Data: We use the burglary data (FBI code 05) for year 2014. There are 14306 ee!ts" each #$th t$%e a!d locat$o! ( x i , y i ) . Model : λ ( x , y , t )=μ( x , y )+
t i
∑ λ ( x − x , y − y ) λ (t −t ) r
i
i
t
i
i
#$th the stat$o!!ary bac&grou!d rate de!s$ty μ ( x , y )=
1 2
2 π L T
∑
−√ ( x − xi )2 +( y − y i )2 / L
ai e
#here T $s
i
the total durat$o! of the dataset (here 365 days). The t#o &er!els λ t a!d λ r " as #ell as the bac&grou!d #e$ghts ai " are to be $!erted. The s%ooth$!g le!gth L $s also to be o't$%$ed. We here follo# the a''roach of arsa! a!d *e!gl$!+ (200,) a!d use a s$%'le h$stogra% d$str$but$o! for the t#o &er!els- λ t ( t )=b k for T k ≤t < T k +1 " a!d λ r ( r )= c k for Rk ≤ r < Rk + 1 .We use the follo#$!g d$scret$at$o! $! t$%e a!d d$sta!ce T ={ 0 ; 0.1 ; 0.2 ; 0.5 ; 1 ; 2 ; 3 ; 4 ; 5 ; ; 10 ; 15 ; 20 ; 30 ; 50 ; 100 } days" a!d R= { 0 ; 0.1 ; 0.2 ; 0.3 ; 0.4 ; 0.5 ; 0. ; 1 ; 1.5 ; 2 ; 3 ; 5 ; 10 ; 20 } &%. Expectation-Maximization algorithm : /!o#$!g L " the 'ara%eters { a i , b k , c k } are $!erted by 'ectat$o!a$%$at$o 'ectat$o!a$%$at$o!. !. The $!flue!ce of ee!t i o! ee!t j $s λ ij =λ r ( x j − xi , y j− yi ) λt ( t j− t i ) " a!d the su% of all the λ ij . The bac&grou!d rate de!s$ty for ee!t j $s $!flue!ces of 'ast ee!ts o! j $s λ j=
∑< i j
μ j=μ ( x j , y j)=∑ μ ij #$th μij = ai e
−√ ( x − x i )2+( y − y i )2/ L
/ 2 π L2 T (!ote that the su%%at$o! $s !o# o!
i
all ee!ts i " thus $!clud$!g ee!ts i > j " a!d ee! j $tself). We We def$!e the 'robab$l$t$es
μ λ ωij = μ +ijλ that j $s causally tr$ggered by i " a!d ω0" ij= μ +ijλ that j $s a bac&grou!d j j j j
ee!t l$!&ed to the bac&grou!d !ode i . These 'robab$l$t$es are !or%al$ed by ω ij + ω0" ij =1 .
∑ < i j
∑ i
The algor$th% $terates the follo#$!g ste's • Expectation - the 'robab$l$t$es ωij a!d ω0"ij 0" ij are co%'uted fro% the est$%ated &er!els. I!$t$ally" I!$t$ally" the 'robab$l$t$es are all ta&e! eual to 1 a!d !or%al$ed accord$!g to the ω ij + ω0"ij !or%al$at$o! 0" ij = 1 .
∑ <
∑
i j
i
• Maximization - &!o#$!g these 'robab$l$t$es" the logl$&el$hood $s the! T − t i
)+ ∑ ω ∑ a −∑ ∫ dt λ (t )+
( a , b , c )=− f (
i
i
∑ω
that ai =
j
t
i
0"ij 0" ij
0
0"ij 0" ij
i, j
l! μ ij +
∑> ω
ij
l! λij . a$%$$!g f g$es
i, j i
Ωk " b k = #here Δi , k =T k + 1 −T k $f T −t i ≥T k +1 " ∑ Δi , k i
Δi , k =T −t i−T k $f T k ≤T −t i< T k +1 " a!d Δi , k =0 $f T −t i < T k " Ω ' k Ωk = ωij " a!d c k = #$th Ω ' k = ∑ ∑ ωij a!d S k ∑ Ω ' i i , j / T ≤t −t < T ,T −t < T i, j / R ≤ r < R k
j
i
k + 1
i
k +1
k
ij
k + 1
i
2
2
S k = π ( R k +1− R k ) . o!erge!ce o!erge!ce $s tested by reu$r$!g that all !o!ero alues bk a!d c k are cha!ged by less tha! l! b k 5 $! logar$th%" e.g." −1 <0.05 " #here bk $s the alue u'dated dur$!g the l! b k '
∣
∣
a$%$at$o! ste'" a!d bk ' $s the alue 'r$or to th$s ste'. The s%ooth$!g le!gth ca! also be o't$%$ed dur$!g the %a$%$at$o! ste'" #$th ω0" ij √ ( x i− x j )2+( y i− y j)2 ij L= . o#eer" do$!g so lead to the tr$$al solut$o! L=0 a!d
∑
2
∑ω
ij
ij
bk =0 " $%'ly$!g that ωij =0 " ω0"ij 0" ij = 0 $f i ≠ j " a!d ω0" ii =1 " #h$ch has !o 'red$ct$e →∞ #he! L → 0 ). We therefore test t#o d$st$!ct alue. Th$s solut$o! $s a global %a$%u% ( f →∞ a''roaches - (model ( model type 1) 1 ) #e %od$fy the %odel by $%'os$!g ω0" ij = 0 $f i a!d j are colocated" a!d $!ert L " s$%$larly to ohler (2014)7 (model ( model type 2) 2 ) #e &ee' L f$ed to a! a 'r$or$ alue" a!d &ee' the best L after co%'ar$!g the %odels #$th a crossal$dat$o! %ethod. The 1st a''roach g$es a best L=0.024 &%. For the 2!d" #e use the burglary data fro% the ,1 f$rst days of 2015 (18182015 to 228382015) a!d co%'ute the logl$&el$hood o! th$s t$%e 'er$od for the $!te!s$ty λ ( x , y , t ) 'red$cted by the 2014 data alo!e. We We also crossal$date the %odels of ty'e 1 to co%'are the t#o a''roaches. We f$!d the best L alue to be 0.1 &% for the %odels of ty'e 2" the crossal$dat$o! g$$!g a better f$t to the 2015 data tha! the %odels of ty'e 1" cf F$gure 1. rossal$dat$o! #$th L=0.024 &% $! the case of the 1st a''roach g$es a 'oor f$t. We sho# $! F$gure 2 the t#o $!teract$o! &er!els λ t a!d λ r " for the ty'e 2 %odel #$th L=0.1 &%. I!teract$o! $s 'ract$cally !egl$g$ble" a'art for !earre'eats #$th$! less tha! a day fro% each other" a!d accou!t$!g for o!ly 1., of all ee!ts. We co%'uted a seco!d set of crossal$dat$o!s" th$s t$%e by also $!clud$!g the 2015 data $! the λ r ( x − x i , y − y i) λ t ( t −t i ) are tr$gger$!g 'art - the %odel 'ara%eters $! λ ( x , y , t )=μ( x , y )+
∑ i
u!cha!ged ($! 'art$cular the bac&grou!d ratede!s$ty μ ( x , y ) $s thus est$%ated fro% the 2014 λr ( x − xi , y − y i ) λt ( t −t i ) data o!ly)" but the tr$gger$!g ter%
∑ i
$s !o# co%'uted by su%%$!g oer both 2014 a!d 2015 data. 9e%ar&ably" the logl$&el$hood $s syste%at$cally fou!d to be lo#er #$th th$s a''roach" see Table Table 1. Th$s $s cou!ter$!tu$t$e" as us$!g %ore rece!t data to u'date the tr$gger$!g ter% $s $ s e'ected to $%'roe the 'red$ct$o!. : closer loo& at the t$%e ser$es (F$gure 3) sho#s that there #ere s$g!$f$ca!tly less ee!ts $! the f$rst ,1 days of 2015 as 'red$cted fro% the 2014 data. ;$!ce $!clud$!g the !e# 2015 ee!ts $! the calculat$o! of the tr$gger$!g ter% result $! a larger 'red$cted !u%ber" do$!g so o!ly stre!gthe! the oerest$%at$o!. oerest$%at$o!.
The oerest$%at$o! of the !u%ber of ee!ts $! 2015 h$ghl$ghts the fact f act that" 'ract$cally s'ea&$!g" o!e #ould l$&e to 'red$ct
∫∫
∑ i
tr$gger$!g ter% of λ ( x , y , t ) $s co%'uted by su%%$!g oer all 'reced$!g ee!ts ($!clud$!g those of 2015). We We sho# $! F$gure 4 that ty'e 2 %odels 'erfor% better tha! ty'e 1" but %ore $%'orta!tly that a s$%'le (e'o!e!t$al) s%ooth$!g of all the 're$ous ee!ts does actually better $! 'red$ct$!g the locat$o! of the !et ee!t" although the $%'roe%e!t $s o!ly %arg$!al. Th$s $s 'art$cularly sur'r$s$!g" s$!ce accou!t$!g for %e%ory $! the syste% should a 'r$or$ $%'roe the 'red$ct$o! co%'ared to a %e%oryless 'red$ct$o! as do!e #$th a s$%'le s%ooth$!g. Th$s $s here due to a cha!ge $! the s'at$al 'ro'ert$es of the burglary ee!ts $! 2015 (co%'ared to 2014)" #h$ch are fou!d
to be %ore d$sta!t of each other - the %ea! d$sta!ce bet#ee! a!y t#o burglar$es #as 13.5, 13.5, &% $! 2014" a!d 14.05 &% $! 2015. For both years" co!secut$e ee!ts te!d be less d$sta!t tha! aerage" but there st$ll e$st a s$g!$f$ca!t d$ffere!ce bet#ee! the t#o t$%e 'er$ods" cf F$gure 5. 'lo$t$!g the te%'oral cluster$!g as do!e #$th our %odels #$ll lead to 'red$cted ee!ts to close to the $%%ed$ately 'reced$!g ('ast) ee!t" #h$le the s$%'le s%ooth$!g #$ll 'red$ct a d$sta!ce sl$ghlty larger" larger" he!ce a better 'red$ct$o!. These results cast stro!g doubts o! the ca'ac$ty of the %odels 'ro'osed here to out'erfor% s$%'le hots'ot %a's obta$!ed by s%ooth$!g" for the dataset a!alyed. The tr$gger$!g co!tr$but$o! to the occurre!ce of future ee!ts $s s%all ($t accou!ts o!ly for 1. for the best %odel). :ccou!t$!g :ccou!t$!g for %e%ory $! the syste% therefore ca! o!ly 'ro$de a ery %odest co!tr$but$o! to the effect$e!ess of the 'red$ct$o! sche%e. ore $%'orta!tly" $%'orta!tly" $t $s assu%ed that the dy!a%$cs of the 'rocess stays the sa%e oer t$%e. =oss$ble !o!stat$o!ar$ty of the 'rocess $s thus clearly a! $ssue" as $t #$ll 'ree!t the use of 'ast $!for%at$o! to 'red$ct the future. f uture. Th$s $s for ea%'le e'er$e!ced $! th$s a!alys$s" as 2015 burglary ee!ts are clearly !ot d$str$buted ($! t$%e a!d $! s'ace) as they #ere $! 2014. Th$s !o! stat$o!ar$ty $s l$&ely due to u!co!troled eolut$o!s $! the #ay these acts are 'erfor%ed" but" $! s$tuat$o!s #ere !e# 'red$ct$o! algor$th%s are set u' a!d e'lo$ted by 'ol$ce 'atrols" could also be a res'o!se by burglars to such a cha!ge. >!l$&e !atural 'rocesses l$&e earthua&es" a!alyses l$&e the o!e 'rese!ted here could therefore hae the ab$l$ty to %od$fy the obsered 'rocess" %a&$!g $t %ore d$ff$cult to correctly 'red$ct future ee!ts.
* (&%)
0.01
0.02
0.05
0.1
0.2
0.4
1
ω̄0
100
??.?
?,.
?,.1
?3.5
3.5
45.,
A1015
A10
0.012
0.01
0.056
0.1
0.33
@$ffere!ce $! logl$&el$hood
Table 1 ! percentae of backro"nd e#ent$ ω̄0 and difference in co$t f"nction −f / % bet&een cro$$#alidation cro$$#alidation &it( and &it(o"t "$e of t(e 2)1* data, f"nction of t(e $moot(in lent( L , for type 2 model$+ n all ca$e$ t(e likeli(ood i$ lo&er &(en incl"din t(e 2)1* data to comp"te t(e trierin inten$ity+
-i"re -i"re 1 ! mean of co$t co$t f"nction . −f / % / for #ario"$ #al"e$ #al"e$ of t(e $moot(in lent( L "$ed to comp"te t(e backro"nd backro"nd rateden$ity approac(ed de$cribed in t(e text, obtained by μ ( x , y ) , for t(e t&o approac(ed cro$$#alidation+ cro$$#alidation+ T(e model of type 1 &it( optimized L (a$ L=0.024 km+
-i"re 2 !nteraction kernel$ kernel$ λ t .top rap($/ and λ r .bottom rap($/ for model type 2 &it( L=0.1 km+ T(e t&o da$(ed line$ $(o& po&erla&$ &it( exponent$ 1+* .for λ t / and 0 .for λ r /+
-i"re ! n"mber of e#ent$ e#ent$ .in bl"e/ and predicted predicted n"mber, n"mber, "$in .maenta/ .maenta/ or not "$in .reen/ .reen/ t(e 2)1* e#ent$ in t(e trierin term $"mmation+
-i"re -i"re ! difference difference in t(e co$t co$t f"nction normalized normalized by t(e n"mber n"mber of e#ent$ to be be predicted, predicted, compared compared to type 2 model model prediction$, prediction$, for t(e t&o $imple $moot(in of t(e 2)1 .bl"e/ and t(e 2)1 and 2)1* data "p to t(e prediction time .red/+ T(e $imple $moot(in i$ done "$in an exponential kernel+ -or $moot(in lent($ "p to )+1 km, t(e $imple $moot(in perform$ better t(an t(e more $op(i$ticated model propo$ed (ere t(at acco"nt$ for memory effect$+
-i"re * ! mean di$tance bet&een pair$ of e#ent$ $eparated $eparated by .n1/ e#ent$, e#ent$, for t(e t&o time period$ analyzed $eparately+
