Visualisation of uncertainty

Kai-Mikael Jää-Aro

Why is this important?

Visualising uncertainty

Means and MethodsScalar dataVector dataVolume data

Generic methods

Let us talk about the weather

A weatherforecast is based on sensor readings, whichare fed into numerical models, the outputfrom which is used to draw weather maps.

Uncertainty and error

At each step in the forecasting process, errors creep in:I Sensors have limited resolution, readings contain noise, their

positions may be uncertain, sampling is sparse in time andspace.

I Converting the raw data to suitable input for numerical modelsmay involve averaging, interpolation, resampling.

I The numerical models are approximate, discrete computationintroduces errors, some algorithms increase errors.

I Visualisation of computation results introduces quantisationerrors, data are interpolated, additional numerical integrationmay be used.

To help interpret the results we should indicate the level ofuncertainty and error in the visualisation.

How can we describe uncertainty?

An uncertainty parameter can be given as (for example):statistic The probability distribution of data.

range Minimum and maximum values for data.error Estimated or known differences between true and

computed values of data.In addition, data may be missing.

Visualising uncertainty

We can treat uncertainty as simply another variable to bevisualised, but it is preferrable if the viewer has to do as few mentaltransformations as possible to understand the image.We have to use the “free parameters” of our visualisation space toindicate the uncertainty, this may occasionally conflict with theindication of other variables.We need to consider how easily the user will interpret what isshown as uncertainty, and how easy it is to detect the quantitativevalues of uncertainty?Do we wish to emphasise uncertain parts of our data set, or shouldthese be downplayed?

One-dimensional scalar data1D scalars have the most effective and well-used methods forindicating uncertainty. (Why should this be?)Simple scatterplots will show data distribution.

We can gain further understanding by indicating statisticalproperties in a box-and-whisker plot.

This plot also indicates the individual data values. The box oftenindicates median and 25th and 75th percentiles, the whiskers 10thand 90th percentiles, but could equally well be average, ± 1 and 2standard deviations, or 90 and 95% confidence intervals orwhatever is meaningful for the statistic.

We can add more statistical indicators by making the box shapemore complex.

On the other hand, box-and-whisker plots do not give anyindication of how successive data ranges are related to each other.We may want to see envelopes of the ranges.

Note that we are introducing uncertainty in the visualisation step.

2D scalar data

As data move up in dimension, we get fewer dimensions in which toshow uncertainty. One option is to use colour.A difference image with a good colour scale shows areas with smalldeviations in black and large deviations in brighter colour.

− =

Over a surface we can indicate increasing uncertainty of data valuesby e g decreasing the saturation or increasing the transparency, butnote that this changes the colour, which may affect itsinterpretation, especially for shaded 3D surfaces.

Vector data

Vectors can have uncertainty in magnitude and direction, whichboth have to be shown. We can generalise standard vectorvisualisation methods to show uncertainty. We will show methodsfrom C. M. Wittenbrink, A. T. Pang, and S. K. Lodha. Glyphs forvisualizing uncertainty in vector fields. IEEE Transactions onVisualization and Computer Graphics, 2(3):266–279, Sept. 1996.

We often use arrow-shaped glyphs to show vector fields, with theobvious mappings for magnitude and direction.

A number of possible generalisations of arrow glyphs are possible.They have to be tested to see which one(s) behave best in a realsituation.

A version where the width of the arrowhead shows the uncertaintyof direction and multiple arrowheads show the uncertainty ofmagnitude makes the glyphs too large.

Remapping so that area indicates magnitude gives better-behavedglyphs.

A side-by-side comparison of “uncertain” and “certain”visualisations.

We can adapt stream ribbons and stream tubes to show uncertaintyin vector fields.

Texture-based methods can be generalised to show uncertainty, e gby “smearing” the texture in proportion to the uncertainty at thatpoint.

R. P. Botchen, D. Weiskopf, and T. Ertl. Texture-basedvisualization of uncertainty in flow fields. In IEEE Visualization2005. IEEE, IEEE Computer Society Press, 2005. URLhttp://doi.ieeecomputersociety.org/10.1109/VIS.2005.97

Volume dataThe simplest way of rendering volume data is by isosurfaces.“Fat surfaces” directly show the range of possible shapes of anisosurface.

Following G. Grigoryan and P. Rheingans. Point-based probabilisticsurfaces to show surface uncertainty. IEEE Transactions onVisualization and Computer Graphics, 10(5):564–573, Sept./Oct.2004 we will look at several methods of isosurface uncertaintyvisualisation.Problem: Finding the boundaries of tumours. Setting a distinctboundary may miss the edges where the tumour blends in withnormal cells.

We begin by looking at a simple isosurface.

We may attempt to indicate uncertainty by colour coding. Butprecisely how uncertain are these surfaces?

Instead, let us show uncertain surfaces by an uncertain surface.Make the surface of points instead of polygons, displace the pointsby a random amount proportional to the uncertainty in that region.

Instead of points, we may use lines to directly show the range ofuncertainty. They can be rendered with varying opacity dependingon the certainty value along the line.

We can speed up rendering and improve the image by combiningpolygonal surfaces and points or lines.

All this can be enhanced with colours showing other parameters,such as cell age.

Volume visualisation

We may also perform direct volume visualisation. FollowingS. Djurcilova, K. Kim, P. Lermusiaux, and A. Pang. Visualizingscalar volumetric data with uncertainty. Computers & Graphics, 26:239–248, 2002, we can map uncertainty to transparency.

Generic methods

Many methods are common for all types of data.Showing the range of possible values in side-by-side displays is oftenthe easiest for the designer, if not always for the viewer.

Animating through the range of values is another option. In thisexample the time at a given value indicate the probability of thatvalue.

If we only know, or care about, the extremes of the range, flickeringbetween them is effective.

Using other modalities

If we have difficulty with fitting in the uncertainty information inavailable dimensions, we can try using haptic or aural presentation.Both these require active probing by the viewer/listener/toucher;this is not necessarily a problem.

Useful links:

I http://www.csee.umbc.edu/~rheingan/I http://www.cse.ucsc.edu/research/slvg/unvis.htmlI http://www1.elsevier.com/homepage/sad/cageo/cgvis/

ehlschl/paper.htmI http://www.geovista.psu.edu/sites/icavis/icavis/

ICAvis_working.html