Nov 212013
crime wave with text

I’m going to show you some interesting things about the way the U.S. crime rate changes from year to year.  My point will be that only harmonic motion can cause such changes.   Harmonic motion in the crime rates has some strange-sounding implications, such as the existence of a restorative force that fights crime.

A Literal Crime Wave

Here is the total number of crimes committed per year in the U.S., from 1938 to 2011, according to the U.S. FBI’s Uniform Crime Reports:

crimes committed per year

Most people, I suppose, would look at that and not see anything interesting.  But it shocks me.  This data is supposed to be the result of conscious, human decision making.  And yet it looks like something you’d see in the window of an FM synthesizer.

three things

There seems to be something mathematical about it.  Just about half way up, starting at 1971, is the first of a series of 4 bumps that grow in size, as demonstrated in the image below.  The first three of these bumps are 5 years apart, the 4th comes 10 years after the 3rd.  And the top point at 1991 is almost exactly ten times greater than it was at the bottom (click image to enlarge).

4 bmps

Overall, the whole thing is just a big sine wave:

long crime sine wave smaller
Researchers use something called the “coefficient of determination,” usually abbreviated r2, to compare mathematical models, such as sine waves, with data, such as crime rates. In this case, r2 is 0.98498167. Theoretically, this means the sine wave above, which is just this simple equation, 'explains' over 98% of the changes in crime rates.  As silly as that sounds, I know there must be another equation that explains the changes even better.  I know that because if we look at the residuals, that is, the differences between the actual data and the sine wave, we see another wave-like pattern.


Notice how the data does not jump around randomly; it goes up and down in a smooth, evolving pattern.  The first 18 years of residuals looks like one cycle of a simple sine wave.  This then represents one 18 year period in which crime rates appear to be a combination of 2 sine waves, the blue one to the right and the big red one above.  But it is more likely a combination of many small waves, than two big ones.   blue sine
In the last half of the residuals you see the shape to the right:   blue harm
Miraculously, I happened to find this very pattern in a book that actually says it results from a combination of harmonically related sine waves.  Harmonically related sine waves are sine waves that share a fundamental frequency.  gray harm
That book is “Acoustics and Psychoacoustics” by D. Howard and J. Angus.  You can click on the image to the right to see the image as it is in the book.  Note that it states the pattern is formed by combining harmonically related sine waves.  This is a 25 year period in which crime rates appear to have a fundamental frequency.  harmonrel


Hiding in plain sight

There are patterns in the crime data that are not obvious at first blush, but are nevertheless quite striking once they have been pointed out.  It starts with a smooth upward curve lasting 30 years.

30 years of exp

After this comes a series of intricate, multi-year patterns that repeat themselves, reverse themselves, or show some other symmetry.   Starting at 1971 is a 6 year period of change that forms a slanted “S” shape, first going down a little, then curving up substantially, then flattening back down again.  Then, starting at 1976, the pattern repeats.

the esses
If this was the only repeating pattern in the data I probably wouldn’t think twice about it.  But virtually every part of the crime wave is involved in some kind of symmetrical repetition like this.  The patterns come one after the other.  The years from 1979 to 1985 form a 4 year pattern that is immediately followed by its own inversion.


From 1983 to 1992 is a 5 year pattern shaped like a hockey-stick, followed by another hockey-stick, same size and shape, but inverted on 2 axes.


Then the entire double-hockey stick thing happens again, only backwards, creating a mirror image of itself.


I’m not sure this image does it justice.  The next chart makes it a little clearer how close the 1992-2000 data comes to the mirror image of the 1983-1991 data.


I realize that if you stare at random data long enough,  sooner or later you’ll find patterns of some kind.  But not like this.  Here, the entire thing is one repeating pattern after another, and it all fits together to form a gigantic sine wave.


Also, I am not claiming that the sine wave can be used to predict future crime rates.  I’m not saying anything about predicting anything.   I’m not saying anything beyond this: harmonic motion is occurring.  An r2 of 0.98 between crime rates and a 70-year long sinusoid cannot be due to chance.  And the patterns in the residuals suggest an even higher r2 between crime rates and some other more-complex formula – a formula consisting of a combination of sine waves: a complex waveform.

So what?

Complex waveforms require oscillators that go up and down, like clockwork, in an organized, coordinated manner.  Not organized as in planned, organized like the molecules of a vibrating tuning fork.  They don’t even know it’s happening.  But like kids on a trampoline, the motion of one affects the motion of them all.   The correct metaphor is a physical medium undergoing harmonic motion.  So, if 70 years of crime data make out a complex waveform, harmonic motion must be happening.  And if harmonic motion is happening, there must be a restorative force in action.

Introducing The Force

All harmonic motion requires a restorative force that resists disturbances.  In the case of a vibrating guitar string, the restorative force is the tension of the string.   In the case of the solar system, the restorative force is gravity.  What might be the restorative force in the case of crime rates?  Karma?  God?  Let’s just call it The Force.  The  Force opposes evil.  The greater the evil, the stronger The Force pushes against it.  As a result, equilibrium is restored in the same amount of time – the same period – regardless of the size, or amplitude, of the disturbance.  But the equilibrium doesn’t last; the medium’s own momentum causes it to distort just as fast and just as far in the other direction.  The force opposes this distortion too, the cycle repeats, the medium vibrates and the distortion propagates out as a wave.


4 bmps   crime wave with text

Sinusoidal Fit: y=a+b*cos(cx+d)
a = 7.62438987706E+000
b = 6.34962282277E+000
c = 7.21400634869E-002
d = -4.93995933452E+001
x= the current year
y=total number of crimes committed that year

Standard Error: 0.6140835
Correlation Coefficient: 0.9924624
r2 is 0.98498167

“coefficient of determination (r2) Definition: A statistical method that explains how much of the variability of a factor can be caused or explained by its relationship to another factor.  Coefficient of determination is used in trend analysis. It is computed as a value between 0 (0 percent) and 1 (100 percent).” 

“The Coefficient of Determination, also known as R Squared, is interpreted as the goodness of fit of a regression. The higher the coefficient of determination, the better the variance that the dependent variable is explained by the independent variable.  The coefficient of determination is the overall measure of the usefulness of a regression. For example, you are looking at an ANOVA table, and you see that your R2 is given at 0.95. This means that the variation in the regression is 95% explained by the independent variable.” –

 Posted by at 10:21 pm

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