Spatial information theory

The spatial information theory literature describes several formal languages called spatial logics which enable reasoning about the representation of systems in space and potentially how this representation changes over time (Aiello et al., 2007, Chapter 9).

Depending on the considered application the employed spatial logic can be qualitative (e.g. (McKinsey and Tarski, 1944; Montanari et al., 2009; Randell et al., 1992; Tarski, 1938)), (semi-)quantitative (e.g. (Condotta, 2000; Xu, 2007)) or a combination thereof (i.e. hybrid) (e.g. (Kor and Bennett, 2013; Liu et al., 2009b)). Due to the uncertainty or lack of precision usually associated with spatial information, qualitative spatial logics are usually employed (Bresolin et al., 2010). Most qualitative spatial logics are defined using constraint based techniques, which were initially developed for temporal reasoning (Aiello et al., 2007, Chapter 4).

The constraints often considered are topology, orientation and distance (Aiello et al., 2007, Chapter 4) considering a 2D representation of space.

Topological qualitative spatial logics enable describing topological relations between spatial entities. One of the most employed topological qualitative spatial

logics is RCC-8 which is an instance of the Region Connection Calculus (RCC) proposed by Randell et al. (Randell et al., 1992). The primitive relation of RCC is C(x, y) read as “region x connects to region y” and which holds when x and y share at least one common point. RCC-8 is an instance of RCC that contains 8 topological relations, namely DC(x, y) (i.e. x is disconnected from y), P O(x, y) (i.e. x and y are partially overlapping), EC(x, y) (i.e. x is externally connected with y), EQ(x, y) (i.e. x is equal to y), T P P (x, y) (i.e. x is a tangential proper part of y), N T P P (x, y) (i.e. x is a non-tangential proper part of y), T P P−1(x, y) (i.e. x is an inverse tangential proper part of y), N T P P−1(x, y) (i.e. x is an inverse non-tangential proper part of y); see Figure 3.11 for a graphical description of these relations.

(a) DC(A, B) (b) P O(A, B) (c) EC(A, B) (d) EQ(A, B)

(e) T P P (A, B) (f) N T P P (A, B) (g) T P P−1(A, B) (h) N T P P−1(A, B)

Figure 3.11: A graphical description of the topological spatial relations in RCC-8. DC(x, y) states that regions x and y are disconnected. P O(x, y) states that x and y are partially overlapping. EC(x, y) states that x and y are externally connected. EQ(x, y) states that x and y are equal. T P P (x, y) states that x is a tangential proper part of y. N T P P (x, y) states that x is a non-tangential proper part of y. T P P−1(x, y) states that x is an inverse tangential proper part of y. N T P P−1(x, y) states that x is an inverse non-tangential proper part of y.

Directional qualitative spatial logics enable reasoning about the relative posi- tioning/orientation of entities in space. An illustrative example of such a logic is Rectangle Algebra (RA). In RA the relative positioning of regions in 2D space is defined with respect to the relative positioning of the regions’ projections on the Ox and Oy axes. The projection of a 2D region on a one-dimensional axis is an interval. To determine the relative positioning of two intervals the set of 13 relations defined in Allen’s Interval Algebra (IA) (Allen and Hayes, 1989) are employed {before, after, meets, met by, overlaps, overlapped by, starts, started by, during, contains, finishes, finished by, equals}; see Table 3.2 for a description of these relations. Since the number of IA relations is 13, the number of relations employed to describe the relative positioning of two regions in 2D space is 13 (considering the Ox axis) · 13 (considering the Oy axis) = 169.

Finally qualitative spatial logics considering distance constraints describe the relative distance between two entities in space using relations such as “very far”,

Table 3.2: Interval Algebra relations defined over two intervals A and B, where A−and A+_{are the}

endpoints of A, respectively B−and B+ _{are the endpoints of B.}

Relation Example Meaning

Bef ore(A, B) A−< A+_{< B}− < B+ Af ter(B, A) A−< A+< B−< B+ M eets(A, B) A−< A+_{= B}− < B+ M etBy(B, A) A−< A+= B−< B+ Overlaps(A, B) A−< B−< A+ < B+ OverlappedBy(B, A) A−< B−< A+ < B+ Starts(A, B) A−= B−< A+ < B+ StartedBy(B, A) A−= B−< A+ _{< B}+ During(A, B) B−< A−< A+ < B+ Contains(B, A) B−< A−< A+ < B+ F inishes(A, B) B−< A−< A+ _{= B}+ F inishedBy(B, A) B−< A−< A+ = B+ Equals(A, B) A−= B−< A+ _{= B}+

“far”, “commensurate”, “close” and “very close”. Depending on the problem considered the number of different distance levels varies. For instance assuming a coarse grained representation of space two distance levels such as “far” and “close” could potentially suffice. An example of a qualitative spatial logic considering distance constraints is described by Clementini et al. (Clementini et al., 1997).

One of the main advantages of qualitative spatial logics is that they enable reasoning about the spatial representation of a system in the presence of un- certainty. Conversely one of the main disadvantages is that qualitative spatial logic descriptions are often imprecise. Therefore it could be potentially difficult using such logics to accurately describe how emergent spatial entities and their properties change over time.

PBLSTL inherently supports only quantitative spatial properties. However qualitative spatial properties can be additionally expressed in PBLSTL if they are rewritten in a quantitative manner. For instance if we would like to specify in PBLSTL that a point-wise region A at position (x, y) in 2D space is a non- tangential proper part of a rectangular region B defined by the points M (0, 10), N (10, 10), O(10, 20) and P (0, 20) (e.g. corresponding to N T P P (A, B) in RCC-8), then we could write that the coordinates of the region A lie within and do not touch the border of region B (i.e. 0 < x < 10 and 10 < y < 20). Moreover high- level spatial functions could be added to PBLSTL to enable encoding quantitative

spatial properties in a compact form (e.g. inside(A, B) if and only if 0 < x < 10 and 10 < y < 20); PBLSTL could be adapted automatically to the new type of spatial functions using the meta model checking concept, which is introduced later in Chapter 5.

Summary

This chapter introduced a novel multidimensional spatio-temporal model checking methodology which enables validating computational models of biological systems with respect to how both their numeric and spatial properties change over time considering a single level of organization. In the beginning SSpDESs are defined as theoretical models for abstractly representing biological systems. Next spatio- temporal analysis modules are introduced for automatically detecting regions or clusters in time series data and computing how their spatial properties (e.g. area, perimeter etc.) change over time. Time series data describing how both numeric and spatial properties change over time are formatted according to the standard representation format STML introduced here. Then the temporal logic (P)BLSTL is defined for encoding the formal specification against which SSpDES models are validated. Afterwards corresponding Bayesian and frequentist, estimate and hypothesis testing based model checking algorithms are described. Moreover a proof is provided illustrating that the multidimensional spatio-temporal model checking problem is well-defined. Implementation details and a concise comparison of the approach with other spatio-temporal model checking methods from the epidemiology and spatial information theory literature are presented in the end.

CHAPTER

4

Validation of multidimensional

computational models of

biological systems

Introduction

This chapter illustrates how the multidimensional spatio-temporal model checking methodology described in Chapter 3 can be employed to validate computational models of biological systems. The biological case studies considered are phase variation in bacterial colony growth and the chemotactic aggregation of cells. Cor- responding computational models have been validated against relevant PBLSTL specifications using the model checker Mudi. Conclusions and limitations of the multidimensional model checking methodology are described at the end.

4.1

Description

The efficiency and expressivity of the multidimensional spatio-temporal model checking methodology was assessed based on two biological case studies encod- ing phase variation patterning in bacterial colony growth, and the chemotactic aggregation of cells.

The corresponding computational models are stochastic and have been encoded using high-level modelling formalisms which can be translated to an equivalent SSpDES representation.

For generalizability purposes the stochastic computational models have been encoded using different high-level modelling formalisms, namely Coloured Stochas-

tic Petri Nets for phase variation in bacterial colony growth, and Cellular Potts and partial differential equations for the chemotactic aggregation of cells.

Results generated via model simulation were processed by the spatio-temporal analysis modules and were translated to STML. The spatial entity types considered for the phase variation and the chemotactic aggregation of cells case studies were regions, respectively clusters.

STML files representing the model behaviour were evaluated against formal PBLSTL specifications. The main purpose of these specifications was to illustrate the expressivity of the methodology and not to test novel biological hypotheses. Probability values considered in the PBLSTL statements are only approximations of corresponding qualitative natural language descriptions (e.g. high probability ⇒ 0.9) and were chosen for illustrative purposes.

For model checking purposes no prior information was employed other than the computational model and the PBLSTL specification. Therefore the frequen- tist rather than the Bayesian statistical model checking approach was employed throughout. Relevant comparisons between different approximate probabilistic model checking approaches are provided in the original papers introducing the approaches (see Subsubsection 2.4.2.3). Since the comparison results are indepen- dent of particular model representations and logic formalisms they will not be restated here.

For reproducibility purposes the generated STML files, and the formal PBLSTL specification corresponding to both computational models have been made avail- able at http://mudi.modelchecking.org/case-studies.