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Long Swings in Homicide
1
222
Outline
Evidence of Long Swings in Homicide
Evidence of Long Swings in Other Disciplines
Long Swing Cycle Concepts: Kondratieff Waves
More about ecological cycles
Models
2
333
Part I. Evidence of Long Swings in Homicide
US Bureau of Justice Statistics
Report to the Nation On Crime and Justice, second edition
California Department of Justice, Homicide in California
3
444
Bureau of Justice Statistics, BJS“Homicide Trends in the United States, 1980-2008”, 11-16-2011
“Homicide Trends in the United States”, 7-1-2007
4
555
Bureau of Justice Statistics
Peak to Peak: 50 years
5
666
Report to the Nation ….p.15
6
7
888
0
2
4
6
8
10
12
14
16
1900 1920 1940 1960 1980 2000
HOMICIDECA HOMICIDEUSA
California
USA
Homicide and Non-negligent Manslaaaughter, Rates Per 100,000
999
2
4
6
8
10
12
14
16
55 60 65 70 75 80 85 90 95 00 05
HOMICIDE
California Homicide rate per 100,000: 1952-2007
1980
9
101010
Executions in the US 1930-2007
http://www.ojp.usdoj.gov/bjs
Peak to Peak: About 65 years
10
111111
0
2000
4000
6000
8000
10000
60 70 80 90 00 10 20 30 40
CAPRISONERS
California Prisoners: 1851-1945
121212
Part Two: Evidence of Long Swings In Other Disciplines
Engineering50 year cycles in transportation technology
50 year cycles in energy technology
Economic DemographySimon Kuznets, “Long Swings in the Growth of Population and Related Economic Variables”
Richard Easterlin, Population, labor Force, and Long Swings in Economic Growth
EcologyHudson Bay Company
131313
Cesare Marchetti
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141414
Erie Canal
151515
0.0
0.1
0.2
0.3
-10 -5 0 5
RAILMILES
FR
EQ
UE
NC
Y
Mean
constructed
90%10%
1859
1890
1921
15
161616
Cesare Marchetti: Energy Technology: Coal, Oil, Gas,
Nuclear52 years 57 years 56 years
16
171717
181818
191919
Richard Easterlin
20 year swings
2020
Canadian Lynx and Snowshoe Hare, data from the Hudson Bay Company, nearly a century of annual data, 1845-1935
The Lotka-Volterra Model (Sarah Jenson and Stacy Randolph, Berkeley ppt., Slides 4-9)
Cycles in Nature
20
212121
2222
What Causes These Cycles in Nature?
At least two kinds of cyclesHarmonics or sin and cosine waves
Deterministic but chaotic cycles
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232323
Part Three: Thinking About Long Waves In Economics
Kondratieff Wave
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242424
Nikolai Kondratieff (1892-1938)Brought to attention in Joseph Schumpeter’s BusinessCycles (1939)
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252525
2008-2014:Hard Winter
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26262626
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Cesare Marchetti“Fifty-Year Pulsation In
Human Affairs”Futures 17(3):376-388
(1986)www.cesaremarchetti.org/arc
hive/scan/MARCHETTI-069.pdf Example: the construction of railroad miles is
logistically distributed
27
282828
Cesare Marchetti
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292929
Theodore Modis
Figure 4. The data points represent the percentage deviation of energy consumption in the US from the natural growth-trend indicated by a fitted S-
curve. The gray band is an 8% interval around a sine wave with period 56 years. The black dots and black triangles show what happened after the graph was first
put together in 1988.[7] Presently we are entering a “spring” season. WWI occurred in late “summer” whereas WWII in late “winter”.29
3030
Part Four: More About Ecological Cycles
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3131
Well Documented Cycles
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3232
Similar Data from North Canada
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333333
Weather: “The Butterfly Effect”
3434
The Predator-Prey Relationship
Predator-prey relationships have always occupied a special place in ecology
Ideal topic for systems dynamics
Examine interaction between deer and predators on Kaibab Plateau
Learn about possible behavior of predator and prey populations if predators had not been removed in the early 1900s
3535
NetLogo Predator-Prey Model
363636
Crime Generation
Crime Control
OffenseRate PerCapita
ExpectedCost ofPunishment
Schematic of the Criminal Justice System: Simultaneity
Causes ?
(detention,Deterrence, Rehabilitation,And revenge)
Expenditures
Weak Link
OF = f(CR, SV, CY, SE, MC)OF = f(CR, SV, CY, SE, MC)
CR = g(OF, L)CR = g(OF, L)
37Source: Report to the Nation on Crime and Justice
Expect
Get
3737
3838
Questions? How to Model?
3939
Part Five: The Lotka-Volterra Model
Built on economic conceptsExponential population growth
Exponential decay
Adds in the interaction effect
We can estimate the model parameters using regression
We can use simulation to study cyclical behavior
4040
Lotka-Volterra ModelLotka-Volterra Model
Vito Volterra Vito Volterra
(1860-1940)(1860-1940)
famous Italian famous Italian mathematicianmathematician
Retired from pure Retired from pure mathematics in 1920mathematics in 1920
Son-in-law: D’AnconaSon-in-law: D’Ancona
Alfred J. Lotka Alfred J. Lotka
(1880-1949)(1880-1949)
American mathematical American mathematical biologistbiologist
primary example: plant primary example: plant population/herbivorous population/herbivorous animal dependent on that animal dependent on that plant for foodplant for food
41414141
Predator-Prey1926: Vito Volterra, model of prey fish and predator fish in
the Adriatic during WWI
1925: Alfred Lotka, model of chemical Rx. Where chemical
concentrations oscillate
41
42424242
Applications of Predator-Prey
Resource-consumer
Plant-herbivore
Parasite-host
Tumor cells or virus-immune system
Susceptible-infectious interactions
42
43434343
Non-Linear Differential Equations
dx/dt = x(α – βy), where x is the # of some prey (Hare)
dy/dt = -y(γ – δx), where y is the # of some predator (Lynx)
α, β, γ, and δ are parameters describing the interaction of the two species
d/dt ln x = (dx/dt)/x =(α – βy), without predator, y, exponential growth at rate α
d/dt ln y = (dy/dt)/y = - (γ – δx), without prey, x, exponential decay like an isotope at rate
43
4444
California Population 1960-2007
0
5,000,000
10,000,000
15,000,000
20,000,000
25,000,000
30,000,000
35,000,000
40,000,000
1960
......
......
....
1962
......
......
....
1964
......
......
....
1966
......
......
....
1968
......
......
....
1970
......
......
....
1972
......
......
....
1974
......
......
....
1976
......
......
....
1978
......
......
....
1980
......
......
....
1982
......
......
....
1984
......
......
....
1986
......
......
....
1988
......
......
....
1990
......
......
....
1992
......
......
....
1994
......
......
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1996
......
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1998
......
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....
2000
......
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2002
......
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....
2004
......
......
....
2006
……
……
..
Year
Po
pu
lati
on
Population Growth: P(t) = P(0)eat
4545
lnP(t) = lnP(1960) + at
16.4
16.6
16.8
17.0
17.2
17.4
17.6
60 65 70 75 80 85 90 95 00 05 10 15
LNCAPOP
LnP(t) = ln[P(1960)e(at)] = lnP(0) + at
year
lnP(t)
4646
CA Population: exponential rate of growth, 1995-2007
is 1.4%Natural Logarithm of California Population Vs Time, 1995-2007
y = 0.0141x + 17.269
R2 = 0.9967
17.26
17.28
17.3
17.32
17.34
17.36
17.38
17.4
17.42
17.44
17.46
17.48
0 2 4 6 8 10 12 14
Time
lnP
9t0
4747
Prey (Hare Equation)Hare(t) = Hare(t=0) ea*t , where a is the exponential growth rate
Ln Hare(t) = ln Hare(t=0) + a*t, where a is slope of ln Hare(t) vs. t
∆ ln hare(t) = a, where a is the fractional rate of growth of hares
So ∆ ln hare(t) = ∆ hare(t)/hare(t-1)=[hare(t) – hare(t-1)]/hare(t-1)
Add in interaction effect of predators; ∆ ln Hare(t) = a – b*Lynx
So the lynx eating the hares keep the hares from growing so fast
To estimate parameters a and b, regress ∆ hare(t)/hare(t-1) against Lynx
4848
Hudson Bay Co. Data: Snowshoe Hare & Canadian
Lynx, 1845-1935
0
20
40
60
80
100
120
140
160
1850 1860 1870 1880 1890 1900 1910 1920 1930
HARE LYNX
HudsonBay Company Data: Snowshoe Hare & Canadian Lynx, 1845-1935
4949
[Hare(1865)-Hare(1863)]/Hare(1864)
Vs. Lynx (1864) etc. 1863-1934{Hare(t+1)-Hare(t-1)]/Hare(t) Vs. Lynx(t), 1863-1934
y = -0.0249x + 0.7677R2 = 0.2142
-5
-4
-3
-2
-1
0
1
2
3
4
0 10 20 30 40 50 60 70 80 90
Lynx
∆ hare(t)/hare(t-1) = 0.77 – 0.025 Lynx
a = 0.77, b = 0.025 (a = 0.63, b = 0.022)
5050
[Lynx(1847)-Lynx(1845)]/Hare(1846)
Vs. Lynx (1846) etc. 1846-1906[Lynx(t+1) - Lynx(t-1)]/Lynx(t) Vs. Hare(t) 1846-1906
y = 0.005x - 0.2412R2 = 0.1341
-1.5
-1
-0.5
0
0.5
1
1.5
0 20 40 60 80 100 120 140 160 180
Hare
∆ Lynx(t)/Lynx(t-1) = -0.24 + 0.005 Hare
c = 0.24, d= 0.005 ( c = 0.27,d = 0.006)
5151
Simulations: 1845-1935
Mathematica http://mathworld.wolfram.com/Lotka-VolterraEquations.html
Predator-prey equations
Predator-prey model
5252
5353
5454
Simulating the Model: 1900-1920
Mathematica a = 0.5, b = 0.02
c = 0.03, d= 0.9
5555
5656
5757
Part Six: A Lotka-Volterra Model For Homicide?
Do other violent crimes move with homicide?
58
4
6
8
10
12
14
16
90 00 10 20 30 40 50 60 70 80 90 00 10
CAHOMICIDEPER100K
20
30
40
50
60
90 00 10 20 30 40 50 60 70 80 90 00 10
CARAPEPER100K
59
0
2
4
6
8
10
12
14
2.75 3.00 3.25 3.50 3.75 4.00 4.25 4.50 4.75 5.00 5.25
Series: RATIORAPETOHOMICIDESample 1890 2012Observations 45
Mean 4.135116Median 4.203704Maximum 5.132353Minimum 2.868217Std. Dev. 0.509489Skewness -0.700057Kurtosis 3.363175
Jarque-Bera 3.922901Probability 0.140654
Distribution of Ratio of Rape to Homicide; Median
= 4.2
602.8
3.2
3.6
4.0
4.4
4.8
5.2
90 00 10 20 30 40 50 60 70 80 90 00 10
RATIORAPETOHOMICIDE
Ratio of Rapes to Homicides
61
100
200
300
400
500
90 00 10 20 30 40 50 60 70 80 90 00 10
CAROBBERYPER100K
4
6
8
10
12
14
16
90 00 10 20 30 40 50 60 70 80 90 00 10
CAHOMICIDEPER100K
6262
Part Six: A Lotka-Volterra Model For Homicide?
Do other violent crimes move with homicide?
We have a measure of the rabbits: homicides. How about a measure for the foxes (coyotes)?
63
0
40,000
80,000
120,000
160,000
200,000
60 70 80 90 00 10 20 30 40 50 60 70 80 90 00 10
CAPRISONERS
California Prisoners 1851-2009
64
2
4
6
8
10
12
14
60 70 80 90 00 10 20 30 40 50 60 70 80 90 00 10
LNCAPRISONERS
1919 1943
1976
Natural Logarithm of California Prisoners, 1851-2009
65
-0.4
0.0
0.4
0.8
1.2
1.6
2.0
2.4
60 70 80 90 00 10 20 30 40 50 60 70 80 90 00 10
DLNCAPRISONERS
Fractional Change in California Prisoners 1852-2009
66
Fractional Change in California Prisoners 1860-2009
Trough to trough 16 years, a half a cycle
67
Fractional Change in California Prisoners 1930-2009
Trough to trough 18 years, a half a cycle
68
0
100
200
300
400
500
90 00 10 20 30 40 50 60 70 80 90 00 10
PRISONPER100K
California Prisoners per 100,000 Population
69
70
71
