Aquanims: Area-Preserving Animated Transitions in Statistical Data Graphics based on a Hydraulic Metaphor
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Aquanims: Area-Preserving Animated Transitions in StatisticalData Graphics based on a Hydraulic Metaphor
Michael Aupetit † Qatar Computing Research Institute, HBKU, Doha, Qatar
Abstract
We propose "aquanims" as new design metaphors for animated transitions that preserve displayed areas during thetransformation. Animated transitions are used to facilitate understanding of graphical transformations between differentvisualizations. Area is a key information to preserve during filtering or ordering transitions of area-based charts like barcharts, histograms, tree maps or mosaic plots. As liquids are incompressible fluids, we use a hydraulic metaphor to conveythe sense of area preservation during animated transitions: in aquanims, graphical objects can change shape, position, colorand even connectedness but not displayed area, as for a liquid contained in a transparent vessel or transferred between suchvessels communicating through hidden pipes. We present various aquanims for product plots like bar charts and histograms toaccomodate changes in data, in ordering of bars or in number of bins, and to provide animated tips. We also consider confusionmatrices visualized as fluctuation diagrams and mosaic plots, and show how aquanims can be used to ease the understandingof different classification errors of real data.
Categories and Subject Descriptors (according to ACM CCS) : I.3.3 [Computer Graphics]: Picture/Image Generation—Line andcurve generation
1. Introduction
Visualization can be used to discover new information or tocommunicate findings. Discovery often requires changing pointsof view, filtering data, modifying the models, selecting differentmetaphors. Communicating requires explaining the process thattransformed the data into insights. In both cases, transformationsoccur between the visualizations. Animated transitions are key tokeep the user understanding the displayed information during thesephases, linking seemingly different representations.A recent survey [CRP ∗
16] showed that animated transitionscan help keep the information in context, be used as a teach-ing aid, improve user experience, support the visual discourseor enhance data encoding for discovery. However animations arenot always guaranteed to improve the visualization performances[HR07] [CDF14]. For instance too many objects moving can bedisturbing and prevent tracking the objects of interest, or occlusionsmay hide some objects during the transition.In statistical graphics, data exploration and understanding aresupported by specific visual metaphors like bar charts, scatterplots,heatmaps or node-link diagrams. We focus on area-based charts † email: [email protected] which use the size channels to encode the primary data. In thesecharts, the area of a graphical element is proportional to the un-derlying count, proportion or probability. Many of them have beenencompassed in the product plot framework [WH11] related tostatistics. Preserving area during the transition is key to complywith the congruence principle [TMB02], i.e.to maintain graphicsvalidity during the transition, and semantic-syntactic mapping us-ing similar metaphors for similar transitions. Minimizing occlusionand maximizing predictability are two other important laws of ani-mations [HR07].However, animated transitions between different area charts dis-tort these areas: complete or partial occlusion modify the perceivedarea; very standard linear interpolation between rectangles with dif-ferent aspect ratio change their area; changing the number of binsof an histogram while maintaining their total area through a contin-uous transition is not trivial too.In this work, we propose a physical metaphor to guide the designof area-preserving animated transitions. It is based on hydraulics.First, because liquids are incompressible fluids that are likely toconvey the sense of volume preservation projected as areas on ourretina. Second because we are very used to manipulate liquids inour everyday life: pouring milk in a glass, filling a sink, emptying aplastic bottle, transferring water between buckets or in a water tank,drinking a soda with a straw... not to mention experimental physics submitted to EUROGRAPHICS 200x. a r X i v : . [ c s . H C ] J a n Michael Aupetit email: [email protected]
Qatar Computing Research Institute, HBKU, Doha, Qatar / Aquanims area-preserving animated transitions at school or in professional environments. We call aquanims theanimated transitions based on the proposed hydraulic metaphor.In section 2, we review the related work. In section 3 we presentthe animation principles based on the hydraulic metaphor and howthey are related to the guiding principles of good animations. Insection 3, we present the hydraulic metaphor and the aquanims fun-damental principles. In section 4 we describe the building-blocksto be used in section 5 to create aquanims for various area-basedcharts. In section 6, we use it to display different facets of a confu-sion matrix to analyze the results of a classification problem. Beforewe discuss in section 7 and conclude in section 8.
2. Related work
The reference work of [HR07] investigates the effectiveness of an-imated transitions between statistical data graphics and connectsthe guiding principles of good visualizations to the ones for goodanimations proposed by Tversky [TMB02]. In particular, transi-tions should avoid occlusion; maintain graphical encoding at eachand every interpolation state; use same semantic-syntactic mappingacross the different visualizations; make change only of the neces-sary syntactic features (expressiveness) to make the animation bestsupport (efficiency) the communication of the semantic changes;use staging when transition is complex, and make it short. Theseauthors also propose a typology of animations, in this work weexplore all the types except the view and substrate transformationones. Wickham and Hofmann [WH11] proposed the product plot framework which encompasses many area-based charts. We usethis framework to design the animated transition between a fluctu-ation plot and a mosaic plots used to visualize confusion matrices.Using metaphors is common in visualization [ZK08] to influencethe representation of information in the mind, like Furna’s gener-alized Fisheye views for instance [Fur86].
Visual Sedimentation isa design metaphor inspired by the physical process of sedimenta-tion. It is especially suited to represent data stream. New data en-ter the visualization as bubbles attracted toward buckets, and pro-gressively accumulate and create strata in form of color-coded areacharts which summarize past data and progressively compress theinformation stream. We are not aware of a metaphor based on liq-uids or hydraulics for information visualization.Animated transitions have been used to explain and support dis-covery with statistical graphics. For instance ScatterDice [EDF08]uses a pseudo rotation of the axes to navigate a scatterplot matrix.Pairing a complex graphics to a simpler one with animated transi-tions has been proposed in [RM15] to help learn how to use com-plex graphics. In LineUp [GLG ∗ aquanims Hydraulics is the branch of science and technology concerned withconveying liquids through pipes and channels, and to use it as a source of mechanical force or control. Typical hydraulic technol-ogy connects cylinders, tanks and pumps with pipes. As liquids arenearly incompressible, they keep their volume constant in the sys-tem, and they can transmit forces between distant containers con-nected by non-elastic pipes. Liquids have a mass so are subject togravity forces, leading to equal surface level equilibrium in possi-bly distant but communicating vessels, as well as in natural artesianwells. Now we detail the analogy between graphics and hydraulicsand derive the fundamental principles of aquanims . aquanims As tanks and vessels in hydraulic systems, area charts have contain-ers like bars or tiles of which the area is measured. We will focuson bars and tiles of rectangular shape with axis-parallel edges inthe sequel, although the analogy is not limited to these shapes andorientations and could be used for pie charts for instance. Bars andtiles are usually the graphical objects whose area must be preservedas they encode the primary data.However in our physical analogy we consider that the equivalentto a bar or a tile is actually the liquid which takes the form of itsgraphical container. By so doing we transfer the semantic of thearea of the graphical container to the one of the graphical fluid itcontains. So we can change the shape of a set of containers, movethe liquid from one container to another, or even let it be sharedbetween several containers without changing its total volume.Thus we derive the very first fundamental principle underlying aquanims : First principle: the liquid encodes the data.
Data are mappedto the fluid content rather than to its container.The second fundamental principle is obvious from the physicalanalogy:
Second principle: the liquid volume is constant , so is its areain the graphical representation. Pipes and containers can change theliquid’s shape but cannot change its total area.As a main design guiding principle: aquanims should be de-signed so as to emphasize the hydraulic metaphor evoking liquid-coded data (First principle), in order to increase the perception ofarea invariance (Second principle).As a consequence, because the container is not encoding datadirectly, the liquid itself must bear some identification mark. Weuse the color channel for this sake (texture could be used also) tofill the rectangles or color its edges.Now we focus on the aquanims building-blocks that simulatehydraulics. aquanims building-block We illustrate the different building-blocks that support the designof aquanims in the figure 1.We distinguish between changing the position (translation) orsize of a container in the underlying space, or the view (pan, zoom)and scale (rescaling) of the underlying space itself, and changing submitted to EUROGRAPHICS 200x. ichael Aupetit email: [email protected]
Qatar Computing Research Institute, HBKU, Doha, Qatar / Aquanims area-preserving animated transitions the container shape (resize, reshape). The latter involved aquanims while the former do not. To comply with animation principles ofcongruence and consistent semantic-syntactic mappings, we distin-guish graphically pan, zoom and rescaling which involve animationof the axes and background and associated changing displayed sizeand position of all the containers and liquids they contain, whiletranslation, resize or reshape occur with no background changes. aquanims modify size, shape and fill using additional graphical el-ements evoking the hydraulic metaphor and area-preserving prop-erty. A container can be filled or emptied . We represent the containerby an empty rectangle and the liquid by a partial filling of it asa smaller rectangle having 3 sides shared with the container. Theanimated transition uses a slow-in/slow-out speed based on a cubicbasis function: u ( t ) = t − t for a regular sequence of values t ∈ [ , ] .In order to comply with the predictibility principle of animatedtransitions, for emptying animations, the final level of the liquid inthe container is indicated by a target line segment throughout thetransition. The initial level being explicitly given by the containeredge. For filling animation, the edges of the final container is dis-played and progressively filled by the liquid, while the initial levelis displayed with a line segment as a reminder. Notice that in thefigure 1 we use bright colors to emphasize edges for explanationsbut when applying aquanims, colors shall be adapted to the paletteused.This filling/emptying animation is identical to the one of standardprogression bars, or volume bars in digital equalizers. The initialand final line segments help the user perceive the transition process,remember the initial state and predict the final one.However this animation alone violates the second principle of aquanims that the liquid area must be preserved, so this building-block is not an aquanim per se but it enables the construction ofaquanims based on communicating containers or segments (seesections 4.4 and 4.5. The liquid in a container can be shifted , i.e. translated along thecontainer axis, so the rectangle representing the liquid is translatedwhile the container remains fixed in the underlying space (Figure1 in intersection of yellow and green frames). This animation isarea-preserving so is an aquanim . A container can be reshaped so the position of more than oneedge is changed (2D brown frame in the figure 1. Compared to thefilled/emptied animation, at least 2 edges oriented in orthogonal di-rections are changing position. As a consequence, the increase inone dimension must be compensated for by a decrease in the otherto preserve area (Figure 2 top).We use a hydraulic cylinder metaphor for these aquanims (Fig-ure 2 center): a bounding box encompassing both the initial and final rectangles is displayed to represent the cylinder; The movingedges under control are prolonged toward the edges of the cylinder,to represent pistons; And remaining edges are either fixed in con-tact to the cylinder, or free edges joining two pistons or a piston andthe cylinder.It must be noticed that a linear interpolation between initial andfinal positions of the rectangle’s vertices does not preserve area(Figure 2 bottom). We linearly interpolate (L code) one of the di-mensions while we compute the remaining one using a hyperbolicinterpolation (H code) which maintains the area of the rectangleinvariant during the transition.The figure 3 show that the 10 cases described in the figure 1right, reduce to 4 cases ignoring symmetry between x and y rectan-gle axes: L . H . , L . HH . , LL . H . and LL . HH . where L and H denotelinear and hyperbolic interpolations respectively, and are duplicatedwhen two edges are animated this way. We assume that linear in-terpolation is more predictible than hyperbolic one, and so as anefficient design guiding principle, the lower the number of movingedges ( L . H . preferred), and the lower the number of H -type onesamong them ( LL . H . preferred to L . HH . ), the better the perceptionof the hydraulic metaphor and so of the area-preserving property. Incases where more than 3 moving edges are required (bottom rows),we might advise to use a staged animation adding a translation ofthe rectangle before or after one of the cases (.c) or (c.).The general formula for these area-preserving animations are: l i ( t ) = ( − u ( t )) l i ( ) + u ( t ) l i ( ) and H ( t ) = A / L ( t ) with t ∈ [ , ] . l i is the position of the piston edge i along theaxis orthogonal to it and controlled directly by the slow-in/slow-outspeed transition. L ( t ) = | l i ( t ) − l i ( t ) | and H ( t ) = | h j ( t ) − h j ( t ) | are the dimensions of the rectangle at step t . h j is the position ofthe free edge j along its orthogonal axis and controlled by the area-preserving condition regarding the area A = L ( ) H ( ) = L ( ) H ( ) of both initial and final rectangles. When two edges follow a hy-perbolic interpolation, we can find the h j ! ( t ) and h j ( t ) values byadding a constraint to center the rectangle along the H -animateddimension: c ( t ) = ( − u ( t )) c ( ) + u ( t ) c ( ) where c ( t ) = ( h j ( t ) + h j ( t )) / Two or more containers can be connected by invisible pipes allow-ing continuous liquid transfer between them during the animation(see Figure 1 purple frame). Any amount of liquid which disap-pears from one container must be distributed among the other ones,so the total sum of variation of the connected containers is zero,maintaining the total amount of liquid at a constant value.We have this formula between variation δ k of levels within con-nected containers of respective width w k : ∑ k w k δ k = λ k ( ) and target λ k ( ) levels in each contain-ers are such that total area A = A ( ) = ∑ k w k λ k ( ) = ∑ k w k λ k ( ) = submitted to EUROGRAPHICS 200x. Michael Aupetit email: [email protected]
Qatar Computing Research Institute, HBKU, Doha, Qatar / Aquanims area-preserving animated transitions
Communicating containers/segments Isolated containersTransfer Shift t’ t’ Empty Fillt t’ t’ t t’ (lb) (lc) (lt) (cb) (cc) (ct) (rb) (rc) (rt)Controlled liquidHidden pipe Flow
MovementFinal level Not aquanims due to changing areas
Shift t t’ t’ t … (ex) Figure 1:
Building-blocks of aquanims. Filled gray rectangles show the initial state, and blue empty rectangle the final state of the animations.Both rectangles have the same area except empty/fill animations in the blue frame depicting non aquanims per se but useful blocks to buildthe communicating containers and segments (purple frame) aquanims. All transitions can be reversed. Liquid can move along the containeraxis (1D green frame) or in both dimensions (2D brown frame). Isolated containers (dark yellow frame) have no hidden pipes. The liquid iseither shifted along the container axis (1D) or reshape (2D) by reshaping the container itself as described in more detailed in figure 2 and 3. A ( ) is constant for these two states, then for any linearly interpo-lated state λ k ( t ) = ( − u ( t )) λ k ( )+ u ( t ) λ k ( ) , the area is preserved A ( t ) = ∑ k w k λ k ( t ) = A .So linear interpolation with possible slow-in/slow-out speed orany other speed scheme ensures the second principle of aquanims . In a single container, two or more segments of the controlled liquidseparated by segments of other liquids can be connected by hiddenpipes following the same rules as communicating containers (seeFigure 1 purple frame). The liquid transferred in the pipe changesthe local segment area but preserves global area. The other liquidsare shifted along the container.In the next section we will present different aquanims we de-signed based on these building-blocks to animate histograms, andbar graphs. aquanims design case study Product plots [WH11] is a framework which encompasses manyarea-based charts. We picked some of the most well known ones toillustrate the design of aquanims . We present the case of verticalbars but it is trivial to adapt the animation for horizontal ones. Let-ters refer to the figure mentioned in each section. We developed theanimations using R programming language and Shiny frameworkfor web interfaces, with the ggplot2 and animation
R packages.
A histogram is a type of bar chart with continuous x and y scales,where bars are adjacent to evoke the continuous nature of the x- axis (if vertical bars). We consider the case of equal width bars,which define contiguous intervals on the x-axis. The area of onebar accounts for the proportion of data falling in that interval orbin, the total area of the bars being 1. All our animations are staged,starting and ending with a pan and zoom phase, at first to adapt thesize of the plot so that the subsequent aquanim fits entirely withinthe graphic area, then at the end to maximize the size of the plot inthis area.
Adapt to changing data
We consider the case of adapting an histogram to data filteringwhere some data can appear and some other disappear (Figure 4).The area of the histogram must remain constant.We use a plain red hue for more data in a bar, and plain blue forless data in a bar (b,c,d). The red rectangles are to be filled by redcolor during the aquanim using the single container fill-block ani-mation and the blue rectangles have to be emptied using the empty-block animation. During the filling, we make the gray color of allthe bars reddish or bluish (c,d) to evoke the over or under pressurein terms of total area which transiently becomes higher or lowerthan 1 depending on the overall surplus or lack of data. Then werescale all the bars (d,e,f) so the area comes back to unity and thereddish or bluish color disappears (f).
Adapt to changing number of bars
Histograms are statistical models which approximate the underly-ing data density. There is no straightforward way to select the bestnumber of bars: too many it becomes spiky, and to few all the de-tails are smoothed, but the user might want to test different numbersso we propose an aquanim to smoothly transition between the re-sults. submitted to EUROGRAPHICS 200x. ichael Aupetit email: [email protected]
Qatar Computing Research Institute, HBKU, Doha, Qatar / Aquanims area-preserving animated transitions Linear interpolation not preserving area t t t’ t’ t’ Reshape aquanim
Controled edge moving by linear interpolation of positions at t’ and t’ Fixed edge
What is displayedModel underneathInitial state Final state
Free edge moving to preserve area
Hyperbolic trajectory preserving area
Figure 2:
Design details of Reshape aquanims following a hydraulic cylinder metaphor with cylinder in black, piston in red and free liquidsurface in green. Notice how the free surface is bounded by the piston and the piston bounded by the cylinder (see also figure 3). Top rowshows the aquanim as displayed. A first step add the bounding box of both initial and final state rectangles and the side which is controled(middle vertical line, red color in the second row) is interpolated linearly between initial (t (cid:48) ) and final states (t (cid:48) ). Then this elements areremoved to render the final state. The underneath model (second row) show that here two sides of the inital rectangle are fixed (black dots),one is directly controlled (red line) and one is moved (green line) under the preserving-area constraint. The corners of the reshaped rectanglefollow hyperbolic trajectories (light blue dashed lines). The bottom row shows that linear interpolation (purple box and dashed line) betweeninitial and final positions of the vertices does not preserve area. Adding bars could be done one by one, but would require todecide arbitrarily where to add the new bar (left, right, middle?).It is also not consistent with the semantic-syntactic mapping: barsare discretizations of a continuous axis, so the incremental addi-tion of whole bars hides the continuous nature of the underlyingdata. Adding the bars from one side incrementally also generatesa sliding effect, giving the erroneous impression of a zoom out orrescaling of the x-axis. The same issues occur when removing barsone by one.We use the communicating containers transfer block (Figure 4).We superimpose the contours of the original (light gray) and target(dark gray) histograms. Thin bars can be merged into large bars or alarge bar be split into the thin bars it overlaps. A linear interpolation(a,b,c) between initial and final levels within each bar of the inter-section is used for the transition. We also tried a more physicallyrealistic approach, where we made adjacent bars as communicatingvessels, and set up an adaptive formula that makes each level con-verging toward the average level of its two neighbors, normalizingthe area at each step. The result can be seen in the supplementalmaterial. How realistic the liquid dynamic should be to optimizethe area-preserving perception has not been investigated.
Show animated tip
The idea here is to show the user what proportion of the total dataa set of bars represents (Figure 6). As all the bars may have differ- ent heights, this is not easy to evaluate visually from the originalhistogram.We color the selected bars in yellow to distinguish them from therest (a). We empty the yellow liquid into the bottom of all the bars,raising all them up to preserve the total area (b). We then releasethe gray liquid to equalize the level of the grey bars (c), so the finalgraphic is a vertical stacked bar with a yellow segment at the bot-tom and a gray one at the top (d), where the sought proportion ofselected data is directly visible. This aquanim tip can be designedwith several colors to several sets of bars selected by some cluster-ing algorithm for instance.
Bar charts are bars that stand on a categorical or discrete ordi-nal axis, and whose height maps a continuous variable. Bars canbe segmented to distinguish levels of some independent category.These segments can be stacked on top of each others or groupedside by side. Heer and Robertson studied different transitions be-tween stacked and grouped bars in [HR07].
Stacked bars vertical reordering
We consider the typical task of reordering the segments of stackedbars, as only the bottom ones along the x-axis benefit from thiscommon straight line to ease the comparison of their amount acrossthe levels of the categorical variable mapped to this axis (Figure 7(a,d)). submitted to EUROGRAPHICS 200x.
Michael Aupetit email: [email protected]
Qatar Computing Research Institute, HBKU, Doha, Qatar / Aquanims area-preserving animated transitions (cc) (LxHy) (HxLy)(lc)(rc) (HHxLy)(LLxHy)(ct)(cb) (LxHHy) (HxLLy)(lc)(rt)(rb)(lb) (LLxHHy) (HHxLLy) 2 moving edges3 moving edges4 moving edgesL: linear H: hyperbolic x/y: movement direction(ex)
Figure 3:
Typology of reshape aquanims. The legend and left codescorrespond to the ones given in figure 1. The left column shows thedetailed intermediary state interpolated between initial and finalstates for each of these cases except for (ex) which can be similarto any of the ones of the bottom row except the dark gray and blueboxes do not cross. The middle column shows a schematic repre-sentation of the detailed cases showing which edges are linearlyinterpolated (red) or follow a hyperbolic trajectory (green) to pre-serve area. The two sub-columns show alternative designs wherethe piston moves either along x (left) or y (right) axes. L and H let-ters code for the linear and hyperbolic interpolations respectively,used along the axes. Double LL and HH mean two edges movingin that way. The right column indicates the number of edges mov-ing. We conjecture that the best aquanims in terms of perceptionof area-preservation are the ones for which a minimum number ofedges are moving (first row) or at most a single edge is movingwith a hyperbolic interpolation (cases HxLLy and LLxHy preferredin case of 3 moving edges)
We use the communicating-segments-shift block directly, andapply it to all the bars at once. The liquid in the segments of the se-lected color-coded level progressively fills the bottom part of theirbar (b,c), shifting up the intermediate segments until the originalones vanish from their initial position (d). The selected segmentsare split, preserving their total area in each bar while avoiding oc-clusion that standard animated translations would generate. (a) (b) (c)(d) (e) (f)
Figure 4:
Aquanim of a histogram for adding (red) and deleting(blue) data values (b). In (c),(d) and (e) the bluish color of thebars indicates that the total density is lower than 1 (summing upadded and removed data), the aquanim resizes the bars (e) to finallycompensate for that lack of data (f). (a) (b)(c) (d)
Figure 5:
Aquanim for a decreasing number of histogram bins.From (a) to (d), levels of intersecting bins are linearly interpolated.Reading from (d) to (a) shows the aquanim for increasing numberof bins.
Stacked bars horizontal reordering
We propose an aquanim to translate a bar from an initial to a finalposition without occlusion (Figure 8).We first create an empty space at the final position between levelsD and E (a,b), with the same label G as the initial. Then we use thecommunicating-containers-transfer block, progressively emptyingthe original bar segment by segment starting from the bottom, andfilling the destination bar at the same time by the liquid comingfrom the initial one (c,d). Then we remove the space let by theempty bar at the initial position (e,f). submitted to EUROGRAPHICS 200x. ichael Aupetit email: [email protected]
Qatar Computing Research Institute, HBKU, Doha, Qatar / Aquanims area-preserving animated transitions (a) (b)(c) (d) Figure 6:
Aquanim tip to show the proportion of the total repre-sented by the selected bars (yellow). The animation starts in (a) upto (d) then after a pause or on user action, come back from (d) to(a). (a) (b)(c) (d)
Figure 7:
Aquanim of segments in vertical bars to move the se-lected ones (magenta) at the bottom.
Here again the split of the transferred segments between two barsavoid occlusion and let invariant the visible area of each segmentduring the transition. The same animation could be used to transferbars in a grouped bar chart. Transferring multiple bars at the sametime might be confusing, a staggered animated transition could beused in that case [HR07]. (a) (b) (c)(d) (e) (f)
Figure 8:
Aquanim to reorder vertical bars. The bar G is moved be-tween D and E using the communicating- containers-transfer block.
None Mild SevereNone 1458 48 78Mild 205 102 144Severe 85 34 1666
Table 1:
Confusion matrix
The data table representation of a con-fusion matrix gives all classification results details but fails to givean overview and focus rapidly on the anomalies. The rows are pre-dicted classes, and the columns are observed ground truth classes.For instance there are images with no damage but predictedshowing severe ones.
6. Case study: animation of a confusion matrix
We consider classification results obtained from an automatic clas-sifier. Access to situation-sensitive data through social media net-works provided images of damages after natural disasters. Rapiddamage assessment is important for many humanitarian organiza-tions. A Convolutional Neural Network has been used to classifythese images into 3 categories:
None , Mild and
Severe . CITE PA-PER OF DAT
The confusion matrix shown in Table 1 displays the number ofoccurrences of correct and erroneous predictions of the model as-sessed on an image data set with known ground truth labels pro-vided by human volunteers.A confusion matrix is a contingency table that can be read in dif-ferent ways [WH11] [Fri94]. Each number in a cell counts the datahaving a specific pair of observed (true) and predicted labels. Thisnumber over the total of all numbers in the matrix accounts for theproportion of such cases that we expect to happen in real condi-tions when the classifier is used to classify new unseen data. Thisproportion is an estimator of the joint probability p ( class o , class p ) that class o is observed and class p is predicted.Classifiers are sensitive to class imbalance. For instance we can submitted to EUROGRAPHICS 200x. Michael Aupetit email: [email protected]
Qatar Computing Research Institute, HBKU, Doha, Qatar / Aquanims area-preserving animated transitions
None Mild SevereOBSERVEDNoneMildSeverePREDICTED
Figure 9:
A fluctuation plot to visualize the confusion matrix givenin the table 1. read from the table that about 200 data are from the Mild damageclass, while about ten times more are images of no damage andsevere damage. If the humanitarian organizations have a specificprogram to run for each type of damage, it is important that theclassifier be reliable for any type. Here we expect it will not begood to detect Mild damages.We use a fluctuation diagram to visualize the confusion matrixas in the figure 9. This encodes the numbers as squares whose areasare proportional to the joint probabilities. Squares’ fill color codesfor the predicted class, and edges’ color for the observed class. Wealso use a fluctuation diagram to display the marginal proportions.The area of the yellow filled square on the top right out of the innerframe map the total amount of images predicted with no damagewhatever the ground truth. The area of the gray square at the bottomright account for the total number of images so its area gives theprobability unit.We can decompose the joint probability into the marginal p ( class p ) and conditional p ( class o | class p ) probabilities, all relatedby the Bayes rule: p ( class o , class p ) = p ( class o | class p ) p ( class p ) .Then we see as studied in depth in [WH11], that the areaof the squares in the fluctuation diagram can be decomposedinto the product of a marginal probability and a conditionalone. The joint probability can be conditioned on the predictedclasses as above, or on the observed classes: p ( class o , class p ) = p ( class p | class o ) p ( class o ) We can use a mosaic plot to visualizethese quantities more efficiently. The Bayesian rule ensure than thearea of corresponding cells in both plots is identical, but the shapeof the cells in the mosaic plot are rectangles whose edge length di-rectly encode the marginal and conditional probabilities of interest.We design an aquanim to make clear the relation between thetwo plots.
The animation is displayed in the figure 10. We first move all thesquares on the right side of the plot (a) . The total area of squares (a) (b)(c) (d)(e)
Figure 10:
Aquanim of the confusion matrix from the fluctuationdiagram to the mosaic plot in the first row
N p (predicted None) is equal to p ( No , N p ) + p ( Mo , N p )+ p ( So , N p ) = p ( N p ) . Doing the same for each row, wesee that summing the area of each square in the right margin equals p ( N p ) + p ( M p ) + p ( Sp ) = aquanim tip to show that identityequation by transferring the liquids from the 3 marginal squares inthe right margin inside the gray square below showing they fill itexactly. The same is true for the ones in the bottom margin.We use the single-container reshaping block to reshape the 3squares of the first row into a rectangle with unit length and height p ( N p ) (b)(c). We do the same for the other two rows. By forc-ing the height of these rectangles, we force the incompressible liq-uid say of the left most square to match the p ( N p ) height and soto spread along this unit length band in proportions x such that: p ( N p ) ∗ x = p ( No , N p ) its original invariant area. By Bayes rule x = p ( No | N p ) so we get the Bayes decomposition we need foreach segment in these bands.Each horizontal band is then piled up (d). All three form a squaremosaic plot of unit length edges, of same size as the gray square.We resize this mosaic plot (e,f) to fit in the full matrix so it is easierto read. submitted to EUROGRAPHICS 200x. ichael Aupetit email: [email protected] Qatar Computing Research Institute, HBKU, Doha, Qatar / Aquanims area-preserving animated transitions Now two patterns become obvious: the blue class is predictedabout ten times less than the other ones. The segments of the bluebar are roughly in proportions 50%, 20%, and 30%, so given theblue Mild class has been predicted, there is only 20% chance thatthere are actually Mild damages.
7. Discussion
We proposed aquanims for all types of transitions described in[HR07] except for the view transformations like pan or zoom whichare global or relative area-preserving respectively, and substratetransformations (non-linear scaling) whose generated space dis-tortions make more difficult aquanims design. In the latter case,we should define first what area preserving means in these non-euclidean spaces.We did not explore aquanims with treemaps, but our study ofmosaic plots involving the need for free space between the tilesshowed that designing aquanims to animate changes in denselypacked treemaps would be challenging. A possibility would beto explode the tiles to get free space between them for reshapingsome of them with aquanims then collapse them back in final posi-tion. It seems possible to design aquanims with Voronoi treemaps[BDL05] as the cells’ geometry is less constrained by rigid align-ments as in standard treemaps or mosaic plots. In that case howeverthe challenge would be to evoke the hydraulic metaphor with par-tially filled Voronoi cells for instance.In communicating-containers or communicating-segmentsblocks (Sections 4.4 and 4.5), the decreasing area of some cellssynchronized with the increasing area of other ones so the sum oftheir areas is invariant, seems to be key. The role of this dynamicfeature to support the perception of area-preservation is still tobe investigated. In isolated-container-reshape blocks (Section4.3), the role of numbers and types of moving edges is also to beinvestigated.We also saw that color can be used to code for non area pre-serving animations in combination with aquanims (Section 5.1)so color coding compensate for that issue REPRENDRE CETTEPHRASE.
8. Conclusion
We presented aquanims animations based on a hydraulic metaphor.The first principle of aquanims states that counts or proportion dataare mapped to liquids volume instead of containers in area-basedcharts. The second principle states that liquids are incompressiblesupporting the design of aquanims as area-preserving animatedtransitions. We present aquanims building-blocks and show howthey can be used to design various aquanims that avoid occlusionsand preserve areas.How effective are these animations for the user remains to beevaluated. Studying how aquanims could be used in treemaps andother non-rectangle area-based charts is let as a future work. [BDL05] B
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