long swings in homicide
DESCRIPTION
Long Swings in Homicide. 1. 1. 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. 2. Part I. Evidence of Long Swings in Homicide. US Bureau of Justice Statistics - PowerPoint PPT PresentationTRANSCRIPT
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Long Swings in Homicide
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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
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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
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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
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Bureau of Justice Statistics
Peak to Peak: 50 years
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Report to the Nation ….p.15
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77
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
88
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
8
99
Executions in the US 1930-2007
http://www.ojp.usdoj.gov/bjs
Peak to Peak: About 65 years
9
1010
0
2000
4000
6000
8000
10000
60 70 80 90 00 10 20 30 40
CAPRISONERS
California Prisoners: 1851-1945
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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
1212
Cesare Marchetti
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1313
Erie Canal
1414
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
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1515
Cesare Marchetti: Energy Technology: Coal, Oil, Gas,
Nuclear52 years 57 years 56 years
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1818
Richard Easterlin
20 year swings
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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
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2020
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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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Part Three: Thinking About Long Waves In Economics
Kondratieff Wave
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Nikolai Kondratieff (1892-1938)Brought to attention in Joseph Schumpeter’s BusinessCycles (1939)
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2008-2014:Hard Winter
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252525
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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
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Cesare Marchetti
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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”.28
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Part Four: More About Ecological Cycles
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Well Documented Cycles
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Similar Data from North Canada
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Weather: “The Butterfly Effect”
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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
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NetLogo Predator-Prey Model
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Questions? How to Model?
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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
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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
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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
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Applications of Predator-Prey
Resource-consumer
Plant-herbivore
Parasite-host
Tumor cells or virus-immune system
Susceptible-infectious interactions
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404040
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
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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
......
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....
1962
......
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1964
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1966
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1968
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1970
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1972
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1974
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1976
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1978
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1980
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1982
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1984
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1986
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1988
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1990
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1992
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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
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2006
……
……
..
Year
Po
pu
lati
on
Population Growth: P(t) = P(0)eat
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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)
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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
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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
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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
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[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)
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[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)
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Simulations: 1845-1935
Mathematica http://mathworld.wolfram.com/Lotka-VolterraEquations.html
Predator-prey equations
Predator-prey model
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Simulating the Model: 1900-1920
Mathematica a = 0.5, b = 0.02
c = 0.03, d= 0.9
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Part Six: A Lotka-Volterra Model For Homicide?