of data only suggests the correlation within the range of the data. Caution should be used when making assumptions outside of the range of that data.

Figure 4

If the regression had been to the hyperbolic secant squared, a*sech²b*x, then the amplitude, a, would be ½, b would be 1, which is consistent with a period of 2πi which only repeats in the imaginary plane, the Coefficient of Determination would be 1 and the Correlation Coefficient would also be 1. A good correlation of the data with a deterministic equation, in this case almost 0.78, could mean that the data is actually random and will not repeat, even if the regression assumes that it will repeat.

 

Ties

 

Let’s Hear It For The Boy

Let's hear it for the boy
Let's give the boy a hand
Let's hear it for my baby
You know you gotta understand
Oh, maybe he's no Romeo
But he's my lovin' one-man show
Oh, whoa-oa-oa
Let's hear it for the boy

Let's hear it for the TIE!

In an elimination game, there is no such thing as a tie. That is why there is sudden death, extra innings, extra time, shoot outs, tie breakers, etc. In reality this is only because the regular contest ended in a tie. The result of any game is a win, a loss, or a tie (or a push for the sports bettors among you). A tie is NOT like kissing your sister. A tie is a legitimate and necessary outcome of every contest. The GAME, the Harvard-Yale Football Game of 1968, ended in a dramatic 29-29 tie, and the Harvard fans in attendance at the Harvard Bowl were jubilant, which  I hope was not how they would feel about kissing their sister.

If there was not the possibility of a tie, then there would never be be a game. In the group stage of the FIFA World Cup, a win is awarded 2 points, a tie is awarded 1 point and a loss is awarded no points. Life is a lot like that group stage. If there was not a reward for a tie, then why play? A win is only the result of a single contest. If you tie, also known as a draw, then you did something that is NOT losing.

Let's add win, loss, tie as among those trinities, a subject of previous blogs.

  https://dbeagan.blogspot.com/2023/01/interest_30.html, https://dbeagan.blogspot.com/2022/09/triads.html. 

Pretending that there are no ties, or that a tie is the same as a loss is arguably why the US is in the condition that it is currently. And maybe that is why soccer...er, fútbol, is the most popular team sport in the world, because it rewards and acknowledges a tie. Viva la tie! Don’t ever change.

Planning II

 

Who’s Afraid Of The Big Bad Wolf

Number three said nicks-on-tricks
"I will build my house with bricks."
He had no chance to sing and dance
'Cause work and play don't mix

And you wonder why there are engineering codes.

Practical Pig built his house of bricks perhaps because the Engineering Building Code considered that there might be the huffing and puffing of a Big Bad Wolf. It would be much cheaper, and less work, to build his house of hay or twigs like his Brother Pigs, but as the story goes, neither of those houses could withstand the blowing of the Big Bad Wolf

And this is why the bean counters at Southwest Airlines should be ashamed of themselves. They built their system to withstand normal operations, but it would fail when confronted with just a little bit of abnormal operations like a winter storm over a holiday season. The measures that might have anticipated and dealt with a winter storm over the holidays might have been more expensive, just like bricks, but eliminating those measures and pocketing those savings is like building your house out of hay or twigs.

Any operation that plans only for normal conditions is NOT an operation at all. Stuff happens. You better be able to deal with it.  Planning on being lucky is NOT a plan.

Planning

 

I Want To Be Happy

I'm a very ordinary man
Trying to work out life's happy plan
Doing unto others as I'd like to have them doing unto me

In life's happy  plan, are things random or deterministic?

By definition, a random number has two parameters: its location, μ, and its scale, σ. To define a deterministic number, the scale must be equal to zero, which is equivalent to saying that there is only one parameter, its location.

If the scale is non-zero, the system is random. If the scale is zero, the system is deterministic. Is the universe in which we exist random, or deterministic? By this it is meant the universe as a whole, not an object within that universe. The properties of an object within the universe can be deterministic even when the shape of the universe is random. The scientific standard for the properties of an object is 3 Sigma, 3σ, that is setting σ as close to zero as possible. This means that you are 99.7% certain that the location is the only value. In particle physics, an even more strict standard applies, 5 Sigma, 5σ. This means that you are 99.99994% certain that the location is the only value.

These percentages are due to the 68/95/99 rule of normal distributions. That is

·        68% of the observations are within 1 Standard Deviation, σ, of the mean, μ; that is μ ± σ,

·        95% of the observations are within 2 Standard Deviations, 2σ, of the mean, μ, that is μ ± 2σ,

·        99.7% of the observations are within 3 Standard Deviations, 3σ, of the mean, μ, that is μ ± 3σ.

There are distributions which have only one parameter. The most common of these is the Exponential Distribution. Its Probability Density Function, PDF, can be stated using a single parameter, λ, as

 λ*e^-λ*x for x>0 and x∈ R.

Its Cumulative Distribution Function, CDF, which is the integral of its PDF, is

1- e^-λ*x for x>0 and x∈ R.

Its mean is 1/λ. Its median is 1/λ.   Its variance, which is the square of the Standard Deviation, is 1/λ². Its skewness is 2. It does not meet the 68/95/99 requirement and is therefore not normal.

This can be translated on the x- axis from an origin of (0,0) to an origin of (μ,0). Then it has two parameters, λ and μ. Its PDF can be stated as y= λ*e^-λ*(x-μ) for x>μ and x∈ R. With these two parameters, the mean is μ+1/λ. Its median is μ+1/λ. But its variance remains 1/λ² and its skewness remains 2. And it still is not normal.

The Gaussian Normal Distribution also has two parameters, μ and σ. Its PDF is conventionally stated as

1/(σ√(2π)) e^{-½*((x-μ)/σ)} for x∈ R.

With these two parameters, the Gaussian Normal Distribution’s mean is μ. Its median is μ. Its variance, whose square is the Standard Deviation, is σ. Its skewness is 0.

The Exponential Distribution is often used in analyses where the location, µ, is known to be zero, for example, the average travel distance from the home in any direction. In this case the home is defined as at location = 0. Fitting the observations of the trip length with respect to the home to an Exponential Distribution gives the mean and median trip lengths.

However when the location is not zero, then an Exponential Distribution should not be used, for example, observations of the size of widgets that are being manufactured. In this case, you would like the observations to tell you the location and standard deviation of the observations and see if those meet the location and variance requirements for the size.

To combine the features of the exponential and random distributions, it has been proposed that there be an Exponentially Modified Gaussian distribution, EMG, which includes all three parameters, but the Probability Density Function and the Cumulative Distribution Function become very complex. The PDF is

λ/2 e^(λ/2 (2*μ+λ*σ^2-2x) ) erfc((μ+λ*σ^2-x)/(√2 *σ))

where erfc(x), the complementary error function, is 1-erf(x), or

(2/√π)* ∫_x^∞ e^(-(t^2) )

The Cumulative Distribution Function of the EMG, the EMG’s CDF, the integral of its PDF, is

Φ(x, μ, σ)-1/2 e^(λ/2 (2μ+λ*σ^2-2x) ) erfc((μ+λ* σ^2-x)/(√2* σ))

where Φ(x, μ, σ) is the is the CDF of a Gaussian distribution.

The mean of an EMG is μ+1/λ. Its median is the value of x at which the EMG’s CDF is 50%. The variance is σ²+1/λ². Its skewness is (2/(σ³λ³)) (1+1/(σ²λ²))^-3/2.  By inspection, the skewness takes on a value between 0.00 and 0.31.

The Gaussian Distribution is not the only Normal Distribution. The Logistics Distribution, also known as the Sech Squared, the square of the hyperbolic secant, Distribution, is a normal distribution which has a PDF of

(1/(4*s))* sech² ((x-μ)/(2*s))

where s is a scale parameter which is equal to √3/π *σ. Its CDF is

(1/2)+(1/2)*tanh((x-μ)/(2*s))

Its mean is μ. Its median is μ. Its variance is s²π²/3. Its skewness is zero.

The CDF of the Logistics Distribution is identical to a logit choice of 0 or 1 at a location of μ where the odds of making a choice of 1 are 50%. In this case the value of s is 0.5, 50%.

A doubling of the logit choice CDF is almost identical to the CDF of a translated exponential distribution. A Standard Normal Logit Choice Distribution is proposed to always have a
σ = √3/0.5π= .91 and have a PDF of sech²(√(√3/π) (x-μ)) for x>μ and x∈ R. This means that it has a CDF of tanh(√((√3)/0.5π) (x-μ)) for x>μ and x∈ R. 

The plot of an exponential association, which is the CDF a conventional exponential distribution, and, the CDF of a Standard Normal Logit Choice Distribution, where μ=0, is shown in Figure 1. The correlation between the two equations between x=0 and x=3 is 0.999. 

Figure 1 Exponential Association and  CDF of Standard Normal Logit Distribution

This suggests that the universe is random, and not deterministic, because the universe has a σ  = 0.91.

New Ideas

 

You’re So Vain

Well, you're where you should be all the time
And when you're not, you're with some underworld spy
Or the wife of a close friend, wife of a close friend, and
You're so vain
You probably think this song is about you
You're so vain (so vain)
I bet you think this song is about you
Don't you, don't you, don't you?

I am so vain.

An article by Ethan Siegel asked why scientist are so hostile to new ideas. https://medium.com/starts-with-a-bang/the-good-reasons-scientists-are-so-hostile-to-new-ideas-27aa237c0375. I have the bad feeling that I triggered this article, I am so vain, because I sent an email to Dr. Siegel about whether the universe could be hyperbolic instead of flat, admittedly a new idea.

Dr Siegel asked some questions in his article.

“What is the problem you’re considering that motivated this idea?

How does this idea compare to the prevailing theory when applied to this specific phenomenon?

How does this idea compare to the prevailing theory when applied to the other major successes of the prevailing theory?

And what are some critical tests that you can legitimately perform (with current or near-future technology) to further discern your idea versus the prevailing theory?

As Richard Feynman once put it so eloquently, “The first principle is that you must not fool yourself — and you are the easiest person to fool.””

Challenge accepted. I will try to answer those questions.

What is the problem you’re considering that motivated this idea?

To start at the very beginning, I was born in Providence, Rhode Island. Oh, not that beginning. How about this one? In my senior year at Brown University, I took a class that made me embark on a career in Transportation Engineering. But that was over 50 years ago, so what was I looking at lately? I was trying to address a problem with the relationship between speed and volume, commonly known as congestion. There was an equation that was simple, but wrong, proposed in the 1930s. There was another equation that was much more complex, at least the relationship was not understandable to me. It was, to use Dr. Siegel’s words, “like Johannes Kepler, who threw away his “beautiful” theory of nested spheres and perfect solids before settling on his “ugly” theory of elliptical orbits that fit the data better,” and I was looking for an ugly theory that might fit the data better. The problem was that the data fit in the normal domain very well where volume on a road is less than the capacity on that road but did not fit the observations in over capacity conditions. While most of traffic engineering is concerned with making things better before capacity is reached, i.e. operations/tactics; I work in the branch of traffic engineering that deals, when, as the great philosopher Jimmy Buffet puts it “Shit hits the fan”, traffic volumes exceed capacity, i.e. planning/ strategy.

In trying to improve, or at least understand, the relationship between speed and volume, I wanted to examine the idea that the relationship, which appeared to be a rotated asymmetrical parabola, might instead be a rotated asymmetrical hyperbola.

This was not the first time that I encountered hyperbolas. The firm at which I have worked for almost 25 years, became known for its ability to explain choices.  In fact one of the early employees of that firm, Daniel McFadden, won the Nobel Prize in Economics for his advances on the theory of choices. A normal distribution based on the percentage making a choice is the logistics distribution. One of alternative names for the logistic distribution is the hyperbolic secant squared distribution. The Cumulative Distribution Function, CDF, of this hyperbolic secant squared distribution is also a scaled version of the hyperbolic tangent.

The hyperbolic tangent looks almost like an exponential association, which is itself the CDF of an exponential distribution, but the exponential association is not normal. I had previously proposed that the relationship between reliability, the ability to achieve an on-time performance, and the mean time on a road was an exponential association.

I was also tasked with looking into improving the way that traffic makes a choice among competing routes. This is when I am forced to employ the old chestnut that coincidences are an example of God’s sense of humor. Remember that course that influenced my career choice. It was taught by the late Dr. Stella Dafermos, who proposed the current method used in those route choices. Here is where I get into a second coincidence. Dr. Dafermos was a member of the Applied Math (popularly known by the students as Apple Math) Department at Brown University. My late mother retired from a position as a clerk in what was effectively the mail room of that department (I was not admitted due to nepotism. She started working there after I graduated). I thus felt a special affinity for that Department. In the mid 1980s I was working at a public agency that had an exceptionally good library. I went looking for articles published by those in that Department at Brown that were also in my field. I came across an article written by a post-doctoral student of Dr. Dafermos, Anna Nagurney. In that article, she explained that the impedance function used in route choice, which worked best with Dr. Dafermos’ method, was a fourth power function of volume, i.e. a parabola, but could not explain why. Shortly before this, the Bureau of Public Roads, the predecessor of the Federal Highway Administration before the formation of the US Department of Transportation, USDOT, had proposed an empirical volume diversion equation, known in the profession as the BPR curve, which was a fourth power function of volume. Despite the warnings of Dr. Alan Horowitz, that this equation had only been observed in conditions less than capacity and should not be used in over capacity conditions, the USDOT’s Travel Model Improvement Program, TMIP, recommended the use of a variation of this curve in the traffic models based on Dr. Dafermos’ method in all conditions.

I participated in a study that related the choices with observations of mean travel time and the planning time (the mean time plus reliability, expressed in units of time). We found that route choice was better correlated with planning time, than with mean time alone. So I was faced with choice (which was based on a hyperbola), mean time (and that is simply the inverse of speed, which appeared to be a hyperbola) and reliability, which also seemed to follow a hyperbola. Given that, I did a Google search on hyperbola to find where else it was used and found a link a paper by Dr. Mabkhout which suggests that the shape of the universe was hyperbolic.

So I did not propose that the universe is hyperbolic. I came across the ideas because I was studying relationships in traffic engineering.

How does this idea compare to the prevailing theory when applied to this specific phenomenon?

The idea that the shape of the universe is hyperbolic means that it is open, i.e. at infinity it will also be infinite, but also that it is not flat, i. e. has a curve. It means that it also should begin at a single point, e.g. a Big Bang. This hyperbolic shape explains cosmic inflation in the early universe, as well as the continued expansion of the universe. It is an “ugly” theory because it explains these without resorting to the “beautiful” additions of Dark Energy and Dark Matter. 

How does this idea compare to the prevailing theory when applied to the other major successes of the prevailing theory?

The hyperbolic universe is consistent with the age and size of the observable universe.  It is also consistent with small scales, i.e. the Planck length is consistent with the Planck energy.

A hyperbolic shape also explains the observations of galaxy rotation while a flat universe does not.

And what are some critical tests that you can legitimately perform (with current or near-future technology) to further discern your idea versus the prevailing theory?

I have been involved in using GPS data as traffic observations. The GPS records the latitude and longitude of a point. The distance between observations is not a straight line, because the earth is not flat, it is the Great Circle Distance, a non-Euclidean solution, because the earth only appears locally to be flat but is in fact a sphere. This same phenomena also have a bearing the shape of the universe. If the universe appears to be flat locally, but is in fact hyperbolic, then Pythagoras’ Theorem, which only applies in flat space, should not be used. Instead, the shortest distance between two points in a hyperbolic universe should use Pythagoras’ hyperbolic theorem. Any rotation by n/2π to n/π where n is odd in a flat universe using conventional trigonometry may change a real number to an imaginary number. However a rotation in a hyperbolic universe will never result in changing a real solution to an imaginary solution because the hyperbolic trigonometry used does not change the set of the solutions. Thus the Lorentz Transform, which in a flat Euclidean universe is √(1-(v/c)²⁾, in a hyperbolic universe would be 1+ln(cosh(v/c)±sinh(v/c)) The results are not appreciably different until the ratio of velocity to the speed of light, v/c, exceeds 85%. ( Acrtually 82.2%, but close enough for government work.)

If the universe is hyperbolic, then it would be random, not deterministic. Determinism requires a flat universe where at a Standard Deviation from the mean of zero, there is only one solution. In a hyperbolic universe, the square root of the variance, which is usually thought to be the Standard Deviation from the mean should always be apprimtely 0.91. You can approach zero but can never reach zero, and thus there is not always a single solution.

If the universe is hyperbolic, curved, then gravity might be an apparent force, where two masses are just following a geodesic to a point of lower entropy. It only appears to be a force because in a flat frame of reference, the masses would not appear to move unless they were acted on by a force.

Speaking of entropy, many of the solutions that appear deterministic in my field are in fact maximum entropy solutions.  That is they are the mesostate that has the greatest number of microstates.   This sounds very complex but is actually very simple.  In a game of craps (i.e. a macrostate), the most probable roll (i.e. a mesostate) is a seven because there are more combinations (i.e. microstates) that total seven than any other roll.  The fact that there is entropy means that there is at least one microstate.  Only when there are no microstates will the be no entropy. The fact that we exist (are a microstate) AND there is entropy, must mean that the universe is random.