Growth Rates

 

What Kind of Fool Am I

What kind of fool I am
What kind of mind is this
An empty shell
A lonely cell in which
An empty heart must dwell

What kind of growth is this?

It is common to express growth in terms of a Compound Annual Growth Rate, CAGR.  Savings accounts, loans, mortgages, inflation, and the Gross Domestic Product of the economy are often expressed this way.  This implicitly assumes that growth is geometric and follows the formula: 

Future= Base *(100%+CAGR)^(periods, years, between future and base).

This is considered a marked improvement over assuming a simple growth rate:

Future=base * (Period, years, between future and base)*(100% + simple growth rate).

The compound rate is also called a geometric growth rate.  It assumes a constant, but non-negative, growth rate (first derivative).  A problem with continuous growth is that it can surpass an absolute amount in the future.  An equally valid growth rate is exponential growth.  It is more commonly encountered as radioactive decay with a half life. For growth, the mirror of decay, the rate is often expressed as a doubling period.  Exponential growth will approach an absolute but will never exceed it.  It thus has a variable, not a constant, growth rate (first derivative).  This is consistent with a constant rate within an exponential function.

Future=base*(100% +exp(rate* period, years, between base and future))

where exp() is the exponential function.

Over the short term, less than 20% of the periods to the absolute, there is virtually no difference between exponential and compound growth.  Exponential growth is higher than compound growth, but less than 5% higher, over the medium term, 20% to 83% of the periods to the absolute.  It is only in the periods near the absolute ( 83% to 100% of the periods to the absolute), that the exponential growth becomes significantly larger.  However unlike compound growth, its future value will never exceed the absolute amount.

It's Relative III

 Dead Man's Curve

(Dead Man's Curve) is no place to play

(Dead Man's Curve) you'd best keep away

(Dead Man's Curve) I can hear 'em say

Won't come back from Dead Man's Curve

If the universe is curved, what does that do to randoneess and choices?

Variance is the dispersion of the data, X,  in a statistical distribution. Mathematically its formula is

Var(X)=∑ (p_i*(x_i-µ))² for i =1 to n variables

The variance is traditionally defined as the square of a parameter of a random distribution,  σ², where σ is also known as the standard deviation, SD.  The other parameter of a random distribution, µ, the mean, which is equal to the median of a normal distribution, is

µ=∑ p_i(x_i-µ)*x_i for i=1 to n variables.

However it is not correct to say that the variance, which is the square of the standard deviation, SD, can be solved by Pythagoras’ formula. This is only true on a flat surface.  In fact on a flat surface, rather than a summation, this should be expressed as a product as

SD(X)=cos^-1(∏ cos(p_i(x_i-µ))) for i=1 to n variables.

On a flat surface, this is identical to  

SD(X)=√( (p_i(x_i-µ))² for i =1 to n variables or √( (x_i-µ))²/n).

That Bessel’s correction, n/(n-1), is necessary for the SD is an indication that the surface is not flat.

On a spherical surface, the formula should be

SD(X)=R*cos^-1(∏ cos(p_i(x_i-µ)/R)) for i=1 to n variables.

Here R is the radius of the spherical surface.  As R approaches infinity, e.g. the sphere become very large compared to any p_i(x_i-µ), this becomes identical to the formula for a flat surface.

On a hyperbolic surface, the formula should be

SD(X)=cosh^-1(∏ cosh(p_i(x_i-µ))) for i=1 to n

The variance, σ², of a random logistics distribution, where the parameters are µ and s, is s²*π²/3.  Because this is the summation of a square, it is correct to say that σ²=s²*π²/3, but it is NOT correct to say that because of this s=σ√3/π.  This is only true for a flat surface.

If the surface is hyperbolic, then the correct formula is

s=cosh^-1(cosh(σ√3/π))

Because cosh^-1(x)=ln(x±√(x²-1)), cosh²(x)-sinh²(x)=1, cosh(x)=½*(e^x+e^-x), and sinh(x)= ½*(e^x-e^-x), this can be expressed as

s=ln(cosh(σ√3/π) ±√( cosh²(σ√3/π)-1))

s=ln(cosh(σ√3/π) ± sinh(σ√3/π))

s=ln(½*(e^σ√3/π +e^-σ√3/π) ± ½*(e^σ√3/π -e^-σ√3/π))

s= σ√3/π ± σ√3/π= 2* σ√3/π or 0

This is because the variance, the square of the standard deviation, is a summation, NOT a single variable.

Only if the surface is flat is it true that s=σ√3/π.

This means that the random normal logistics distribution, the hyperbolic secant distribution,  

PDF = (1/(4s))*sech²((x-µ)/2s); CDF = ½*tanh((x-µ)/2s)+ ½

can be expressed in terms of the parameters µ and σ as

PDF = (π /(8* σ√3))*sech²(π *(x-µ)/(2* σ√3)); CDF = ½*tanh( π *(x-µ)/(2* σ√3))+ ½.

When the standard deviation, σ, is π/(4√3) then this simplifies to

PDF = ½*sech²(x-µ); CDF = ½*tanh(x-µ)+ ½ which is identical to saying that  s=½.

In this case, when x=µ, then the PDF is 50% and the CDF is also 50%. If π is the absolute, and negative numbers are not allowed, then, according to Pearson’s Second Coefficient of Skewness, µ=π/2.  If s=½, then σ²=(½)²π²/3=0.822467.

If the universe is hyperbolic, as proposed by Mabkhout (Mabkhout, 2012), the absolute is defined as π, x is an absolute measurement which means that 0 is an absolute zero, numbers x<0 are not defined, AND the universe is random, then if the universe is normal, it has both a mean and a variance which can be expressed in terms of its absolute. That universe also allows choice, s=½, where there is one choice: the absolute or absolute zero.

If the universe is hyperbolic and random, then random events do not repeat except on an imaginary plane. Random events may appear to repeat, be cyclical, but it is not in the real surface where the coefficient of the imaginary axis is 0.  I.e., Mark Twain was correct.  History does NOT repeat, but it sure does rhyme.

Mabkhout, S. (2012). The infinite distance horizon and the hyperbolic inflation in the hyperbolic universe. Phys. Essays, 25(1), p.112.

Choices

 

I’m Putting All My Eggs in One Basket

I'm putting all my eggs in one basket
I'm betting everything I've got on you
And, I'm giving all my love to one baby
Lord help me if my baby don't come through

This might be a good strategy in love, but it is NOT a good betting strategy.

We are warned not to put all of our eggs in one basket.  The reason being that if anything happens to that basket, then we will have no eggs.  IOW, Henry Ford’s famous choice for the color of a Model T in the 1920s, “Any color, as long as it is black”, is not a choice at all.  It might have been good for Ford’s assembly process since black paint dried faster, but it was not necessarily good for his customers. The company store in the song Sixteen Tons is also not a choice since you either bought at the company store or did without. In both cases having only one choice is putting all your eggs in one basket.

Having a choice between two things is also not a choice.  Game Theory proves that having at least three choices is necessary, https://dbeagan.blogspot.com/2021/05/tough-but-fair-beats-always-being-nasty.html assuming that the parties/players offering those choices are not colluding.  The song should be “I’m Putting All My Eggs in THREE Baskets”

Baseball

 

Take Me Out To The Ball Game

Katie Casey was baseball mad,
Had the fever and had it bad.
Just to root for the home town crew,
Ev’ry sou Katie blew.
On a Saturday her young beau
Called to see if she’d like to go
To see a show, but Miss Kate said “No,
I’ll tell you what you can do

 

Take me out to the ball game,

Take me out with the crowd;

Just buy me some peanuts and Cracker Jack,

I don’t care if I never get back.

Let me root, root, root for the home team,

If they don’t win, it’s a shame.

For it’s one, two, three strikes, you’re out,

At the old ball game. 

Next time you sing the chorus, remember the verse!

The American League regular baseball season is 162 games, at least since 1961.  That is what the asterix after Roger Maris’ regular season home run record was all about. The National League increased its season to 162 games in 1962.

An average season since the mid-1960s is  81 wins and 81 losses, a .500 record.  For the standings to be normally distributed, no team can win more than 162 games or lose more than 162 games.  Thus the record has to be such that 81 minus standard deviation/3 ≥0.  That means that the standard deviation of a 162-game season is 81 divided by 3 ≤ 0 or less than 27 games.  The variance is the square of the standard deviation or less than 729.  The team with the best regular season record, before the bias of the playoffs, was 100% the DOMINATE team, but each game was randomly decided, no fixes, so that the CERTAINTY of its being best team is: 68% with a record of 108 and 54; 95% with a record of 135 and 27; and over 99% with a record of 162 and 0.

However, since the introduction of the championship, division, and wildcard playoff series, the league regular season champions might be sitting their best players once the team has clinched a spot in the playoffs, and is no longer trying to win every game at the end of the season. Thus, the playoffs can determine the most DOMINATE team in those playoffs, but the CERTAINTY that this is also the best team can no longer be established.  The regular season is only a means to determine the seeding for the playoffs.

Each playoff series can determine a dominate team for that series, but can not also determine that the DOMINATE team in each series is the BEST team.  In a 3-game series, a team that wins 2 games is dominate but it could have been expected to have a 3-0 or a 2-1 record if all three games had been played.  But the CERTAINTY of the dominate team being the best team is only 34% (it is 88% for a 3-0 record but that is only 12% of the possible outcomes!)   In a 5-game series, a team that wins 3 games can be expected to have a 5-0, 4-1, or 3-2 record.  The winning team has a certainty of being the best team of 38%.   In a seven-game series, a team that wins 4 games can be expected to have a 7-0, 6-1, 5-2, or a 4-3 record.  This is a certainty of 40%.  However, the winner of the wild card or division round was not necessary the higher seed, so if the lower seed wins its certainty should be reduced by being in the previous round. 

The playoffs determine the most DOMINATE team in each series with an odd number of games, but they do not determine the best team with CERTAINTY.  It is the record of the losing team that does that. If the series is stopped after a majority of the wins have been achieved  (e.g. four games in a seven-game series), there is no way to determine what the record would have been in the games which were not played.

That is why the playoffs might be more exciting, but they are not, can not, be certain. If we wanted to be certain, we would have to play ALL of the games. In other words, Play Ball!

Free Speech on Social Media

 

This Land Is Your Land

There was a big, high wall there that tried to stop me
A sign was painted said "Private Property"
But on the backside, it didn't say nothing
This land was made for you and me

This Land was made not for you and me as individuals, but for us as a group.

Woody Guthrie wrote the song This Land Is Your Land in response to the philosophy in Irving Berlin’s God Bless America.  In it, he stated his philosophy that there was no such thing as Private Property in the seldom sung fourth stanza.  The problem is that not only is there public property, Our Land, but there is also private property. And public property might also have no trespassing signs.

The fact that you are a member of a group does NOT give you the right to use the property of the group.  And the fact that you are not an owner of private property, means that the owner of that property, not you, can control that property. 

The problem on the Far Right is that those who are controlled from Public Property, think that because they are a member of the public, they have control on that public propety.  E.g. the issue of private grazing on public land.  The problem on the Far Left is that those who are forbidden from private property, insist that there is no Private Property.  E.g. the issue of occupying private spaces.  Both are wrong.

The public can control access to public property, whether it is forbidding grazing on certain public land, or licensing the public airwaves.  Private owners can control who speaks, carries guns, assembles, etc on their property.  Those spaces are private spaces over which the public has no control.  The fact that you were invited into a private space did not change it into a public space.

Thus while you have a right of free speech in public places, X (formerly known as Twitter), Facebook, Instagram, etc. are NOT public spaces.  An individual’s public rights have no standing in these private spaces.  Every thing that is not your private property is not public property.  It can be private property owned by another.  And the fact that you are a member of a group does not mean that the group’s property has become your private property.  

And yes, Donald Trump, us, a group, is U.S.  IOW, United we stand, divided we fall! So stop trying to divide us.

Acceptance

 

Smile 

Smile though your heart is aching
Smile even though it's breaking
When there are clouds in the sky, you'll get by
If you smile through your fear and sorrow
Smile and maybe tomorrow
You'll see the sun come shining through for you 

In other words, say the Serenity Prayer!

The Vinegar Tasters is a traditional subject in Chinese religious painting. The painting depicts the three founders of China's major religious and philosophical traditions: K'ung Fu Tzu, who has been latinized as Confucius; Buddha; and Lao Tzu, the founder of Taoism.  

K'ung Fu Tzu has a an expression of resolve, because the vinegar is sour and needs to be dominated, by the following of strict rules. 

The Buddha has a sad expression because vinegar is sour like life, but like life it must be endured. 

Lao Tzu has a smile because he has accepted that vinegar should taste sour like vinegar. 

Smile!

New and Improved!

 

This Old House

Ain't got time to fix the shingles
Ain't got time to fix the floor
Ain't got time to oil the hinges
Nor to mend no window panes
Ain't gonna need this house no longer
I’m gettin' ready to meet the saints

But if it ain’t broke, DON’T fix it!

A basic rule in engineering is “If it ain’t broke, don’t fix it”.  Doctors follow the same instructions in the Hippocratic Oath but it sounds better there,  “First, do no harm”.  Unfortunately beginning with Thomas Watson, Jr., IBM and all tech companies, have been violating these rules.

Tech companies have constantly “improved” their offerings with frequent updates.  While this was supposedly to benefit the users of these programs, all too often it is at the expense of dropping features that were previously included, or putting features that peviously existed behind a pay wall or other “protections”. Also there is always a user cost to these improvements.  There is a learning curve to unlearn the old ways, and learn the new way, of doing things.  The more you know, then the more you have to unlearn, so the cost is borne particularly by long-time users.

Does anyone remember the “New” Coke marketing fiasco of the 1990s.  Coca-Cola corporate honchos though it would be a good ideas to introduce a “New” formula for their soft drink.  There was nothing wrong with the Old Formula.  They just wanted to improve it.  Their customers revolted and forced the reintroduction of Classic Coke and eventually New Coke disappeared.  

I could do without the new upgrades that are made by tech companies where they are forcing their users to accept that it wasn’t broke, but they fixed it anyway.  Thanks for nothing tech companies.  I guess the tech companies had the time because they aren’t “gettin’ ready to meet the saints.”