Topological Data Analysis Ball Mapper for Finance

Herein the data being visualised consists of five variables, with each drawn randomly from a normal distribution of given mean and variance. To assess the properties of BM graphs there will be two clouds considered. In the first the draws for each axis are made from standard normal distributions of mean 0 and variance 1. This cloud is essentially noise since there are no relationships between any of the axis variables. We may thus refer to it as the “noise cloud”. Our second cloud consists of five smaller noise clouds each with a different mean. To almost separate the points the means are set at 0,2,4,6 and 8, whilst the variance remains 1. Because of it’s composition from five smaller clouds we refer to the second cloud as being the “five-part cloud”. In the analyses that follow we explore the impact of imposing correlation structures within these noise clouds, changing the number of points and changing the number of variables. Unless otherwise indicated the number of points will be 500 and the number of dimensions will be 5. Consequently each sub cloud of the five-part cloud has 100 observations contained within it. Use of normal distributions in this paper follows from the commonality of a normal distribution assumption in empirical finance. Qualitatively similar results emerge if we use a uniform distributions to develop the point clouds.

The BM graph also requires an outcome variable. For this purpose we will simply assume that the outcome is the sum of all of the co-ordinates plus a noise term. Formally for point $i$ the outcome $Y_{i}$ is given by:

where $X_{ji}$ is the value of $X_{j}$, $j\in[1,5]$ associated with point $i$. Where applied $\nu_{i}$ is a random term drawn from a normal distribution of mean 0 and variance 0.1. The presence of this noise avoids the overly simplistic summing of all axis values. We may also impose other rules on the colouration and this is discussed in Section 6.

When tasked with introducing a data set, convention dictates that authors report summary statistics for each variable. Typically this means the mean and standard deviation as measures of place and dispersion. These may be supplemented by measures of the minimum and maximum, or quantiles of the variables, as a means to indicate the distribution of the variables. In Finance it is also common to report the skewness and kurtosis owing to the importance of distributions within the discipline. Our exemplar data for both the noise cloud and five part cloud is normally distributed, but the skewness and kurtosis are reported for completeness.

Table 1 reports summary statistics for the data set defined in the opening of this section. Panel (a) shows the noise cloud to have the expected properties in that the mean of all of the $X$ variables is 0 and the standard deviation is 1. Minima and Maxima for the respective variables are also similar, reflecting the fact that all of the values are drawn from the same distribution. There is little within panel (a) which suggests that any variable is not normally distributed, the skewness is close to 0 and the kurtosis is close to 3.

Panel (b) of Table 1 represents our second point cloud, the five part cloud. Here the focus is on five sub clouds, each of which would have similar properties to panel (a). However, because each sub cloud has a different mean when we bring everything back together the overall distribution no longer has normal properties. In this case we cannot infer much from the means and standard deviations since there are inevitably very low and very high values within the sample. The large variation in $Y_{1}$ is likewise not particularly informative. An unintended consequence of the data construction is that kurtosis is lower than the noise cloud. Panel (b) is suggestive of the limitations of simply understanding data from the summary statistics.

A further useful insight comes from the correlation statistics. We know that the noise cloud should have zero correlation between the $X$ variables, but the comparatively small sample size means that some non-zero correlation is likely. Panel (a) of Table 2 informs us that $X_{1}$ is correlated with $X_{2}$ and $X_{3}$ at 0.08, which is still very close to 0. The process of drawing random clouds from the normal distribution means that another draw could easily produce -0.08 as the correlation. Panel (a) gives broad support to the idea that cloud 1 is the noise cloud. In panel (b) the way that the variables comprise five sub clouds creates a strong correlation when aggregated together. Each sub cloud would have a similar correlation matrix to panel (a) otherwise. These correlation matrices provide useful information on top of the summary statistics table, but there is still a benefit to understanding more about the respective data clouds.

Directly related to point clouds are our two dimensional scatter plots from which we draw inference on relationships, correlations and underlying distributions. Beyond merely showing the locations of each data point these plots can be extended with colours and shapes to convey more details about the observation(s) they represent. Figure 1 plots examples from the three data sets using $X_{1}$ and $X_{5}$

As a second consideration let us apply colouration to the data points to understand more of the process that generates the cloud. Let us also apply a shape distinction by having squares represent any point where $X_{3}$ is less than 0. Figure 2 is designed to show the data generation process for the five part cloud. By colouring the points according to which of the five groups the individual point is drawn from, we are able to see clearly the different means. We may understand also that there is some overlap between each group. Panel (a) is coloured by the outcome variable $Y_{1}$, lighter colours again showing the higher values. Panels (b) and (c) show $X_{1},X_{3}$ and $X_{1},X_{5}$ respectively, with the colouration being the sub cloud to which the point belongs. It is further apparent from panels (b) and (c) that although the overall cloud has stronger correlation the points within each of the five groups bear strong resemblance to the full point cloud.

It is further understood that the means of each part of the cloud are only a distance of 2 apart for each variable. Since the standard deviation of the distribution from which points are drawn is 1 it is very possible to have overlap between the colours in panels (b) and (c) of Figure 2. On any axis the distance between two means of the sub-clouds is 2, and hence with a standard deviation of 1 there is an overlap whenever a point in the lower sub-cloud is 1 standard deviation above the mean, and a point in the higher sub-cloud is 1 standard deviation below the mean. The design of the five part cloud is to illustrate how BM can recover information on true group membership amongst multi-dimensional point clouds.

Data may be grouped with the aim of understanding the properties of the outcomes of specific groups. Later the similarity between BM and clustering is explored, here we simply look at potential clustering within the three artificial data sets. The k-means clustering algorithm of Hartigan and Wong, (1979) is used, as implemented within the base package of R (R Core Team,, 2020). K-means targets the division of the data set into a set number of clusters, $k$, such that the distance from every point to the average point within the cluster is minimised. Choice of $k$ is the primary input, but over time a number of algorithms have been devised that suggest optimal levels and optimal centre points (Khan and Ahmad,, 2004; Steinley,, 2006; Chiang and Mirkin,, 2010). Our aim in this paper is to discuss data visualisation, the lack of overlap within k-means clustering renders it unsuitable for the construction of graphs of the BM type. Inclusion of k-means here is therefore as a demonstration of the differentiation of the approach from BM.

In the first data set, our noise cloud, there are no natural clusters; all variables are centered on 0 and have normal distributions. Segregation of the data follows simply because the algorithm is targeting a user specified number of clusters. By contrast, there is natural clustering within the second data set and this is picked up by the k-means clustering algorithm. The elbow plot⁴⁴4Elbow plots here are constructed using the function wssplot described in https://www.r-bloggers.com/2013/08/k-means-clustering-from-r-in-action/. of panel (b) shows the within cluster variation falls quickly as the number of clusters increases towards 5. After 5 the variation falls much slower. This contrasts markedly with the elbow plot for the noise cloud. Within a small distance of the origin you may find points in any one of the five clusters. This is a feature of K-means clustering that we may contrast with the BM representations later. For this demonstration it is panel (d) which recovers much of the information of the underlying data generation process that k-means has the most value to represent the point cloud.

Here we may also wish to retain knowledge about the values of $Y_{1}$ that are associated with each data point. There are many ways to do so, but the simplest is simply to colour according to the average value within a cluster. Because each of these is just a single number for the cluster the effect is qualitatively similar to panels (c) and (d), but with the colour now representative of a specific value. To aid interpretation Table 3 reports the colour and associated average $Y$ values for each of the five identified clusters on each cloud. We may note the correspondence of the five part cloud to the underlying means in that the five clusters were specified to have average input values of 0, 2, 4, 6 and 8 respectively, making the average sum of the axes five times these values. Hence we would expect the average $Y$ in the limit to be 0, 10, 20, 30, and 40 respectively. Since the sub-clouds contain just 100 points the estimates seen in Table 3 are consistent. The values for the noise cloud are less informative since there is no natural segmentation and the algorithm is producing groups with high values for some axes and low for others; the sum therefore depends precisely on the combination.

BM is often likened to a clustering algorithm, though there are some very important differences. Clustering algorithms target segmentation of the data and allow large differentials in cluster size. Further clustering targets inputs, such as the minimisation of the within sum of squares in k-means, and therefore may not identify all of the interesting pictures in the outcomes. BM by contrast captures the features on the input data set in a consistent way that is agnostic to the distances between points. The next section of this paper will set out the BM algorithm to highlight these key differentials and the benefits brought to analysis.

A BM graph consists of a series of balls, the number of which is a function of the properties of the underlying data set and the radius selected for the BM algorithm. In assessing the properties of these balls we may measure size, connectivity and colouration as well as the number of balls within the graph. To capture size we consider both the average size as well the minimum and maximum size observed within the data set. Often the smallest ball is just a single point, but the largest balls vary greatly depending on the density of the data set. Between the smallest and largest is then a range, which then mirrors closely the maximum size. In the analysis that follows the range of sizes is therefore not included. To capture connectivity we use both the total number of connections and the average number of connections amongst connected balls. Choice of measuring only amongst connected balls is that it helps us understand the behaviour of the map absent of any outlier ball. An outlier ball is defined as being one which is not connected into any other. Both Figures 4 and 5 demonstrated unconnected balls so we know that for $\epsilon=2$ there are such zero connection balls. Finally, for the colouration we may also capture minimum, maximum and the range. As previously noted BM requires a random selection of landmark points from the uncovered set and therefore the order of the dataset matters. To account for any variation we run 10,000 repetitions of the code and report the average and 95% confidence interval there around.

Connectivity between a pair of landmarks in BM graphs exists when there are points in the intersection of the balls that surround that landmark pair. Where there are more data points it stands to reason that the probability of a point existing in the intersection of two balls is much higher. However, there is also a lower probability that any given point is selected as a landmark. If different landmark selections produce different BM graphs then the statistics will necessarily be slightly different, but the extra density of the areas around the landmark will moderate the difference. Changing the number of points within a data set will not necessarily extend the domain over which observations are seen. Consequently, we would not expect significant variation in the colouration of balls. More points do mean greater potential for outliers as well as meaning that there will be fewer “holes” within the main cloud which do not contain points. A modest increase in the number of balls may be anticipated. Intuitively, more points means bigger balls. Such increases will come both from larger number of points falling within the balls at the centre of the distribution, as well as a similar proportional increase in other parts of the space. Eventually we may expect the end of single points within a ball, even amongst the tails of the respective axes. In this subsection we change the number of data points to show exactly how the number of balls, colourations and connections are affected.

In this section we limit our attention to point numbers between 200 and 2000, taking us up to 4 times the number of points as in Section 4. Such an increase is sufficient to chart the effects of point numbers on the BM graph. Intervals of 100 points are used with selected results reported in Table 4. For brevity only selected values are given and the two clouds are reported within the same table. What we see is that the hypothesised impacts do indeed materialise. However, pattern spotting from these tables is not as simple as if graphs are drawn. Figure 6 and 7 provide plots of the key measures over the range of data set sizes studied. Within the plots vertical dotted lines are added at 200, 500 and 1000. These are the values used to generate the example plots in Figure 8.

Figure 6 shows the results of 10000 bootstraps of the BM graph with numbers of points from 200 to 2000 in intervals of 100. An increasing number of balls accompanying the increased number of points is clear from panel (a). Whilst the assumed distributions of the variables do not change we may see here that there are more points appearing in the extremes of the distribution. Further as this happens there are also increasing number of points to create connections between former outliers and the main shape. Here it follows that the there will be more connections and a downward pressure on the number of zero connected balls. Panel (b) informs this is the case, with the number of zero connection balls holding constant as the number of points increases. Likewise because the colouration is simply the sum of the $X$ variables, the increase in the number of points means that, as hypothesised, the colouration itself does not change by much. Both the highest (black), and lowest (red), values hold comparatively constant in panel (c).

As the number of balls has increased so too has the ball size. The largest balls are around 0 on each axis, these being at the centre of the normal distribution where each variable is at its most dense. As more points are added the size of these main balls will inevitably grow. Progressively the confidence interval around the mean ball size becomes larger. Meanwhile the lowest values, shown in red, remain just one observation in every case. Not shown here, but an inevitable consequence of this rise, the range of ball size also increases. Finally panel (e) shows total connections increasing as more points are added, reaffirming the idea that points are appearing in previous gaps in the less dense clouds. Average connections amongst those balls that have at least one connection are also shown to rise.

A similar message is seen for the five part cloud, but here there is a different pattern to be observed in the colouration owing to the higher range. An effect of the maximum values always being above 40, and the lowest below 0, is that the confidence intervals appear small and the lines more horizontal than in Figure 6. Whilst further variation may arise from alternative distribution assumptions the way that more points facilitate more connections and give rise to higher ball numbers is clear.

To illustrate the way the increased number of points affects the BM representation, Figure 8 contains three columns, one with 200 points, one with 500 points and one with 1000 points. Panels (a) to (c) correspond to the noise cloud, with panels (d) to (f) being the five part cloud. The shape of each is familiar from Section 4 and Figures 4 and 5. With just 200 points we still see the shapes but the five part cloud in panel (d) lacks the point that connects the third and fourth, green and light blue, sub-clouds meaning that there are now two parts to the main shape. the restoration of the link in the 500 point case of panel (e) is clear. Moving through 500 points to 1000 points there is much more consistency. As noted in the commentary above the presence of more points may induce connectivity between previous outliers and the main shape, but it may equally add more points in the tails of distributions that do not connect. In panel (c) there are more zero connected balls than in panel (b) for this reason. Meanwhile in the case of the five part cloud we note more connectivity forming with fewer outliers in panel (f) than (d).

Consistently the addition of more points to the cloud has shown an ability to obtain a better impression of the underlying shape of the data; this is fully consistent with the standard statistical result that more points on a single variable better capture the distribution thereof. More points induces more connectivity and adds points to existing balls. However, there is limited impact on the number of outliers as there will always be points in the tail. Although not fully uniform, there is limited impact on the colouration either since the points being added to balls necessarily have similar outcome values as their peers. In an applied setting the message remains that more data is desirable.

Within any modelling design the choice of explanatory variables is key, balancing the desire to consider all possible alternative drivers of the observed outcomes and maintenance of tractability. Within BM graphs the primary impact from increasing the number of variables is that the radius must now be spread over more dimensions. Consider the two dimensional case where the ball is simply a circle of radius $\epsilon$. The maximum distance from the centre of the ball on one axis is $\epsilon$ and that may be achieved only when the other variable is identical. Adding more axes simply means that it would be more exceptional to find two points where the only variation is on one of the axes. In describing the effect of adding more dimensions we therefore consider that the average distance from points to the centre of their ball on any one dimension gets smaller. In turn we can expect more disconnection of balls as the number of axes grows. Averaging should also lead to lower total connections, smaller balls and a larger diversity of colouration for the balls.

Table 4 provides a summary of the estimates for the two clouds used as exemplars in this paper. We use larger radii for the balls than in the previous section as the addition of axes will quickly reduce connectivity; a larger $\epsilon$ allows that reduction to be better showcased. The increasing number of balls, and subsequent increase in the number of zero connected balls is clear for both panels. By 50 axes the noise cloud already shows almost 100% of BM graphs to have 500 individual balls. For the five part cloud the increase is slower, with the average reaching 461.5 at 50 axes. From an early stage in the five part cloud we see the average difference between the smallest and largest balls falling; from a high almost 9 when there are 5 axes the difference is already approaching 5 at 20 axes, and 2 at 50 axes. For the noise cloud the pattern is slightly different since the difference actually rises before falling rapidly to 0 by 50 axes. Average number of connections in the final column shows how adding axes at a given epsilon first increases connection numbers amongst the higher numbers of balls, before this falls down to zero. By contrast the five part cloud the average number of connections does not rise, instead falling steadily as more axes are added. For the five part cloud average connections remain around 0.53, rather than falling to 0 as they do for the noise cloud. In order to see these patterns more clearly we again include graphs.

Illustrating the evolution of the respective measures over the increasing number of axes, Figure 9 shows the increases in the number of balls, and number of balls without any connections to other balls, in panels (a) and (b) respectively. The latter curve is much steeper than the former, still being close to 0 at 20 axes but then hitting 500 around 40 axes. Declining ball sizes are also fully in line with expectation. Total connections and average connections display notable hump shapes, with the connectivity rising as the increasing axis numbers split up the original large balls. Once the largest ball sizes start to approach the smallest, seen from panel (d), the peak of the hump is passed and total connections fall. Average connections in panel (f) can be seen to fall first, the peak being around 18 axes versus 22 for the total connection number. Although we may note some small variations, the confidence intervals that result from the 10000 repetitions are narrow and indicate that the shape of the BM graph is indeed affected by the changes in numbers of axes, rather than any variation at a given number of axes. To see the impact of the number of axes we use 5, 10 and 20 axes for the example plots, these are indicated in Figure 9 as vertical dotted lines.

Results for the five part cloud in Figure 10 are fully consistent with this message also. Indeed the numbers of balls and number of zero connection balls, panels (a) and (b) are only differentiated in the time that it takes to reach 500 balls. Both fall just short at 50 axes but already show the flattening of the curve seen around 20 axes in the noise cloud. Colouration varies, a much larger range appearing when the number of axes gets larger. Such is consistent with the way in which the full cloud comprises five sub-clouds with their own assumptions about mean colouration. Maximum ball size in panel (d) falls towards 1 in the same way as it does for the noise cloud, but again we see that has not quite reached 1 at 50 axes. Total connections and average connections, panels (e) and (f), are where the biggest difference is found. Where the noise cloud showed a hump shape the five part cloud only shows falling numbers of connections, this is suggestive that the BM graphs are already past the hump. Further suggestion of this comes in the ball size of panel (d) where the smallest balls are already just 1 point, contrasting with the noise cloud which has 15 axes before the smallest ball size is 1. Choosing different radii for the five part cloud is likely to yield stronger similarity. In the case of the five part cloud we will again give examples with 5, 10 and 20 axes and these are illustrated by the vertical dashed lines on the plot once more.

The impact of adding more axes is easily visualised in the two rows of Figure 11. The two balls identified in the first figure contrast with the dense cloud of connected balls in the last figure on each row. Note here that the larger ball radius, $\epsilon=5$, is being used compared to the other examples in this paper where the five axis cloud is plotted with $\epsilon=2$. For the five part cloud we see that the impact of increasing the number of axes is to create a split between the various sub-clouds. Moving from panel (d) to (f) this split is highly apparent. With $\epsilon=5$ panel (e) of Figure 8 shows the connection of five sub-clouds on five axes, the increase of $\epsilon$ to panel (d) of Figure 11 is discussed in the next section. Panel (e) has five different colourations, although the yellow and orange do merge into the red and there is strong overlap between blue and green. In panel (f) the split on the sub-clouds is achieved by those points which formerly sat in the overlap between the sub-clouds now becoming outliers owing to the number of axes over which the radius must be split. We see outliers with all colourations to confirm this. Each sub-cloud has a similar shape to the noise cloud.

Numbers of points, and numbers of axes are typically defined by the available data and are therefore not the choice of the modeller applying BM. For the BM algorithm the single parameter of choice is the radius of the balls that will be used, $\epsilon$. Our final exploration is then the choice of $\epsilon$. Because smaller balls will inevitably contain fewer points it follows that there will be more required to provide a cover for the same data set. Likewise we may expect that there would be lower within ball variation in colour, but that the between ball variation would rise. As $\epsilon$ increases we would expect the ball numbers to fall and the colouration range to narrow. A further consequence of the ball radius increasing is that balls which were previously separate from the main shape come within range; zero connection ball numbers fall. Connections, and hence average connections have counter veiling effects acting upon them. More radius means more balls will connect, but the fact that there will be fewer balls means the total number of connections may also fall.

Summarising the results, Table 6 informs that the predicted relationships can indeed be seen. It is noteworthy that the maximal radius considered here for the noise cloud is 4. Although we may obtain plots for $\epsilon>4$ there are cases where the algorithm selects an initial point from which all of the points are covered by a single ball. This is no problem in our exposition of the number of axes, but for the purposes of this discussion it is not useful. Hence we restrict attention in the noise cloud to radii of 4 or lower. For the five part cloud we are able to obtain 10000 estimations up to, and including, $\epsilon=11$ so we feature more rows in panel (b). For both examples the rate at which zero connected balls fall to zero is very quick, by $\epsilon=3$ there are no balls that are not connected to another ball. At this point there are still multiple balls and it is only as $\epsilon$ pushes higher that we see the number of balls approaching 1. As noted we do not include cases where there are only one ball, or where the algorithm cannot produce 10000 repetitions with more than one ball. We may again see more by plotting these numbers.

Figure 12 plots the ball radius as the horizontal axis, capping the range at 1 and 4. Dotted vertical lines at 1, 2 and 4 show where the example plots in Figure 14 fit. Panel (a) shows that ball numbers fall smoothly through the range, whilst the number of zero connected balls plotted in panel (b) falls faster. This is as may be hypothesised. Panels (c) and (d) show both the lowest and highest colouration and ball size respectively. What we see as the radius increases is that the homogeneity of the balls increases, particularly the colouration which has a much reduced range of values as $\epsilon$ rises. In panel (d) the biggest ball shows large variation in the mid range of radii, whilst a similar pattern is only evident in the lowest ball size as $\epsilon$ gets close to 4. Patterns in the connection numbers show greatest diversion from the hypothesised effect since we first observe the connections growing before then subsequently contracting. Here we see the balls connecting at low radii before the density of the noise cloud’s centre starts to see fewer balls needed to create a cover. In turn these fewer balls require less connections and the total falls. Meanwhile the connecting, and then subsuming, of points on the outside of the cloud will further raise then reduce connections. Given the falling number of balls, and falling number of zero connection balls, the average connections per connected ball hold longer, but the hump shape is still evident in panel (f) of Figure 12.

Figure 13 presents similar graphs for the five part cloud, showing again the quick decay of ball numbers and zero connected balls as the radius increases. Once more it is evidenced that the larger balls encompass a greater range of outcome values resulting in a narrowing of the colouration range in panel (c). With the growth of the balls comes not only a large maximum size but also a larger minimum size as well. By the highest radii there is overlap of the 95% intervals around the mean from the bootstraps. Both the total number of connections in panel (e), and the average number per connected ball in panel (f), display hump shapes as initially the radius increase creates more overlap between balls. When the radius becomes even larger the reduction in ball numbers duly reduces the total number of connections and the number of possible balls a remaining ball may connect to. Here we evidence all of the hypothesised effects from radius increasing, and confirm the humped relationship from the noise cloud.

To see what this means for the BM graphs themselves we may showcase examples from both clouds. In the case of the noise cloud the restricted radius range means we show radii of 1, 2 and 4 in panels (a), (b) and (c) of Figure 14 respectively. The way that the balls come together to form the main mass is clear, as is the creation of some larger balls in panel (b). When comparing across plots it is not possible to hold the size constant, rather the sizes are on a scale from the largest to smallest. Although we see some similar sized balls in (a) it cannot be stated that these contain as many points as the light blue ball so prominent in (b). Likewise when we move to panel (c) the balls may appear small but they are in fact containing many more points than any in (a) or (b). What we see additionally is the connectivity increasing and then becoming complete in (c). For the five part cloud, panel (d) is the familiar result from the earlier examples of $\epsilon=2$, whilst the simplification to panel (e) produces no outliers. The five different averages remain present, but there is some blurring as a result of the connection through the overlapping points. When the radius reaches 10 and there are only two balls they will naturally be at the opposite ends of the colour spectrum. This reminds on the need to read the scale bar, so doing reveals that the range of colours in panel (f) is much smaller than that in (e) or (d).

Nothing in this section says what the optimal radius is. Indeed there is no way to say that having more balls, or more connectivity is better. In this case having a radius of 2 shows the shape of the clouds well but does not produce a BM graph that is easy to read. Jumping to a radius of 4 makes the graphs simple to discuss but is losing some of the detail of the underlying datasets. The conclusion from this review is thus that including more than one radius would be good practice, with consideration at least given to the impact of different ball radii on results. Section 8 puts these thoughts into practice for two example studies.

Assumption of a normal distribution for the artificial data makes natural appeal to the datasets studied within finance. However, the distribution of points is important to the shape of the resulting point cloud in exactly the way it is when visualised on a scatter plot. As a final empirical exercise the role of radius is considered with draws from the random uniform distribution. We contrast the results with the role of radius in a normal distribution as used for the noise cloud of previous subsections. For comparability the variables are all scaled onto [0,1] prior to running the BM algorithm. Results are presented for five dimensional case, in line with the earlier examples of this paper.

Table 7 provides summary statistics for the two clouds, with Figure 15 showing the key differential between the uniform and normalised noise cloud. Concentration of the noise cloud around the mean, 0.5, is evident, whilst the few points in the tails of the normal distribution also stand out. By contrast the more even spacing of the uniform points is also immediate. Hence the distances between points in the uniform cloud are higher, but the transitions to full connectivity are also quicker because there are no tail observations located far from the main mass of data. In Table 7 these effects can be seen in the larger interquartile range within the uniform distribution. Because only 500 points are used the statistics are not equal to their theoretic values, although most are within a short range. Means range from 0.496 to 0.527, and medians likewise are not precisely 0.5.

Using these two clouds we apply the BM algorithm at radii between 0.1 and 1 in intervals of 0.01. For each radii the algorithm is repeated 10,000 times with random re-ordering of the cloud data between each repetition. The results are summarised in Table 8 and plotted in Figure 16. From the summary statistics it is hypothesised that the number of balls will fall faster for the normal distribution, that the connectivity will be higher in the normal distribution and that the ball sizes will increase quicker for the normal distribution. Because the colouration of each point is derived from the sum of it’s co-ordinates we would expect the maximum and minimum values of colouration to be higher in the case of the uniform cloud.

Evidence form Table 8 and Figure 16 shows that the hypothesised relationships do indeed appear. The number of balls falls as the radius increases, as it does in all cases. However, the rate of fall is much faster for the normal distribution. Panel (a) of Figure 16 shows that at a radius of 0.01 there are still almost 500 individual balls, whilst the normal distribution has already seen multiple points in one ball. This message is reinforced in the range of sizes where the normal distribution has 3.10 but the uniform distribution produces a maximum range of just 1. The message on connection also comes through clearly, beginning with the much more rapid fall in the number of zero connection balls in the normal cloud. In panel (b) of Figure 16 we see both distributions reaching 0, but the faster progress of the normal distribution case to that point is evident. Closer inspection reveals that although the normal cloud does fall fast the transition to 0 is actually slower as it takes higher radii to capture those last few outliers. Panels (e) and (f) support the faster construction of the connections in the normal cloud, both cases having their peak value below the corresponding peaks of the uniform cloud. Panel (c) helps us understand the impact on colouration, with the normal distribution illustrated as solid lines. We see that the highest and lowest colourations converge much quicker than the corresponding uniform lines. By the highest radius of 1, the uniform maximum and minimum colouration are still not within each others confidence interval.

When dealing with data in practice, the distributions of the points are given and need not be either uniform or normal. The message from this section is that if data is spread out then BM is able to better show what is happening than when the data becomes focused on a very narrow part of the space. Where data is normally distributed the solution is to introduce smaller radii in the first instance and focus on the messaging within the densest part of the space. A second plot with higher radius can then comment on the connectivity of the sparser regions near the tails. In many instances the fact that the balls are not connected to the main mass will be sufficient for inference. Within the finance literature it is common to perform winzorisation on a dataset to reduce outliers. In such cases the values of each of the axis variables that are above (below) a stated quantile are replaced by that stated quantile. For BM winzorisation will mean that there are multiple points with identical co-ordinates and may result in larger balls at the ends of the distributions. This should be born in mind by any researcher applying winzorisation prior to the BM algorithm.

Figure 17 provides BM plots of the normal and uniform noise clouds used within this section at three radii. Through these plots the density of the normal distribution near the centre of the space comes through clearly, as does the rapid reduction in the number of balls when the radius increases. At the radius of 0.25 the uniform cloud in panel (d) of Figure 17 does not show any value, whilst the normal cloud is already emerging as a dense mass with some outliers in panel (a). Continuing to the radius of 0.5, we see that the normal distribution produces larger balls that are nearer the middle value and then smaller balls whose colouration is at the ends of the range. The colour bar associated with panel (b) covers a much shorter range, approximately 2.35 to 3.45, compared to the range of 1.4 to 4.1 on the uniform plot in panel (e). Although we see some larger balls, these cover many different colouration levels so do not depict a central mass. Likewise we note that the BM algorithm scales ball size according to the range of number of points within the ball; the number of points within a ball in panel (b) will be much higher than one which is given the same size on the plot in panel (e). Finally by panel (c) the normal distribution is producing just three balls and each is similar in colouration. The uniform case by contrast still has many balls and a colouration range of 1.5 to 3.4, far wider than the normal produced at the radius of 0.50.

A natural question emerging from the comparison of normal and uniform distribution clouds in Figure 17 concerns the similarity of panels (b) and (f). These are produced by the normal cloud with radius 0.50 and the uniform cloud with radius 0.75 respectively. Closer inspection reveals that the larger balls in panel (b) have similar colouration values, whilst those in panel (f) do not. We see all of the balls in (b) around the 2.4, 2.5 level, whilst in panel (f) the two largest balls have colourations of 2.4 and 2.7. There is a large blue ball with colouration 3.0 in panel (f) but no large ball in panel (b) comes close to this. To see BM as a way of distinguishing between distributions it is necessary to apply similar radii, but even with different radii the message can be taken from colouration. Were we to colour by the co-ordinates on any one of the axes then the differences between (b) and (f) would appear strongly again.

This section serves as a brief overview of the impact of distributions on BM plots. It reminds that our data will behave differently according to the underlying distribution of each of the axis variables. Extension may be made to look at how skewness and kurtosis change the pictures. Assuming colouration continues to be the sum of the co-ordinates of a point, the impact of such is perhaps intuitive since skewness creates a long tail and leaves the colouration of the largest balls nearer to one end of the overall range. Kurtosis meanwhile concentrates the peak and means that the tails are longer on both sides. For application in Finance the distribution is important, but it is the other factors that have been reviewed here, such as numbers of points, axes and the ball radius that are the important application decisions.

In the case of a binary variable the average value is the proportion of points within the ball that have the given characteristic. For example we may employ the binary variable for $X_{3}<0$, which was used in Section 2 to show how scatter plots can use different shape points to showcase further information. In this case let us instead create a dummy variable in our dataset for $X_{3}>0$ and then use that variable in the colouration of the BM plot. Figure 18 presents the resulting BM graph.

Panel (a) shows how the lack of correlation between the axes of the noise cloud means that there is a reasonable spread of $X_{3}>0$ across the full range of values of other variables. Despite this BM is able to concentrate the higher values within the overall cloud, those balls in the blues and purples being proximate in most instances. Where we do see value from BM is in the colouration of the five part cloud in panel (b). Here the 0 mean cloud is where we see cases of $X_{3}<0$; the colouration is not purple. However we can also see how there are some balls in the next block that do not have all of their member observations with $X_{3}>0$. There is a green ball to the bottom right of the plot that shows such well. Recalling that BM preserves knowledge of which points are in which ball we may query to find out more about either this green ball or the outliers that do not have coloration 1.

In most instances where a binary variable is used the inference would be for something stronger than just the value of one axis. In the two applications for Section 8 the outcome variables are binary, being firm failure and the stock market moving in an upward direction. Note that for these two example clouds there is no meaningful dummy that is not related directly to one of the axes.

Understanding average values of an outcome variable in a space is akin to plotting the distribution across one variable, the size of the ball corresponding to the density. There is inherent value in seeing the multidimensional surface created by the axis variables. However, we do not need to restrict ourselves to the average value within a ball. One obvious area of interest is the amount of variability of the outcome variable within the ball. To capture such we may consider the standard deviation of the outcome variable as an alternative colouring function. The results are presented in Figure 19

Within the core of each sub-cloud cluster the main large balls of Figure 19 are green in colour, showing the variation we might expect from having so many points. In our case the outcome variable is the sum of all inputs so it is no surprise that the lowest standard deviations are in smaller balls. When the outcome is a noisy function of the axis variables, or there is limited dependence, then we may expect high standard deviations within balls. High within variation may therefore be used as a further summary of the data and hypothesised relationships. For the five point cloud, Figure 19 shows that when the radius increases we find balls that cover more than one sub-cloud and inherently this gives them a larger standard deviation. At the extreme, there are balls in the middle of the shape that have the potential to cover three sub-clouds and these will have the highest standard deviation. Here again the evidence from panel (b) of Figure 19 supports the position with the highest colouration being towards the centre of the shape.

Our first example follows Qiu et al., (2020), taking a point cloud from the five financial ratios in the Altman, (1968) z-score model. In Qiu et al., (2020) we demonstrated that firm failures are constrained within a small subset of the financial ratio space. Here we demonstrate the robustness of that result under the metrics for analysing BM graphs developed within this paper. Our demonstration focuses on 2015 data, being the most recent example in Qiu et al., (2020).

Five variables form the explanatory set. First is the ratio of working capital to total assets, this is denoted as $X_{1}$. Secondly the ratio of retained earnings to total assets is incorporated. In what follows the retained earnings to assets ratio is $X_{2}$. Both of these give an impression of financial health and the ability to generate future income. Third is the ratio of earnings before income and tax to total assets. This is the return on assets and measures the productivity of the firm. High profitability means that the firm should be safe from bankruptcy. $X_{3}$ is the return on assets. Fourth is the ratio of the market value of equity to total liabilities; higher values mean the market values the firm sufficiently relative the amount it owes others. We denote this ratio as $X_{4}$. Finally, the total sales to assets ratio informs how productively the firm is making use of its assets to generate revenue. $X_{5}$ is used for the sales to assets ratio. Later versions of Altman, (1968) have used more ratios, but here we stick with these original set¹⁰¹⁰10See the recent review paper by Altman et al., (2017) for more.. Failure in this case is defined as the firm being removed from Compustat citing bankruptcy as the reason.

From the summary statistics in Table 9 it may be seen that $X_{4}$, the market value of equity to total liabilities ratio has an average value far higher than the others. There is also variation in the standard deviations and ranges of observed values that mean we may confirm these variables to be on different scales. For this reason variables are normalised onto the range $[0,1]$ before the BM is run. This is the approach adopted in Qiu et al., (2020). When performing BM analysis it is imperative that the scales of variables are similar. Having normalised the range of potential axis values is much smaller and so the range of ball radii to consider must include very small values. In the analysis we use $\epsilon=0.1$ as the minimum and $\epsilon=1$ as the maximum. Correlations in panel (b) show that there is more relationship between the variables than there is in the noise cloud, but the majority of correlations are still very low.

Once again we vary the ball radius and take 10000 BM graphs for each radius considered. Consequently we may take summary statistics on the results from these repetitions to inform about the behaviour of the firm failure cloud. Table 10 demonstrates the same properties over increasing $\epsilon$ as the artificial cases, the ball number falling rapidly and the size increasing accordingly. Connectivity follows the inverted u-shape as well, the average connectivity amongst connected balls peaking at 5.29 when $\epsilon=0.4$. It may be noted that the number of zero connection balls has almost reached zero at this stage. Given the application of this model to identifying failing firms the difference in colour being 1 at $\epsilon=0.1$ confirms there are some balls in which all firms fail. However, the average number of balls being 926 means that this is not practical from a visualisation perspective. In the examples that follow we use $\epsilon=0.2$, but this already shows a maximum colour difference of just 0.16. First though we may apply the metrics to produce graphs which summarise the 10000 repetitions of the BM algorithm.

Each of the six panels in Figure 21 resembles the shapes from the artificial examples, especially the role of increasing ball radius in reducing ball numbers. However, the colouration picture is different owing to the application. Where the artificial examples took colour from the sum of the axes in this case it is the proportion of firms within a ball that fail. As there are always balls with no failures the minimum colouration stays at 0, whilst the maximum colouration falls quickly. The nature of the data set means that there are small balls for much of the considered range, only above $\epsilon=0.7$ do we see larger values for the minimum ball size in panel (d). The total number of connections is falling throughout the range, which does differ from the artificial cases. There are many potential causes for this pattern, bu all confirm the cloud for this section does not share the properties of the noise cloud. On all plots we add vertical lines at $\epsilon=0.2$, $\epsilon=0.4$ and $\epsilon=0.6$ to show the radii for which example plots are developed. These example plots follow in Figure 22.

Our example graphs are shown without the ball number in panels (b) and (c) of Figure 22 for clarity. Given the very low proportion of firm failures it is immediate that the majority of the balls have 0 failed firms and are coloured red. A result identified in Qiu et al., (2020) is that the failed firms occupy a very small subspace of the whole cloud, and this is repeated here in those few balls that do have a proportion of failing firms. In the larger panel (a) failures may be found in balls 2, 7 and 10, which are connected at the top of the shape. Those points in the overlap of 7 and 6, 11, and 5 are not failures as the colouration of those latter three balls is all 0. A concern for the identification of the fail region from $\epsilon=0.6$ is that there are some failing firms in balls 1 and 3, their colour is not quite as red as the others. The proportion is below 0.0005 but that may still be seen as conveying some risk to lenders. Of more use is the $\epsilon=0.4$ that is considered in Qiu et al., (2020). Reducing the radius can produce a lot more balls as panel (b) shows. Here again the separation of the failing firms is strong, including one ball where 20% of the constituent firms failed. However looking at the BM graph it is hard to fully visualise the message from the data.

Within the Altman, (1968) example the ability of BM to identify regions of the characteristic space in which firms fail is of high value. This information may then inform further modelling, or simply act as a useful guide to help lenders identify potentially safe firms from within the distress zone that Altman, (1968) suggests.

For the second example we consider the use of financial variables to forecast one month ahead stock movements. Our example follows the discussion in Nyberg, (2011) and Nyberg, (2013) and makes use of the 3 month treasury bill rate, 10 year constant maturity treasury bond rate and the first differences thereof. We denote these variables as 3M, 10Y, $\Delta$3M and $\Delta$Y10. Further use is made of the term spread (term), the dividends to price (DP) and earnings to price (EP) ratios and the volatility (Vol) of the market returns¹¹¹¹11Following Christoffersen and Diebold, (2006) this is calculated as the sum of the squared daily returns for the period. As such we have a 8 dimensional point cloud. Colouration is then a binary indicator which takes the value one if the return on the S&P 500 exceeds the three month treasury bill rate in the subsequent month. Our sample runs from January 1960 to September 2020, with the market return exceeding the risk free rate in 57% of these periods.

Summary statistics in Table 11 help understand the data that we are considering here. Panel (a) gives the mean, variance and range information for each of the variables. Meanwhile panel (b) shows that the correlations between some of the variables in the cloud is high. Dividends to price and earnings to price are correlated at 0.882, whilst the three month and ten year interest rates have a correlation of 0.923. Such high correlation is problematic for the inclusion of all variables within a regression model but does not create a challenge for BM. These high correlations are similar to those for the five part artificial cloud considered earlier. We may take differences in mean value and ranges from panel (a) of Table 11 that make it optimal to first normalise the data before applying the BM algorithm.

BM is run on the normalised data for intervals of $\epsilon$ of 0.05 over the range 0.1 to 0.8. For each radius 10000 BM graphs are constructed and the summary statistics reported in Table 12. Measures are those developed in Section 5. As in all other cases the increasing of the radius causes a reduction in the number of balls and an increase in the size of balls. In Table 12 we note an increase in the difference between the largest and smallest ball as captured in $\Delta$Size. A positive excess return to the S&P 500 index is a binary variable and therefore a ball with all increases has a colouration of 1 and a ball with only decreases has a colouration of 0. As the radius increases the difference between the highest and lowest colouration moves from close to 1 to much closer to 0. This may be seen in the move from $\epsilon=0.5$ to $\epsilon=0.8$. Intuitively the number of disconnected balls falls again. Because the number of balls is falling we see a reduction in the average number of connections per connected ball.

Where Table 12 is useful in clarifying the broad picture, the graphical representations developed in Section 5 can provide a more direct visualisation of the robustness of the BM graph. Figure 23 reveals many consistencies with the artificial examples as the number of balls falls rapidly with even a small increase in the ball radius. This reduction comes with a reduction in the number of zero connection balls, the fall in numbers being faster than the overall ball number. The hump shape of panels (e) and (f) in Figures 12 and 13 can be seen at the very low radii, but the effect is not as pronounced. Recalling that the range of possible outcomes is $[0,1]$, the horizontal line on the maximum in panel (c) of Figure 23 reveals that many of the balls in which the market always increases in the following month do not connect into other balls until $\epsilon$ is sufficiently large. Results for the minimum colouration remain at 0 until a similar radius. We note that the rise in the minimum colouration is large compared to the fall in the maximum. Panel (d) completes the picture by showing how the largest balls grow steadily, whilst there remain balls with just one region in until a high radius. This, and the results on colouration, remind that there are some periods where the market characteristics are very different from normal times.

Through the example plots we can see that the increases in the S&P 500 index are more concentrated at one end of the main shape, this comes through strongly in all three panels. Panel (a), with $\epsilon=0.3$, shows the most obvious grading of the colour scheme across the space. High correlations identified in Table 11 manifest in the narrow shape of the central collection of balls. At the high end of this we see the proportion of rises in the subsequent month being 70% or higher. Indeed given that across the whole sample 56% of months have increases. It is also informative how many balls at the lower end of the shape have proportions below 40%. The lowest colouration is around 0.2. Hence there is no ball where the market never rises, and indeed there is none where the market always rises. However, the information from the data shows that these predictors combine to give a very good signal of the market movement. When the radius is made larger we see that the biggest ball is the red with a low proportion of increases. This connects into blue balls with 70% of cases having a subsequent rise. The overlap balls that create the edge contribute greatly to the greater colouration relative to the lowest balls at smaller $\epsilon$.

Considering the examples in Figure 24 prompts natural interest in the characteristics in that upper part of the cloud. As a next stage we may plot the same cloud but colour according to the eight axes that make up the cloud. We plot these without ball numbers, but the correspondence to Figure 24 is immediate.