Uncertainty and Parameter Space Analysis in Visualization -

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1 Uncertaint and Parameter Space Analsis in Visualiation - Session 4: Structural Uncertaint Analing the effect of uncertaint on the appearance of structures in scalar fields Rüdiger Westermann and Tobias Pfaffelmoser Computer Graphics & Visualiation Specific literature [1] C. Johnson and A. Sanderson, A net step: Visualiing errors and uncertaint, Computer Graphics and Applications, IEEE, vol. 23, no. 5, pp. 6 10, [2] A. Pang, C. Wittenbrink, and S. Lodha, Approaches to uncertaint visualiation, The Visual Computer, vol. 13, no. 8, pp ,1997. [3] T. Pfaffelmoser, M. Reitinger, and R. Westermann, Visualiing the positional and geometrical variabilit of isosurfaces in uncertain scalar fields, in Computer Graphics Forum, vol. 30, no. 3. Wile Online Librar, 2011, pp [4] T. Pfaffelmoser and R. Westermann, Visualiation of global correlation structures in uncertain 2d scalar fields, in Computer Graphics Forum, vol. 31, no. 3. Wile Online Librar, 2012, pp [5] T. Pfaffelmoser and R. Westermann, Correlation Visualiation for Structural Uncertaint Analsis, in Journal of Uncertaint Quantification, 2012 Facult of Informatics Session 4: Structural uncertaint Part 1: Structural uncertaint Part 1: Structural uncertaint [3] [5] Stochastic modelling of uncertaint Variance as an uncertaint indicator Limitations of variance as an uncertaint indicator Definition of structural uncertaint and correlation Part 2: Local and global correlation visualiation [4] [5] Requirements and challenges Local correlation tensor Glph based correlation visualiation Global correlation clustering Concluding remarks Uncertain scalar fields Uncertain scalar field Y: M R on compact domain M R n Discretel sampled on nodes i I with values Y i I We assume a stochastic uncertaint model: Y i I are (correlated) Gaussian distributed random variables 1, 2, 3... Described b a multivariate Gaussian distribution 1 p = ep ( 0.5 μ T Σ 1 ( μ)) 2π detσ with mean μ = (μ1,, μn) and covariance matri Σ Y i = μ i is the most likel configuration Multivariate Gaussian distribution Uncertaint parameters in the multivariate case The degree of uncertaint is modeled b the Σ 1 p = 2π detσ ep ( 0.5 μ T Σ 1 ( μ)) For 3 grid points: Σ = 2 σ 1 σ 12 σ 13 σ 12 2 σ 2 σ 23 σ 13 σ 23 2 σ 3 σ i 2 = var Y i = E Y i μ i 2 σ ij = Cov Y i, Y j = E( Y i μ i (Y j μ j )) 1

2 Multivariate Gaussian distribution Compute μ i, σ ij from a given ensemble of data sets Ensemble members are treated as realiations of a corresponding multivariate random variable ensemble μ i σ i Specif μ i, σ ij and map independent normall distributed random number vectors to a corresponding realiation Σ Gaussian covariance matri Variance information is often used as primar indicator of the uncertaint (square of the standard deviation σ) Σ = 2 σ 1 σ 12 σ 13 σ 12 2 σ 2 σ 23 σ 13 σ 23 2 σ 3 Measure of the amount of variation of the values of a random variable Often visualied directl via confidence regions, uncertaint glphs, or specific color or opacit mappings Variance as uncertaint indicator Eample: an ensemble of 2D seismic tomograph wave velocities Relative velocities are color coded from blue (negative) to red (positive)) ensemble μ i σ i Variance as uncertaint indicator Uncertaint mapping in 3D via Stochastic Distance Functions μ i θ Ψ θ i ma σ i, σ min Mahalanobis distance: in numbers of standard deviation to the level- surface in the mean field ( Two prominent circular features are observed in the mean values Mapping the standard deviations to colors shows low, and almost constant uncertaint in both regions SDF surfaces are level-sets in the Mahalanobis distance field, enclosing the volume which contains the uncertain surface with a certain probabilit For Gaussian distributions, the less standard deviations an observation is from the mean, the higher the probabilit of this observation Variance as uncertaint indicator Uncertaint visualiation for an iso-surface in the mean values of a 3D temperature ensemble Variance as uncertaint indicator Uncertaint visualiation for an iso-surface in the mean values of a 3D temperature ensemble Color coding of stochastic distance Color coding of Euclidian distance along normal curves [3] Iso-surface in the mean values of a 3D temperature field Positional variabilit of the surface due to uncertaint 2

3 The standard deviation describes the local uncertaint but does not allow inferring on possible variations at different positions relative to each other e.g., if mean values and standard deviations at two adjacent points are identical, but the realiations of the random variables at both points behave independentl, it cannot be predicted whether there is a positive, ero, or negative derivative of the data between the two points In general, the effect of uncertaint on structures which depend on random values at multiple points cannot be predicted from the mean values and standard deviations alone The effect is to a large etent arbitrar if the realiations of the random values are stochasticall independent, while a structure s shape can be assumed stable if the realiations are stochasticall dependent Variations of an uncertain curve due to uncertaint Curve points Y i = Y i are modelled via a multivariate Gaussian random variable with smoothl varing mean values (green curve) and a constant standard deviation (blue curves show corresponding confidence interval) realiations realiations In a) and b), high and low stochastic between the values at adjacent i was modelled between scalar values was modelled eplicitl : between scalar values was modelled eplicitl : (5) along (5) along between scalar values was modelled eplicitl : between scalar values was modelled eplicitl : (5) along (5) along 3

4 between scalar values was modelled eplicitl : between scalar values was modelled eplicitl : (5) along (5) along between scalar values was modelled eplicitl : between scalar values was modelled eplicitl : (5) along (5) along between scalar values was modelled eplicitl : between scalar values was modelled eplicitl : (5) along (5) along 4

5 Random values in the 3D scalar field shown before were generated using a multivariate Gaussian distribution with constant standard deviation and linearl increasing mean (along ) Conclusion: just b looking at the standard deviation it is impossible to infer on the stabilit (or confidence) of structures in the mean values A structure is ver likel to change its shape due to uncertaint, if the stochastic between the data values making up the region is low A structure is rather unlikel to change (it is stochasticall stable), if the stochastic between the data values making up the structure is high, even if the standard deviations are high (a) Iso-surface in the mean values (b) containing all points that belong to the surface with a certain probabilit (c) Occurrence of the surface in (a) for one possible realiation of the random values Thus, one important goal in uncertaint visualiation is to conve the stochastic and conclude on the stabilit of structures in the data Structural uncertaint We call the effect of uncertaint in the data values on structures depending on the values at multiple points a structural uncertaint It is associated with the occurrence of particular structures in the data which are affected b the degree of between the values at two or more data points Indicators for structural uncertaint are given b the subdiagonals of Σ, which contain covariance information on relative uncertainties between random variables Structural uncertaint For Gaussian distributed random variables, the stochastic of random values at two points is given b the correlation: ρ ij = Σ ij Σ ii Σ jj = σ ij σ i σ j, 1 ρ ij 1 Positive correlation: ρ ij > 0 Σ = 2 σ 1 σ 12 σ 13 σ 12 2 σ 2 σ 23 σ 13 σ 23 2 σ 3 Stochastic in: ρ ij = 0 Negative/ correlation: ρ ij < 0 Variabilit of a scalar value around its mean position: Variabilit of a scalar value around its mean position: 5

6 Variabilit of a scalar value around its mean position: Variabilit of a scalar value around its mean position: Variabilit of a scalar value around its mean position: Variabilit of a scalar value around its mean position: Variabilit of a scalar value around its mean position: Correlation describes relative variabilit of one or more random variables Standard Deviation Confidence Interval Mean Position Positive Correlation Negative Correlation 6

7 Correlation describes relative variabilit of one or more random variables Correlation describes relative variabilit of one or more random variables Positive Correlation Negative Correlation Positive Correlation Negative Correlation Correlation describes relative variabilit of one or more random variables Part 2: Local and global correlation visualiation 2.1 Local Correlation Visualiation Positive Correlation Negative Correlation Structural uncertaint Structural uncertaint Remember the eample in Part 1: locall varing structural variations of an iso-surface in different realiations of an uncertain 3D scalar field Caused b different correlation structures Resulting in stochasticall stable and unstable surface parts Cannot be revealed b mean and standard deviation values High structural uncertaint Low structural uncertaint Problems addressed in this part of the tutorial: 1. How can one visualie correlations to anale the structural uncertaint of particular features in 2D and 3D scalar fields, given a set of realiations or a (Gaussian based) stochastic uncertaint model Two different approaches will be discussed, both aim at conveing correlation structures in the data: 1. Local correlation analsis using characteristic correlation tensors 2. Global correlation analsis showing longe-range dependencies such as inverse correlations via correlation clusters 7

8 Correlation Correlation Local correlation analsis Assumption of a distance dependent correlation model: higher correlations between the data values at points with shorter Euclidean distance Local correlation analsis Assumption of a distance dependent correlation model implies that correlation strength depends on grid resolution Weak correlation Strong correlation Domain of a 2D scalar data set Low correlation to 1-ring neighborhood High correlation to 1-ring neighborhood Distance dependent correlation model The Gaussian correlation function relates correlation to spatial distance Distance dependent correlation model Correlation strength parameter τ relates correlation to distance and is independent of grid resolution for local correlation analsis 1 τ 1 τ 2 τ 3 correlation strength parameter 1 τ 1 τ 2 τ 3 0 Euclidean distance 0 Euclidean distance Correlation anisotrop Problem: Correlation values and thus correlation strength parameters depend on direction, ielding a directional correlation distribution at each grid point τ 2 τ 3 τ 1 Correlation strength tensor Idea: The (anisotropic) distribution of the correlation strength parameter is modelled b a rank-2 tensor T The correlation strength tensor T models the correlation strength to 8 (2D) or 26 (3D) neighbors Given T, correlation strength τ at position into direction r can be computed via tensor-vector multiplication A single correlation value can not represent the stochastic s! 8

9 Correlation strength tensor Given the tensor T, the distance and direction dependent correlation becomes Correlation strength tensor computation T is smmetric and has 3 (2D) or 6 (3D) free values to be determined T models the correlation strength to 8 (2D) or 26 (3D) neighbors T is obtained for ever grid point b solving an over-determined linear sstem using a least squares approach At ever grid point a correlation tensor is computed i.e., correlation data is transformed into a distance dependent tensor model Reduces memor requirments because it avoids storing (global) correlation values eplicitl T is independent of the grid resolution, because correlation is set in relation to Euclidean distances Correlation visualiation on iso-surfaces Goal: Visualiing the structural uncertaint of specific features in the mean data, i.e. an iso-surface Requires correlation visualiation with respect to the mean surface s geometric shape Correlation visualiation on iso-surfaces Approach: Correlation values are etracted for the two principal correlation directions in the surface s tangent plane (highest and lowest strength) and normal direction [4] Basis transformation is necessar to obtain correlation strength parameters in principal tangential directions (τtan _ma) and (τtan _min), and normal direction (τnormal) at ever surface point B interactivel specifing an Euclidean radius d, correlation values along the three principal directions are computed as T Idea: Distinguish between correlations in the surface s local tangent planes and normal direction, and visualie these correlations via glphs and used to model the appearance of circular correlation glphs Iso-surface Correlation glphs Correlation glphs encode the correlation values in the first and second pricipal tangent and surface normal direction Correlation glphs Correlation ratio between first and second tangent direction is encoded in the shape of the elliptic one (2) Glph coloring wrt correlation in surface Glph coloring wrt correlation in surface (1) normal direction (2) first principal tangent direction (3) second principal tangent direction (1) normal direction (2) first principal tangent direction (3) second principal tangent direction 0 Correlation 1 0 Correlation 1 9

10 Correlation glphs Absolute and relative correlation anisotrop in the surface tangent plane are encoded via color differences Structural uncertaint analsis Surface realiation and glph based correlation visualiation on mean surface 0 Correlation 1 Surface realiation Correlation glphs Structural uncertaint analsis Structural uncertaint analsis High and low correlation in vertical and horiontal direction High and low correlation in horiontal and vertical direction Structural uncertaint analsis Structural uncertaint analsis Low correlation in both vertical and horiontal direction Low correlation especiall in normal direction 10

11 Structural uncertaint analsis Structural uncertaint analsis Mean surface in a simulated temperature ensemble Glph based correlation visualiation Structural uncertaint analsis Local correlation analsis summar Local distance dependent correlation tensor model allows memor reduction to O(n) for n grid points Visualiation of local correlation anisotrop possible Simultaneous visualiation of correlation ratios as well as absolute correlation values Region with high structural uncertaint Interactive analsis of anisotrop in surface tangent plane and between tangent plane and surface normal direction Local correlation analsis summar No integration of absolute uncertaint information (e.g. standard deviations) Part 2: Local and global correlation visualiation 2.2 Global correlation visualiation Global s are ignored correlation cannot be visualied as correlation model onl accounts for correlation strengths (magnitudes) 11

12 Cardinal number Global correlation analsis To anale long-range spatial dependencies between the data values in uncertain scalar fields, in principle the full covariance matri needs to be visualied This is problematic due to the following reasons For n spatial grid points it requires O n 2 memor Entries in Σ are not in a spatial conte, making it difficult to infer on spatial dependencies Uncertaint information has to be visuall separated from correlation information Σ Global correlation analsis Requirements due to the aforementioned problems The correlation information first has to be condensed Requires to define and seek for the most prominent short- and long-range dependencies Correlation structures have to be embedded into standard visualiations providing spatial relationships Approach Definition of a measure for the degree of dependenc of a random variable to its local and global spatial surrounding Spatial clustering of random variables based on this measure Color coding of data points wrt. cluster IDs Eample data set 2D temperature ensemble simulated b the European Centre for Medium-Range Weather Forecasts Correlation strength indicator Assumption: the more correlation partners a particular point in the domain has to which the correlation is larger than a given level, the more important this point is The sets of partners spread anisotropicall across the domain The epansion in different directions is directl related to the correlation distribution in the respective region Mean values Mean values as heightfield Standard deviations For a given level and point i, the number of partners indicates the degree of dependenc between the random variable Y( i ) to its local and global spatial surrounding It counts the most prominent partners of i, independent of their position in the domain The number of partners is used as the correlation strength indicator Correlation strength indicator Correlation strength indicator Definitions Correlation neighborhood: η ρ + i j D ρ Y i, Y j ρ + } D: data domain ρ 1 : correlation level Eample showing relation between correlation neighborhoods and cardinal numbers high Cardinal number: η ρ + i Correlation neighborhoods of 2 grid points Cardinal numbers low 12

13 Cardinal number Cardinal number Spatial correlation clustering Correlation clustering algorithm Basic idea: use correlation neighborhoods as clusters Problem: correlation neighborhoods of different points can overlap Results in ambiguities in the assignment of points to clusters Solution: process correlation neighborhoods sequentiall and eclude those neighborhoods which overlap an previousl processed neighborhood The strateg avoids overlapping clusters but, depending on selection order, will eclude large clusters which overlap small cluster Thus, neighborhoods are selected in descending order of cardinalit such that the largest clusters are alwas selected first Step 1: Select correlation level ρ + Step 2: Compute cardinal numbers for all points in the 2D domain Step 3: Select domain point with largest cardinal number as first cluster center Step 4: Create cluster (correlation neighborhood) high Mean values Cardinal numbers low Correlation clustering algorithm continued Correlation clustering results Step 5: In the set of points not et assigned to a cluster, search for point p with highest cardinal number whose correlation neighborhood c does not overlap an eisting cluster Step 6: Create cluster using p and c and repeat with Step 5 generate custers for different levels ρ + (multilevel clustering) high Visualiation of correlation clusters and cluster centers on the mean height surface for a particular correlation level ρ + Mean values Cardinal numbers low Correlation clustering results for varing levels dependencies stochastic dependencies tpicall eist between spatiall separated regions; ie. pairs which are inversel correlated to each other correlation structures are global features and cannot be modeled b distant dependent correlation models

14 dependencies stochastic dependencies tpicall eist between spatiall separated regions; ie. pairs which are inversel correlated to each other correlation structures are global features and cannot be modeled b distant dependent correlation models dependencies stochastic dependencies tpicall eist between spatiall separated regions; ie. pairs which are inversel correlated to each other correlation structures are global features and cannot be modeled b distant dependent correlation models dependencies stochastic dependencies tpicall eist between spatiall separated regions; ie. pairs which are inversel correlated to each other correlation structures are global features and cannot be modeled b distant dependent correlation models correlation clustering algorithm Step 1: Select negative correlation level ρ Step 2: Compute correlation neighborhoods (ρ Y i, Y j ρ ) and corresponding cardinal numbers for all points Step 3: Select point with largest cardinal number as first cluster center correlation clustering algorithm Step 1: Select negative correlation level ρ Step 2: Compute correlation neighborhoods (ρ Y i, Y j ρ ) and corresponding cardinal numbers for all points Step 3: Select point with largest cardinal number as first cluster center Step 4: Assign negativel correlated points to first cluster correlation clustering algorithm Step 1: Select negative correlation level ρ Step 2: Compute correlation neighborhoods (ρ Y i, Y j ρ ) and corresponding cardinal numbers for all points Step 3: Select point with largest cardinal number as first cluster center Step 4: Assign negativel correlated points to first cluster Step 5: Select point in cluster with largest number of inversel correlated points 14

15 correlation clustering algorithm Step 1: Select negative correlation level ρ Step 2: Compute correlation neighborhoods (ρ Y i, Y j ρ ) and corresponding cardinal numbers for all points Step 3: Select point with largest cardinal number as first cluster center Step 4: Assign negativel correlated points to first cluster Step 5: Select point in cluster with largest number of inversel correlated points correlation clustering algorithm Step 1: Select negative correlation level ρ Step 2: Compute correlation neighborhoods (ρ Y i, Y j ρ ) and corresponding cardinal numbers for all points Step 3: Select point with largest cardinal number as first cluster center Step 4: Assign negativel correlated points to first cluster Step 5: Select point in cluster with largest number of inversel correlated points correlation clustering algorithm Step 1: Select negative correlation level ρ Step 2: Compute correlation neighborhoods (ρ Y i, Y j ρ ) and corresponding cardinal numbers for all points Step 3: Select point with largest cardinal number as first cluster center Step 4: Assign negativel correlated points to first cluster Step 5: Select point in cluster with largest number of inversel correlated points Step 6: Assign both clusters to one pair correlation clustering algorithm Step 1: Select negative correlation level ρ Step 2: Compute correlation neighborhoods (ρ Y i, Y j ρ ) and corresponding cardinal numbers for all points Step 3: Select point with largest cardinal number as first cluster center Step 4: Assign negativel correlated points to first cluster Step 5: Select point in cluster with largest number of inversel correlated points Step 6: Assign both clusters to one pair Step 7: Proceed with Step 3, taking into account that clusters do not overlap dependencies Each pair of inversel correlated spatial regions gets assigned one distinct color Clusters in each pair are distinguished b differentl oriented stripe patterns correlation dependencies for varing negative levels ρ

16 Eample: An uncertain 2D scalar field with constant means and standard deviations. Specific orrelation structures have been enforced: Strong positive correlation within the sets of points colored blue. Zero correlation between these sets correlation between the two sets in the upper left region Mean surface Random realiation Random realiation Random realiation 16

17 Random realiation Random realiation Random realiation Random realiation Random realiation Random realiation 17

18 Correlation clustering algorithm identifies positivel correlated regions correctl Correlation clustering algorithm identifies inversel correlated regions correctl Validation Validation Global correlation analsis summar Correlation clustering allows analing global correlation structures such as inverse correlations, requiring an amount of memor that is linear in the number of data points The clusters distributions reveal anisotropic correlation effects Uncertaint information can be integrated easil, for instance, b etruding clusters depending on standard deviation [5] Etension to 3D is possible, but special projection or restriction schemes are required when used to anale correlation structures on iso-surfaces Future work and challenges in correlation visualiation Approaches for visualiing correlation structures in 3D Correlation analsis/visualiation for other data tpes (e.g. vector fields) Integration of correlation information in eisting uncertaint visualiation approaches (e.g. positional uncertaint of features) Quantification of the effect of correlation on the occurence of differential quantities and higher order features, like critical points Modeling and interpretation of stochastic dependencies in non- Gaussian distributed random fields 18