Spatial analysis of bone types with Point Pattern Analysis methods

7/27/2019 Spatial analysis of bone types with Point Pattern Analysis methods

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Spatial analysis of bone types with Point Pattern Analysismethods

Limitsios George1. Introduction

The complete dataset of bones consists of the locations and attributedata of human and animal bones found in the tomb, but for the currentanalysis a very small part was used, which fulfilled some preconditions forthe statistical analysis to be valid. Since every point in the statistical

analysis is accounted as a whole and independent event in a point pattern,bones that were fragmented into smaller pieces cause a significant

problem. Therefore only complete bones are selected1. The bone typeswith the most occurrences marked as complete were the navicular bone,

marked as Naviculare and the talus bone, marked as Talus.Furthermore, this subset of data was split into right and left foot bones,

marked as Dxtand Sin, respectively. But only the two point patternswere finally used (see next):

Talus, Dxt Talus, Sin

Figure 1. Density surface for each point pattern and overlay of the point locations. Theright and left Talus bones are spatially distributed in almost the same way as the resultof the same point process and are highly inter-dependent as the analysis next will show.

The two bone types are part of the Tarsus of the feet and articulatewith each other. This means that both types should exhibit similar

behaviour when statistically analysed. The reason for this lies in the factthat their distribution and spatial structure is a result of the same pointprocess and this is something for which we are sure of. But, after

1 These which were marked as Komplett in the STATUS field of their

attribute data table.


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extensive testing the point patterns associated with the navicular boneswere found not to comply with the expected behaviour so they were

rejected as well2.On the next image we can see the density layer for each point pattern 3.

As it can be seen, the density for the point patterns is substantially higher

on the same location, as expected. The number of events/bones for eachpoint pattern is around thirty and are plotted on top of the density layers.

2. Point Pattern Analysis methods

Point pattern analysis methods, rarely seen in archaeological analysis,were used to analyze the combination of the point patterns, transformingthem into a markedpoint pattern. The marks in this case are the left and

right foots Talus bone, referred to as Sin and Dxt.Key method is the Pair Correlation Function, g(r), which is a second

order summary statistic for a point pattern and analogous to the first

derivative of the K function (Illian, 2008). It can be defined as

g(r)=

d

drK(r)

2rfor r0

for a stationary4 and isotropic point pattern. It can be thought of as the

probability of observing a pair of points separated by distance r , divided

by the corresponding probability for a Poisson process (Baddeley, 2011).

When the locations of events are entirely independent of each other,

g(r)=1, but when the function is above or below 1, for distance, r it

means that the points of the pattern are attracted or repulsed,

respectively. For a non-stationary point pattern the function is weighted by

the local intensity at each point, (x), and not an average intensity for

the whole analysis area, (X). The actual relation of the

aforementioned probability with the pair correlation function for a pair of

points (x,y) of a non-stationary point pattern is

p (r)=(x)(y )g(r)dxdy. The case of non-stationarity, or spatially

varying intensity, is more appropriate for the bone patterns.

Also, the contribution of each point to the empirical pair correlationfunction of the pattern can be computed by the Local Inhomogeneous Pair

Correlation Function defined for every point i asg(r)=

a

2 nji

k(dijr),

2 The navicular bones for the left and right foot were behaving differently when

analysed with the same methods, which means that some other process/-esaffected the spatial structure of the point patterns for each foot in different ways.

e.g. taphonomy conditions, bones classification, sampling.3 The analysis window was approximately reproduced. Small differences

between the actual and the reproduced boundaries of the tomb cannotsignificantly affect the results of the analyses, in this case.

4 Spatially homogeneous point pattern where the intensity is uniformly

distributed across space.


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where a is the area of the analysis window, n the number of points,

k a kernel to give some weight to the points j whose distance from

point i is dijr. This can focus on the correlation of each point with the

rest and often is considered as a Local Indicator of Spatial Association as

well. In an experimental effort, the abstract concept of correlation wasmapped around each point and the overlaying values of correlation at

every location were averaged5.

The next method is the Partial Pair Correlation Function or Cross Pair

Correlation Function. (Illian, 2008) This is used with marked point patterns

and in this case the marks were the left and right side. The behaviour is

exactly the same as in the original Pair Correlation Function and except for

its use in exploratory analysis of the point patterns marks, it can be used

as a test for independence between the marks. In that latter case, it can

moreover lead to the type of independent marking of a point pattern. And

a few lines for the subject of independent marking follows.Random Labelling means that the point patterns are created as a whole,

without marks, and afterwards their individual points are marked by a

point pattern process. Also referred to as a posteriori marking. An examplecould be the point pattern of all bones and the work of a process thatdestroyed some of them but not the rest, so it marked them as incompleteand complete respectively. In this case the Pair Correlation Function and

the Partial PCFs should be equal, g(r)=g11(r)=g22(r)=g12(r) .Random Superposition means that a pattern actually consists of two

other patterns joined together into this bivariate pattern. Also referred to

as a priori marking. An example could be the pattern of cranium and ulna

bones, because in case the cranium bones were re-located, this would

consist a separate process for the distribution of this type of bones and

would not apply for the distribution of the ulna bones. Then the Partial Pair

Correlation Function should be approximately equal to one, g12(r)1

(Illian, 2008).

Obviously, only the second type of independent marking can relate tothe pattern of the right/left sides for Talus bone type and should be tested.

And the last method is the Mark Connection Function p ij (r) which canbe used to analyze qualitative marks. It can be interpreted as the

conditional probability that points of marks i and j can be found at distancer, given that these points belong to the point process N. In terms of the

pair correlation function it can be defined as

p ij (r)=pi p jgij(r)g(r)

for r > 0.

Lack of correlation is estimated by the asymptotic behaviour of the markconnection function as r tends to infinity and is useful for the estimation of

the correlation range, which ends at that distance r where the function

5 The mean was used as the simplest statistic to summarize the distribution ofcorrelation values at each location. It might prove to be not the best choice, but

for the moment seems as a simple and quite logical solution.


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starts to be asymptotic and/or fluctuates around the values of p (r) asr and is estimated as:

limr

pii (r)=p i2

and limr

pij (r)=2pi pj for i j.

3. Analysis and results

Before moving to the analyses results, the questions that are going to beanswered should be presented shortly. The point pattern datasets shouldbe consistent with each other, meaning that the point patterns come in

pairs from almost the same point process so their statistical functionsshould not differ significantly, because the difference between them is only

a locational shift. For example, the right and left side bone patterns cantdiffer because they were placed in space together as parts of the same

human body, according to the plan of the buriers.One of the main goals of these analyses is to find the typical distances

between the buried individuals, but also between the left and right foot ofeach body. Moreover, it would be of great benefit if we could have an asdetailed as possible view of the spatial structure of the point pattern,meaning the relationships in the pattern and how much these points were

dependent on each other and consequently the buried individuals.And finally, which method or procedure would be the most appropriate

for this kind of analysis.

Pair Correlation Function and Partial PCF

First the Partial Pair Correlation Function and the PCFs for the marksSin and Dxt are presented because the spatial structure of the specificpoint patterns can be presented more clearly, although theyre not themost robust statistical functions, especially in the case of small number of

events/points. This holds true for the Talus type right and left side marksand should be noted that the values of the PCFs are exaggerated but thenature of the spatial structure can be clearly seen and is still valid.


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Figure 2. Cross (Partial) - Pair Correlation Function g12(r) (black line) of marksm1==Sin and m2==Dxt, Pair Correlation Function g11(r) for mark Sin (blueline) and Pair Correlation Function g22(r) for mark Dxt (green line). The dashedhorizontal line corresponds to the PCF for the random Poisson point process. The points

of focus here are the almost equal global maxima of the g12(r) which correspond toshort range correlation in small distance among the left-right bones of the sameindividual and the short range correlation at bigger distance among the left-right bone of

different individuals, the repulsion (gii(r)


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But before that, the next schematic will try to simplify the spatialstructure of the point pattern and the relationship between left and right

bones.In an ideal case, the

point pattern should

look like the pattern inthe schematic, wherethe points 5 and 4 arethe left and right footbones of the same

individual and theirdistance range

between 0.03 and0.10. The points 5 and3 are the left and rightfoot bones of different

individuals and areseparated by distances

around 0.18. So,normally we would assume that the shorter distances should also be thenearest neighbours distances, but actually the distribution of the cross-nearest neighbours distances is bi-modal around 0.10 and 0.18 (Figure 3).

Some extrinsic factors might have disturbed the point patterns of left andright foot so the pairs are broken up. Maybe taphonomic factors areresponsible for this disturbance or maybe some bones were drawn awayduring the consecutive burials6.

Figure 3. Histogram of Cross-

Nearest Neighbours distancesfor left and foot bones. Twomodes can be distinguishedaround 0.10 and 0.18. Somebones e.g. of the left foot donthave a right foot NN at distanceless than 0.10 belonging to thesame individual but reach out tothe next closest right foot boneat distances around 0.18 whichbelongs to a different individual.

The gap between the two maxima reveals a small degree of local

regularity of the process. This is the range of distances which is typicallyempty. Bones of the other mark, neither of the same nor of the close byindividual, are unlikely to be found between the two maxima and this isshown as well in the histogram of the cross-NN. The empty space is shown

in the schematic for point 7. The small circles around it represent the first

6 It should be noted that only complete bone occurrences were used for the analyses, so themissing NN could be the fragmented bones excluded at the beginning or maybe they were lost

in their local natural context.


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short range of distances at the first maximum where the other sides boneof the same individual can be typically found, the biggest circles represent

the second short range of distances where the other sides bone of not thesame individual can be typically found and the circles in the middle are therange of distances where no bones are found and are typically empty.

In the actual point pattern this range of distances should have beenempty but instead is populated with a small number of uncorrelatedoccurrences of bones.

As mentioned before, the second range of distances closely resemblesthe distance between individuals. But these distances link e.g the left bone

of one individual with the right bone of another, so for the actual distancebetween individuals the distance linking the same foots bones should be

analysed. In the schematic, such pairs are the 1-2 and 5-6, representingthe right-right and left-left distances.

For this purpose, the Pair Correlation Functions were calculatedseparately for the left and right foot bones and plotted as blue and green

lines, respectively, in Figure 2

7

. At short distances, the two PCFs seem todiffer but both are clearly and significantly under the Poisson process line

and reveal the repulsion between the bones of the same side. This meansthat when starting a search from e.g. a left bone, at these shortdistances,it is less probable than a random process to find another leftbone. But, at distances 0.237 and 0.232, which are extremely similar, both

functions have their local maxima as the peaks of short range correlation,approximately between 0.21 and 0.27. This is the estimated typicaldistance between individuals buried in this tomb.

Shortly after, approximately around distance 0.35, two second local

maxima are found and my assumption is that these correspond to the nextafter the immediate neighbour individual. This is represented in the

schematic by the points 6 and 8. Because the point pattern isnt on aregular grid with fixed distances between individuals, the distance of 6 to 8isnt twice the distance of 6 to 5 but shorter, rather like the long side of atriangle. So, the spatial structure of the marks of the point pattern I

consider to be distinguished for distances shorter than 0.40. After thatpoint, I think that the functions are invalid and just fluctuate.

Mark Connection Function

The next statistical function that was used is the Mark Connection

Function p ij (r), specifically for the spatial analysis of qualitative marks.More robust than the Partial Pair Correlation Function, even for smallnumber of points, and its values are estimated in terms of conditionalprobability of marks. Three functions were calculated again, the MCF of

left-right foot bones, of the left-left and right-right. The outcome of thisanalysis comes to support the PCFs.

7 The Partial Pair Correlation Function between the points of the same mark is

actually the Pair Correlation Function of this marks point pattern and is referred

here as gii

(r), gsin

sin(r) or gDxtDxt

(r) .


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Figure 4. Mark Connection Function p12(r) (black line) of marks m1==Sin andm2==Dxt, Mark Connection Function p11(r) for mark Sin (blue line) and MarkConnection Function p22(r) for mark Dxt (green line). The dashed horizontal linescorresponds to the MCFs for the random Poisson point process

8.

By examining this graph, we can see that the MCFs for the right or leftfoot are indicating clearly the difference of the patterns at short distances

but also the repulsion which ends for both patterns nearly at the samedistance, 0.2 m. Also, the MCF for the right and left foot is high above thePoisson line and starts to decrease rapidly near the distance of 0.18.Unfortunately, the two short ranges of correlation shown in the PPCF are

not clear enough here and this might be due to the smoothing functionapplied on the curves. This highlights the value of the methodological

pluralism in statistical analyses.

Mapping Local Correlation

The concept of correlation is quite abstract when it comes to defining itsspatial extent. In an experimental effort to map this concept, I used the

results of the Local Pair Correlation Functions for every point of the pointpatterns and combined them into maps.

As explained earlier, the LPCF estimates the contribution of every singlepoint to the PCF and by this we have a way to observe how every point

relates to the other across distance, without the aggregation effects. In thecase of the particular point patterns of bones this is not very useful

because the distribution of the bones have been disturbed in some way in

8 The red dashed line is plotted as the random case of psinDxt

(r) , the black one.


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the past, so now we can find most value preferably in summary statisticswhich depend on the aggregated power of statistical analysis of all points

together. But after the LPCFs have been combined into a layer, they canserve as a map of high interdependence areas, as shown in the nextfigures.


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Figure 5. On the first row, two examples of the LPCF of two points. Correlation isestimated and mapped as isotropically spread. Also, bands of correlation can bedistinguished, which correspond to the two main short ranges of correlation shown in thePPCFs and MCFs.

On the second and third rows, for the right and left sides respectively, the combinedLPCFs mapping locally the areas of higher interdependence between points of the

patterns and their perspective view.

The LPCFs of the right and left side point patterns mainly arecharacterised by these bands of correlation which differ relatively to the

dependence of the point with the other. Also, the estimation is made asisotropical, but this doesnt stand true in reality. Though, the combinationof the LPCFs gives the advantage to areas where high correlation valuesare common. In other words, someone could say that the maps on the

second and third rows, first column, are density maps of correlationvalues.

In the usual density maps9 we can see how points are clustered together

but this depends on the first order effects of the point pattern and thenumber of points clustered. The LPCF maps are related to second ordereffects, which also means that the high valued areas are not necessarily

populated by a great number of points, but there is a great degree ofdependence between points. I think that this is more useful in generalbecause real-world point patterns tend to lose points for various reasonsand this method can specify the areas that really matter for the point

pattern process under research.

4. Conclusions

Summarizing, these statistical analyses were conducted on the pointpatterns of complete human bones of the type Talus for the left and rightfoot. The fragments of bones couldnt have been used because these

methods use counts of occurrences for different spatial scales.After conducting the analyses, we can conclude that the results are

credible and the consistency in the statistical behaviour of the pointpatterns holds true. The point patterns of the left and right side bones aresimilar and come from the same point pattern process.

Numerically speaking, the distance between buried individuals isestimated to be around 0.23 m. The second closest individuals are foundat a distance of about 0.35 m.The spatial structure of the point pattern

presents some weak regularity by locational shifts on both directions of thecartesian coordinate system. The distance between the left and right footis estimated to lie approximately between 0.03 and 0.10 m.

By mapping the LPCFs, we could visualize the aforementioned spatial

structure as repeating circles of higher correlation around most points,starting from the short range correlation between the left - right foot

9

A comparison can be made with the density maps of Figure 1.


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bones of the same individual and then the second short range ofcorrelation between the left-right foot bones of neighbouring individuals.

As to what method would be the most appropriate, I would argue that nomethod should be used by itself, but, in contrast, there should be acombination of valid methods which offer to the analysis different

perspectives and work supplementarily to each other.

5. Bibliography

Baddeley, A., 2006. Case studies in spatial point process modeling. Birkhuser.Baddeley, A., 2011. Analysing spatial point patterns in R, in: Workshop Notes. p. 232.Baddeley, A., Turner, R., 2005. Spatstat: an R package for analyzing spatial point

patterns. Journal of Statistical Software 12, 142.Bailey, T.C., Gatrell, A.C., 1995. Interactive spatial data analysis. Longman Scientific

& Technical.Cressie, N.A.C., 1993. Statistics for spatial data. J. Wiley.

Illian, J., 2008. Statistical analysis and modelling of spatial point patterns. John Wiley& Sons.Law, R., Illian, J., Burslem, D.F.R.P., Gratzer, G., Gunatilleke, C.V.S., Gunatilleke, I.

a. U.N., 2009. Ecological information from spatial patterns of plants: insightsfrom point process theory. Journal of Ecology 97, 616628.

Orton, C., 2004. Point pattern analysis revisited. Archeologia e Calcolatori 299315.Ripley, B.D., 1981. Spatial statistics. Wiley Online Library.

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Spatial analysis of bone types with Point Pattern Analysis methods