Statistische Verfahren zur Berechnung von Sicherheitszonen ... · Einf uhrung und Problemstellung...

Statistische Verfahren zur Berechnung vonSicherheitszonen aus kartierten Bombentrichtern

Monia Mahling, Michael Hohle, Helmut Kuchenhoff

4. Dezember 2009

Einfuhrung und ProblemstellungDatenbeispiele

MethodikVergleich der Methoden

Zusammenfassung und Ausblick

Uberblick

1 Einfuhrung und Problemstellung

2 Datenbeispiele

3 Methodik

4 Vergleich der Methoden

5 Zusammenfassung und Ausblick

Monia Mahling, Michael Hohle, Helmut Kuchenhoff 2/33




Einfuhrung und Problemstellung

Auch mehr als 60 Jahre nach Ende des zweiten Weltkrieges stellenBombenblindganger noch immer eine Gefahr dar:

haufige Bombenfunde bei Bauarbeiten (2008 alleine in Bayern 121Spreng- und Splitterbomben), die teilweise Evakuierungen vonWohnhausern und Sperrung von Straßen oder Bahnlinien notigmachen

vereinzelt schwere Unfalle durch Detonationen





Einfuhrung und Problemstellung

Konsequenz: systematische Suche in Risikogebieten vor Beginn vonBauarbeiten

Anhand von Luftbildern konnen die Lokationen der detoniertenSprengbomben bestimmt werden. Hierzu stellte die OberfinanzdirektionHannover drei typische Datenbeispiele zur Verfugung.

Ziel der Analyse: Ausweisung von Sicherheitszonen, außerhalb derermit hoher Sicherheit keine Blindganger auftreten, mit Hilfe vonKoordinaten der Explosionstrichter detonierter Sprengbomben





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Methode 1Methode 2Methode 3

Methode 1: Verwendung eines vorgegebenen Radius

Kreis mit einem fest vorgegebenen Radius um alle Ereignispunkte

Bestimmung des Abstandes zum nachsten Ereignis fur jeden Punktin der betrachteten Region






Methode 1

Die Sicherheitszone besteht aus allen Punkten, fur die der Abstand zumnachsten Ereignis kleiner ist als der vorgegebene Radius, was derVereinigung aller Kreise entspricht.






Methode 2: Verwendung eines aus der Verteilung desNachste-Nachbarn-Abstandes bestimmten Radius

Kreise um alle Ereignispunkte werden zu einer Sicherheitszonevereinigt.

Radius: empirisches Quantil der Verteilung desNachste-Nachbarn-Abstandes

0 50 100 150 200

0.0

0.2

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1.0

Beispiel 7

Entfernung (Radius r) in m

D(r

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Beispiel 8


D(r

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00.

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Beispiel 9


D(r

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Methode 2

Ansonsten entspricht das Vorgehen dem Vorgehen bei Methode 1.






Methode 3: Konstruktion basierend auf derIntensitatsfunktion des Punktprozesses

Punktprozess-Methodik (Illian et al. (2008), Møller & Waagepetersen(2007)):

Beobachtungsfenster W

raumlicher Punktprozess X (Lokationen aller Bomben)

ausgedunnte Version Y des Prozesses beobachtet

Blindgangerprozess Z = X\YAnnahme: Blindgangerwahrscheinlichkeit q homogen uber W






Methode 3

Annahme: Prozess Y inhomogener Poisson-Punktprozesses mitIntensitatsfunktion λ(s)

Fur gegebenen Schwellenwert c > 0 kann eine Sicherheitszone Rc auffolgende Art konstruiert werden:

Rc = {s ∈W : λ(s) ≥ c}.

Frage: Wie ist c zu bestimmen, damit ein gewisses Sicherheitsniveauerreicht wird?

Fur die Lokation eines einzelnen Punktes x in A ⊆W gilt:

P(x ∈ A) =

∫Aλ(u)du∫

Wλ(u)du

=Λ(A)

Λ(W )






Methode 3

Fur die Anzahl der Blindganger NZ (W ) gilt (Hunter & Griffiths, 1978):

NZ (W ) ∼ NegBin(NY (W ) + 1, 1− q).

● ● ● ● ● ● ● ● ● ● ●

Versuch

1 2 3 4 5 6 7 8 9 10 11






Methode 3

Seien nz Punkte unabhangig voneinander in W zu platzieren.

Anzahl von Punkten in einer Region A ⊆W :

N(A) | nz ∼ Bin

(nz ,

Λ(A)

Λ(W )

)Somit Anzahl von Punkten außerhalb der Sicherheitszone:

N(W \Rc) | nz ∼ Bin

(nz ,

Λ(W \Rc)

Λ(W )

)






Methode 3

Betrachte die Wahrscheinlichkeit, dass fur nz Punkte in W mindestenseiner in W \Rc liegt:

P(N(W \Rc) > 0 | nz) = 1− P(N(W \Rc) = 0|nz)

= 1− P(N(Rc) = nz | nz)

= 1−(

Λ(Rc)

Λ(W )

)nz

Damit lasst sich die Wahrscheinlichkeit dafur angeben, dass mindestensein Punkt des Blindganger-Prozesses Z außerhalb der Sicherheitsregionliegt:

P(N(W \Rc) > 0) =∞∑

nz=0

P(N(W \Rc) > 0|nz) · P(NZ (W ) = nz)






Methode 3

Vorgabe einer sehr hohen Sicherheitswahrscheinlichkeit derSicherheitszone Rc

P(NZ (W \Rc) = 0) = 1− P(NZ (W \Rc) > 0) > 1− α,

wobei 0 ≤ α ≤ 1 die Sicherheitstoleranz angibt.

Bestimmung eines Schwellenwertes c , so dass

arg maxc≥0

{P(NZ (W \Rc) > 0) ≤ α

},

damit die Flache der Region moglichst klein ist.

Die Sicherheitszone ist dann durch das so bestimmte c festgelegt.






Methode 3

In einem ersten Schritt wird mittels Kerndichteschatzung eine Schatzungder Intensitatsfunktion vorgenommen. Dazu muss die Bandweite geeignetgewahlt werden.






Methode 3

Im zweiten Schritt wird dann der Cutoff-Wert c numerisch so bestimmt,dass P(NZ (W \Rc) > 0) ≤ α gerade erfullt ist.





Aufbau der SimulationQuantilsbasierte MethodePunktprozess-Methode

Aufbau der Simulation I

Vergroßerung des Beobachtungsfensters um 15 m

je 100 Iterationen:

Ausdunnen des beobachteten Punktmusters mittels bernoulliverteilterZufallszahlen (mit Blindgangerwahrscheinlichkeit q)Bestimmung der Sicherheitszonen auf Basis des ausgedunntenProzesses

Dabei wurden mehrere Parameter variiert:

zwei verschiedene Werte fur die Blindgangerwahrscheinlichkeit q

vier verschiedene Cutoff-Werte fur Radius bzw. Quantil

zwei verschiedene Werte fur die Sicherheitstoleranz α

automatisch durch Kreuzvalidierung bestimmte Bandweite(Silverman, 1992) sowie das Doppelte und das Dreifache diesesWertes






Aufbau der Simulation II

Beurteilung der Sicherheitszonen:

Bestimmung der FlacheBestimmung des Anteils der ubersehenen Blindgangern an allenBlindgangern






Quantilsbasierte Methode

Verwendung von Information aus den Daten durch Festlegung desRadius uber Quantil des Nachste-Nachbarn-Abstandesrelativ gute Eigenschaften in den drei SimulationsbeispielenEs werden hauptsachlich solche Blindganger ubersehen, die weit vonallen anderen Ereignissen entfernt liegen.

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Datensatz 7

Fläche der Sicherheitszone

Ant

eil d

er ü

bers

ehen

en a

n al

len

Blin

dgän

gern

90%95%99%99.9%99% mit q=0.15

Abbildung: Ergebnisse fur Beispieldatensatz 7Monia Mahling, Michael Hohle, Helmut Kuchenhoff 23/33






Bei kleinem Stichprobenumfang (Beispieldatensatz 8) problematisch:stark schwankende Flache der Sicherheitszonen bei hohererBlindgangerwahrscheinlichkeit q = 0.15 (ganz rechts).

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90% 95% 99% 99,9% 99%

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+06

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+06

1.0e

+07

1.2e

+07






Punktprozess-Methode

Nutzung der Information aus den Daten zur Festlegung derSicherheitszoneEs werden fast ausschließlich solche Blindganger ubersehen, die weitvon allen anderen Ereignissen entfernt liegen.

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4e+06 6e+06 8e+06 1e+07

0.00

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0.20

Datensatz 8


Ant

eil d

er ü

bers

ehen

en a

n al

len

Blin

dgän

gern

αα=1e−05, einf. BWzehnfaches ααdoppelte Bandweitedreifache Bandweiteαα=1e−05, einf. BW, q=0.15







mogliche Verbesserung des Verfahrens durch

genauere Betrachtung der Bandweite (Bestimmung der optimalenBandweite im Sinne einer – bei gleichem Anteil ubersehenerBlindganger – moglichst kleinen Sicherheitszone)

Schatzung der Blindgangerwahrscheinlichkeit anhand von Daten uberbekannte Blindganger und Kovariablen wie Bodenbeschaffenheitoder Bebauung an Stelle einer Homogenitatsannahme





ZusammenfassungAusblick

Zusammenfassung

Verwendung eines festen Radius als generelles Vorgehen nichtgeeignet

Entscheidung zwischen der quantilsbasierten Methode und derPunktprozess-Methode anhand der erhaltenen Anteile derubersehenen Blindganger und der Flachen der Sicherheitszonenschwierig (Vergleich in zwei Dimensionen), entwickelteAnalyseverfahren jedoch ausbaufahig

Analysen und Aussagen zur Zeit auf Vergleich von dreiBeispieldatensatzen beschrankt

erstellte R-Programme zur Berechnung und quantitativen Analyseder Sicherheitszonen leicht auf andere Daten anwendbar;umfassenderer und direkter Vergleich der quantilsbasierten Methodeund der Punktprozess-Methode moglich






Zusammenfassung

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Datensatz 9


Ant

eil d

er ü

bers

ehen

en a

n al

len

Blin

dgän

gern







Zusammenfassung

Quantitative Analyse dreier Methoden zur Bestimmung vonSicherheitszonen aus kartierten Bombentrichtern

Neuentwicklung eines auf einem Punktprozess-Ansatz aus derraumlichen Statistik basierenden Verfahrens

Entwicklung eines simulationsbasierten Verfahrens zur statistischenRisikoabschatzung von Sicherheitszonen






Ausblick

Simulation als allgemein anwendbares Prinzip zur Risikoabschatzung

Kalibrierung des Verfahrens notig

extreme Ausreißer problematisch

Große der Kandidatenflachen

Modellierung als mogliches weiteres Vorgehen






Literaturverzeichnis

Baddeley, A., Møller, J. & Waagepetersen, R. (2000). Non- andsemi-parametric estimation of interaction in inhomogeneous pointpatterns. Statistica Neerlandica 54, 329–350.

Hunter, A. J. & Griffiths, H. J. (1978). Bayesian Approach toEstimation of Insect Population Size. Technometrics Vol. 20, No. 3(Aug., 1978), 231–234.

Illian, J., Penttinen, H., Stoyan, H. & Stoyan, D. (2008). Statisticalanalysis and modelling of spatial point patterns, Wiley, Chichester,West Sussex, Eng.

Møller, J. & Waagepetersen, R. P. (2007). Modern Statistics forSpatial Point Processes. Scandinavian Journal of Statistics 34 (4),643–684.






Literaturverzeichnis

Silverman, Bernard W. (1992). Density estimation for statistics anddata analysis, Chapman & Hall, London.

R Development Core Team (2009). R: A language and environmentfor statistical computing. R Foundation for Statistical Computing,Vienna, Austria. ISBN 3-900051-07-0, URL:http://www.R-project.org

A. Baddeley and R. Turner (2005). Spatstat: an R package foranalyzing spatial point patterns. Journal of Statistical Software 12(6), 1–42. ISSN: 1548-7660, URL: http://www.jstatsoft.org


Statistische Verfahren zur Berechnung von Sicherheitszonen ... · Einf uhrung und Problemstellung...

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Transcript of Statistische Verfahren zur Berechnung von Sicherheitszonen ... · Einf uhrung und Problemstellung...