2107.04974

Commonly n-D points are mapped to 2-D points for visualization, which can approximate n-D information partially preserving it in the form of similarities between n-D points by using MDS, SOM, t-SNE, PCA and other unsupervised methods [6,11,12,14]. Moreover, both n-D similarities injected by these methods and their 2-D projections for visualization can be irrelevant to a given learning task [1]. In [12] t-SNE is used to visualize not only the input data but also the activation (output) of neurons of a selected hidden layer of the MLP and CNN trained models. This combines unsupervised t-SNE method with the information learned at this layer. However, projection of the multidimensional activation information of the layer to 2-D for visualization is lossy. It only partially preserves the information of that layer. Moreover, the layer itself only partially preserves n-D input information. Thus, some information that is important for classification can be missed.

An alternative methodology is mapping n-D points to 2-D graphs that preserves all n-D information [1,2,5,13]. Elliptic Paired Coordinates belong to the later. In both methodologies the respective visual representations of n-D data are used to solve predictive classification tasks [7].

The advantages of EPC include: (1) preserving all n-D information, (2) capturing nonlinear dependencies in the data, and (3) requiring fewer visual elements than methods such as parallel and radial coordinates. This chapter expands our prior work [10] on EPC in: (1) successful testing the EPC approach and methodology in additional experiments with both real and simulated data, (2) generalizing EPC to the Dynamic Elliptic Paired Coordinates (DEPC), (3) generalizing EPC and DEPC by incorporating the weights of coordinates to optimize the visual discovery in EPC/DEPC, alternating side ellipses and introducing the concept of incompact machine learning methodology based on EPC/DEPC.

The chapter is organized as follows. Section 2 describes the concept of elliptic paired coordinates. Section 3 presents a visual knowledge discovery system based on the elliptic paired coordinates. Section 4 describes the results of experiments with multiple real data sets. Section 5 presents experiments with synthetic data. Section 6 provides generalization options for elliptic paired coordinates and Section 7 concludes the paper with its summary and future work.

In EPC coordinate axes are located on ellipses (see Fig. 1). In [1] the EPC shows in Fig. 1 is called EPC-H, in this chapter we omit H. In Fig. 1, a short green arrow losslessly represents a 4-D point P = (0.3,0.5,0.5,0.2) in EPC, i.e., this 4-D point can be restored from it. The number of nodes in EPC is two times less than in parallel and radial coordinates with less occlusion of lines. For comparison see point P in Fig. 2 in radial and parallel coordinates with 4 nodes instead of 2 nodes in EPC. The dark blue ellipse CE in Fig. 1 contain four coordinate curves X1-X4. It is called the central ellipse. Fig. 1 shows only one of the options how we can split an ellipse or a circle to the segments. Here each coordinate starts at the horizontal red marks on the right or on the left edges of the central ellipse, where X1, X4 go up and X2, X3 go down from respective points. An alternative sequential way of directing coordinates is that X1 starts from the top, ends in the right middle point, where X2 starts and go down to the bottom point, where X3 starts as so on. This is a simpler way when we have more than 4 attributes to be mapped to the ellipse. In general, the direction of each coordinate can be completely independent of the directions of the other coordinates. While all these possibilities exist for EPC, we use either one which is shown in Fig. 1 It means that we can reverse these steps to restore the 4-D point P from (P1→P2) arrow. Visually we slide the red right-side ellipse to cross point P1. Its crossing point with the central blue ellipse is x1 = 0.3. Similarly, we get x2 = 0.5 with the right-side blue ellipse. The same process allows restoring P2, with red and blue left-side ellipses.

Below we present steps of EPC algorithm for n-D data referencing Fig. 1 and notation introduced in it:

1. Create the central ellipse, CE, with a vertical line M bisecting it. 5. Pair data points by dimension, starting at (x1, x2). If there is an odd number of pairs of coordinates, use the horizontal bisector N for the middle pair instead of the vertical bisector M.

2. Create the red and blue side ellipses, such that they touch the line M. Have the top of red ellipses to intersect the data point xi on the central ellipse and the bottom of blue ellipses to intersect the next data point on the central ellipse and then alternate between red and blue for each coordinate in sequence.

3. Find the intersects between each pair, connect them.

Steps 6 of EPC algorithm requires constructing thin ellipses. Their width W and heigh H are equal to them for the central ellipse CE, but it requires computing their centers. Let point (A,B) be the center of such ellipse. It can be found by soling for (A,B) the general equation for an ellipse,

with given the W, H, x-and y-coordinates of the point on the ellipse. Then we can calculate the intercepts of the ellipses to form the points P1 and P2. Let the center of the central ellipse be the point (cx, cy) then formulas for A and B for side ellipses are as follows.

Right and Left Ellipses. The +/-assignment is dependent on the ellipse being to the right or left of the line M, respectively, 𝐴𝐴 = 𝑐𝑐𝑥𝑥 ± 𝑊𝑊 ∕ 2

The +/-assignment (red/blue respectively) depends on the ellipse intersecting at the top or bottom of the thin ellipse.

Top and Bottom Ellipses. If the number of coordinates n is even, but the number of pairs of coordinates n/2 is odd (e.g. 5 pairs from 10 coordinates), a horizontal line N across the center ellipse is used similarly to M for placing two ellipses for the selected pair (xi,xj). These ellipses touch line N (being above or below line N) and the intersect point of these ellipses is computed. If the selected pair is (xn/2, xn/2+1), e.g. (x5, x6) for n=10 then the bottom ellipse is constructed with

If the selected pair is (x1,xn), e.g. (x1,x10) for n=10 then then the top ellipse is constructed with

For the value A the +/-assignment depends on the ellipse being to the right or left of the line M, respectively.

The pseudo-code is as follows.

// Duplicate the last coordinate to make n even. Normalize(x(i)): x(i) = x(i) -min(i) / (max(i) -min(i)) // normalize each x(i) of // Coordinate Xi, where min(i) and max(i) are min and max for coordinate Xi in data.

For i= 1:n DrawDataPointOnEllipse(x(i)) // Draw data point value x(i) along CE’s axis on sector(i).

If n is even then Consider a set of n-D point visualized in EPC, a class Q and a rectangle R in the EPC, then the forms of the rule r that we explore in this chapter are:

A point rule captures a non-linear relation of 2 attributes that are encoded in the graph node in R. An intersect rule captures a non-linear relation of 4 attributes that form a line that crosses R. This line is the edge that connects two nodes of the graph x*.

We discover both types of rules using the DDR algorithm. The steps of the DRR algorithm are:

1. Visualize data in EPC; 2. Set up parameters of the dominant rectangles: coverage, dominance precision thresholds and type (intersect or point); 3. Search for dominant rectangles: 3.1. Automatic search: setting up the size of the rectangle, shift rectangle with a given step over the EPC display, compute coverage and precision for each rectangle, record rectangles that satisfy thresholds; 3.2. Interactive search: draw a rectangle of any size and at any location in the EPC display, collect coverage and precision for this rectangle; 4. Remove/hide cases that are in the accepted rules and continue search for new rules if desired.

The automatic search involves the dominance search parameters: coverage (recall in the class) and precision thresholds. A user gets only rules that satisfy these thresholds. We typically used at least 10% coverage and 90% precision.

In EllipseVis users can interactively conduct visual analysis of datasets in EPC and discover classification rules. EllipseVis is available at GitHub [3]. A user can more easily detect patterns to classify data by moving the camera, and hiding various elements of the visualization.

In the automatic mode EllipseVis can do calculations to find classification patterns. EllipseVis checks if most data points within the rectangle belongs to a single class when dividing the EPC space into smaller rectangular sections. We call such rectangles class dominant rectangles or shorter as dominant rectangles. Respectively rules that correspond to dominance rectangles discovered by EllipseVis are called dominance rules. The quality of rules is defined by the total number of points in the rectangle (rule coverage) and the percentage of dominant class compared to the others (precision of classification). In the interactive mode a user creates the dominance rules in EllipseVis. After a rule is created, the data points that satisfy the rule, i.e., inside the rectangle, are removed from the rest of the data and further rules are discovered without them.

• Camera move (pan and zoom) • Setting of a dominance rectangle: minimum coverage and precision; rectangle dimensions; • Toggle whether the rectangles are calculating data on their points or intersect lines;

Automatically find dominance rectangle rules, Clear rectangles. • Hide or show elements: intersect lines, dominance rectangles, cycle between showing all lines, lines not within rules, and line within rules, Side, thin red and thin blue ellipses.

The goal of experimentation is exploring efficiency of EPC on several benchmark datasets from UCI ML repository [4]. The rule r1 covers 52 cases (50 correct 2 misclassified cases), the rule r2 covers 48 cases (all 48 correct cases) and the rule r3 covers 50 cases (all 50 correct cases). These rules correctly classified 148 out of 150, i.e., 98.67%. Often each individual rule discovered in EPC cover only a fraction of all cases of the dominant class. For such situations we use the weighted precision formula to compute the total precision of k rules

where pi is precision of rule ri and ci is the number of cases covered by ri

Verification of discovered models is an important part of machine learning process. This section presents a simplified visual rule verification approach that has advantages over traditional k-fold cross validation (CV) is as we show with the Iris example.

A random split data to training and validation data to 10 folds is conducted in a common 10-fold CV approach to test the quality of prediction accuracy on the validation data. This random split does not guarantee that the worst-case split will be discovered and evaluated that is important for many applications with high cost or errors. In other words, cross validation can provide of the predictive model. EllipseVis allows to simplify or even eliminate cross validation.

Consider Iris data in Fig. 3 and randomly select 90% of lines to training and remaining 10% of them to validation data. Let these 90% of lines (training cases) cover the overlap area shown in Fig. 4, then we build rules r1-r3 as shows in Fig. 3 using these training data. The accuracy of these rules is 100% on validation data because these validation data are outside of the overlap area, confirming these rules.

In the opposite situation, when the training data do not include lines in the overlap area, but validation data include them, the discovered dominance rules (rectangles) can differ from r1-r 3. Rule r1 can be shifted lower or r2 can be shifted higher and misclassify some validation cases. These cases are misclassified because the training data are not representative for these validation data. This is a worst-case cross validation split of data to training and validation when the training data are not representative for the validation data.

The visual EPC representation allows to find and see the worst split, and use this worst split, say, for selecting 15 Iris cases (10%) in the overlap area to a worst fold that include all overlap cases. A trivial lower bound of worst-case classification accuracy of this fold is 0 when all cases that are in this fold are misclassified. In contrast 9 other folds of 10-fold CV will likely be recognized with 100% accuracy by a well-designed and trained rules/models when considered as validation folds, because these folds are outside of the overlap area. Thus, the average 10-fold CV accuracy will be 90%. So, finding accurate rules such as r1-r3 on the full dataset likely indicates that 10-fold CV result will be similar. Thus, instead of CV we can focus on finding visually the worst split and evaluate its accuracy on validation data [1]. It will likely be greater than 0 in general as it is the case for the iris data. Therefore, the 10-fold average will be greater than 90%. A more general and detailed treatment of the worstcase visual cross validation based on the Shannon function can be found in [9].

The goal of this experiment is testing EPC capabilities for the Wisconsin Breast Cancer (WBC) dataset [4] that consists of 683 full 9-D cases: 444 benign (red) and 239 malignant (green) cases as shown in Figs. 6789in EPC. To get an even number of coordinates we doubled coordinate X9 creating X10. Table 1 and Figs. 678show the automatically discovered rules in the EllipseVis system. Together five simple rectangular rules cover 96.34% of cases with weighted precision 95.13%. Fig. 9 shows an example where interactive rule discovery is less efficient than automatic requiring two times more rules and larger rectangles.

The 10-D multi-class Glass Identification dataset consists of 214 cases of 6 imbalanced classes of glass [4] used in criminal investigations by the U.S. Forensic Science Service. All other classes vs. class 5. Table 2 shows all three rules discovered. They cover 87.06% of cases of all other classes with weighted precision 98.29%. All other classes vs. class 6. Fig. 11 shows the result of rule discovery to separate cases of all other classes from class 6. The EllipseVis system found three rectangles, R1-R3. and respective rules r1-r3. Total all three rules cover 99.51% of cases with weighted precision 95.59% (see Table 3). 4). In addition, EllipseVis discovered rule r1 of class 7 vs. all others that covered 79.31% of all cases of class 7 with 91.30 % precision (see Table 5). These rules are intercept-based and respectively, capture the nonlinear relations between 4 attributes, which form a line that intersects the dominance rectangle), while point-based rules capture them between 2 attributes. 9 present the results for Abalone data [4] on full data and Table 10 shows in split 70%-30% to training and validation data for two classes (green and red). This experiment is to explore abilities to build EPC rectangular rules on a large skin segmentation dataset [4], which contain 245,000 cases of three dimensions and two classes. Below we present discovered rules for these data based on two methods: (1) the number of points within the rectangle and (2) the number of lines that intersects the rectangle.

We use the following notation: skin class (class 1, red) and non-skin class (class 2, green). In Figs. 17-20, the darker green color shows the cases of the green class which have already been included in previous rules. The current version of EPC requires an even number of attributes. Therefore, a new attribute x4 is generated to be equal to 1.0 for all cases on the normalized [0,1] scale. The rendering 245,000 cases in EPC is relatively slow in the current implementation: approximately 20 seconds, compared to less than one second for much smaller WBC data with less than 700 cases. Approximately 50,000 cases are skin class.

The discovered point-based rules and intersect line-based rules are presented, respectively, in Table 10 and Figs. 17-18, and Table 11 and Figs. 19-20. Fig. 17 and19 shows non-skin cases (green) in front of the skin cases (red) with automatically and sequentially discovered rectangles for green class rules. Figs. 18 and 20 show the opposite order of the green and red cases.

While the precision or rules for Skin segmentation data shown in Tables 10 and11 is quite high (94.62%, 97.78%), the coverage is relatively low (61.4%, 42%). It is likely related to the large size of this dataset (245,000 cases) where the data are hardly can be homogeneous to be covered by few rectangles, which cover over 10% of the data each.

The higher dimensions (Ionosphere data) is another likely reason why that data are nonhomogenous requiring more rectangles. Multiple imbalanced classes with some extreme classes which contain a single case (Abalone data) is another possible reason why that data are non-homogenous requiring more rectangles too. These issues suggest areas of further development and improvement for EPC. We explore generalizations of EPC in Section 6 to address these issues in the future work. The goal of this experiment is to explore the abilities of EPC to represent data as simple easy recognizable shapes like straight horizontal lines, rectangles, and others by using synthetic data.

Experiment S1: complex dependence. This experiment was conducted with a set A of 9 synthetic 8-D data points x1=(0.9, 0.1, 0.9, 0.1,…), x2 =(0.8, 0.2, 0.8, 0.2,…), …., x9= (0.1, 0.9, 0.1,0.9,…), which have complex dependencies within each 8-D points and between points. The non-linear dependence within each 8-D point is: These data are shown in EPC and parallel coordinates Fig. 21 where it is visible that their pattern is more complex in parallel coordinates than in EPC. Fig. 22 shows these 4-D points as lines of different colors in EPC and parallel coordinates. In parallel coordinates, 9 parallel lines show these 4-D points. These lines do not overlap and respectively simpler for visual analysis than EPC. This is expected because each of these 4-D points satisfy simple linear dependences, while EPC are designed for nonlinear dependencies. There is also a difference between EPC and parallel coordinates that is especially visible when x1 and x9 belong to the same class and we would like to have a visualization where they are close to each other in this visualization. Parallel coordinates do not do this. In Fig. 22, they are far away in parallel coordinates, while in EPC x1 and x9 are next to each other. Thus, this example shows that EPC allows visualizing cases of non-compact classes next to each other in contrast with parallel coordinates.

Since the graph is dependent on the central ellipse, changing the horizontal and/or vertical dimensions of this ellipse results in a stretched/shrunk graph. This helps to make the data points and lines more distinct in the respective axis being stretched, i.e. increased, and conversely compresses data for a decreased axis. Increased data point distinction within the ellipse is currently done by zooming in using the camera controls; effectively increasing both axes’ dimensions.

Experiment S3: complex dependence. For this experiment, we generated a set C of nine 4-D points x1=(0.1, 0.9, 0,1, 0.9), x2=(0.2, 0.8, 0.2, 0.8), x3=(0.3, 0.7, 0.3, 0.7), x4=(0.4, 0.6, 0.4, 0.6), x5=(0.5, 0.5, 0,5, 0.5), …, x9=(0.9, 0.1, 0,9, 0.1). This is an inverse dependence within each 4-D point. The relation within and between these 4-D points are more complex than in the set B. Fig. 23 shows these data in EPC and parallel coordinates The importance of this example is in the fact that all these 4-D points are located in the straight horizontal line producing a simple preattantive pattern. This is beneficial if all of them belong to a single class. In contrast in parallel coordinates on the right in Fig. 23 thue do not form a simple compact patter, but cover almost the whole area. In contrast with the Fig. 23 all these 4-D points are located vertically in EPC. If these points represent one class and points in Fig. 23 represent another class, then we easily see the difference between classes –one is horizontal another one is vertical overlap that can be observed quickly pre-attentively.

Fig. 6 also shows this data in parallel coordinates. The patterns in in parallel coordinates are more complex. Each 4-D point is represented as a polyline that consists of 4 points and three segments that connect them while in EPC each 4-D point requires two points and one segment to connect them. Next, the polylines in parallel coordinates in this figure are not straight lines and therefore cannot be observed pre-attentively.

In addition, these polylines cross each other cluttering the display. In contrast these data in EPC in Fig. 24 are simple straight horizontal lines that do not overlap that can be observed pre-attentively.

Comparing these data in parallel coordinates in Figs. 23 and 24 allows to see the difference clearly when these two images are put side-by-side. However, when all data of both classes are in the single parallel coordinates the multiple lines cross each other and the difference between classes is muted. In contrast in EPC the lines of two classes overlap only in single the middle point. This section presents two generalizations of elliptic paired coordinates: the dynamic elliptic paired coordinates (DEPC) and weights of coordinates. The first one allows to represent more complex relations and the second one allows making selected attributes more prominent that can reflect their importance.

In DEPC the location of each next point xi+1 depends on the location of the previous point xi. DEPC differs from EPC that is static, i.e., the location of points xi does not depend on the location of the other points. Fig. 1 shows a 4-D point x = (x1,x2,x3,x4) = (0.3, 0.4, 0.2, 0.6) in the dynamic elliptic paired coordinates. in Fig. 26 the value of x1 is located at 0.3 on the blue central ellipse. The value of x2 is located at 0.3+0.4=0.7 on the same blue ellipse starting from the origin, the value of x3 is located at 0.7+0.2=0.9, and the value of x4 is located at 0.9+0.6=1.5, i.e., each next coordinate value starts at the location of the previous point. Fig. 27 shows the same 4-D points in the static EPC.

The difference from the static EPC is that we do not need to create separate sections for each attribute on the EPC central ellipse, but start from the common origin at the top of the central ellipse (the grey dot in Fig. 26) and add value for the next attribute to the location of the previous one. Then the process of construction of points P1 and P2 is the same as in the static EPC. The advantages of DEPC is that it allows discovering more complex nonlinear relations than EPC. 3b shows the case when all ellipses go down and Fig. 28c shows the case when all ellipses go up. Respectively points P1 and P2 are differently located relative to the central ellipse as shows in these figures. Both points P1 and P2 are in the central ellipse in Fig. 3a, P2 is outside in Fig. 3b and P1 is outside in Fig. 3c. In Fig. 1 all side ellipses go up (centers above the xi point), and in Fig. 2 red ellipses go down, and blue ellipses go up with P1, P2 within the central ellipse, while in Fig. 1 only P2 is in the central ellipse.

EPC coordinates have been defined in section 2. Below we present alternative definitions. Such alternative definitions expand the abilities to represent n-D data in EPC differently opening an opportunity to find representation that will be most appropriate for particular n-D data and specific machine learning task on this data.

The EPC defined in Section 2 require an even number of coordinates. For the odd number of coordinates, we artificially add another coordinate either by copying one of coordinates or set up an additional coordinate be equal to a constant for all cases.

However, there is a way to visualize in EPC the odd number of coordinates as This idea of using a mixture of top, bottom, left and right ellipses is expandable to the situations with any number of coordinates odd or even. Some of these ellipses can be used for some coordinates while others for other coordinates in multiple possible combinations. The number of these combinations is quite large resulting in a wide variety of visualizations with an opportunity to optimize the visualization for both human perception and machine learning.

The introduction of weights of the attributes xi allows to build EPC visual representations where some attributes will be more prominent than others. A base option is multiplying every value of xi by its weigh wi with using wixi in EPC instead of original xi to build a visual representation of n-D point x. This option can fail when the value of wixi, will be out of the range of the ellipse segment assigned to the coordinate xi. It can also exceed the whole ellipse. This can happen when the values of weights are not controlled.

In a controlled static option, the length of segments of the ellipse associated with each coordinate xi are proportional to its weight wi. For example, consider, four segments for attributes x1-x4 with respective weigh wi as 4, 2, 6, and 5. Then 4/17, 2/17, 6/17 and 5/17 will be fractions of the ellipse circumference assigned to x1-x4, respectively. If x1=0.3 then it’s location xiw will be 0.3*4/17 fraction of the ellipse circumference with a general formula:

𝑥𝑥 𝑖𝑖𝑖𝑖 = 𝑤𝑤 𝑖𝑖 𝑥𝑥 𝑖𝑖 /(∑ 𝑤𝑤 𝑖𝑖 𝑛𝑛 𝑖𝑖=1

)

A controlled dynamic option is using the values xiw from (1) instead of xi in DEPC without creating separate segments for each coordinate.

Assigning and optimizing weights. Weights can be assigned by a user interactively or can be optimized by the program in the following way by first making all weights wi =1 and then adjusting them with steps δ(wi). After each adjustment of the set of weights, the program computes accuracy of classification and other quality indicators on the training data in search for the best set of weights that maximize the selected quality indicators. The adjustments can be conducted randomly, adaptively, by using genetic algorithms and others. The compactness of location of cases each class and far away from case of other classes is one of such quality indicators.

The two generalizations presented above open the opportunity for solving incompact machine learning tasks -tasks with cases of the same class that are far away from each other in the feature space. A traditional assumption in machine learning is the compactness hypothesis that points of one class are located of next each other in the feature space, in other words, similar real-world objects have to be close in the feature space [8]. This is a classical assumption in k-nearest neighbor algorithm, Fisher Linear Discrimination Function (LDF) and other ML methods.

Experiment S2 in Section 5 had shown an example of n-D points x1 and x9 that are far away from each other in the feature space are close to each other in EPC space. Thus, EPC has capabilities needed for incompact ML. The use of weights for attributes can enhance these capabilities. Thus, we want to find area A1 in the EPC where the cases of a one class will be compactly located in spite likely being far away in the original feature space and cases another class will concentrate in another area A2. Similarly, as we had shown in Figs. 18, 5 and 6 in section 5.

While we have these positive these examples, it is desirable to prove mathematically that EPC and DEPC have enough power for learning such distinct areas A1 and A2 with optimization of weights for the data with a wide range of properties. For instance, this can be data where each class consists of the cases that belong to multidimensional normal distributions located far away from each other.

The rectangles we discovered for real data in section 4 partially satisfy this goal. It is not a single rectangle for each class but several rectangles. Next, these rectangles contain only a part of each n-D point. In contrast the compactness hypothesis expects that full n-D points will be in some local area of the feature space. In other words, the compactness hypothesis assumes a space where each n-D point is a single point in this n-D space. In contracts in EPC each n-D point is graph in 2-D EPC space. Thus the compactness hypothesis in EPC space must differ from the traditional formulation requiring only a part of the graphs of cases of a class localized in the rectangle. Next the minimization of the number of rectangles is another important task, while reaching a single rectangle per class can be impossible. Also, these rectangles may not cover all cases of the class as we have seen in Section 4. All these rectangles are not in the predefined locations, because they discovered with fixed weights without optimization of weights. Optimization of weighs can make the compactness hypothesis richer allowing rectangles to be in the predefined locations by adjusting weighs combined with dynamic EPC.

The results of all experiments with EPC using the EllipseVis are summarized in Table 12 showing precision of rules from 91% to 100% with recall from 42% to 100% and 1-8 rules per experiment. They produced a small number of simple, visual rules which show cases clearly where a given class dominates. The end users and domain experts who are not machine learning experts can discover these rules themselves using EllipseVis doing end-user self-service. The visual process does not require mathematical knowledge of ML algorithms from users to produce these visual rules in the EllipseVis system. In addtion, the accuracy of the presented results are competitive with reported in [1,5] and exceed some of the other published resutls, while the goal is not competing with them in accuracy, but showing opprtunity for the the end users and domain experts to discover understandable rules themselves as self-service without programming and studying mathematical intricacies of ML algorithms. Another brenefit for the end-users is ablities to Find the visually the worst split of the data for training and validation and evaluating the accuracy of classification on this split getting the worst case estimate for algorithm accuracis it is demonstrated with the Iris data. Several options of the further development are outlined in section 6 that include generalization of elliptic paired coordinates to the dynamic elliptic paired coordinates, incorporating and optimizing weights of these coordinates and intorducing incompact machine learning tasks. As development and evaluation of EllipseVis and EPC continues, we will be able to gain a better understanding of EPC’s capabilities to construct a data visualization for visual knowledge discovery.

GLASS RULES FOR CLASS 7 VS. ALL OTHERS (INTERSECT-BASED).

.5.Experiment with Car data Figs. 12, 13 and Table6show the results for Car data with total coverage

.5.Experiment with Car data Figs. 12, 13 and Table6

.5.Experiment with Car data Figs. 12, 13 and Table

.5

.

Fig 14. Ionosphere data with rules r1 (green) and r3 (red).

8-D ABALONE DATA 70%:30% SPLIT (POINT BASED)