CrimeStat III
P a rt II: S p a ti a l D e s c rip ti o n
Chapter 4
Spatial Distribution
In t h is ch a pt er , t h e spa t ia l dis t r ibu t ion of cr im e in ciden t s w ill be d iscu ss ed. Th e
st a t ist ics t h a t a r e u sed in describing t h e spa t ial dist r ibut ion of cr ime in ciden t s will be
exp la in ed a n d will be illu st r a t ed wit h exa m ples fr om Crim eS tat ® III. For t he exam ples,
crim e in ciden t da t a from Ba lt im ore Coun t y a n d Ba lt im ore Cit y will be u se d. F igu r e 4.1
sh ows t h e u ser in t er face for t h e spa t ia l dis t r ibu t ion st a t ist ics in Crim eS tat. F or ea ch of
t h ese, t h e st a t ist ics will fir st be pr esen t ed followed by exa m ples of t h eir u se in crim e
an alysis.
Ce n t ro g ra p h i c S ta t is t ic s
Th e m ost ba sic t ype of descript or s for t h e spa t ial dist r ibut ion of cr ime in ciden t s a r e
cen trograph ic statistics. Th ese a r e in dices wh ich est im a t e ba sic p a r a m et er s a bou t t h e
dis t r ibu t ion (Lefever , 1926; F u r fey, 1927; Ba ch i, 1957; Neft , 1962, H u lt qu is t , Br own a n d
H olmes , 1971; E bdon, 1988). They inclu de:
1.
2.
3.
4.
5.
6.
Mea n cen t er
Media n cen t er
Cent er of m inim u m dist a n ce
St a n da r d deviat ion of X a n d Y coor din a t es
St a n da r d dis t a n ce devia t ion
Sta nda rd deviationa l ellipse
Th ey a r e ca lled cen t r ogr a ph ic in t h a t t h ey a r e t wo dim en sion a l cor r ela t es t o t h e
ba sic s t a t is t ica l m om en t s of a sin gle-va r ia ble dis t r ibu t ion - m ea n , s t a n da r d devia t ion ,
sk ewn ess, a n d k u r t osis (see Bachi, 1957). They ha ve been a pplied t o cr ime a n a lysis by
St eph en son (1980) a n d, m ore r ecen t ly, by La n gwor t h y a n d J efferis (199 8).
Beca u se t wo dim en sion s a dds com plexit y not seen in one dim en sion , th ese
st a t istica l momen t s h a ve been m odified to be ap pr opr iat e. Figur e 4.2 sh ows h ow th e
cen t r ogr a p h ic s t a t is t ics a r e s elect ed in Crim eS tat.
Me a n Ce n t e r
Th e sim plest descript or of a dist r ibut ion is t h e m ean center. Th is is m er ely t h e
mean of th e X an d Y coordina tes. It is somet imes called a center of gravity in t h a t it
r epr esen t s t h e poin t in a dis t r ibu t ion wh er e a ll ot h er poin t s a r e ba la n ced if t h ey exist ed on
a pla n e a n d t h e m ea n cent er wa s a fulcr u m (Ebdon , 1988; Bu r t a n d Ba r ber , 1996).
F or a sin gle var iable, th e m ea n is t h e poin t a t wh ich t h e su m of a ll differ en ces
bet ween t h e m ea n a n d a ll ot h er poin t s is zer o. U n for t u n a t ely, for t wo va r ia bles, s u ch a s
t h e loca t ion of cr im e in cid en t s, t h e m ea n cen t er is n ot n ecessa r ily t h e poin t a t wh ich t h e
su m of a ll dist a n ces t o a ll ot h er poin t s is m inim ized. Tha t pr oper t y is a t t r ibut ed t o t h e
4.1
Figure 4.1:
Spatial Distribution Screen
Figure 4.2:
Selecting Centrographic Statistics
cen t er of min im u m dis t a n ce (s ee be low). H owever , t h e m ea n cen t er ca n be t h ou gh t of as a
poin t wh er e bot h t h e su m of a ll differ en ces bet ween t h e m ea n X coor din a t e a n d a ll ot h er X
coor d in a t es is zer o a n d t h e s u m of a ll d iffer en ces bet ween t h e m ea n Y coor d in a t e a n d a ll
ot h er Y coordin a t es is zer o.
The form ula for t he mea n center is:
_
X
Xi
-------i=1
N
_
N
=
E
Y
Yi
--------i=1 N
N
=
E
(4.1)
wh er e Xi an d Yi a r e t h e coor din a t es of in divid u a l loca t ion s a n d N is t h e t ot a l n u m ber of
poin t s.
To ta k e a sim ple exa m ple , t h e m ea n cent er for bu r gla r ies in Ba lt im ore Coun t y h a s
sph er ica l coor din a t es of lon git u de -76.608482, lat itu de 39.348368 an d for r obber ies
lon git u de -76.620838, lat itu de 39.334816. Figu r e 4.3 illust r a t es t h ese t wo m ea n cen t er s.
We ig h te d Me a n Ce n t e r
A weigh t ed m ea n cen t er ca n be pr odu ced by weigh t ing ea ch coor din a t e by an ot h er
va r ia ble, W i . F or exa m p le, if t h e coor d in a t es a r e t h e cen t r oid s of cen s u s t r a ct s , t h en t h e
weigh t of ea ch cen t r oid cou ld be t h e p op u la t ion wit h in t h e cen s u s t r a ct . F or m u la 4.1 is
ext en ded s ligh t ly t o inclu de a weigh t .
_
X
=
N
W i Xi
_
---------N
Y
i=1
E
=
N
W i Yi
i=1
---------N
E
(4.2)
Th e a dva n t a ge of a weigh t ed m ea n cen t er is t h a t poin t s a ssocia t ed wit h a r ea s ca n
h a ve th e ch a r a ct er ist ics of t h e a r ea s in clud ed. For exa m ple, if t h e coor din a t es a r e t h e
cent r oids of cens u s t r a cts , t h en t h e weight of ea ch cen t r oid cou ld be t h e popu la t ion wit h in
t h e cen su s t r a ct . This will pr odu ce a differ en t cen t er of gra vit y th a n , say, th e u n weight ed
center of all census t ra cts. Crim eS tat a llows t h e m ea n t o be weigh t ed by eith er t h e
weigh t in g va r ia ble or by t h e in t en sit y va r ia ble. U ser s sh ou ld be ca r efu l, h owever , n ot t o
weigh t t h e m ea n wit h both t h e weight in g a n d in t en sit y var ia ble u n less t h er e is a n explicit
distinction being ma de between weight s an d int ensities.
To t a k e a n exa m ple, in t h e six ju r is dict ion s m a k in g u p t h e m et r opolit a n Ba lt im or e
a r ea (Ba ltim or e City, an d Ba ltim or e, Car r oll, H a r for d, H owa r d a n d An n e Ar u n del
cou n t ies), t h e m ea n cent er of a ll cen su s block gr oup s is lon git u de -76.619121 , lat it u de
39.304344. This would be a n un weighted m ea n cen t er of t h e block gr ou ps. On t h e ot h er
h a n d, th e m ea n cen t er of t h e 1990 popu lat ion for t h e Balt imore m et r opolita n a r ea h a d
coor din a t es of lon git u de -76.625186 a n d la t it u de 39.304186, a posit ion sligh t ly sou t h west of
t h e u n weight ed m ea n cen t er . Weigh t ing t h e block gr ou ps by m edia n h ou seh old incom e
4.4
Figure 4.3: Burglary and Robbery in Baltimore County
Comparison of Mean Centers
Mean center of burglaries
Mean center of robberies
Miles
0
2
4
'
pr odu ces a m ea n cen t er wh ich is st ill m or e sout h west . Figur e 4.4 illust r a t es t h ese t h r ee
mean center s.
Weigh t ed m ea n cen t er s ca n be u sefu l beca u se t h ey d escr ibe spa t ia l d iffer en t ia t ion
in t h e m et r opolita n a r ea a n d factors t h a t m a y cor r elat e with cr ime d ist r ibut ion s. An ot h er
exa m ple is t h e weigh t ed m ea n cen t er s of differ en t et h n ic gr ou ps in t h e Ba lt im or e
m et r opolit a n a r ea (figu r e 4.5). Th e m ea n cen t er of th e Wh it e popu la t ion is a lm ost id en t ica l
t o t h e u n weigh t ed m ea n cen t er . On t h e ot h er h a n d, t h e m ea n cen t er of th e Afr ica n Am er ica n /Bla ck pop u la t ion is sou t h wes t of t h is a n d t h e m ea n cen t er of t h e H is pa n ic/La t in o
popula t ion is con sider a bly sout h of t h a t for t h e Whit e popula t ion . In oth er wor ds, differ en t
et h n ic groups t en d t o live in differ en t pa r t s of t h e Balt imore m et r opolita n a r ea . Wh et h er
t h is h a s a n y im pa ct on cr ime d ist r ibut ion s is a n em pir ica l quest ion . As we will see, th er e
is n ot a sim ple sp a t ia l corr ela t ion bet ween t h ese weight ed m ea n cent er s a n d p a r t icula r
crime distr ibut ions.
Wh en t h e M csd box is checked, Crim eS tat will r u n t h e r ou t in e. Crim eS tat ha s a
st a t u s ba r t h a t in dica t es h ow m u ch of th e r ou t in e h a s be en r u n (F igu r e 4.6).1 The results of
t h ese st a t ist ics a r e sh own in t h e M csd out pu t t a ble (figur e 4.7).
Me d ia n Ce n t e r
Th e m edia n cen t er is t h e int er section bet ween t h e m edia n of t h e X coor din a t e a n d
t h e m edia n of t h e Y coor din a t e. Th e con cept is s im ple . H owever , it is n ot s t r ictly a
m edia n . For a sin gle va r ia ble, su ch a s m edia n h ous eh old in com e, t h e m edia n is t h a t point
a t wh ich 50% of t h e ca ses fall below a n d 50% fa ll a bove. On a t wo dim en sion a l plan e,
h owever , t h er e is n ot a sin gle m edia n beca u se t h e loca t ion of a m edia n is d efin ed by t h e
wa y t h a t t h e a xes a r e dr a wn . For exa m ple, in figu r e 4.8, t h er e a r e eigh t in ciden t point s
sh own . F ou r lin es h a ve been dr a wn wh ich divid e t h ese eigh t poin t s in t o t wo gr ou ps of four
ea ch . However , th e fou r lin es do not iden t ify an exact loca t ion for a m edia n . Ins t ea d, th er e
is a n a r ea of n on-u n iqu en ess in wh ich a n y pa r t of it cou ld be con sid er ed t h e ‘m edia n
cen t er ’. This violat es one of t h e bas ic pr oper t ies of a st a t ist ic is t h a t it be a u n ique va lue.
Never t h eless, a s lon g a s t h e a xes a r e n ot r ot a t ed, t h e m edia n cen t er ca n be a u sefu l
st a t ist ic. The Crim eS tat rout ine out put s th ree sta tistics:
1.
2.
3.
Th e sa m ple size
Th e m ed ia n of X
Th e m ed ia n of Y
The tabu lar out put can be print ed and t he median center can be out put as a
gr a ph ica l object t o Ar cView ‘sh p’, Ma pIn fo ‘m if’ or At la s*GIS ‘bn a ’ files. A r oot n a m e
sh ou ld be pr ovid ed. Th e m edia n cen t er is out pu t a s a poin t (Md n Cn t r <r oot n a m e>).
4.6
Figure 4.4: Center of Baltimore Metropolitan Population
Mean Center of Block Groups Weighted By Selected Variables
Baltimore County
City of Baltimore
Population
Unweighted
HH Income
Howard County
Miles
0
2
4
Anne Arundel County
'
Figure 4.5: Center of Baltimore Metropolitan Population
Mean Center of Block Groups Weighted By Selected Variables
Baltimore County
City of Baltimore
Unweighted
Black
White
Latino
Howard County
Miles
0
2
4
Anne Arundel County
'
Figure 4.6:
CrimeStat Calculating A Routine
Figure 4.7:
Mean Center and Standard Distance Deviation Output
Figure 4.8: Non-Uniqueness of a Median Center
Lines Splitting Incident Locations Into Two Halves
Baltimore County
City of Baltimore
Area of non-uniqueness
Howard County
Miles
0
2
4
Anne Arundel County
'
Ce n t e r of Mi n im u m D i s ta n c e
An ot h er cen t r ogr a p h ic s t a t is t ic is t h e cen ter of m in im u m d istan ce. U n for t u n a t ely,
t h is st a t ist ic is somet imes a lso ca lled th e m edian center, wh ich ca n m a k e it con fu s in g s in ce
t h e a bove st a t ist ic ha s t h e sa m e n a m e. N ever t h eles s, u n lik e t h e m edia n cent er a bove, t h e
cen t er of m in im u m d is t a n ce is a un iqu e s t a t is t ic in t h a t it d efin es t h e p oin t a t wh ich t h e
su m of t h e dist a n ce t o a ll ot h er poin t s is t h e sm a llest (Bur t a n d Ba r ber , 1996). It is defined
as:
Cen t er of
Min im u m Dist a n ce = C =
N
Gd
ic
is a m in im u m
(4.3)
i=1
wh er e d ic is t h e dist a n ce bet ween a sin gle point , i, an d C, t h e cen t er of m in im u m d is t a n ce
(wit h a n X a n d Y coor d in a t e). U n for t u n a t ely, t h er e is n ot a for m u la t h a t ca n ca lcu la t e t h is
loca t ion.
In st ead , an iter a t ive algor ith m is used t h a t ap pr oxim a t es t h is loca t ion (Ku h n an d
Ku en n e, 1962; Bur t a n d Ba r ber , 1996). Depend ing on wh et h er t h e coor din a t es a r e
sp h er ical or pr oject ed, Crim eS tat will calcula t e dist a n ce as eit h er Gr ea t Cir cle (sp h er ical)
or E u clid ea n (p roject ed ), a s d is cu ss ed in t h e p reviou s ch a pt er .2 Th e r esu lt s a r e sh own in
t h e M cm d out pu t t a ble (figur e 4.9).
Th e imp or t a n ce of t h e cen t er of m inim u m dist a n ce is th a t it is a loca t ion wh er e
dist a n ce t o a ll t h e defin ing incident s is t h e sm a llest. Sin ce Crim eS tat on ly m ea su r es
dis t a n ces a s eit h er dir ect or in dir ect, actu a l t r a vel t im e is n ot bein g ca lcula t ed. Bu t in
m a n y jur isd ictions, t h e m in im u m dis t a n ce to a ll point s is a good a pp r oxim a t ion t o th e poin t
wh er e t r a vel dis t a n ces a r e m in im ized. F or exa m ple, in a police pr ecinct , a pa t r ol car cou ld
be st a t ion ed a t t h e cen t er of m in im u m dis t a n ce t o a llow it t o r espon d qu ick ly t o ca lls for
ser vice.
F or exa m ple, figu r e 4.10 m a ps t h e cen t er of min im u m dis t a n ce for 1996 a u t o t h eft s
in bot h Ba lt im or e Cit y a n d Ba lt im or e Cou n t y a n d com p ar es th is to bot h t h e m ea n cen t er
a n d t h e m edia n cen t er st a t is t ic. As seen , bot h t h e cen t er of m in im u m d is t a n ce a n d t h e
m edia n cen t er a r e sou t h of t h e m ea n cen t er , in dica t in g t h a t t h er e a r e sligh t ly m or e
in cid en t s in t h e s ou t h er n pa r t of t h e m et r op olit a n a r ea t h a n in t h e n or t h er n pa r t .
H owever, t h e differ en ce in t h ese t h r ee st a t ist ics is very sm a ll, especially t h e m edia n cen t er
a n d t h e cen t er of m inim u m dist a n ce.
S ta n d ard D ev ia ti on o f t h e X a n d Y Co o rd in a te s
In a dd it ion t o th e m ea n cent er a n d cen t er of m in im u m dis t a n ce, Crim eS tat will
ca lcu la t e va r iou s m ea s u r es of s pa t ia l d is t r ibu t ion , wh ich d es cr ibe t h e d is per s ion ,
or ient a t ion , an d sh a pe of t h e dist r ibut ion of a var iable (H a m m on d a n d McCullogh 1978;
E bdon 1988). Th e sim plest of t h ese is t h e r a w st a n da r d devia t ion s of t h e X a n d Y
4.12
Figure 4.9:
Center of Minimum Distance Output
Figure 4.10: 1996 Metropolitan Baltimore Auto Thefts
Mean Center and Center of Minimum Distance for 1996 Auto Thefts
Miles
0
2
4
Mean Center
Center of Minimum Distance
'
coordina tes, respectively. The form ulas used ar e the sta nda rd ones foun d in m ost
element ar y sta tistics books:
_
N
SX
=
(Xi - X )2
G --------- ]
SQRT[
i=1
(4.4)
N-1
_
N
SY
=
(Yi - Y )2
G --------- ]
SQRT[
i=1
(4.5)
N-1
_
_
wh er e Xi an d Yi a r e t h e X a n d Y coor din a t es for ind ividu a l point s, X a n d Y a r e t h e m ea n X
a n d m ea n Y, a n d N is t h e t ot a l n u m ber of poin t s. N ot e t h a t 1 is su bt r a ct ed fr om t h e
n u m ber of poin t s t o pr odu ce a n u n bia sed est im a t e of t h e st a n da r d devia t ion .
Th e st a n da r d devia t ion s of t h e X an d Y coordin a t es in dica t e t h e degr ee of
dis per sion . F igu r e 4.11 sh ows t h e st a n da r d devia t ion of th e coordin a t es for a u t o t h eft s a n d
r epr esen t s t h is a s a r ecta n gle. As s een , t h e dist r ibu t ion of a u t o th efts s pr ea ds m ore in a n
ea st -west dir ect ion t h a n in a n or t h -sou t h dir ect ion .
S t a n d a r d D i s ta n c e D e v i a t i o n
Wh ile t h e st a n da r d devia t ion of t h e X an d Y coordin a t es pr ovides som e in for m a t ion
a bou t t h e dis per sion of t h e in cid en t s, t h er e a r e t wo pr oblem s wit h it . F ir st , it does n ot
pr ovide a sin gle su m m a r y st a t ist ic of t h e disp er sion in t h e in ciden t locat ion s a n d is
a ctu a lly two sep a r a t e st a t ist ics (i.e., disp er sion in X a n d d isp er sion in Y). Second , it
pr ovides m ea su r em en t s in t h e u n it s of t h e coor din a t e sys t em . Th u s, if sph er ica l
coor din a t es a r e being us ed, th en t h e u n its will be decima l degrees.
A m ea s u r e wh ich over com es t h es e p r oblem s is t h e stan d ard dista n ce deviation or
st an d a rd d ist an ce, for s h or t . Th is is t h e s t a n da r d devia t ion of t h e d ist an ce of ea ch p oin t
from t h e m ea n cent er a n d is expr essed in m ea su r em en t u n it s (feet , m et er s, m iles). It is
t h e t wo-dim en sion a l equ iva len t of a st a n da r d devia t ion .
Th e for m u la for it is
N
S XY
=
Sqr t
[
(d iM C )2
G {----------------} ]
i=1
N-2
4.15
(4.6)
Figure 4.11: 1996 Metropolitan Baltimore Auto Thefts
Mean Center and Standard Deviations of X and Y Coordinates
2
4
0
Miles
'
wh er e d iM C is t h e dis t a n ce bet ween ea ch poin t , i, a n d t h e m ea n cen t er a n d N is t h e t ot a l
n u m ber of poin t s. Note t h a t 2 is su bt r a ct ed from t h e n u m ber of poin t s t o pr odu ce a n
u n bia sed est im a t e of st a n da r d d ist a n ce sin ce th er e a r e t wo cons t a n t s fr om wh ich t h is
dis t a n ce is m ea su r ed (m ea n of X, mea n of Y). 3
Th e st a n da r d dis t a n ce ca n be r epr esen t ed a s a sin gle vect or r a t h er t h a n t wo vect or s
a s wit h t h e st a n da r d deviat ion of t h e X a n d Y coor din a t es. Figur e 4.12 shows t h e m ea n
cent er a n d s t a n da r d d ist a n ce devia t ion of both r obber ies a n d bu r gla r ies for 1996 in
Baltimore Coun ty represented as circles. It is clear th at th e spatial distr ibut ions of th ese
t wo t ypes of cr ime va r y with r obber ies being sligh t ly m or e con cen t r a t ed.
S t a n d a rd D e v i a ti o n a l E lli p s e
Th e st a n da r d dis t a n ce d evia t ion is a good sin gle m ea su r e of th e dis per sion of th e
in cid en t s a r ou n d t h e m ea n cen t er . H owever , wit h t wo dim en sion s, d is t r ibu t ion s a r e
fr equ en t ly sk ewed in on e dir ect ion or a n ot h er (a con dit ion ca lled anisotropy). Instea d,
t h er e is an ot h er st a t ist ic wh ich gives disper sion in t wo dim en sion s, t h e stan d ard
d eviat ion al ellipse or ellipse, for sh ort (Ebdon , 1988; Cr omle y, 1992).
Th e st a n da r d d evia t iona l ellipse is d er ived from t h e biva r ia t e dist r ibu t ion (F u r fey,
1927; Neft, 1962; Bachh i, 1957) a n d is defin ed by
Biva r ia t e
=
Dis t r ibu t ion
[ F2 x + F2 y ]
SQRT ------------2
(4.7)
Th e t wo st a n da r d deviat ion s, in t h e X a n d Y dir ect ion s, ar e or t h ogon a l to each oth er
a n d defin e a n ellip se. E bdon (1988) r ot a t es t h e X an d Y axis so t h a t t h e su m of squ a r es of
dist a n ces bet ween poin t s a n d a xes ar e m inim ized. By con vent ion , it is sh own a s a n ellipse.
Aside from t he mea n X an d mean Y, th e form ulas for t hese stat istics ar e as follows:
1.
The Y-axis is rota ted clock w ise t h r ou gh a n a n gle, 2, wher e
_
_
2
2 = ARCTAN {( E(Xi -X ) - E(Yi -Y )2 ) +
_
_
_
_
_
_
[(E(Xi -X)2 - E(Yi -Y )2 )2 + 4(E(Xi -X )(Yi -Y ))2 ]1 /2 }/(2 E(Xi -X )(Yi -Y ))
wh er e a ll su m m a t ion s a r e for i=1 t o N (E bdon , 1988).
4.17
(4.8)
Figure 4.12: 1996 Baltimore County Burglaries and Robberies
Comparison of Mean Centers and Standard Distance Deviations
SDD of Burglaries
SDD of Robberies
MC Burglaries
MC Robberies
Miles
0
2
4
'
2.
Two st a n da r d d evia t ion s a r e calcu la t ed, on e a lon g t h e t r a n sp osed X-axis a n d
one a lon g t h e t r a n sp osed Y-axis.
G{ (Xi - X ) Cos 2
Sx =
Sy =
- G(Yi - Y ) Sin 2 }2
SQRT[ 2*-------------------------------------------------- ]
N-2
_
_
G{ (Xi - X ) Sin 2 - G(Yi - Y ) Cos 2 }2
SQRT[ 2*-------------------------------------------------- ]
N-2
(4.9)
(4.10)
wh er e N is t h e n u m ber of poin t s. N ot e, a ga in , t h a t 2 is su bt r a ct ed fr om t h e
n u m ber of poin t s in bot h den om in a t or s t o pr odu ce a n u n bia sed est im a t e of
t h e st a n da r d devia t ion a l ellip se sin ce t h er e a r e t wo con st a n t s fr om wh ich t h e
dis t a n ce a lon g ea ch a xis is m ea su r ed (m ea n of X, mea n of Y). 4
3.
4.
Th e X-a xis a n d Y-a xis of t h e ellipse a r e defin ed by
Len gt h x = 2S x
(4.11)
Len gt h y = 2S y
(4.12)
The area of th e ellipse is
A = BS x S y
(4.13)
Figur e 4.13 shows the out put of th e ellipse rout ine and figur e 4.14 ma ps th e
st a n da r d devia t ion a l ellip se of a u t o t h eft s in Ba lt im or e Cit y a n d Ba lt im or e Cou n t y for
1996.
Ge o m e t ri c Me a n
The mean center rout ine (Mcsd) includes two additiona l mean s. First, th ere is th e
geom et r ic m ea n , which is a m ea n a ssociat ed wit h t h e m ea n of t h e logar ith m s. It is defin ed
as:
N
Geom et r ic Mea n of X = GM(X) =
A
(Xi )1 /N
(4.14)
(Yi )1 /N
(4.15)
i=1
N
Geom et r ic Mea n of Y = GM(Y) =
A
i=1
4.19
Figure 4.13:
Standard Deviational Ellipse Output
Figure 4.14: 1996 Metropolitan Baltimore Auto Thefts
Mean Center and Standard Deviational Ellipse
Miles
0
2
4
'
wh er e A is t h e pr odu ct t er m of ea ch poin t valu e, i (i.e., t h e valu es of X or Y a r e m u ltiplied
t imes ea ch ot h er ), an d N is t h e sa m ple size (E verit t , 1995). The equ a t ion ca n be evalu a t ed
by logar ithm s.
1
1
Ln [GM(X)] = ---- [ Ln (X1 ) + Ln (X2 ) + ..... + Ln (XN ) ] = ----- G Ln (Xi )
N
N
(4.16)
1
1
Ln [GM(Y)] = ---- [ Ln (Y1 ) + Ln (Y2 ) + ..... + Ln (YN ) ] = ----- G Ln (Yi )
N
N
(4.17)
GM(X) =
e L n ( GM (X)
(4.18)
GM(Y) =
e L n ( GM (Y)
(4.19)
Th e geomet r ic m ea n is t h e a n t i-log of t h e m ea n of t h e logar it h m s. Beca u se it fir st
con vert s a ll X a n d Y coor din a t es in t o logar ith m s, it h a s t h e effect of discou n t ing extr em e
va lu es . Th e geom et r ic m ea n is ou t pu t a s p ar t of t h e Mcs d r ou t in e a n d h a s a ‘Gm ’ p refix
befor e t h e u ser defin ed n a m e.
H a rm o n i c Me a n
Th e h a r m onic mea n is a lso a m ea n wh ich discou n t s ext r em e va lu es, bu t is
ca lcu lat ed differ en t ly. It is defined a s
H a r m on ic m ea n of X = H M(X) =
N
--------------G (1/Xi )
(4.20)
H a r m on ic m ean of Y = HM(Y)
N
--------------G (1/Yi )
(4.21)
In ot h er wor ds, t h e h a r m on ic m ea n of X a n d Y respect ively is t h e in ver se of th e
m ea n of th e in ver se of X a n d Y respect ively (i.e., t a k e t h e in ver se; t a k e t h e m ea n of th e
in ver s e; a n d in ver t t h e m ea n of t h e in ver s e). Th e h a r m on ic m ea n is ou t p u t a s p a r t of t h e
Mcsd r ou t ine a n d h a s a ‘H m ’ pr efix befor e t h e u ser defin ed n a m e.
Th e geom et r ic a n d h a r m on ic m ea n s a r e d is cou n t ed mea n s t h a t ‘h u g’ t h e cen t er of
t h e dist r ibut ion . They differ fr om t h e m ea n cen t er wh en t h er e is a ver y skewed
dist r ibut ion . To con t r a st t h e differ en t m ea n s, figu r e 4.15 below sh ows five differ en t m ea n s
for Ba lt im or e Cou n t y m ot or veh icle t h eft s:
4.22
Figure 4.15:
Five Mean Centers for 1996 Baltimore Vehicle Thefts
Five Different Means Compared
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1.
2.
3.
4.
5.
Mea n cen t er ;
Cent er of m inim u m dist a n ce;
Geom et r ic m ea n ;
H a r m on ic m ea n ; a n d
Tria n gulat ed m ean (discu ssed below)
In t h e exa m ple, t h e m ea n cen t er , geom et r ic m ea n , a n d h a r m on ic m ea n fa ll a lm ost
on t op of ea ch ot h er ; h owever , t h ey will n ot a lwa ys be so. Th e cen t er of m in im u m dis t a n ce
a ppr oxim a t es t h e geogra ph ica l cen t er of t h e dist r ibut ion . The t r ian gula t ed m ea n is defin ed
by th e angularity and dista nce from t he lower-left a nd u pper-right corn ers of th e data set
(see be low).
Cen t r ogra ph ic des crip t ors can be ver y power ful t ools for exa m in in g sp a t ia l pa t t er n s.
Th ey a r e a fir st st ep in a n y sp a t ia l a n a lysis , bu t a n im por t a n t one. Th e a bove exam ple
illust r a t es h ow th ey can be a ba sis for decision -ma k in g, even wit h sm a ll sa m ples. A coup le
of oth er exam ples can be illust ra ted.
Av e r a g e Den sity
The avera ge density is th e num ber of incidents divided by th e area. It is a measu re
of t h e a ver a ge n u m ber of even t s per u n it of a r ea ; it is som et im es ca lled t h e intensity. If t h e
a r ea is defin ed on t h e m ea su r em en t pa r a m et er s pa ge, t h e r ou t ine u ses t h a t valu e;
ot h er wis e, it t a k es t h e r ect a n gu la r a r ea d efin ed by t h e m in im u m a n d m a xim u m X a n d Y
va lu es (t h e bou n din g r ect a n gle).
Ou t pu t Fi le s
Ca lc u l a ti n g t h e S t a ti s t ic s
On ce t h e st a t is t ics h a ve been select ed, t h e u ser clicks on Com pute t o r u n t h e
r ou t ine. The r esu lts a r e sh own in a r esu lts t a ble.
Ta b u l ar Ou t p u t
F or ea ch of th es e st a t ist ics, Crim eS tat p r od u ces t a bu la r ou t p u t . In Crim eS tat, all
t a bles a r e la beled by s ym bols , for exa m ple Mcsd for t h e m ea n cen t er a n d st a n da r d dis t a n ce
deviat ion or Mcm d for t h e cen t er of m inim u m dist a n ce. All t a bles pr esen t t h e sa m ple size.
Graph ical Objec ts
Th e six cen t r ogra ph ic st a t ist ics can be ou t pu t a s gr a ph ica l object s. The m ea n cen t er
a n d cen t er of min im u m dis t a n ce a r e out pu t a s sin gle poin t s. Th e st a n da r d devia t ion of th e
X a n d Y coor d in a t es is ou t p u t a s a r ect a n gle. Th e s t a n da r d dis t a n ce d evia t ion is ou t p u t a s
a circle a n d t h e st a n da r d deviat ion a l ellipse is ou t pu t a s a n ellipse.
4.24
Crim eS tat cu r r en t ly su ppor t s gr a ph ica l ou t pu t s t o ArcView ‘.s h p’ files, t o M apIn fo
‘.m if’ a n d t o Atlas*GIS ‘.bn a ’ files. Befor e r u n n in g t h e ca lcu la t ion , t h e u ser sh ou ld select
t h e desir ed out pu t files an d sp ecify a r oot n a m e (e.g., P r ecinct1Bu r gla r ies). Figur e 4.16
sh ows a dia log box for select in g for t h e GIS pr ogr a m ou t pu t . F or M apIn fo ou t p u t on ly, t h e
u ser h a s t o a lso ind ica t e t h e n a m e of t h e pr oject ion , th e pr oject ion n u m ber a n d t h e da t u m
n u m ber . These can be fou n d in t h e M apIn fo u se r s gu ide . By defau lt , Crim eS tat will use
t h e st a n da r d pa r a m et er s for a sph er ica l coor din a t e sys t em (E a r t h pr oject ion , p r oject ion
n u m ber 1, a n d d a t u m n u m ber 33). If a u se r r equ ir es a differ en t coord in a t e syst em , t h e
a ppr opr ia t e va lu es sh ou ld be t yp ed in t o t h e spa ce. F igu r e 4.17 sh ows t h e select ion of th e
M apIn fo coor din a t e p ar a m et er s .
If r equ est ed, th e ou t pu t files ar e sa ved in t h e specified directory u n der t h e specified
(r oot ) n a m e. F or ea ch st a t is t ic, Crim eS tat will a dd p r efix lett er s t o t h e r oot n a m e.
MC<root> for t h e m ea n cen t er
Mdn Cn t r <root > for t h e m edia n cen t er
Mcmd<root> for cen t er of m in im u m d is t a n ce
XYD<root> for t h e st a n da r d deviat ion of t h e X a n d Y coor din a t es
SDD<root> for t h e st a n da r d dis t a n ce devia t ion
S DE <root> for t h e st a n da r d deviat ion a l ellipse.
Th e ‘.s h p’ files ca n be r ea d dir ect ly in t o ArcView a s t h em es. The ‘.mif’ a n d ‘.bna ’
files h a ve t o be im por t ed in t o M apIn fo an d Atlas*GIS , r es pectively. 5
S ta tis tic al Te s tin g
Wh ile t h e cu r r en t ver sion of Crim eS tat d oes n ot con d u ct s t a t is t ica l t es t s t h a t
com pa r e t wo dist r ibut ion s, it is possible t o con du ct su ch t est s. App en dix B pr esen t s a
dis cu ssion of t h e st a t is t ica l t est s t h a t ca n be u sed. In st ea d, t h e dis cu ssion h er e will focu s
on u s in g t h e ou t p u t s of t h e r ou t in es wit h ou t for m a l t es t in g.
De cision -ma king Withou t Forma l Tes ts
F or m a l s ign ifica n ce t est in g h a s t h e a dva n t a ge of pr ovidin g a con sis t en t in fer en ce
a bout wh et h er t h e differ en ce in t wo dis t r ibu t ions is lik ely or u n lik ely t o be du e t o ch a n ce.
Almost a ll form al tests compa re th e distr ibut ion of a st at istic with th at of a r an dom
dis t r ibu t ion . H owever , p olice depa r t m en t s fr equ en t ly h a ve t o m a k e decis ion s ba sed on
sm a ll sa m ples, in wh ich ca se t h e for m a l tes t s a r e less u seful t h a n t h ey wou ld with lar ger
sa m ples. St ill, th e cent r ogr a ph ic st a t ist ics ca lcula t ed in Crim eS tat ca n be u sefu l a n d ca n
h elp a police depa r t m en t m a ke d ecision even in t h e a bsen ce of for m a l tes t s.
Ex am ple 1: J u n e an d J u ly Auto The fts in P re cin ct 11
We wan t t o illust r a t e th e us e of t h ese st a t istics t o m a ke decision s with t wo
exa m ples. Th e fir st is a com pa r is on of cr im es in sm a ll geogr a ph ica l a r ea s. In m ost
m et r opolit a n a r ea s, m ost a n a lyst s will con cen t r a t e on pa r t icu la r su b-a r ea s of th e
4.25
Figure 4.16:
Outputting Objects to A GIS Program
Figure 4.17:
MapInfo Output Options
ju r is dict ion , r a t h er t h a n on t h e ju r is dict ion it s elf. In Ba lt im or e Cou n t y, for in s t a n ce,
a n a lysis is d one both for t h e ju r isd iction a s a wh ole as w ell a s by in divid u a l pr ecincts .
Below in F igu r e 4.18 ar e t h e st a n da r d deviat ion a l ellipses for 1996 au t o t h eft s for J u n e a n d
J u ly in P r ecin ct 11 of Ba lt im or e Cou n t y. As ca n be seen , t h er e wa s a spa t ia l s h ift t h a t
occur red between J un e and J uly of th at year, th e result most pr obably of increased
va cat ion t r a vel t o th e Ch esa pea k e Ba y. While t h e com pa r ison is ver y sim ple, involving
look in g a t t h e gr a ph ical object crea t ed by Crim eS tat, such a mont h t o mont h compa rison
can be useful for police depar tm ents becau se it points t o a sh ift in incident pa tt erns,
a llowin g t h e police depa r t m en t t o reorie n t t h eir pa t r ol un it s.
Exam ple 2: Serial Bu rglaries in B altimore City an d Baltim ore Coun ty
Th e secon d exa m ple illu st r a t es a r a sh of bu r gla r ies t h a t occu r r ed on bot h sid es of
t h e bor der of Balt imore Cit y an d Ba ltim or e Cou n t y. On on e h a n d t h er e wer e t en
r esiden t ia l bu r gla r ies t h a t occur r ed on t h e west er n edge of t h e Cit y/Coun t y bord er wit h in a
sh ort t im e per iod of ea ch oth er a n d, on t h e oth er h a n d, t h er e wer e 13 com m er cial
bu r gla r ies t h a t occu r r ed in t h e cen t r a l p a r t of t h e m et r opolit a n a r ea s. Bot h police
depa r t m en t s su spect ed t h a t t h ese t wo set s wer e t h e wor k of a ser ia l bu r gla r (or gr ou p of
bu r gla r s). Wha t t h ey were n ot su r e a bou t wa s wh et h er t h e t wo set s of bur gla r ies wer e
done by th e same individua ls or by different individua ls.
Th e n u m ber of in cid en t s in volved a r e t oo sm a ll for sign ifica n ce t est in g; on ly on e of
t h e pa r a m et er s t es t ed wa s s ign ifica n t a n d t h a t cou ld ea sily be du e t o ch a n ce. However , t h e
police do h a ve to ma ke a guess a bou t t h e possible per pet r a t or even wit h lim ited
inform at ion. Let’s use Crim eS tat t o tr y a n d m a k e a decision a bout t h e dist r ibu t ions .
F igu r e 4.19 illust r a t es t h ese dist r ibut ion s. The t h irt een com m er cial bu r gla r ies a r e
sh own a s squ a r es wh ile t h e t en r esiden t ial bu r gla r ies a r e sh own a s t r ian gles. Figur e 4.20
plot s t h e m ea n cen t er s of t h e t wo dis t r ibu t ion s. Th ey a r e clos e t o ea ch ot h er , bu t n ot
ident ica l. An initia l hu n ch wou ld suggest t h a t t h e robber ies ar e com m itt ed by two
per pet r a t ors (or gr oup s of per pet r a t ors ), but t h e m ea n cent er s a r e n ot d iffer en t en ough t o
t r u ly confirm t h is exp ecta t ion . Sim ila r ly, figur e 4.21 p lot s t h e cent er of m in im u m
dis t a n ce. Aga in , t h er e is a differ en ce in t h e dist r ibu t ion , bu t it is n ot gr ea t en ough t o tr u ly
r u le out t h e sin gle per pet r a t or t h eor y.
F igu r e 4.22 plot s t h e r a w st a n da r d deviat ion s, expres sed a s a r ect a n gle by
Crim eS tat. The disp er sion of inciden t s over lap s t o a sizeable exten t a n d t h e a r ea defin ed
by t h e r ect a n gle is a ppr oxim a t ely t h e sa m e. In ot h er wor ds, t h e sea r ch a r ea of th e
per pet r a t or or per pet r a t ors is a pp r oxim a t ely t h e sa m e. Th is m ight a r gu e for a sin gle
per pet r a t or , r a t h er t h a n t wo. F igu r e 4.23 sh ows t h e st a n da r d dis t a n ce d evia t ion of th e
two sets of incidents. Again, there is sizeable overlap an d th e sear ch r adiuses ar e
a pp r oxima t ely t h e sa m e.
On ly with t h e st a n da r d deviat ion a l ellipse, however , is t h er e a fu n da m en t a l
differ en ce bet ween t h e t wo dist r ibut ion s (figu r e 4.24). The pa t t er n of com m er cial r obber ies
is fa llin g a lon g a n or t h ea s t -s ou t h wes t or ien t a t ion wh ile t h a t for r es id en t ia l r obber ies a lon g
4.28
Figure 4.18:
Vehicle Theft Change in Precinct 11
Standard Deviational Ellipses for June and July 1996
Features
Arterial
Baltimore County
City of Baltimore
Baltimore County
Police Precinct 11
Street
Precinct 11
June Auto Thefts
City of Baltimore
July
June
Miles
0
1
July Auto Thefts
2
Figure 4.19:
Identifying Serial Burglars
Incident Distribution of Two Serial Offenders
Features
"
County
Baltimore County
"
Commercial Burglaries
#
Residential Burglaries
#
"
#
#
#
"
"
#
#
"
"
#
#
"
#
"
"
#
#
#
"
#
City of Baltimore
Miles
0
2
4
Anne Arundel County
Figure 4.20:
Identifying Serial Burglars
Mean Centers of Incidents for Two Serial Offenders
Features
"
County
Baltimore County
#
"
#
#
#
"
"
#
#
"
"
#
"
MC Commercial Burglaries
# MC Residential Burglaries
#
"
#
"
"
#
#
#
"
#
City of Baltimore
Miles
0
2
4
Anne Arundel County
"
Commercial Burglaries
#
Residential Burglaries
"
Mean Center of Residential Burglaries
#
Mean Center of Group B
Figure 4.21:
Identifying Serial Burglars
Center of Minimum Distances for Incidents for Two Serial Offenders
Features
"
County
Baltimore County
#
"
#
#
#
"
"
#
#
MDN Commercial
Burglaries
"
%
"
#
- MDN
Residential Burglaries
#
"
#
"
"
#
#
#
"
#
City of Baltimore
Miles
0
2
4
Anne Arundel County
"
Commercial Burglaries
#
Residential Burglaries
%
Median Center of Group A
-
Median Center of Group B
Figure 4.22:
Identifying Serial Burglars
Standard Deviations of Incidents for Two Serial Offenders
Features
"
Standard Deviation of Commercial Burglaries
Baltimore County
Standard Deviation of Residential Burglaries
#
County
SD Commercial Burglaries
"
#
#
"
Commercial Burglaries
#
Residential Burglaries
#
"
"
#
#
"
"
#
#
"
#
"
"
#
#
SD Residential Burglaries
#
"
#
City of Baltimore
Miles
0
2
4
Anne Arundel County
Figure 4.23:
Identifying Serial Burglars
Standard Distance Deviation of Incidents for Two Serial Offenders
Features
"
Standard Distance Deviation of Commercial Burglaries
SDD Commercial Burglaries
Baltimore County
Standard Distance Deviation of Residential Burglaries
#
County
"
#
#
"
Commercial Burglaries
#
Residential Burglaries
#
"
"
#
#
"
"
#
#
"
#
"
"
SDD Residential Burglaries
#
#
#
"
#
City of Baltimore
Miles
0
2
4
Anne Arundel County
Figure 4.24:
Identifying Serial Burglars
Standard Deviational Ellipse of Incidents for Two Serial Offenders
Features
"
Standard Deviational Ellipse of Commercial Burglaries
Baltimore County
SDE Commercial Burglaries
Standard Deviational Ellipse of Residential Burglaries
#
County
"
#
#
"
Commercial Burglaries
#
Residential Burglaries
#
"
"
#
#
"
"
#
SDE Residential Burglaries
#
"
#
"
"
#
#
#
"
#
City of Baltimore
Miles
0
2
4
Anne Arundel County
a n ort h wes t -sout h ea st a xis. In oth er words , wh en t h e orien t a t ion of t h e in ciden t s is
exa m in ed, as d efined by t h e st a n da r d d evia t ion a l ellips e, t h er e a r e t wo comp let ely opposit e
pa t t er n s. U n les s t h is d iffer en ce ca n be expla in ed by a n obviou s fa ctor (e.g., th e
dist r ibut ion of com m er cial est a blishm en t s), it is pr oba ble th a t t h e t wo set s of r obber ies
were comm itted by two different perpetra tors (or groups of perpetra tors).
D i re c t io n a l Me a n a n d Va ri a n c e
Cen t r ogr a ph ic st a t ist ics u t ilize t h e coor din a t es of a poin t , d efin ed a s a n X a n d Y
va lu e on eit h er a sp h er ical or pr oject ed/Car t es ia n coord in a t e syst em . Th er e is a n oth er t ype
of m et r ic th a t can be u se d for iden t ifying in ciden t loca t ions , n a m ely a polar coordinate
sys t em . A vector is a lin e wit h dir ect ion a n d len gt h . In t h is sys t em , t h er e is a r efer en ce
vector (u su a lly 0 0 du e N ort h ) an d a ll locat ion s a r e defined by a n gu la r devia t ion s fr om t h is
r efer en ce vect or . By con ven t ion , a n gles a r e defin ed a s devia t ion s fr om 0 0 , clockwis e
t h r ou gh 360 0 . Note t h e m ea su r em en t sca le is a circle wh ich r et u r n s ba ck on it se lf (i.e. 0 0 is
a lso 360 0 ). P oin t loca t ion s ca n be r ep res en t ed as vect or s on a pola r coor din a t e s ys t em .
With su ch a syst em , or din a r y st a t ist ics can n ot be us ed. For exa m ple, if t h er e a r e
five points which on t he nort hern side of th e polar coordina te system an d ar e defined by
th eir a ngular deviat ions a s 0 0 , 100 , 150 , 3450 , an d 350 0 from t h e r efer en ce vect or (m ovin g
clock wis e fr om du e N or t h ), t h e s ta t is t ica l m ea n will p rod uce a n er r on eou s es t im a t e of 144 0 .
Th is vect or wou ld be sou t h ea st a n d will lie in a n opposit e dir ect ion fr om t h e dis t r ibu t ion of
poin t s.
In st ea d, s t a t is t ics h a ve t o be ca lcu la t ed by t r igon om et r ic fu n ct ion s. Th e in pu t for
su ch a sys t em is a set of vect or s, d efin ed a s a n gu la r devia t ion s fr om t h e r efer en ce vect or
a n d a dis t a n ce vect or . Bot h t h e a n gle a n d t h e dis t a n ce vect or a r e defin ed wit h r espect t o
a n or igin. The r ou t ine can ca lcu lat e a n gles dir ect ly or ca n con vert a ll X a n d Y coor din a t es
in t o a n gles wit h a bea r in g fr om a n or igin . F or r ea din g a n gles dir ect ly, t h e in pu t is a s et of
vectors , define d a s a n gu la r devia t ions from t h e r efer en ce vector . Crim eS tat calculat es the
m ean direction an d t h e circular var ian ce of a s eries of point s defin ed by th eir a n gles. On
t h e pr im a r y file scr een , t h e u se r m u st select Dir ect ion (a n gles) as t h e coord in a t e syst em .
If t h e a n gles a r e t o be ca lcu lat ed from X/Y coor din a t es, th e u ser m u st defin e a n
origin locat ion . On t h e r efer en ce file pa ge, t h e u ser can select a m ong t h r ee or igin point s:
1.
Th e lower -left cor n er of t h e da t a set (th e m in im u m X a n d Y valu es). Th is is
t h e defa u lt set t in g.
2.
Th e u pper -r igh t corn er of th e da t a set (t h e m a xim u m X an d Y va lu es); an d
3.
A u ser -defined p oin t .
Us er s sh ou ld be ca r efu l about ch oosin g a pa r t icu lar loca t ion for a n or igin, eith er
lower -left, u pp er -righ t or u se r -defin ed. If th er e is a poin t a t t h a t origin , Crim eS tat will
dr op t h a t cas e sin ce an y calcu la t ions for a poin t wit h zer o dist a n ce ar e in det er m in a t e.
4.36
User s s h ould check t h a t t h er e is n o poin t a t t h e desir ed origin . If th er e is, t h en t h e origin
sh ould be a dju st ed sligh t ly so th a t n o poin t falls a t t h a t locat ion (e.g., t a k in g sligh t ly
sm a ller X an d Y va lu es for t h e lower -left cor n er or sligh t ly la r ger X an d Y va lu es for t h e
u pp er r igh t cor n er ).
Th e r ou t in e con ver t s a ll X an d Y poin t s in t o a n a n gu la r devia t ion fr om t r u e Nor t h
r elat ive t o t h e specified or igin a n d a dist a n ce fr om t h e or igin. The bea r ing is ca lcu lat ed
wit h d iffer en t for m u la e d ep en d in g on t h e qu a d r a n t t h a t t h e p oin t fa lls wit h in .
F irs t Qu a dra n t
Wit h t h e lower -left cor n er a s t h e or igin , a ll a n gles a r e in t h e fir s t qu a d r a n t . Th e
clockwise a n gle, 2 i is ca lcu lat ed by
2i =
Abs(Xi - XO )
Arct a n [------------------]
Abs(Yi - YO )
(4.22)
wh er e Xi is t h e X-va lu e of t h e point , Yi is t h e Y-va lu e of t h e point , XO is t h e X-va lu e of th e
origin, an d YO is t h e Y-va lu e of t h e origin .
Th e a n gle,
2i, is in
r a dia n s a n d ca n be conver t ed t o polar coord in a t e degr ees u sin g:
2i (degr ees ) = 2i (r a dia n s) * 180/B
(4.23)
Th ird Qu a dra n t
Wit h t h e u p per -r igh t cor n er a s t h e or igin , a ll a n gles a r e in t h e t h ir d qu a d r a n t . Th e
clockwise a n gle, 2i , is calcu lat ed by
2i =
B+
Abs(Xi - XO )
Ar ct a n [-------------------]
Abs(Yi - YO )
(4.24)
wh er e t h e a n gle, 2i , is a gain in r a dia n s. Since th er e a r e 2 B r a dia n s in a circle, B r a dia n s is
180 0 . Again , th e a n gle in r a dia n s can be con vert ed int o degrees with for m u la 4.23 above.
Se con d an d Fou rth Quadran ts
When t h e or igin is u ser -defin ed, each p oint m u st be evalu a t ed a s t o wh ich qua dr a n t
it is in . Th e secon d a n d four t h qu a dr a n t s d efin e t h e clockwise a n gle, 2i , differ en t ly
4.37
Using Spatial Measures of Central Tendency with Network Analyst
to Identify Routes Used by Motor Vehicle Thieves
Philip R. Canter
Baltimore County Police Department
Towson, Maryland
Motor vehicle thefts have been steadily declining countywide over the last 5
years, but one police precinct in southwest Baltimore County was experiencing
significant increases over several months. Cases were concentrated in several
communities, but directed deployment and saturated patrols had minimal impact. In
addition to increasing patrols in target communities, the precinct commander was
interested in deploying police on roads possibly used by motor vehicle thieves. Police
analysts had addresses for theft and recovery locations; it was a matter of using the
existing highway network to connect the two locations.
To avoid analyzing dozens of paired locations, analysts decided to set up a
database using one location representing the origin of motor vehicle thefts for a
particular community. The origin was computed using CrimeStat’s median center for
motor vehicle theft locations reported for a particular community. The median
center is the position of minimum average travel and is less affected by extreme
locations compared to the arithmetic mean center. The database consisted of the
median center paired with a recovery location. Using Network Analyst, a least-effort
route was computed for cases reported by community. A count was assigned to each
link along a roadway identified by Network Analyst. Analysts used the count to
thematically weight links in ArcView. The precinct commander deployed resources
along these routes with orders to stop suspicious vehicles. This operation resulted in
27 arrests, and a reduction in motor vehicle thefts.
Distance Analysis
Man With A Gun Calls For Service
Charlotte, N.C., 1989
James L. LeBeau
Administration of Justice
Southern Illinois University – Carbondale
Hurricane Hugo arrived on Friday, September 22, 1989 in Charlotte, North
Carolina. That weekend experienced the highest counts of Man With A Gun calls for
service for the year. The locations of the calls during the Hugo Weekend are
compared with the following New Year’s Eve weekend.
CrimeStat was used to compare the two weekends. Compared to the New
Year’s Eve weekend: 1) Hugo’s mean and median centers are more easterly; 2)
Hugo’s ellipse is larger and more circular; and 3) Hugo‘s ellipse shifts more to the
east and southeast. The abrupt spatial change of Man With A Gun calls during a
natural disaster might indicate more instances of defensive gun use for protection of
property.
2i =
2i =
0.5 B
Abs(Yi - YO )
+ Ar ct a n [-------------------]
Abs(Xi - XO )
(4.25)
1.5 B
Abs(Yi - YO )
+ Ar ct a n [--------------------]
Abs(Xi - XO )
(4.26)
On ce a ll X/Y coor din a t es a r e con vert ed int o a n gles, th e m ea n a n gle is ca lcu lat ed.
Me a n A n g le
With eith er a n gula r inp u t or con vers ion fr om X/Y coor din a t es, th e M ean An gle is
t h e r esu lt a n t of a ll in divid u a l vect or s (i.e., poin t s defin ed by t h eir a n gles fr om t h e r efer en ce
vect or ). It is a n a n gle t h a t s um m a r izes t h e m ea n d ir ect ion . Gr a p h ica lly, a resultant is t h e
su m of a ll vect ors a n d ca n be s h own by la ying ea ch vect or en d t o end . St a t ist ically, it is
d efin ed a s
G d i s in 2i
_
Mea n a n gle
= 2 =
Abs { Arct a n [--------------------------] }
(4.27)
G d i cos 2i
wh er e t h e su m m a t ion of sin es a n d cosin es is over t h e t ot a l nu m ber of poin t s, i, defin ed by
t h eir a n gles, 2 i . Ea ch a n gle, 2 i , can be weigh t ed by t h e len gt h of t h e vect or, d i . In a n
u n weigh t ed a n gle, d i is a ssu m ed t o be of eq u a l len gt h , 1. Th e a bsolu t e va lu e of t h e r a t io of
t h e su m of t h e weigh t ed sin es t o t h e su m of t h e weigh t ed cosin es is t a ken . All a n gles a r e
in r a dia n s. In det er m in in g t h e m ea n a n gle, t h e qu a dr a n t of t h e r es ult a n t m u st be
identified:
_
1.
If G s in 2i >0 an d G cos 2i >0, th en 2 can be u sed d ir ectly as t h e m ea n a n gle
_
2.
If G s in 2i >0 an d G cos 2i <0, th en th e mean an gle is B/2 + 2.
_
3.
If G s in 2i <0 an d G cos 2i <0, th en th e mean an gle is B + 2.
_
4.
If G s in 2i <0 an d G cos 2i >0, t h en t h e m ea n a n gle is 1.5 B + 2.
F or m u la s 4.22, 4.24, 4.25 a n d 4.26 a bove a r e t h en u sed t o con ver t t h e dir ect ion a l
m ea n ba ck t o an X/Y coord in a t e, depen din g on wh ich coord in a t e it falls wit h in .
4.40
Ci rc u l a r Va r ia n c e
Th e disp er sion (or va r ia n ce) of t h e a n gles a r e a lso defin ed by t r igon omet r ic
fu n ct ion s. The u n st a n da r dized var ian ce, R, is somet imes ca lled th e sam ple resultant
length sin ce it is t h e r es u lt a n t of a ll vectors (an gles).
R=
SQRT [ (G d i s in
2 i )2
+ (G d i cos
2 i )2 ]
(4.28)
wh er e d i is t h e len gt h of vect or , i, wit h a n a n gle (bea r in g) for t h e vect or of 2 i . F or t h e
u n weigh t ed sa m ple r es u lt a n t , d i is 1.
Beca u se R in cr ea ses wit h sa m ple size, it is st a n da r dized by d ivid in g by N t o pr odu ce
a m ean resultant length .
_
R
=
R
----------N
(4.29)
wh er e N is t h e n u m ber poin t s (sa m ple size).
F ina lly, th e a vera ge dista n ce fr om t h e or igin, D, is calcu lat ed a n d t h e circular
v aria n ce is ca lcu lat ed by
_
1
R
_
R
Cir cu la r va r ia n ce
=
------ {D - ---------} = (D - R)/D = 1 - ------(4.30)
D
N
D
Th is is t h e st a n da r dized va r ia n ce wh ich va r ies fr om 0 (n o va r ia bilit y) t o 1
(m a xim u m var iability). The det a ils of t h e der iva t ion s can be fou n d in Bur t a n d Ba r ber
(1996) a n d G a ile a n d Ba r ber (1980).
Me a n D i s t a n c e
_
Th e m ea n dis t a n ce, d, is calcu la t ed dir ectly fr om t h e X a n d Y coor din a t es. It is
iden t ified in r ela t ion t o th e defined or igin.
D i re c t i o n a l Me a n
Th e dir ect ion a l mea n is ca lcu lat ed a s t h e int er section of t h e m ea n a n gle a n d t h e
m ea n dist a n ce. It is n ot a u n ique position sin ce dist a n ce a n d a n gula r ity a r e indep en den t
dim en sion s. Th u s, t h e dir ection a l m ea n calcula t ed u sin g t h e m in im u m X a n d m in im u m Y
loca t ion a s t h e r efer en ce or igin (t h e ‘lower left cor n er ’) will yield a differ en t loca t ion fr om
t h e dir ect ion a l m ea n ca lcu la t ed u sin g t h e m a xim u m X an d m a xim u m Y loca t ion a s t h e
origin (th e ‘u pp er r igh t cor n er ’). Th er e is a weigh t ed a n d u n weigh t ed dir ect iona l m ea n .
4.41
Th ou gh Crim eS tat calculat es the locat ion, users should be awar e of th e non-un iqueness of
t h e loca t ion . The u n weight ed dir ect ion a l mea n ca n be ou t pu t with a ‘Dm ’ pr efix. The
weight ed directiona l mean is not out put .
Tr ia n g u l a t e d Me a n
Th e t r ia n gu la t ed m ea n is defin ed a s t h e in t er sect ion of t h e t wo vect or s, on e fr om
t h e lower -left cor n er of t h e st u dy a r ea (t h e m in im u m X an d Y va lu es) a n d t h e ot h er fr om
t h e u pper -r igh t cor n er of t h e st u dy ar ea (t h e m a xim u m X a n d Y valu es). It is ca lcu lat ed by
est im a t in g m ea n a n gles fr om ea ch origin (lower left a n d u pp er r ight cor n er s), t r a n sla t in g
t h ese in t o equa t ion s, a n d fin din g t h e poin t a t wh ich t h ese equ a t ion s in t er sect (by set t in g
t h e t wo fun ction s equ a l t o each ot h er ).
D ire c ti on a l Me a n Ou tp u t
The directiona l mean rout ine out put s nine sta tistics:
1.
2.
3.
4.
5.
6.
7.
8.
9.
Th e
Th e
Th e
Th e
Th e
Th e
Th e
Th e
Th e
sa m ple size;
u n weight ed m ea n a n gle;
weigh t ed m ea n a n gle;
u n weight ed cir cu lar var ian ce;
weigh t ed cir cu lar var ian ce;
m ea n dist a n ce;
in t er sect ion of th e m ea n a n gle a n d t h e m ea n dis t a n ce;
X an d Y coor din a t es for t h e t r ia n gu la t ed m ea n ; a n d
X a n d Y coor din a t es for t h e weigh t ed t r ian gula t ed m ea n .
The directiona l mean an d tr iangulated mean can be saved as a n ArcView ‘shp’,
M apIn fo ‘m if’, or Atlas*GIS ‘bn a ’ file. Th e u n weigh t ed dir ect ion a l m ea n - t h e in t er sect ion
of t h e m ea n a n gle a n d t h e m ea n d is t a n ce is ou t p u t wit h t h e p r efix ‘Dm ’ wh ile t h e
u n weight ed t r ian gula t ed m ea n loca t ion is ou t pu t with a ‘Tm ’ pr efix. The weight ed
t r ian gula t ed m ea n is ou t pu t with a ‘Tm Wt’ pr efix. The dir ect ion a l mea n ca n be sa ved as a n
ArcView ‘sh p’, M apIn fo ‘m if’, or Atlas*GIS ‘bn a ’ file. Th e let t er s ‘Dm ’ a r e pr efixed t o t h e
u ser defined file na m e. See th e exa m ple below.
F igu r e 4.25 sh ows t h e u n weigh t ed t r ia n gu la r m ea n for 1996 Ba lt im or e Cou n t y
r obber ies a n d compa r es it t o t h e t wo dir ect ion a l m ea n s ca lcu la t ed u sin g t h e lower -left
cor n er (Dm ea n 1) a n d t h e u pper -r igh t cor n er (Dm ea n 2) r espectively a s origin s. As can be
seen , t h e t wo dir ect ion a l m ea n s fa ll a t differ en t loca t ion s. Lin es h a ve been dr a wn fr om
ea ch or igin p oint t o t h eir r espective directiona l mea n s a n d a r e exten ded u n t il t h ey
in t er se ct. As s een , t h e t r ia n gu la t ed m ea n falls a t t h e loca t ion wh er e t h e t wo vectors (i.e.,
m ea n a n gles) int er se ct.
Beca u se t h e t r ia n gu la t ed m ea n is ca lcu la t ed wit h vect or geom et r y, it will n ot
n ecessa r ily ca pt u r e t h e cen t r a l t en den cy of a dis t r ibu t ion . Asym m et r ica l d is t r ibu t ion s ca n
cau se it t o be pla ced in per iph er a l locat ion s. On t h e oth er h a n d, if th e dist r ibu t ion is
4.42
Figure 4.25:
Triangulated Mean for Baltimore County Robberies
Defined by the Intersection of Two Mean Angles
# Upper right corner
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"Triangulated mean
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Dmean2
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Miles
Lower left corner
0
5
10
r elat ively ba lan ced in ea ch dir ect ion , it ca n ca pt u r e t h e cen t er of or ient a t ion per h a ps bet t er
th an oth er mea ns, as figur e 4.25 shows.
Appen dix B in clu des a dis cu ssion of h ow t o for m a lly t est s t h e m ea n dir ect ion
between two different distr ibut ions.
Con ve x H u ll
Th e con vex h u ll is a boun da r y dr a wn a r oun d t h e dist r ibu t ion of point s. It is a
r ela t ively sim ple con cept , a t lea st on t h e su r fa ce. In t u it ively, it r epr esen t s a polygon t h a t
cir cu m scr ibes a ll t h e poin t s in t h e dis t r ibu t ion su ch t h a t n o poin t lies out sid e of th e
polygon .
Th e com plexit y com es becau se t h er e a r e differ en t wa ys t o defin e a con vex h u ll. The
m ost ba sic a lgor ith m is t h e Graham scan (Gr a h a m , 1972). Sta r t ing with on e point kn own
t o be on t h e con vex h u ll, typically th e poin t wit h t h e lowest X coord in a t e, t h e a lgor it h m
sor t s t h e r em a in in g poin t s in a n gu la r ord er a r oun d t h is in a cou n t er clockwise m a n n er . If
t h e a n gle for m ed by th e n ext point a n d t h e las t edge is less t h a n 180 degrees , th en t h a t
poin t is ad ded t o t h e h u ll. If t h e a n gle is grea t er t h a n 180 degrees , th en t h e ch a in of n odes
st a r t in g fr om t h e la st edge m u st be delet ed. Th e r ou t in e pr oceeds u n t il t h e h u ll clos es ba ck
on it self (de Ber g, va n Kr eveld , Over m a n s, a n d Sch wa r zk opf, 2000).
Man y alter n a t ive algor ith m s h a ve been pr oposed. Am on g th ese ar e th e ‘gift wra p’
(Ch a n d a n d Ka pu r , 1970; Sk iena , 1997), th e Qu ick H u ll, t h e “Divide a n d con qu er ”
(P r epa r a t a a n d H on g, 1977), a n d t h e in cr em en t a l (Ka lla y, 1984) a lgor it h m s. E ven m or e
com p lexit y h a s been in t r od uced by t h e m a t h em a t ics of fr a ct a ls wh er e a n a lm os t in fin it e
n u m ber of bor der s cou ld be defin ed (La m a n d De Cola , 1993). In m ost im plem en t a t ion s,
t h ough , a sim plified a lgor it h m is u sed t o pr odu ce th e con vex h u ll.
Crim eS tat im p lem en t s a ‘gift wr a p ’ a lgor it h m . S t ar t in g wit h t h e p oin t wit h t h e
lowes t Y coor d in a t e, A, it s ea r ch es for a n ot h er p oin t , B, s u ch t h a t a ll ot h er p oin t s lie t o t h e
left of t h e lin e AB. It t h en fin d s a n ot h er p oin t , C, s u ch t h a t a ll r em a in in g p oin t s lie t o t h e
left of th e lin e BC. It con t in u es in t h is wa y un t il it r ea ches t h e origin a l point A a ga in . It is
like ‘wr a pp in g a gift ’ a r oun d t h e out sid e of t h e poin t s.
The rout ine out put s th ree sta tistics:
1.
2.
3.
Th e sa m ple size;
Th e n u m ber of point s in t h e con vex h u ll
Th e X a n d Y coor din a t es for ea ch of t h e poin t s in t h e con vex h u ll
Th e con vex h u ll ca n be sa ved as a n Ar cView 'sh p', MapIn fo 'mif', or At las *GIS 'bn a '
file wit h a 'Ch u ll' pr efix.
F igu r e 4.26 sh ows t h e con vex h u ll of Ba lt im or e Cou n t y r obber ies for 1996. As seen ,
t h e h u ll occu pies a r elat ively sm a ller pa r t of Balt imore Coun t y. Figu r e 4.27, on t h e ot h er
4.44
Figure 4.26:
Convex Hull of Baltimore County Robberies: 1996
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Figure 4.27:
Convex Hull of Baltimore County Burglaries: 1996
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h a n d, s h ows t h e con vex h u ll of 1996 Ba lt im ore Coun t y bu r gla r ies. As s een , t h e con vex h u ll
of t h e bur gla r ies cover a m u ch lar ger a r ea t h a n for t h e r obber ies.
Us e s a n d Lim ita tio n s o f a Con ve x H u ll
A convex h u ll can be u se ful for dis pla yin g t h e geogr a ph ical ext en t of a dis t r ibu t ion.
Sim ple com pa r is on s, s u ch a s in figu r es 4.26 a n d 4.27, ca n sh ow wh et h er on e dis t r ibu t ion
h a s a gr ea t er ext en t t h a n a n ot h er . F u r t h er , a s we sh a ll see, a con vex h u ll ca n be u sefu l for
describing t h e geogra ph ica l spr ea d of a cr ime h ot spot, essen t ially in dica t ing wh er e t h e
crimes ar e distr ibut ed.
On t h e oth er h a n d, a con vex h u ll is vu ln er a ble t o extr em e va lu es. If on e in ciden t is
isolat ed, th e h u ll will of n ecessit y be la r ge. The m ea n cen t er , too, is in fluen ced by extr em e
valu es bu t n ot t o t h e sa m e exten t sin ce it a vera ges t h e loca t ion of a ll poin t s. The con vex
h u ll, on t h e ot h er h a n d, is defined by th e m ost extr em e poin t s. A com pa r ison of differ en t
cr im e t yp es or t h e sa m e cr im e t yp e for differ en t yea r s u sin g t h e con vex h u ll m a y on ly sh ow
t h e va r ia bilit y of t h e ext r em e va lu es, r a t h er t h a n a n y cen t r a l p r oper t y of t h e dis t r ibu t ion .
Th er efor e, cau t ion m u st be u sed in in t er pr et in g t h e m ea n in g of a h u ll.
S p a t i a l Au t o c o r re l a t io n
Th e con cep t of spatial au tocorrelation is on e of t h e m os t im p or t a n t in s pa t ia l
st a t ist ics. Spa t ia l in d ep en d en ce is an a r r a n gemen t of inciden t loca t ion s su ch t h a t t h er e a r e
n o spa t ia l r ela t ion sh ips bet ween a n y of th e in ciden t s. Th e in t u it ive con cept is t h a t t h e
loca t ion of an in cid en t (e.g., a st r eet r obber y, a bu r gla r y) is u n r ela t ed t o t h e loca t ion of an y
ot h er in cid en t . Th e op pos it e con dit ion - s pa t ia l a u t ocor r ela t ion , is an a r r a n gem en t of
in cid en t loca t ion s wh er e t h e loca t ion of p oin t s a r e r ela t ed t o ea ch ot h er , t h a t is t h ey a r e n ot
st a t ist ically in depen den t of one a n oth er . In oth er words , sp a t ia l a u t ocorr ela t ion is a
spat ial ar ra ngement wh ere spatial independence has been violated.
Wh en even t s or people or fa cilit ies a r e clus t er ed t oget h er , we r efer t o th is
ar ra ngement a s positive spa t ial a u t ocor r elat ion . Con vers ely, an a r r a n gemen t wh er e
people, even t s or fa cilit ies a r e dis per sed is r efer r ed t o a s negative sp a t ia l a u t ocorr ela t ion ; it
is a r a r er a r r a n gem en t , bu t does exis t (Levin e, 1999).
Ma n y, if n ot m ost , social p h en ome n a a r e spa t ia lly a u t ocorr ela t ed. In a n y la r ge
m et r opolit a n a r ea , m ost socia l ch a r a ct er is t ics a n d in dica t or s, s u ch a s t h e n u m ber of
per sons, incom e levels, eth n icity, edu ca t ion , emp loymen t , an d t h e loca t ion of fa cilit ies a r e
n ot s pa t ia lly in depen den t , bu t t en d t o be con cent r a t ed.
Th er e a r e pr a ct ica l con sequ en ces. P olice a n d cr im e a n a lyst s kn ow fr om exp er ien ce
t h a t in cid en t s fr equ en t ly clu st er t oget h er in wh a t a r e ca lled ‘h ot spot s’. Th is n on -r a n dom
a r r a n gem en t a llows p olice t o ta r get cert a in a r ea s or zones w h er e t h er e a r e h igh
con cen t r a t ion s a s well a s pr ior it ize a r ea s by t h e in t en s it y of in cid en t s . Ma n y of t h e
in ciden t s a r e com m it t ed by t h e sa m e in dividu a ls. F or exa m ple, if a pa r t icula r
n eigh borh ood h a d a con cent r a t ion of st r eet r obber ies over a t im e per iod (e.g., a yea r ), ma n y
4.47
of t h ese r obber ies will h a ve been com m it t ed by t h e sa m e per pet r a t or s. St a t is t ica l
depen den ce bet ween event s oft en h a s com m on ca u ses.
St a t is t ica lly, h owever , n on -spa t ia l in depen den ce su ggest s t h a t m a n y s t a t is t ica l
t ools a n d in fer en ces a r e in a ppr opr ia t e. F or exa m ple, t h e u se of cor r ela t ion coefficien t s or
Or din a r y Lea st Squ a r es r egr ession (OLS) t o pr edict a con sequ en ce (e.g., t h e cor r ela t es or
predictors of bur glar ies) assu mes t ha t t he observations h ave been selected r an domly. If
t h e obser vat ion s, however, ar e spa t ially clus t er ed in s om e wa y, t h e est ima t es obta ined
fr om t h e cor r elat ion coefficient or OLS est ima t or will be biased a n d over ly pr ecise. They
will be biased beca u se t h e a r ea s wit h h igh er con cen t r a t ion of event s will h a ve a gr ea t er
imp a ct on t h e m odel est ima t e a n d t h ey will overes t ima t e pr ecision beca u se, since even t s
t en d t o be con cen t r a t ed, t h er e a r e a ct u a lly fewer n u m ber of in depen den t obs er va t ion s t h a n
a r e bein g a ss u m ed. Th is con cept of sp a t ia l a u t ocorr ela t ion u n der lies a lm ost a ll t h e spa t ia l
st a t ist ics t ools t h a t a r e in clud ed in Crim eS tat.
In d i c e s o f S p a t ia l Au t o c o r re l a ti o n
Th er e a r e a n u m ber of for m a l st a t ist ics wh ich a t t em pt t o mea su r e spa t ia l
a u t ocor r ela t ion . Th is in clu de sim ple in dices, s u ch a s t h e Mor a n ’s I” or Gea r y’s C s t a t is t ic;
der iva t ives in dices, s u ch a s Ripley’s K st a t is t ic (Rip ley, 1976) or t h e a pplica t ion of Mor a n ’s
I t o in dividu a l zon es (An selin , 1995); an d m u lt iva r ia t e in dices, s u ch a s t h e u se of a sp a t ia l
a u t ocorr ela t ion pa r a m et er in a biva r ia t e r egr ession m odel (Cliff an d Or d, 1973; Gr iffit h ,
1987) or t h e u se of a spa t ia lly-la gged depen den t va r ia ble in a m u lt ip le va r ia ble r egr ession
m odel (An selin , 1992). Th e sim ple in dices a t t em pt t o iden t ify wh et h er sp a t ia l
a u t ocor r elat ion exist s for a sin gle var iable, while th e m or e com plica t ed ind ices a t t em pt t o
est ima t e t h e effect of spa t ial a u t ocor r elat ion on ot h er var iables.
Crim eS tat in clu des t wo globa l in dices - Mor a n ’s I st a t is t ic a n d Gea r y’s C s t a t is t ic,
a n d a n a pplica t ion of Mor a n ’s I t o differ en t dist a n ce int er vals. Mor a n a n d Gea r y ar e global
in t h a t t h ey r ep r es en t a s um m a r y va lu e for a ll t h e d a t a p oin t s . Th ey a r e a ls o ver y s im ila r
ind ices a n d a r e oft en u sed in con jun ct ion . The Mor a n st a t ist ic is sligh t ly m or e r obu st t h a n
t h e Gea r y, bu t t h e Gea r y is oft en u sed a s well.
Moran ’s I Statistic
Mora n ’s I st a t ist ic (Mor a n , 1950) is one of t h e oldest in dicat ors of sp a t ia l
a u t ocor r elat ion . It is a pplied t o zon es or poin t s wh ich h a ve con t inu ou s var iables a ssociat ed
wit h t h em (int en sit ies ). For a n y con t in u ous va r ia ble, Xi , a m ea n ca n be ca lcu la t ed a n d t h e
d evia t ion of a n y on e obs er va t ion fr om t h a t m ea n ca n a ls o be ca lcu la t ed . Th e s t a t is t ic t h en
com p a r es t h e va lu e of t h e va r ia ble a t a n y on e loca t ion wit h t h e va lu e a t a ll ot h er loca t ion s
(E bdon, 1985; Gr iffith , 1987; An selin, 1992). Form a lly, it is defin ed a s
4.48
_
GG
I
=
_
N
i
j W ij (Xi - X )(Xj - X )
--------------------------------------_
2
( i
j W ij )
i (Xi - X )
GG
(4.31)
G
wh er e N is t h e n u m ber of cas es , Xi is t h e var iable_ va lu e a t a pa r t icula r loca t ion, i, Xj is t h e
va r ia ble va lu e a t a n oth er locat ion (wher e i =/ j), X is t h e m ea n of t h e var iable a n d W ij is a
weigh t a pp lied t o th e com pa r ison bet ween loca t ion i a n d locat ion j.
In Mora n ’s in it ia l for m u la t ion, t h e we igh t va r ia ble, W ij, was a cont iguity ma tr ix. If
zone j is a dja cent t o zon e i, t h e in t er a ction r eceives a weigh t of 1. Ot h er wis e, t h e
in t er a ct ion r eceives a weigh t of 0. Cliff a n d Or d (1973) gen er a lized t h ese defin it ion s t o
in clud e a n y t ype of weigh t . In m ore cur r en t u se , W ij is a dis t a n ce-ba sed weigh t wh ich is
t h e in ver se dis t a n ce bet ween loca t ion s i a n d j (1/d ij). Crim eS tat u ses t h is in t er pr et a t ion .
Essent ially, it is a weighted Mor a n ’s I wh er e t h e weigh t is an inver se dist a n ce.
Th e weigh t ed Mora n ’s I is sim ilar t o a cor r elat ion coefficient in t h a t it com pa r es t h e
su m of t h e cross-pr odu cts of va lu es a t differ en t loca t ions , t wo a t a t im e we igh t ed by t h e
inver se of t h e dist a n ce bet ween t h e loca t ion s, with t h e var ian ce of t h e var iable. Like t h e
cor r ela t ion coefficien t , it t yp ica lly va r ies bet ween -1.0 a n d + 1.0. H owever , t h is is n ot
a bsolu t e a s a n exa m ple la t er in t h e cha pt er will sh ow. When n ea r by poin t s h a ve sim ila r
valu es, th e cr oss-pr odu ct is high . Con vers ely, when n ea r by point s h a ve dissim ilar valu es,
t h e cross -pr odu ct is low. Cons equ en t ly, an “I” va lu e t h a t is h igh in dica t es m ore s pa t ia l
a u t ocorr ela t ion t h a n a n “I” t h a t is low.
H owever, u n like t h e cor r elat ion coefficient , th e t h eor et ica l va lue of t h e index does
n ot equa l 0 for lack of spa t ial depen den ce, but ins t ea d a n u m ber wh ich is nega t ive but very
close t o 0.
E(I)
=
1
- --------N-1
(4.32)
Valu es of “I” a bove th e t h eore t ical m ea n , E(I), indica t e posit ive sp a t ia l
a u t ocorr ela t ion wh ile va lu es of “I” below t h e t h eore t ical m ea n in dica t e n ega t ive sp a t ia l
a u t ocor r ela t ion .
Ad ju s tm e n t fo r S m all D is ta n ce s
Crim eS tat calcu la t es t h e we igh t ed Mora n ’s I for m u la u sin g equ a t ion 4.31.
H owever, t h er e is on e pr oblem wit h t h is for m u la t h a t ca n lead t o u n r eliable res u lts . The
dis t a n ce weigh t s bet ween t wo locat ions , W ij, is defin ed a s t h e r ecip r oca l of t h e dis t a n ce
between th e two points:
4.49
W ij
=
1
---------d ij
(4.33)
Un fort un at ely, as d ij becom es sm a ll, t h en W ij becom es ver y lar ge, a pp r oach in g
in finit y a s t h e dist a n ce bet ween t h e point s a pp r oach es 0. If t h e t wo zon es wer e n ext t o
ea ch ot h er , wh ich wou ld be t r u e for t wo a dja cen t block s for exa m ple, t h en t h e pa ir of
obser va t ion s would h a ve a ver y high weigh t , su fficien t t o dist ort t h e “I” va lu e for t h e en t ir e
sa m ple. Fu r t h er , th er e is a s ca le pr oblem t h a t a lter s t h e valu e of t h e weigh t . If t h e zon es
a r e police pr ecincts , for exam ple, th en t h e m inim u m dist a n ce bet ween pr ecincts will be a
lot la r ger t h a n t h e m in im u m dis t a n ce bet ween a sm a ller t yp e of geogr a ph ica l u n it , s u ch a s
blocks . We n eed t o t a ke in t o a ccou n t t h ese differ en t scales.
Crim eS tat in clud es a n a dju st m en t for sm a ll dis t a n ces s o th a t t h e m a xim u m weigh t
can n ever be gr ea t er t h a n 1.0. Th e a dju st m en t sca les dis t a n ces t o on e m ile, wh ich is a
t yp ica l d is t a n ce u n it in t h e m ea su r em en t of cr im e in cid en t s. Wh en t h e sm a ll dis t a n ce
a djus t m en t is t u r n ed on, th e m inim a l dist a n ce is au t om a t ica lly scaled t o be on e m ile. The
form ula used is
on e m ile
W ij = --------------------on e m ile + d ij
(4.34)
in t h e u n it s a r e specified . For exa m ple , if th e dist a n ce un it s, d ij, ar e ca lcu lat ed a s feet,
t h en
5,280
W ij = --------------------5,280 + d ij
wh er e 5,280 is t h e n u m ber of feet in a m ile. Th is h a s t h e effect of ins u r in g t h a t t h e we igh t
of a p ar t icu la r p a ir of p oin t loca t ion s will n ot h a ve a n u n du e in flu en ce on t h e over a ll
st a t ist ic. Th e t r a dit ion a l m ea su r e of “I” is t h e defau lt con dit ion in Crim eS tat (figu r e 4.28),
bu t t h e u ser ca n t u r n on t h e sm a ll dist a n ce a djus t m en t .
Te s t i n g t h e S i g n i fi c a n c e o f t h e We i g h t e d Mo r a n ’s I
Th e em pir ica l dist r ibut ion ca n be com pa r ed wit h t h e t h eor et ica l dist r ibut ion by
divid in g by a n est im a t e of t h e t h eor et ica l s t a n da r d devia t ion
Z(I)
=
I - E(I)
-----------SE(I)
(4.35)
wh er e “I” is t h e em pir ica l va lu e ca lcu la t ed fr om a sa m ple, E (I) is t h e t h eor et ica l m ea n of a
ra ndom distr ibut ion a nd S E ( I ) is t h e t h eor et ical s t a n da r d d evia t ion of E(I).
4.50
Figure 4.28:
Selecting Spatial Autocorrelation Statistics
Th er e a r e sever a l in t er pr et a t ion s of t h e t h eor et ica l s t a n da r d devia t ion wh ich a ffect
t h e pa r t icu la r st a t is t ic u sed for t h e den om in a t or a s well a s t h e in t er pr et a t ion of th e
s ign ifica n ce of t h e s t a t is t ic (An s elin , 1992). Th e m os t com m on a s su m p t ion is t o a s su m e
t h a t t h e st a n da r dized va r ia ble, Z(I), h a s a sa m plin g d is t r ibu t ion wh ich follows a st a n da r d
n or m a l dist r ibut ion , th a t is with a m ea n of 0 an d a var ian ce of 1. This is ca lled th e
norm ality a s su m p t ion . 6 A s econ d in t er p r et a t ion a s su m es t h a t ea ch obs er ved va lu e cou ld
h a ve occur r ed a t a n y locat ion , t h a t is t h e loca t ion of t h e va lu es a n d t h eir sp a t ia l
a r r a n gemen t is as su m ed t o be un r elat ed. This is called t h e ran d om ization a s su m p t ion a n d
h a s a sligh t ly differ en t form u la for t h e t h eor et ica l s t a n da r d devia t ion of I. 7 Crim eS tat
ou t p u t s t h e Z-va lu es a n d p -va lu es for bot h t h e n or m a lit y a n d r a n d om iza t ion a s su m p t ion s
(figu r e 4.29).
E x a m p l e 3: Te s t i n g Au t o Th e f t s w i t h t h e We i g h t e d Mo r a n ’s I
To illu s t r a t e t h e u s e of Mor a n ’s I wit h p oin t loca t ion s r equ ir es da t a t o h a ve
int en sit y va lues a ssociat ed wit h ea ch poin t . Since m ost cr ime in ciden t s a r e r epr esen t ed a s
a sin gle poin t , th ey do n ot n a t u r a lly h a ve ass ociat ed int en sit ies. It is n ecessa r y, t h er efor e,
t o ad a pt crim e da t a t o fit t h e for m r equ ir ed by Mor a n ’s I . On e wa y t o do th is is a ss ign
crim e in ciden t s t o geogr a ph ical zon es a n d coun t t h e n u m ber of in ciden t s p er zone.
F igu r e 4.30 sh ows 1996 m otor veh icle th eft s in both Ba lt im ore Coun t y a n d
Ba lt im or e Cit y by in divid u a l block s. Wit h a GIS pr ogr a m , 14,853 veh icle t h eft loca t ion s
wer e overla id on t op of a m a p of 13,101 cens u s blocks a n d t h e n u m ber of m otor veh icle
t h eft s wit h in ea ch block wer e cou n t ed a n d t h en a ssign ed t o t h e block a s a var iable (see t h e
‘As sign pr im a r y p oin t s t o s econ da r y p oin t s’ r ou t in e in ch a pt er 5). Th e n u m ber s va r ied
fr om 0 in cid en t s (for 7,675 block s) u p t o 46 in cid en t s (for 1 block ). Th e m a p sh ows t h e plot
of t h e n u m be r of a u t o t h e ft s p er b lock .
Clea r ly, a ggr ega t in g in cid en t loca t ion s t o zon es, s u ch a s block s, elim in a t es som e
in for m a t ion sin ce all in ciden t s wit h in a block a r e a ss igned t o a s in gle locat ion (th e cent r oid
of t h e block ). Th e u s e of Mor a n ’s I, h owever , r equ ir es t h e d a t a t o be in t h is for m a t . U sin g
dat a in th is form , Mora n’s I was calculat ed using th e small dista nce adjust ment becau se
m a n y block s a r e ver y close t ogeth er . Crim eS tat calculat ed “I” as 0.012464 an d th e
t h eore t ical va lu e of “I” a s -0.000076. Th e t est of sign ifican ce us in g th e n orm a lit y
a ssu m pt ion gave a Z-valu e of 125.13, a h igh ly significa n t valu e. Below a r e t h e
calcu la t ions .
Z(I)
I - E(I)
0.012464 - (-0.00076)
= ------------ = ------------------------------SE ( I )
0.000100
=
125.13 (p #.001)
In oth er wor ds , m otor t h eft s a r e h igh ly a n d p osit ively spa t ia lly a u t ocorr ela t ed.
Blocks with m a n y in ciden t s t en d t o be loca t ed close t o blocks wh ich a lso h a ve ma n y
in cid en t s a n d, con ver sely, block s wit h few or n o in cid en t s t en d t o be loca t ed clos e t o blocks
wh ich a lso h a ve few or n o incide n t s.
4.52
Figure 4.29:
Moran's I Statistic Output
H ow does t h is com pa r e wit h oth er dis t r ibu t ion s? F in din g posit ive sp a t ia l
a u t ocorr ela t ion for a u t o th eft s is n ot s u r pr isin g given t h a t t h er e is su ch a h igh
con cen t r a t ion of p op ula t ion (a n d, h en ce, m ot or veh icles ) t owa r ds th e m et r op olit a n cen t er .
F or compa r is on , we r a n Mor a n ’s I for t h e popu la t ion of th e blocks (F igu r e 4.31).8 With
t h ese da t a , Mor a n ’s I for popu la t ion wa s 0.001659 wit h a Z-va lu e of 17.32; t h e t h eor et ica l
“I” is t h e sa m e sin ce t h e sa m e n u m ber of blocks is being us ed for t h e st a t ist ic (n =13,101).
Com pa r in g t h e “I” va lu e for m ot or veh icle t h eft s (0.012464) wit h t h a t of popu la t ion
(0.00166) su ggest s t h a t m ot or vehicle th eft s a r e sligh t ly m or e con cen t r a t ed t h a n wou ld be
exp ect ed on t h e ba sis of t h e popu la t ion dis t r ibu t ion . We ca n set u p a n a ppr oxim a t e t est of
t h is h ypot h esis. Th e join t sa m plin g dist r ibu t ion for t wo var ia bles , su ch a s m otor veh icle
t h eft s a n d popu la t ion , is n ot kn own . H owever , if we a ssu m e t h a t t h e st a n da r d er r or of th e
dis t r ibu t ion follows a sp a t ia lly r a n dom dis t r ibu t ion u n der t h e a ss u m pt ion of nor m a lit y,
th en equat ion 4.35 can be applied:
Z(I)
IM V - IP
0.012464 - 0.001659
= ------------ = ------------------------------- = 108.05 (p #.001)
SE ( I )
0.000100
wh er e I M V is t h e “I” va lu e for m otor veh icle th eft s, I P is th e “I” value for populat ion, and S E ( I )
is t h e st a n da r d d evia t ion of “I” u n der t h e a ss u m pt ion of n orm a lit y. The h igh Z-va lu e
su ggest s t h a t m ot or veh icle t h eft s a r e m u ch m or e clu st er ed t h a n t h e clu st er in g of
popu la t ion . To pu t it a n ot h er wa y, t h ey a r e m or e clu st er ed t h a n wou ld be exp ect ed fr om
t h e popu la t ion dis t r ibu t ion . As m en t ion ed, t h is is a n a pp r oxim a t e t est sin ce th e join t
dist r ibut ion of “I” for t wo emp irica l distr ibut ion s of “I” is not kn own .
Gea ry’s C Statistic
Gea r y’s C st a t ist ic is sim ila r t o Mor a n ’s I (Gea r y, 1954). In t h is ca se , h owever , t h e
intera ction is not t he cross-product of th e deviations from t he mea n, but th e deviations in
int en sit ies of ea ch obser vat ion loca t ion with on e a n ot h er . It is defined a s
GG
C
=
(N -1) [ i j W ij (Xi - Xj)2 ]
-------------------------------------_
2
2( i
j W ij )
i (Xi - X )
GG
(4.36)
G
Th e va lu es of C t ypically var y bet ween 0 a n d 2, a lt h ough 2 is n ot a st r ict u pp er lim it
(Gr iffith , 1987). The t h eor et ica l va lue of C is 1; t h a t is, if valu es of a n y on e zon e a r e
spa t ially un r elat ed t o a n y ot h er zon e, th en t h e expect ed valu e of C wou ld be 1. Va lues less
t h a n 1 (i.e., bet ween 0 a n d 1) t ypically ind icat e posit ive sp a t ia l a u t ocorr ela t ion wh ile
va lu es gr ea t er t h a n 1 in dica t e n ega t ive sp a t ia l a u t ocorr ela t ion . Th u s, t h is in dex is
in ver sely rela t ed t o Mor a n ’s I . It will n ot p r ovide iden t ical in feren ce becau se it
em ph a sizes t h e differ en ces in valu es bet ween pa irs of obser vat ion s com pa r ison s r a t h er
t h a n t h e covar iat ion bet ween t h e pa irs (i.e., pr odu ct of t h e deviat ion s from t h e m ea n ). The
4.54
Figure 4.30:
1996 Baltimore Region Motor Vehicle Thefts
Number of Vehicle Thefts Per Block Group
Auto Thefts
10 or fewer thefts
Baltimore County
11-20 thefts
21-30 thefts
31-40 thefts
41-50 thefts
51 or more thefts
City of Baltimore
unty
Miles
0
2
4
Figure 4.31:
1990 Baltimore Population Density
Number of Persons Per Square Mile by Block
Harford
County
Persons
Per Sq Mi
Less than 1,000
1,000-1,999
2,000-2,999
Baltimore County
3,000-3,999
4,000 or more
City of Baltimore
Howard County
D
Miles
0
2
4
Mor a n coefficien t gives a m or e globa l in dica t or wh er ea s t h e Gea r y coefficien t is m or e
sen sit ive t o differ en ces in sm a ll n eigh bor h oods.
Ad ju s tm e n t fo r S m all D is ta n ce s
Like Mor a n ’s I, t h e weigh t s a r e defin ed a s t h e inver se of t h e dist a n ce bet ween t h e
paired points:
W ij
=
1
---------d ij
(4.33)
r ep ea t
H owever, t h e weigh t s will t en d t o increa se su bst a n t ially as t h e dist a n ce bet ween
poin t s decrea ses. Con sequ en t ly, a sm a ll dist a n ce a djus t m en t is allowed wh ich en su r es
t h a t n o weight is gr ea t er t h a n 1.0. Th e a dju st m en t sca les t h e dist a n ces t o on e m ile
on e m ile
W ij = --------------------on e m ile + d ij
(4.34)
r ep ea t
in t h e u n it s a r e specified. Th is is t h e defau lt con dit ion a lt h ough t h e u ser can calcula t e a ll
weight s a s t h e r ecipr oca l dist a n ce by tu r n ing off t h e sm a ll dist a n ce a djus t m en t .
Te s t i n g t h e S i g n i fi c a n c e o f Ge a r y ’s C
Th e em pir ica l C distr ibut ion ca n be com pa r ed wit h t h e t h eor et ica l dist r ibut ion by
divid in g by a n est im a t e of t h e t h eor et ica l s t a n da r d devia t ion
Z(C)
=
C - E (C)
-----------SE(C)
(4.37)
wh er e C is t h e em pir ica l va lu e ca lcu la t ed fr om a sa m ple, E (C) is t h e t h eor et ica l m ea n of a
ra ndom distr ibut ion a nd S E ( C ) is t h e t h eor et ica l st a n da r d deviat ion of E (C). The u su a l tes t
for C is t o a s su m e t h a t t h e s a m ple Z follows a st a n d a r d n or m a l d is t r ibu t ion wit h m ea n of 0
a n d va r ia n ce of 1 (n orm a lit y a ss u m pt ion). Crim eS tat on ly ca lcu la t es th e n or m a lit y
a ssu m pt ion t h ou gh it is possible t o ca lcu la t e t h e st a n da r d er r or u n der a r a n dom iza t ion
a ssu m pt ion (Ripley, 1981).9 Figur e 4.32 illust ra tes th e out put .
E x a m p l e 4: Te s t i n g Au t o Th e f t s w i t h Ge a r y ’s C
Us ing t h e sa m e da t a on a u t o t h eft s for Balt imore Coun t y an d Ba ltim or e City, th e C
va lu e for a u t o t h eft s wa s 1.0355 wit h a Z-va lu e of 10.68 (p #.001) wh ile t h a t for popu la t ion
wa s 0.924811 wit h a Z-va lu e of 122.61 (p #.001). The C va lu e of m otor veh icle t h efts is
great er th an th e theoretical C of 1 and su ggests negative spa t ial a u t ocor r elat ion , ra t h er
t h a n posit ive spa t ia l a u t ocorr ela t ion. Th a t is, t h e in dex su ggest s t h a t block s w it h a h igh
4.57
Figure 4.32:
Geary's C Statistic Output
Global Moran’s I and Small Distance Adjustment:
Spatial Pattern of Crime in Tokyo
Takahito Shimada
National Research Institute of Police Science
National Police Agency, Chiba, Japan
Crimestat calculates spatial autocorrelation indicators such as Moran’s I and
Geary’s C. These indicators can be used to compare the spatial patterns among
crime types. Moran’s I is calculated based on the spatial weight matrix where the
weight is the inverse of the distance between two points. There is a problem that
could occur for incident locations in that the weight could become very large as the
distance between points become closer. In Crimestat, the small distance adjustment
is available to solve this problem. The adjustment produces a maximum weight of 1
when the distance between points is 0.
The number of reported crimes in Tokyo increased from 1996 to 2000
although the city is generally very safe. For this analysis, 68,400 cases reported in
the eastern parts of Tokyo were aggregated by census tracts (N=350). Then
Crimestat calculated Moran’s I for each crime type with and without the small
distance adjustment.
The “I” value for most crime types, including burglary, theft, purse snatching,
showed significantly positive autocorrelation. The results with and without the
small distance adjustment were generally very close. The Pearson’s correlation
between the original and adjusted Moran’s I is .98. Among 10 crime types, relatively
strong spatial patterns were detected for car theft, sexual assaults, and residential
burglary.
Calculated Moran’s I by Crime Types
Spatial Patterns of
Residential Burglary:
Moran’s I = 0.023. z=7.58
Preliminary Statistical Tests for Hotspots:
Examples from London, England
Spencer Chainey
Jill Dando Institute of Crime Science
University College
London, England
Preliminary statistical tests for clustering and dispersion can provide insight
into what types of patterns will be expected when the crime data is mapped. Global
tests can confirm whether there is statistical evidence of clusters (i.e. hotspots) in
crime data which can be mapped, rather than mapping data as a first step and
struggling to accurately identify hotspots when none actually exist.
Using CrimeStat, four statistical tests were compared for robbery, residential
burglary and vehicle crime data for the London Borough of Croydon, England. For
the incident data, the standard distance deviation and nearest neighbor index were
used. For crime incidents aggregated to Census block areas, Moran’s I and Geary’s
C spatial autocorrelation indices were compared. The crime data is for the period
June 1999 – May 2000.
Crime type
Robbery
Residential
burglary
Vehicle crime
Number
of crime
records
1132
Standard
distance
NN
Index
3119.5 m
3104
9314
Crime type
All crime
Robbery
Residential
burglary
Vehicle crime
Evidence of
Clustering?
0.47
z-score
(test
statistic)
-34.2
3664.6 m
0.46
-57.5
Yes
3706.2 m
0.26
-137.0
Yes
Moran’s I
0.0067
0.0078
0.014
Geary’s C
1.14
1.15
0.99
0.0082
1.08
Yes
With the point statistics, all three crime types show evidence of clustering.
Vehicle crime shows the more dispersed pattern suggesting that whilst hotspots do
exist, they may be more spread out over the Croydon area than that of the other two
crime types. For the two spatial autocorrelation measures, there are differences in
the sensitivities of the two tests. For example, for robbery, there is evidence of
global positive spatial autocorrelation (overall, Census blocks that are close together
have similar values than those that are further apart). On the other hand, the
Geary coefficient suggests that, at a smaller neighbourhood level, areas with a high
number of robberies are surrounded by areas with a low number of robberies.
n u m ber of a u t o t h eft s a r e a dja cen t t o blocks wit h a low n u m ber of a u t o t h eft s or wit h low
popu la t ion den sit y. Th e C va lu e of popu la t ion , on t h e ot h er h a n d, is below t h e t h eor et ica l
C of 1 an d point s t o positive spa t ial a u t ocor r elat ion . Thu s, Gea r y’s C pr ovides a differ en t
inference from Mora n’s I regar ding the spat ial distr ibut ion of th e blocks.
In t h e exa m ple a bove, Mor a n ’s I in dica t ed posit ive spa t ia l a u t ocor r ela t ion for bot h
a u t o th eft s a n d p opu la t ion d en sit y. An in sp ect ion of figu r e 4.30 a bove sh ow however , t h a t
t h er e a r e lit t le ‘pea ks ’ a n d ‘valleys’ a m on g th e blocks . Severa l blocks h a ve a h igh n u m ber
of au to th efts, but a re sur roun ded by blocks with a low num ber of au to th efts.
In oth er wor ds , t h e Mor a n coefficien t h a s in dica t ed t h a t t h er e is m ore posit ive
spa t ial a u t ocor r elat ion for m ot or vehicle th eft s a m on g th e 13,101 blocks wh ile t h e Gea r y
coefficient ha s empha sized the irregular pat tern ing am ong the blocks. The Geary index is
m ore s en sit ive t o loca l clu st er in g (secon d-ord er effects) t h a n t h e Mor a n in dex, wh ich is
bet t er seen a s m ea su r ing firs t -or der spa t ial a u t ocor r elat ion . This illu st r a t es h ow t h ese
ind ices h a ve to be used with ca r e a n d can n ot be gener a lized by th em selves. Ea ch of t h em
em ph a sizes sligh t ly differ en t in for m a t ion r ega r din g sp a t ia l a u t ocorr ela t ion , yet n eit h er is
su fficient by it se lf. They sh ould be u se d a s p a r t of a la r ger a n a lysis of sp a t ia l pa t t er n in g. 1 0
Mo r a n Co r re l o g ra m
Mor a n ’s I a n d Gea r y’s C ind ices a r e su m m a r y tes t s of globa l au t ocor r elat ion . Tha t
is , t h ey s u m m a r ize a ll t h e da t a a n d don ’t dis t in gu is h bet ween spa t ia l a u t ocor r ela t ion for
differ en t su bset s. In su bsequ en t ch a pt er s, we will exa m ine pa r t icu lar su b-set s of t h e da t a
t h a t a r e spa t ia lly a u t ocorr ela t ed, su ch a s ‘h ot s pot s’, ‘cold s pot s’ or s pa ce-t im e clust er s.
On e sim ple a pp lica t ion of Mora n ’s I is a plot of t h e “I” by differ en t dis t a n ce int er va ls
(or bins). Called a Moran Correlogram , th e plot ind ica t es h ow con cen t r a t ed or dist r ibut ed
is t h e spa t ia l a u t ocorr ela t ion (Cliff a n d H a gget t , 1988; Ba iley a n d G a t r ell, 1995).
E ssen t ia lly, a ser ies of concen t r ic cir cles is over la id over t h e poin t s a n d t h e Mor a n ’s I
st a t ist ic is ca lcula t ed for only t h ose poin t s fa lling wit h in t h e circle. Th e r a diu s of t h e circle
ch a n ges fr om a sm a ll cir cle t o a ver y la r ge on e. As t h e cir cle in cr ea ses, t h e “I” ca lcu la t ion
a ppr oa ch es t h e globa l va lue.
I n Crim eS tat, th e u ser ca n specify how ma n y dista n ce int er vals (i.e., circles) a r e t o
be ca lcu lat ed. The defau lt is 10, but t h e u ser ca n ch oose a n y ot h er int eger va lue. The
r ou t in e t a k es t h e m a xim u m dis t a n ce bet ween poin t s a n d divid es it in t o t h e n u m ber of
sp ecified dis t a n ce int er va ls, a n d t h en calcula t es t h e “I” va lu e for t h ose poin t s fa lling wit h in
th at ra dius.
Ad ju s tm e n t fo r S m all D is ta n ce s
If checked, sma ll dista nces are adjusted so th at th e maximu m weight ing is 1 (see p.
49 a bove). Th is en su r es t h a t t h e “I” va lu es for in divid u a l d is t a n ces won 't becom e
excess ively lar ge or excessively s m a ll for p oin t s t h a t a r e close t oget h er . Th e defau lt va lu e is
n o a dju st m en t .
4.61
Si m u lat io n of Con fide n ce Int e rva ls
A Mon t e Ca r lo s im u la t ion ca n be r u n t o es t im a t e a p pr oxim a t e con fid en ce in t er va ls
a r ou n d t h e "I" valu e. Ea ch sim u lat ion inp u t s r a n dom da t a a n d calcu lat es t h e “I” valu e.
Th e dis t r ibu t ion of t h e r a n dom “I” va lu es pr odu ce a n a ppr oxim a t e con fid en ce in t er va l for
t h e a ct u a l (em pir ica l) “I”. To r u n t h e sim u lat ion , specify th e n u m ber of sim u lat ion s t o be
ru n (e.g., 100, 1000, 10000). The defau lt is no simu lations.
Ou tp u t
The out put includes:
1.
2.
3.
4.
5.
Th e
Th e
Th e
Th e
Th e
sa m ple size
m a xim u m d is t a n ce
bin (in t er val) n u m ber
m idp oin t of t h e dist a n ce bin
"I" va lu e for t h e dis t a n ce bin (I[B])
a n d if a s im u la t ion is r u n :
6.
7.
8.
9.
10.
11.
Th e
Th e
Th e
Th e
Th e
Th e
m in im u m "I" valu e for t h e dist a n ce bin
m a xim u m "I" valu e for t h e dist a n ce bin
0.5 p er cent ile for t h e dist a n ce bin
2.5 p er cent ile for t h e dist a n ce bin
97.5 p er cent ile for t h e dist a n ce bin
99.5 p er cent ile for t h e dist a n ce bin.
Th e t wo pa irs of per cen t iles (2.5 an d 97.5; 0.5 an d 99.5) cr ea t e a n a ppr oxim a t e 5%
a n d 1% con fiden ce int er val. The m inim u m a n d m a xim u m "I" valu es crea t e a n en velope.
Th e t a bu la r r esu lt s ca n be pr in t ed, s a ved t o a t ext file or sa ved a s a '.dbf' file. F or t h e
lat t er , specify a file na m e in t h e "Sa ve resu lt t o" in t h e dia logue box. The dbf file ca n be
im p or t ed in t o a s pr ea d sh eet or gr a p h ics pr ogr a m t o m a k e a gr a p h .
Gr a p h i n g th e "I: v a lu e s b y D i s t a n c e
A quick gra ph is pr odu ced t h a t sh ows t h e "I" valu e on t h e Y-a xis by th e dist a n ce bin
on t h e X-a xis . Click on t h e "Gr a p h " bu t t on . Th e gr a p h dis pla ys t h e r ed u ct ion in s pa t ia l
a u t ocorr ela t ion wit h dis t a n ce. The gr a ph is u seful for selectin g t h e t ype of ker n el in t h e
Sin gle- a n d Du el-ker n el in t er pola t ion r ou t ines wh en t h e pr ima r y va r iable is weight ed (see
In t er pola t ion).
E x a m p l e : Mo r a n Co r re l o g ra m o f 20 00 B a lt i m o re P o p u l a ti o n
I’ll illu st r a t e t h e Mor a n cor r elogra m wit h t h e 2000 Ba lt im ore r egiona l popu la t ion.
Un lik e figu r e 4.31 a bove, d a t a by Tr a ffic An a lysis Zon es (TAZ) we r e u sed. Th ese a r e zon es
u sed t yp ica lly for t r a vel d em a n d m odelin g (s ee ch a pt er 12). Th e r ea son for u sin g TAZ’s,
4.62
h owever, is t h a t da t a on bot h em ploymen t a n d popu lat ion a r e a vailable a n d it ’s possible to
com pa r e t h em . Th e TAZ da t a wer e obt a in ed from t h e Ba lt im ore Met r opolita n Coun cil, th e
Met r op olit a n P la n n in g Or ga n iza t ion for t h e Ba lt im or e Met r op olit a n a r ea .
F igur e 4.33 s h ows a m a p of t h e 2000 Ba lt im ore p opu la t ion by TAZ’s. Th er e is a
h igher con cent r a t ion of popu la t ion in t h e Cit y of Balt im ore, t h ough som e of t h e out lying
TAZ’s a lso h a ve a la r ge popu la t ion (pr im a r ily beca u se t h ey a r e la r ge in a r ea ).
Never t h eless, t h e dis t r ibu t ion of popu la t ion by TAZ’s fa lls off a t a r ela t ively slow r a t e fr om
t h e cen t er .
F igu r e 4.34 sh ows t h e Mor a n cor r elogr a m for t h e 2000 TAZ popu la t ion a n d
com p a r es it t o t h e m a xim u m a n d m in im u m va lu es fr om a Mon t e Ca r lo s im u la t ion of 100
r u n s. As seen , t h e “I” va lu e a t sh or t dis t a n ces of less t h a n a m ile is qu it e h igh , 0.78. As
t h e dist a n ce bet ween zon es in crea se (i.e., th e sea r ch circle ra diu s get s la r ger ), t h e “I” va lu e
dr ops off un t il about 8 m iles w h er eu pon it a pp r oach es t h e global “I” va lu e. H owever , for a ll
dis t a n ce int er va ls, t h e em pir ical “I” va lu e is h igher t h a n t h e m a xim u m sim u la t ed “I” va lu e
wit h r a n dom da t a . In ot h er wor ds, it is h igh ly u n lik ely t h a t t h e “I” va lu es obt a in ed for
ea ch of t h e dist a n ce in t er va ls wa s du e t o ch a n ce ba sed on t h e dist r ibu t ion of r a n dom “I”
values.
Now, let ’s look a t t h e dist r ibut ion of em ploymen t (figu r e 4.35). In t h is ca se,
employment is mu ch m ore concentr at ed in a ha ndful of TAZ’s. In most metr opolita n a reas,
employment is usu ally more concentr at ed tha n populat ion. A nu mber of TAZ’s in
down t own Ba lt im ore h a ve a h igh con cent r a t ion of em ploymen t a s d oes a cor r idor lea din g
n or t h wa r d a lon g Cha r les St r eet . In Ba ltim or e Cou n t y, t h er e a r e st r et ch es of h igh er
em ploym en t bu t , a ga in , t h ey t en d t o be lim it ed t o a h a n dfu l of TAZ’s. In ot h er wor ds,
com p ar ed to t h e d is t ribu t ion of p op ula t ion , t h e d is t ribu t ion of em p loym en t is m or e
clustered.
F igu r e 4.36 com pa r es t h e Mor a n cor r elogr a m of em ploym en t wit h t h a t of
popula t ion . As seen , emp loymen t h a s a very h igh “I” valu e for sh or t dist a n ces, m u ch h igh er
t h a n for popula t ion . As m en t ion ed a bove, t h e Mor a n I t ypica lly fa lls between -1.00 an d
+1.00, bu t t h is is n ot gu a r a n t eed. If t h e differ en ces in va lu es bet ween zon es is m u ch
gr ea t er t h a n t h e a ver a ge dist a n ce wit h in zones, t h en t h e “I” va lu e can exceed 1.0. In t h e
cas e of figu r e 4.36, it a pp r oach es 3.0. Never t h eles s, a s t h e dist a n ce incr ea ses, t h e “I” va lu e
dr ops qu ick ly a n d becom es lower t h a n popu la t ion for la r ger dis t a n ce sepa r a t ion s.
U s e s a n d L im i t a t io n s o f t h e Mo r a n Co r re l o g ra m
In ot h er wor ds, t h e Mor a n cor r elogr a m pr ovides in for m a t ion a bou t t h e sca le of
spa t ia l a u t ocor r ela t ion , wh et h er it is diffu se over a la r ger a r ea (e.g., a s wit h t h e popu la t ion
exam ple) or is m or e con cen t r a t ed (e.g., as wit h t h e em ploymen t exam ple). This ca n be
u sefu l for ga u gin g t h e ext en t t o wh ich ‘h ot spot s’ a r e t r u ly is ola t ed con cen t r a t ion s of
in ciden t s or wh et h er t h ey a r e by-pr odu cts of sp a t ia l clus t er in g over a la r ger a r ea . In
ch a p t er 6, we will exa m in e a h ier a r ch ica l clu s t er in g a lgor it h m t h a t exa m in es a hier a r ch y
4.63
Figure 4.33:
Baltimore Region Population: 2000
By Traffic Analysis Zones
Baltimore County
Beltway
Baltimore County
City of Baltimore
Population
Less than 1000
1000 - 1999
2000 - 2999
3000 - 3999
4000 - 20228
City of Baltimore
N
W
E
S
0
10
20 Miles
Figure 4 .3 4 :
Moran Correlogram of Baltimore Population: 2000
0.80
" I" v a lu e
0.60
0.40
Population
Minimum random "I"
0.20
Maximum random "I"
0.00
0
4
7
11
14
18
22
25
-0.20
-0.40
Distance interval
29
32
Figure 4.35:
Baltimore Region Employment: 2000
By Traffic Analysis Zones
Baltimore County
Beltway
Baltimore County
City of Baltimore
Employment
Less than 1000
1000 - 1999
2000 - 2999
3000 - 3999
4000 - 20228
City of Baltimore
N
W
E
S
0
10
20 Miles
Figure 4 .3 6 :
Moran Correlogram of Baltimore Employment & Population: 2000
2.80
" I" v a lu e
2.40
2.00
1.60
Population
Employment
1.20
0.80
0.40
0.00
0
4
7
11
14
18
22
Distance interval
25
29
32
of clu s t er s (e.g., fir s t -or d er clu s t er s wh ich a r e wit h in la r ger s econ d -or d er clu s t er s wh ich , in
t u r n , a r e wit h in even la r ger t h ir d-or der clu st er s). Th e Mor a n cor r elogr a m pr ovides a qu ick
sn a psh ot of t h e exten t of spa t ial a u t ocor r elat ion a s a fu n ct ion of scale.
An ot h er u se for t h e Mor a n cor r elogr a m is to es t im a t e t h e t yp e of k er n el fu n ct ion
t h a t will be u sed for in t er pola t ion . In cha pt er 8, t h is m et h odology will be expla in ed in
det a il. But , th e key decision is t o select a m a t h em a t ica l fu n ct ion t h a t will int er pola t e da t a
from point locat ions t o grid cells. The shape of th e Mora n corr elogra m a nd t he spread is a
good indicat or of th e type of ma th emat ical fun ction t o use.
On t h e oth er h a n d, lik e a ll global sp a t ia l a u t ocorr ela t ion s t a t ist ics, t h e cor r elogra m
will n ot in dica t e wh er e t h er e is clu st er in g or dis per sion , on ly th a t it exist s. F or t h a t , we’ll
h a ve t o exa m in e t ools t h a t a r e m or e focu sed on t h e loca t ion of concen t r a t ion s of even t s (or
t h e opposit e, t h e loca t ion of a la ck of even t s).
To explor e t h is fu r t h er , we will n ext exa m in e pr oper t ies of dis t a n ces be t ween poin t s.
Ch a pt er 5 will exa m in e t ools for m ea su r in g secon d -ord er effect s u sin g t h e p r op er t ies of t h e
dista nces between incident locat ions.
4.68
En dn ot e s for Ch ap te r 4
1.
H in t . Th er e a r e 40 ba r s in dica t ed in t h e st a t u s ba r wh ile a r ou t in e is r u n n in g. F or
lon g r u n s, u ser s ca n est im a t e t h e ca lcu la t ion t im e by t im in g h ow lon g it t a k es for
t wo ba r s t o be displa yed an d t h en m u ltiply by 20.
2.
Crim eS tat’s im plem en t a t ion of t h e Ku h n a n d Ku en n e a lgor it h m is a s follows (fr om
Bu r t a n d Ba r ber , 1996, 112-113):
A.
Le t t be t h e n u m ber of th e it er a t ion . F or t h e fir st it er a t ion only (i.e ., t =1) t h e
weigh t ed m ea n cen t er is t a k en a s t h e in it ia l est im a t e of t h e m edia n loca t ion ,
Xt an d Yt .
B.
Ca lcu la t e t h e d is t a n ce fr om ea ch p oin t , i, t o t h e cu r r en t es t im a t e of t h e
m edia n loca t ion, d ict , wh er e i is a sin gle point a n d ct is t h e cur r en t est im a t e
of t h e m ed ia n loca t ion du r in g it er a t ion t .
C.
a.
If th e coordina tes ar e spherical, th en Great Circle dista nces are used.
b.
If th e coordina tes ar e projected, then E uclidean dista nces are used.
Weigh t ea ch ca se by a weigh t , W i , an d ca lcula t e
K it
=
W i e -d (ict )
wh er e e is th e base of th e nat ur al logar ithm (2.7183..) an d d (ict ) is a n
a lter n a t ive wa y to writ e d ict .
a.
If n o weigh t s a r e defin ed in t h e pr im a r y file, Wi is as su m ed t o be 1.
b.
If weigh t s a r e defin ed in t h e pr im a r y file, Wi ta kes th eir values.
Note t h a t a s t h e dist a n ce, d ict , a p pr oa ch es 0, t h en e -d (ict ) becom es 1.
D.
Ca lcu la t e a n ew es t im a t e of t h e cen t er of m in im u m dis t an ce fr om
Xt+ 1
=
G K it Xi
----------------- for i=1...n
G K it
Yt+ 1
=
G K it Yi
----------------- for i=1...n
G K it
4.69
wh er e Xi an d Yi a r e t h e coor din a t es of poin t i (eit h er la t /lon for sph er ica l or
fee t or m et er s for pr oject ed).
E.
Ch eck t o s ee h ow m u ch ch a n ge h a s occu r r ed sin ce t h e la st it er a t ion
ABS| Xt+1 - Xt | # 0.000001
ABS| Yt+1 - Yt | # 0.000001
3.
a.
I f eit h er t h e X or Y coor d in a t es h a ve ch a n ged by gr ea t er t h a n
0.000001 bet ween iter a t ion s, su bst itu t e Xt+1 for Xt an d Yt+1 for Yt an d
repeat steps B t h r ou gh D.
b.
If both t h e cha n ge in X a n d t h e cha n ge in Y is les s t h a n or equ a l t o
0.000001, th en th e estima ted Xt an d Yt coor d in a t es a r e t a k en a s t h e
cen t er of m edia n dist a n ce.
Wit h a weigh t for a n obser va t ion, w i , th e squa red distan ce is weight ed and t he
for m u la becom es
G w (d
)2
SQRT ---------------i
S XY
=
(
iM C
G w ) -2
i
Both su m m a t ions a r e over a ll poin t s, N .
4.
F orm u la s for t h e n ew a xes p r ovided by E bdon (1988) a n d Cr omley (1992) yield
st a n da r d d evia t ion a l ellips es t h a t a r e t oo sm a ll, for t wo differ en t r ea son s. F ir st ,
t h ey pr odu ce tr a n sform ed a xes t h a t a r e t oo sm a ll. If th e dist r ibu t ion of point s is
r a n dom a n d even in a ll dir ect ion s, ideally th e st a n da r d deviat ion a l ellipse sh ou ld be
equ a l to th e st a n da r d dist a n ce deviat ion , since S x = S y . The for m u la u sed h er e h a s
t h is pr oper t y. Sin ce t h e form u la for t h e st a n da r d dis t a n ce d evia t ion is (4.6):
_
_
G(Xi - X )2 + G(Yi - Y )2
SDD = S QRT[ ------------------------------ ]
N-2
_
_
If S x = S y , t h en G(Xi - X )2 = G(Yi - Y )2 , t h er efor e
_
G(Xi - X )2
SDD = S QRT[2* -------------------- ]
N-2
Sim ila r ly, t h e form u la for t h e t r a n sfor m ed a xes a r e (4.9, 4.10):
4.70
_
_
- G(Yi - Y ) Sin 2 }2
SQRT[ 2*-------------------------------------------------- ]
N-2
_
_
G{ (Xi - X ) Sin 2 - G(Yi - Y ) Cos 2 }2
SQRT[ 2*-------------------------------------------------- ]
N-2
G{ (Xi - X ) Cos 2
Sx =
Sy =
H owever , if S x = S y , t h en 2 = 0, Cos0 = 1, Sin0 = 0 an d, th er efor e,
_
G(Xi - X )
Sx = Sy =
SQRT[ 2* ---------------]
N-2
wh ich is t h e sa m e a s for t h e st a n da r d dist a n ce deviat ion (SDD) un der t h e sa m e
con dit ion s. The for m u las u sed by E bdon (1988) a n d Cr om ley (1992) pr odu ce a xes
wh ich a r e SQ RT(2) tim es t oo sm a ll.
Th e secon d pr oblem wit h t h e E bdon a n d Cr om ley for m u la s is t h a t t h ey d o n ot
cor r ect for d egr ees of fr eed om a n d , h en ce, p r od u ce t oo s m a ll a s t a nd a r d d evia t ion a l
ellip se . Sin ce th er e a r e t wo cons t a n t s in ea ch equ a t ion, Mea n X a n d Mea n Y, t h en
t h er e a r e on ly N-2 degrees of fr eedom. The cum u lat ive effect of u sin g tr a n sfor m ed
a xes t h a t a r e t oo sm a ll a n d n ot cor r ect in g for degr ees of fr eedom yield s a m u ch
sm a ller ellipse t h a n t h a t u sed h er e.
5.
I n M apIn fo, th e comm an d is T a ble Im p or t <M apin fo interch an ge file>. With
Atlas*GIS , th e comm an d is File Op en <bou n d ary (*.bna) file>. Wit h t h e DOS
ver sion of Atlas*GIS , t h e At las Im port-E xport p r ogr a m h a s t o be u s ed t o con ver t t h e
‘.bn a ’ ou t pu t file t o a n Atlas*GIS ‘.agf’ file.
6.
Th e t h eor et ica l s t a n da r d devia t ion of “I” u n der t h e a ssu m pt ion of n or m a lit y is (fr om
E bdon , 1985):
SE(I)
7.
=
N 2 Gij w ij2 + 3(Gij w ij)2 - N Gi (Gj w ij)2
SQRT[ -------------------------------------------------]
(N 2 -1) ( Gij w ij)2
Th e for m u la for t h e t h eor et ica l s t a n da r d devia t ion of “I” u n der t h e r a n dom iza t ion
a ssu m pt ion is (fr om E bdon , 1985):
N{(N 2 +3-3N)Gij
w ij2 +3(Gij w ij)2 -N Gi (Gj w ij)2 }-k((N 2 -N)Gij w ij2 +6(Gij w ij)2 -2N( Gi (Gj
2
w ij) )
S E ( I ) = S QRT[-----------------------------------------------------------------------------------------------------------]
(N-1)(N -2)(N -3)( Gij w ij)2
4.71
8.
We cou ld h a ve com p a r ed Mor a n ’s I for a u t o t h eft s wit h t h a t of p op u la t ion , r a t h er
t h a n popula t ion den sit y. H owever, sin ce t h e a r ea s of blocks t en d t o get la r ger t h e
far t h er t h e dist a n ce fr om t h e m et r opolit a n cent er , t h e effect of t est in g on ly
popu la t ion is pa r t ly bein g m in im ized by t h e cha n gin g sizes of th e blocks .
Con sequ en t ly, p opu la t ion den sit y wa s u sed t o pr ovide a m or e a ccu r a t e m ea su r e of
popu la t ion con cent r a t ion . In a n y ca se, Mora n ’s I for popu la t ion is a lso highly
sign ifica n t : I = 0.00166 (Z=17.32).
9.
Th e t h eor et ica l s t a n da r d devia t ion for C u n der t h e n or m a lit y a ssu m pt ion is (fr om
Ripley, 1981):
SE(I)
10.
=
(2 Gij w ij2 + Gi (Gj w ij)2 )(N-1) - 4(Gij w ij)2
SQRT[ -------------------------------------------------]
2(N +1) (Gij w ij)2
An selin (1992) p oin t s ou t t h a t t h e r es ult s of t h e t wo in dices ar e d et er m in ed to a
lar ge ext en t by th e t ype of weigh t ing u sed. In t h e or igina l for m u lat ion , wher e
a dja cent weigh t s of 1 an d 0 a r e u se d, t h e t wo indices a r e lin ea r ly r ela t ed, t h ough
m ovin g in opposite d irection s (Griffith , 1987). Thu s, on ly a djacent zon es h a ve an y
imp a ct on t h e index. Wit h inver se dist a n ce weigh t s, however, zon es far t h er
r em oved ca n influ en ce t h e overa ll ind ex so it is possible to ha ve a sit u a t ion wh er eby
a djacent zon es h a ve similar valu es (hen ce, ar e positively au t ocor r elat ed) wher ea s
zon es far t h er a wa y could h a ve dis sim ila r va lu es (hen ce, ar e n ega t ively
au tocorr elat ed).
4.72