Statistical analysis of spatial point patterns. - Uni Ulm

Statistical analysis of spatial pointpatterns.

Applications to 3D heterochromatin structures ininterphase nuclei and to root systems of pure

stands of European beech and Norwegian spruce

Frank Fleischer

[email protected]

University of Ulm

Department of Applied Information Processing

Department of StochasticsKarlsruher Stochastik-Tage 2004, Frank Fleischer 1

mailto:[email protected]

Overview

1. Overview

2. Spatial Point Processes and their Characteristics

3. Examination of Heterochromatin Structures inInterphase Nuclei

4. Analysis of the 2D-Distribution of Tree Roots

5. Summary and Outlook

Karlsruher Stochastik-Tage 2004, Frank Fleischer 2

OverviewHeterochromatin structures in interphase nuclei

3D-reconstruction of NB4 cell nuclei (DNA shown in graylevels) and centromere distributions (shown in red)


OverviewAnalysis of the 2D-Distribution of Tree Roots

Root systems of a) spruce [Kiefer] and b) beech [Buche]


Spatial Point Processes and their Characteristics

Let X = {X1, X2, ...} be a point process in Rd, whered ∈ {2, 3}

Let X(B) = #{n : Xn ∈ B} denote the number ofpoints Xn of X located in a set B

The intensity measure Λ is defined as

Λ(B) = EX(B)

In the stationary case

Λ(B) = λ|B|



The pair correlation function g(r) is given as

g(r) =ρ(2)(r)

λ2,

where ρ(2)(r) is the second product density function

Let C1 and C2 be two spheres with infinitesimalvolumes dV1 and dV2 and midpoints x1 and x2

respectively

The probability for having in each sphere at least onepoint of X is approximately equal to ρ(2)(x1, x2)dV1dV2



In the case of complete spatial randomness gPoi(r) ≡ 1

Therefore g(r) > 1 (g(r) < 1) indicates that there is aclustering (repulsion) effect for point pairs of such adistance



The K-Function is defined by

λK(r) = E∑

Xn∈B

X(b(Xn, r)) − 1

λ|B|

In the Poisson case KPoi(r) = bdrd, where bd is the

volume of the d-dimensional unit sphere



Estimation of λK(r). Edge effects are occuring



The L-function L(r) is defined as

L(r) = d

√K(r)

bd

In the Poisson case L(r) = r

Often the slope of L(r) − r is regarded


Spatial Point Processes and their CharacteristicsOther Characteristics

Let δ(Xi) be the distance of a point Xi ∈ X to itsnearest neighbor in X and let λ(r) be the intensity ofthe point process consisting of those points Xi ∈ X

with δ(Xi) ≤ r

The nearest-neighbor distance distribution D(r) isdefined by

D(r) =λ(r)

λ


Spatial Point Processes and their CharacteristicsOther Characteristics

The spherical contact distribution function Hs(r) isgiven by

Hs(r) = 1 − P (X(b(0, r)) = 0)

The J-function is defined by

J(r) =1 − Hs(r)

1 − D(r)


Examination of Interphase Nuclei

A leukemia cell line NB4 has been observed beforeand after differentiation, resulting in 3D confocalmicroscopy images

Position of Centromeres in interphase nuclei have beendetected by automated segmentation methods

Data is given as 3D-coordinates of the centromeremidpoints

28 cell nuclei from undifferentiated cells with 68.25chromocenters on average

27 cell nuclei from differentiated cells with 57.26chromocenters on average



3D-reconstruction of NB4 cell nuclei (DNA shown in graylevels) and centromere distributions (shown in red)



Projections of the 3D chromocenter distributions of anundifferentiated (left) NB4 cell and a differentiated (right)

NB4 cell onto the xy-plane



g(r)

r

0.5

1

1.5

2

500 1000 1500 2000

Averaged estimated pair correlation functions forundifferentiated cells and for differentiated cells



L(r)-r

r

–300

–200

–100

0

100

200

500 1000 1500 2000

Averaged estimated functions L̂(r) − r for undifferentiatedcells and for differentiated cells


Examination of Interphase NucleiResults

Number of chromocenters is significantly decreasedduring differentiation

Hardcore effects for both groups, no point pairdistances of less than 350 nm. Reasons are both theapplied detection algorithm as well as the natural sizeof the chromocenters


Examination of Interphase NucleiResults

Especially the number of point pairs with a distancebetween 350 nm and 700 nm is reduced. While the cellsbefore differentiation show clustering effects, the cellsafter differentiation show even slight signs of repulsion.

Centromeres are clumping during differentiation, theirdistance is reduced below the detectable range(350 nm)


Analysis of the 2D-Distribution of Tree Roots

Adjoining pure stands of European beech andNorwegian spruce are examined

Two-dimensional vertical profile walls with an area of2 m × 1 m have been obtained

The position of small roots with diameters between2 mm and 5 mm have been marked

20 profile walls of European beech with 52.7 points onaverage

16 profile walls of Norwegian spruce with 81.19 pointson average



a) b)

Samples of small root patterns for a) European beech andb) Norwegian spruce



Data is inhomogeneous with respect to the vertical axis

A vertical transformation has to be performed

For beeches (spruces) the depth coordinates havebeen assumed to be gamma (exponentially) distributed

The transformed depth htran is given as

htran =F ∗(horig)

F ∗(htot)htot,

where horig is the original depth, htot is the total depthof the sampling window and F ∗(x) is the assumeddepth distribution



Depth distribution of a) beech and b) spruce



a) b)

Samples of small root patterns for European beech,a) original, b) transformed

c) d)

Samples of small root patterns for Norwegian spruce,c) original, d) transformed



1

2

3

4

5

6

7

10 20 30 40 50

Radius r

Averaged estimated pair correlation functions forNorwegian spruce (·) and European beech (♦)



0

0.5

1.0

1.5

2.0

2.5

10 20 30 40 50

Radius r

Averaged estimated functions L̂(r) − r for Norwegianspruce (·) and European beech (♦)



Both tree root systems show clear signs of clusteringfor short (less than ≈ 10 cm) point pair distances

The clustering effect in the case of the Norwegianspruces (European beeches) seems to be stronger(weaker) in a smaller (larger) region of custering

A Matern-cluster model has been fitted to thetransformed data


Analysis of the 2D-Distribution of Tree RootsMatern-cluster model

A Matern-cluster process XMC is based on a Poissonprocess with intensity λp, whose points are calledparent points

Around each parent point a disc with fixed radius R istaken, in which the points of XMC are scattereduniformly

The number of points in such a disc is Poissondistributed with parameter πR2λd, where λd is the meannumber of points per unit area generated by a singleparent point in a disc of radius R


Analysis of the 2D-Distribution of Tree RootsMatern-cluster model

Parameters of the Matern-cluster processes have beenestimated using minimum-contrast methods applied tothe theoretical and estimated pair correlation function

Results are λsprucep = 0.00690, λbeech

p = 0.00603,λ

spruced = 0.00774, λbeech

d = 0.00253, Rspruce = 4.9 cm andRbeech = 7.4 cm

Afterwards the models have been validated bysimulation-based goodness-of-fit tests for differentpoint process characteristics



A retransformation of the data has to be performed

The inverse transformation is given by

horig = (F ∗)−1(F ∗(htot)

htothtran)

The inhomogeneous Matern-cluster models arisinghave inhomogeneous Poisson processes as parentprocesses with intensity functions

λp(x, y) = λp(y) = λpf∗(y)

F ∗(htot)htot,

where f∗(x) is the density function of the suitabledistribution function F ∗(x)



The cluster regions are given by

{(x, y) : (x − xp)2 + (F ∗(y) − F ∗(yp))

2(htot

F ∗(htot))2 ≤ R2}

They are no longer circles, but the images of thesecircles under the inverse transformation



Realizations with cluster regions of the homogeneousMatern-cluster model and of the retransformed

inhomogeneous Matern-cluster model for Norwegianspruce


Summary and Outlook

For the interphase nuclei clear differences between thetwo examined groups have been detected

Next steps are a fitting and validating of an appropriatepoint process model

For the tree root patterns a Matern-cluster model hasbeen fitted to the data

Mixed stands have already been examined, too.Confirmation of the results for pure standsInteresting results for effects between different treespecies


Cooperation Partners

We want to thank our cooperation partners

Dr. Dr. M. Beil, Department of Internal Medicine I,University of Ulm

Prof. Dr. M. Kazda, Department Systematic Botanyand Ecology, University of Ulm


Literature

Beil, M., Fleischer, F., Paschke, S. and Schmidt, V.,Statistical analysis of 3D centromeric heterochromatin structure ininterphase nuclei. Preprint

Fleischer, F., Eckel, S., Kazda, M., Schmid, I. andSchmidt, V., Statistical analysis of the spatial distribution of treeroots in pure stands of Fagus sylvatica and Picea abies. Preprint(both available at www.geostoch.de)

Stoyan, D., Kendall, W. S. und Mecke, J., StochasticGeometry and its Applications. J. Wiley & Sons, Chichester,2nd edition, 1995


Statistical analysis of spatial point patterns. - Uni Ulm

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