A tutorial in spatial statistics for microscopy data analysis Ed - PowerPoint PPT Presentation

A tutorial in spatial statistics for microscopy data analysis Ed Cohen Department of Mathematics, Imperial College London wwwf.imperial.ac.uk/ ∼ eakc07 QBI 2019

Spatial Statistics Spatial Statistics: Statistical theory and methodology for modelling and analysing spatial data. Fluorescence microscopy is concerned with imaging objects , We are interested in understanding spatial organisation of objects to inform our understanding of biological mechanisms and processes. Therefore we will restrict ourselves to spatial point patterns.

Spatial Point Pattern Data in the form of a set points, irregularly distributed in a region of space are called a spatial point pattern. Arise in many different contexts, e.g. ◮ Location of trees in a forest ◮ Location of ants nests in compact geographical region ◮ Location of a particular protein of interest in a cellular environment.

Spatial Point Pattern Mathematically, we can represent a spatial point pattern as a set of locations Φ = { s 1 , s 2 , ... } , with each event s i belongs to X , a (locally compact subset) of R d . For example: in fluorescence microscopy imaging, X is typically a square region in R 2 , and each fluorophore/event has a true position s i = ( x i , y i ).

Spatial point processes Informally, a point process is a stochastic mechanism that generates a countable set of events - i.e. a spatial point pattern. It is the probabilistic framework that governs how many events there are and where they occur. Analogous to a probability distribution for random variables.

STOCHASTIC MECHANISM REALIZATIONS ROLLING TWO DICE BIVARIATE DISCRETE UNIFORM DISTRIBUTION ON {1,2,3,4,5,6} x {1,2,3,4,5,6} CLATHARIN COATED PITS ON CELL MEMBRANCE POISSON PROCESS WITH INTENSITY !

Describing and characterizing spatial point processes We typically represent a spatial point process by N , where N ( A ) is a random number indicating the number of events within some set A . ! " # = 8 ! & # = 0 ! ( # = 4 It is the probability distribution of N ( A ) for all (nice) sets A that characterizes a spatial point process.

Characterizing spatial point processes Intensity (localized rate of events): E { N ( d s ) } λ ( s ) = lim . | d s | | d s |↓ 0 The second-order intensity of a spatial point process N at points s , u ∈ X is E { N ( d s ) N ( d u ) } γ ( s , u ) = lim . | d s || d u | | d s || d u |↓ 0 The second-order covariance of a spatial point process N at points s , u ∈ X is c ( s , u ) = γ ( s , u ) − λ ( u ) λ ( s ) cov ( X , Y ) = E ( XY ) − E ( X ) E ( Y ) . Pair correlation function: γ ( s , u ) g ( s , u ) = λ ( s ) λ ( u ) .

Characterizing first and second order moments of spatial point processes Homogeneity: λ ( s ) is constant for all s ∈ X . Translates as: the chance of getting an event at any particular point in spaces is the same across X . Stationarity and isotropic: γ ( s , u ) = γ ( || s − u || ) = γ ( r ). Translates as: The covariance between any two points in space having an event or not depends only on the distance between them. These assumptions are not as restrictive as they first seem. If the heterogeneity is itself random then these notions can still hold.

Poisson process Spatial point process N is Poisson if the following hold: For every (nice) subset A ⊂ X , the number of events is Poisson distributed with expected value � µ ( A ) = A λ ( s ) d s For any collection A 1 , ..., A n , the random variables N ( A 1 ) , ..., N ( A n ) are independent of one another. Poisson processes have memoryless property - all events are independent of eachother. Homogeneous Poisson processes are known as completely spatial random (CSR).

Complete spatial randomness

Complete spatial randomness 0 0 1 0 0 0 1 1 0 0 1 1 0 1 0 1 1 0 0 0 1 1 1 1 1 0 1 1 0 0 1 1 1 1 0 0 0 1 1 1 0 1 0 0 1 0 1 1 0 0 0 0 1 0 1 0 0 0 1 0 0 1 0 0 0 1 0 1 0 1 1 0 0 1 0 0 0 1 1 0 1 1 0 0 0 0 1 1 1 1 0 0 1 0 1 0 1 0 1 0 0 1 0 1 1 0 1 0 1 0 0 0 1 0 0 1 1 0 0 1 1 1 1 1 1 1 1 1 0 1 0 1 1 1 0 0 0 0 0 1 1 1 1 0 0 0 0 1 0 0 1 0 0 0 0 1 1 0 1 0 0 0 1 1 1 1 0 0 0 1 1 0 1 0 0 1 0 0 0 0 0 0 0 0 1 1 0 0 1 0 0 0 1 0 0 0 0 1 0 1 1 0 0 0 1 0 1 0 0 0 1 1 1 0 0 0 0 0 1 0 1 1 0 0 0 1 0 0 1 1 1 0 1 0 0 1 1 0 0 0 0 1 1 0 0 0 1 0 1 1 1 0 1 1 1 1 0 0 1 1 0 0 0 0 1 0 1 1 1 1 1 1 1 0 1 1 0 0 0 0 0 0 0 1 1 1 1 0 1 1 0 1 0 0 0 0 1 1 0 0 0 0 0 0 1 1 1 0 0 0 0 1 0 0 1 0 0 1 0 0 1 1 1 0 0 1 0 0 0 0 1 1 0 1 1 1 0 1 0 1 0 0 1 0 1 0 0 1 1 1 1 1 0 0 0 0 1 0 1 0 1 0 1 0 0 0 0 0 1 1 0 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0 0 1 1 0 0 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 0 1 0 1 1 1 1 1 1 1 1 1 1 0 0 0 1 1 1 0 1 1 1 1 1 0 1 0 0 1 0 0 0 0 1 0 1 1 0 0 1 1 0 0 0 0 1 0 1 0 1 0 0 1 1 1 1 1 0 1 0 1 1 1 1 0 0 0 0 1 1 1 1 1 0 0 0 1 1 0 0 0 0 0 0 1 0 1 1 1 1 0 1 0 1 1 0 1 0 0 1 1 0 0 0 1 1 0 0 0 0 0 1 0 1 1 0 1 0 0 1 1 1 0 1 1 1 1 1 1 1 1 0 1 0 0 1 1 0 1 1 0 1 0 0 0 0 1 0 1 0 0 0 1 1 1 0 0 0 0 1 0 0 1 0 1 1 0 1 0 0 0 0 0 0 1 1 0 1 1 1 0 0 0 1 1 0 0 0 1 1 0 0 0 1 0 1 0 1 1 1 0 0 1 1 1 0 0 1 1 1 0 0 1 1 0 1 1 0 1 0 0 0 1 0 0 1 0 1 1 1 0 0 1 0 1 0 0 1 0 1 0 1 1 0 0 1 1 0 1 0 0 0 1 1 0 0 0 0 0 1 0 1 1 1 0 0 0 0 0 0 0 1 1 1 0 0 0 0 1 0 0 1 0 1 1 1 1 0 1 1 1 1 1 0 1 1 1 0 1 0 0 1 0 0 1 0 0 1 1 0 0 0 1 0 0 0 0 1 1 1 1 1 1 0 0 1 1 1 1 1 1 0 1 0 1 1 1 1 1 0 1 0 0 0 1 1 1 0 0 1 1 0 0 1 0 0 1 0 1 1 0 0 1 0 0 0 1 1 1 0 1 1 0 0 1 1 1 0 1 1 0 1 0 0 1 1 0 0 0 0 1 0 1 1 1 0 0 0 0 1 0 1 0 0 0 1 1 0 0 0 0 0 0 0 1 0 1 1 0 1 1 0 1 1 1 1 1 0 1 1 1 1 0 1 0 0 0 0 1 1 0 1 0 0 0 1 0 1 1 1 0 0 0 0 1 0 1 0 0 0 0 0 1 1 0 1 1 1 1 0 0 0 1 0 0 0 0 1 0 1 0 0 0 1 0 1 1 0 0 1 0 0 1 1 1 1 1 0 1 0 1 0 1 0 0 1 1 1 1 1 1 0 0 0 1 0 1 0 0 1 0 0 0 0 0 1 1 1 0 0 1 1 0 0 0 1 0 0 1 1 1 0 0 1 0 0 1 1 0 0 0 0 1 1 0 0 0 1 0 0 0 1 0 0 1 0 0 1 0 1 0 1 1 0 1 1 1 1 0 0 1 1 0 1 0 0 0 0 1 0 0 0 0 0 1 0 0 1 1 0 1 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 1 0 0 1 0 1 1 1 1 1 0 1 1 1 0 1 0 0 0 1 0 0 1 1 1 0 1 0 0 0 1 1 0 1 1 1 1 0 0 0 0 1 1 1 0 0 1 1 1 0 0 1 1 1 0 1 0 1 0 0 0 0 0 1 0 1 1 1 0 1 0 0 0 0 1 0 0 0 1 0 0 0 1 1 1 0 0 0 1 0 0 0 1 1 0 0 0 1 1 0 0 1 1 1 0 0 0 0 1 0 0 1 1 1 1 0 0 0 0 1 0 1 1 1 1 0 1 0 0 0 1 0 0 0 1 0 1 1 0 1 1 1 0 1 1 0 1 0 0 0 1 0 1 0 1 0 1 0 0 1 1 1 0 0 1 0 0 1 1 1 0

CSR vs Clustering vs Inhibition/Regularity CSR Clustered Inhibited

CSR vs Clustering vs Inhibition/Regularity CSR Clustered Inhibited

Ripley’s K -function Ripley’s K-function is used extensively across the sciences, including microscopy to detect and characterize clustering behavior. It is a theoretical property of the point process. � r 2 π r ′ g ( r ′ ) d r ′ K ( r ) ≡ ! 0 = λ − 1 E { number of events within distance r of an arbitrary event } . Its widespread use lies in its interpretability and the ease at which it can be estimated with robust, well studied estimators from a single point pattern. Many of the recent developments in spatial data analysis have this function at their heart.

Worked example: Poisson process The second-order intensity of a homogeneous N at points s , u ∈ X is E { N ( d s ) N ( d u ) } γ ( s , u ) = lim | d s || d u | | d s || d u |↓ 0 E { N ( d s ) }{ N ( d u ) } = lim = λ ( s ) λ ( u ) ! | d s || d u | | d s || d u |↓ 0 γ ( s , u ) g ( s , u ) = λ ( s ) λ ( u ) = 1 K ( r ) = π r 2 � L ( r ) − r ≡ K ( r ) /π − r = 0 .

K -function for different types of process Clustered CSR Inhibited L(r) - r L(r) - r L(r) - r r r r

Estimation REMEMBER: we are interested in knowing the properties of the PROCESS. We need to estimate them from the pattern - typically we only get one pattern with which to estimate them. n n A ˆ � � K ( r ) = w ij I (0 < d ij < r ) n ( n − 1) i =1 j =1

Estimation REMEMBER: we are interested in knowing the properties of the PROCESS. We need to estimate them from the pattern - typically we only get one pattern with which to estimate them. n n A ˆ � � K ( r ) = w ij I (0 < d ij < r ) n ( n − 1) i =1 j =1

Estimation REMEMBER: we are interested in knowing the properties of the PROCESS. We need to estimate them from the pattern - typically we only get one pattern with which to estimate them. n n A ˆ � � K ( r ) = w ij I (0 < d ij < r ) n ( n − 1) i =1 j =1

Estimation REMEMBER: we are interested in knowing the properties of the PROCESS. We need to estimate them from the pattern - typically we only get one pattern with which to estimate them. n n A ˆ � � K ( r ) = w ij I (0 < d ij < r ) n ( n − 1) i =1 j =1

Testing Inference is typically performed through hypothesis testing: H 0 : the process is CSR vs H A : the process is not CSR For this we need a test statistics and its distribution under the null (CSR). r {| ˆ T = max L ( r ) − r |} . Lagache et al, Analysis of the Spatial Organization of Molecules with Robust Statistics, PLOS One, 2013.

Testing 0.03 0.03 0.03 0.02 0.02 0.02 0.01 0.01 0.01 L(r)-r 0 0 0 -0.01 -0.01 -0.01 -0.02 -0.02 -0.02 -0.03 -0.03 -0.03 0 0.01 0.02 0.03 0 0.1 0.2 0.3 0 0.1 0.2 0.3 r r r