This article demonstrates two GIS methods for visualisation of uncertainty associated with spatial prediction of continuous and categorical variables. In the case of continuous variables, the key issue is to visualise both predictions and the prediction error at the same time, while in the case of categorical data, the key issue is to visualise multiple memberships and confusion in-between them. Both methods are based on the Hue-Saturation-Intensity (HSI) colour model and calculations with colours using the colour mixture (CM) concept. The HSI is a psychologically appealing colour model � hue is used to visualise values or taxonomic space and whiteness (paleness) is used to visualise the uncertainty. In the case of continuous variables, a two-dimensional legend was designed to accompany the visualisations � vertical axis (hues) is used to visualise the predicted values and horizontal axis (whiteness) is used to visualise the prediction error. In the case of categorical variables, a circular legend is used � perimeter (hues) defines the taxonomic space and radial distance represents the confusion. The methods are illustrated using two examples: (a) interpolation of soil thickness using regression-kriging and (b) fuzzy k-means classification of landforms classes. For more details, see the following references:

Hengl T., Heuvelink G.M.B., Stein A. 2004. A generic framework for spatial prediction of soil variables based on regression-kriging. Geoderma 122(1-2): 75-93.

Hengl T., Walvoort D.J.J., Brown A. 2004. A double continuous approach to visualisation and analysis of categorical maps. Int. Jou. of Geographical Information Science, 18(2): 183-202.

Hengl T. 2003. Visualisation of uncertainty using the HSI colour model: computations with colours. 7th International Conference on GeoComputation (CD-ROM), pp. 8.

	ILWIS scripts for visualisation of prediction error Download and unzip to "C:\Program Files\ILWIS 3.3 Academic\Scripts\"
	R script for visualisation of prediction error (whitening) Download and install R, download and unzip the scirpt to a working directory.
	ILWIS scripts and sample datasets for visualisation of multiple memberships (209 KB)

DO-IT-YOURSELF

STEP 0: Get the most recent version of ILWIS and install it on your PC. Detailed explanation of GIS operations and ILWIS commands can be found in the ILWIS help documentation or user's guides (both available on-line). If you need more details on how to create and run a script, I advise you to read the ILWIS 3.0 Academic user's guide chapter 12. The ILWIS script consists of set of commands that can be used with up to nine script parameters. These can be either spatial objects, values or textual strings. A script, in principle, consists of two parts: definition of script parameters and list of commands. Sign "//" is used to exclude to insert comments and explanation of formulas.

I. VISUALISATION OF PREDICTION ERROR

There are two options to visualize uncertainty of predictions: by using colour mixing (VIS_error) or by using pixel mixing (VIS_sim). See also Hengl et al. (2002) for more details. For pixel mixing, you will also need to derive simulated values (equiprobable realisation) instead of using the predictions. Follow this guide to learn how to produce realisations using the same prediction model. The advantage of pixel mixing is that the final map will show both how significant is the short-range variation (nugget variation) and the areas where the model is less certain.

STEP 1: Download the and unzip the ILWIS scripts for visualization of prediction error, best in in the default directory of scripts (C:\Program Files\ILWIS 3.3 Academic\Scripts\).

STEP 2: Run a script from the to left menu (operations list) or from the main menu -> Operations -> Scripts. Use the help button to find more information about the algorithm. See example of how to enter parameters for the script.

STEP 3: If you are using colour mixing, you will need to use this legend (legend2D.tif) to create final maps.

TIPS:
* You can manually change the lower and upper values for both prediction and error maps. I advise you to stick to the 0.4 and 0.8 (max 1.0) as the default values for the normalized prediction error values. A satisfactory prediction is when the model explains more than 85% of the total variation (normalized error = 40%). Otherwise, if the values of the normalized error gets above 80%, the model accounted for less than 50% of variability at the validation points and the prediction is unsatisfactory. For more details, see the publications above.

Figure: Visualisation of uncertainty for a quantitative variable (topsoil thickness in cm) interpolated using regression kriging: uncertainty included with whiteness and the accompanying two-dimensional legend.

Figure: Visualisation of uncertainty using simulations and pixel mixing: (a) prediction map, (b) simulated values using the same prediction model, (c) visualization using punctual estimates and (d) block estimates.

VISUALISATION OF UNCERTAINTY USING WHITENING IN R

Here you can obtain the script to visualize the prediction error directly using R computing environment. While in R, you will need to first obtain packages rgdal, maptools and colorspace. Note that the results might slighlty differ between ILWIS and R, which is mainly to somewhat different HSI-RGB conversion algorithms.

 

VIS_sim Purpose of this script is to simultaniosly visualize one realisation and the estimated prediction error. Input parameters: %1.mpr - simulated values, %2 - prediction error, %3 - original sampled variance. Note that the results of visualization can often be quite discouring, depending on the amount of variation explained by the model. To improve the precision of predictions, use block-kriging or try to improve the prediction model.

II. VISUALISATION OF MULTIPLE MEMBERSHIPS

STEP 1: Download and unzip the ILWIS scripts and sample dataset for visualisation of multiple memberships in some working directory (e.g. d:\ILWIS_maps\). This is the Wesepe dataset used in the paper of De Gruijter et al. (Geoderma, 1997).

Figure: Six membership maps for five soil types (c1 � c5) and one extragrade class
(cx). Pixel size is 20 m. The soil classes (Dutch National soil classification system)
are: c1 � beek earth soil (coarse); c2 � enk earth soil; c3 � podzoil soil; c4 � beek
earth soil (fine); c5 � reworked soil; cx � extragrades;.

STEP 2: Select the representation colours and visualise the multiple memberships:

a) Select colours freely and visualize the memberships using the defuzzification (Simple_RGB).
//Created by: T. Hengl ([email protected])
//Input parameters: %1-%8 memberships, %9 - domain where the class names and colours are defined
//Purpose: Calculates the MB class having the highest MB value

Figure: The highest membership per pixel - defuzzification. Colours selected freely.

b) Select the colours freely and use the Pixel Mixture technique (PM_RGB). Note that you can also simulate values and produce an animation such as the one down-below.
//Created by T. Hengl ([email protected])
//Input parameters: %1-%6 membership maps, %7 - domain where the class names and colours are defined
//Purpose: script to calculate mixed pixels, six memberships: c1, c2, c3, c4 ,c5 cx; needs the georeference; domain set at {dom=value;vr=0.000:1.000:0.001}

Figure: 10 simulations of visualisation using the pixel mixture technique.

b) Select the colour representations using some objective method - see Hengl et al. (IJ GIS, 2004). You can then also use the colour wheel that reflects geometrical position of class centres in the attribute space. In this case you first need to estimate the Hue values at the colour wheel for each class (see table HSI_RGB) and then you can run the script (Cont_RGB).
//Created by: T. Hengl ([email protected]), last update June 2004.
//Input parameters: %1-%6 - membership maps; representation values are in the table "HSI_RGB";
//Purpose: Calculation of the R G B bands based on the membership values and representation values; use zero map for missing maps: zero.mpr=iff(isundef(%1),?,0); no need to calculate the sumMB if it is = 1

Figure: Visualisation of multiple memberships using pixel mixture: (a) mixed colours
without any adjustment and (b) mixed colours adjusted for the
thematic confusion with circular legend indicating similarity between
the categories.

 

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