A Scalable Method for Visualising Changes in Portfolio Data

Tim Dwyer

School of Information Technology,University of Sydney,Madsen Building F09University of Sydney

NSW 2006Australia

Email: dwyer@it.usyd.edu.au

Abstract

In this paper techniques from multidimensional scaling andgraph drawing are coupled to provide an overview-and-detailstyle method for visualising a high dimensional dataset whoseattributes change over time. The method is shown to be usefulfor the visualisation of movements within a large set of fundmanager’s stock portfolios.

Keywords: portfolio, multidimensional scaling, prin-cipal component analysis, graph, visualization, stock-market

1 Introduction

Perhaps due to the conservative nature of capital mar-ket research or possibly an unwillingness to publishprofitable techniques, methods for visualising finan-cial and capital market data tend to be unambitious.Most “state-of-the-art” stock market visualisationsare:

• variations on time-series charts which show vari-ations in price of a small number of shares overtime (Tegarden 1997)

• multidimensional scalings showing correla-tions between share price or other attributes(Brodbeck, Chalmers, Lunzer & Cotture 1997)

• glorified pie charts showing market share of sec-tors or stocks (Jungmeister & Turo 1992).

This paper is an attempt to push the boundariesby showing that it is possible to visualise the chang-ing interests of thousands of fund managers over aperiod of time in such a way that an analyst can po-tentially diagnose the state of the market, learn fromthe apparent behaviour or even identify illegal activ-ities. Further more, even though the technique wasconceived for visualising such portfolio data it willhave application in any field where the changing at-tributes of a large set of high dimensional data needsto be studied.

The dataset that inspired this research was UKstock market registry data in which the changingportfolio contents of all the registered fund managersis held at a granularity of approximately one month.The data contains in the order of 3,000 portfolioswhich are made up of a selection of over 2,000 dif-ferent stocks from 54 different market sectors. Toobtain a useful picture of the movements within this

Copyright c©2003, Australian Computer Society, Inc. This paperappeared at the Australian Symposium on Information Visual-isation, Adelaide, 2003. Conferences in Research and Practicein Information Technology, Vol. 24. Tim Pattison and BruceThomas, Eds. Reproduction for academic, not-for profit pur-poses permitted provided this text is included.

dataset at least twelve months’ worth of this datamust be available. Obviously, presenting such a largebody of data to an analyst in a visual form is going tochallenge their “perceptual bandwidth”. In (Dwyer &Eades 2002) I presented a graph-visualisation basedmethod for visualising a relatively small subset of thisdata over a few time periods. However, the methoddid not scale very well to larger samples. (Card,Mackinlay & Schneiderman 1999) describe variousmethods for allowing users to effectively navigatethrough large datasets by showing both an over-allview and a detailed view. In the over-all view, oroverview, large scale patterns can be observed. Userscan then select a portion of the overview and “zoom-in” to obtain a more detailed view. In this paperthis overview-and-detail concept is used to producea more scalable visualisation method. To providethe overview for the dataset a visualisation based onPrincipal Component Analysis (PCA) is introduced.The key difference between this PCA based methodand other multidimensional scaling approaches is thatchanges over time are clearly shown. This overviewcan then be coupled with my earlier graph visualisa-tion based methods to allow the user to “zoom in” tothe finest level of detail available.

Section 2 provides some background on multidi-mensional scaling and the PCA method used in thispaper. In sections 3 and 4 a novel system for visu-alising the results of the PCA dimension reductionis introduced. Section 5 gives a formal definition ofthe graph used in the detailed view and section 6 de-scribes the layout algorithm used to find an embed-ding of the graph for visualisation.

2 Multidimensional Scaling

As mentioned above, the dataset consists of severalthousand portfolios each of which contains a selectionof shares from different companies belonging to differ-ent market sectors. If there are 2,000 different compa-nies one can think of these portfolios as being pointsin a 2,000-dimensional space. Alternatively, they canbe placed by market sector in a 54-dimensional space.Obviously, to make an intelligible visualisation thishigh-dimensional space must be reduced to two orthree dimensions. This is the domain of Multidimen-sional Scaling (Borg & Groenen 1997), the chief aim ofwhich is to find a lower dimensional representation fora high dimensional dataset that preserves, as much aspossible, the relative Euclidean distances between thedata points. Typically, this is achieved by minimisinga stress function for the entire data set. That is, mul-tidimensional scaling seeks to find a mapping froman n-dimensional set of proximities between pairs ofdata points pij to a configuration of the data X in anm-dimensional visualisable space: f : pij → dij(X)

where n >> m. Stress can then be defined as thesum of squared errors between distances in a possibleX and the ideal mapping:

σ =∑(i,j)

(f(pij)− dij(X))2

Usually an iterative approach, such as functionalmajorisation, is used to find X such that σ is min-imised. Obviously, for n data points such a methodis going to require at least O(n2) operations per it-eration to consider the distances between each pairof points. In the dataset considered in this paperthere are in the order of 3,000 portfolios with a datapoint for each of at least 12 monthly samples. There-fore, the multidimensional-scaling approach needs tobe able to scale up to around 36,000 data points. Itshould also have a processing time that will allow forinteractive zooming. For such a large dataset wherethe dimensionality of the data is significantly lowerthan the number of data points it is more practical touse a method based on Principal Component Analy-sis (Borg & Groenen 1997). In PCA the complexityis based on the dimensionality of the data rather thanthe number of points.

PCA aims to find the axes of greatest variance(principal components) through our m-dimensionaldata. The data can then be projected onto the planedefined by two such axes to obtain a two dimen-sional representation which captures this variance. Ofcourse, this does not guarantee to minimise σ buthopefully an adequate visualisation will be obtainedin reasonable time.

The approach used to find the principal compo-nents is fairly standard. There are m possible stocksor market sectors, l different portfolios, k differenttemporal samples for each portfolio and thereforen = k · l data points. The first step is to place all ofthis data in an m×n-matrix A such that the columnsare the dimensions and the rows are the data points.The next step is to find the covariance matrix C ofour data. This is done by transposing the data sothat the barycentre of the points is at the origin:

Bij = Aij −1n

n∑j=1

Aij

and multiplying the resulting matrix B by its trans-pose to find C:

C =1n

BB′

Next, an eigen decomposition is obtained such thatC = QΛQ′ where Q contains the eigenvectors ~q1..mand Λ is a diagonal matrix of the corresponding eigen-values. The two eigenvectors with the largest eigen-values, q1 and q2 are then the principal components.

Finally, we can perform the projection to find xand y coordinates of the n data points in two dimen-sions:

xi = ~pi′q1

yi = ~pi′q2

where i = 1, ..., n.

3 Displaying temporal changes by extrudinginto 3D

The novel visualisation of the 2D PCA projection de-scribed above involves extruding the data into 3Dsuch that time is now represented by the 3rd di-mension. The result is a mass of “worms” crawling

through time. Visualisations in which the third di-mension is used in a significantly different way to theother two dimensions are commonly called 2 1

2D andI will use this terminology in the sequel.

Each of the data points comes from one of k sam-ples, usually taken at regular intervals in time. Eachof the n data points above is now assigned a z coor-dinate such that z = c · (i−k/2) where i = 0...(k− 1)and c is a constant scale factor. When drawing, eachpair of adjacent points representing the same portfo-lio is connected with a line segment to produce theworms.

It is worth noting that in any type of multidimen-sional scaling the position of data points in the re-duced dimensional space relative to the axes is mean-ingless. The important thing is that clusters and out-liers are clearly visible. In the “worm” visualisationan analyst can see how various portfolios move in andout of clusters over time. This also allows colour andthickness of the lines to be used to display additionalattributes as they change over time.

The resulting visualisation is shown in Figure 1.The collection of dots on the right-hand side is a crosssection of the 2 1

2D worm view. The particular timeperiod shown in cross section is selected by movingthe transparent blue disk (the “water-level”) on theleft. Colour and alpha-transparency of the worms andtheir corresponding points in the slice view are used toconvey extra attributes. In the figure colour is usedto indicate whether a portfolio has increased or de-creased in value between time periods. The worms areshaded from dark to light green to show an increase invalue or dark to light red to show loss. Solid black in-dicates the portfolio has broken even. The total valueof the fund is indicated by transparency. In the fig-ure transparency potentially ranges from invisibility,indicating a zero value fund, to complete opacity, in-dicating the fund is worth more than £100,000. Theother user interface features provided by the interfaceare described in more detail in Section 4.

An interesting feature of the data that can be seenin Figure 1 is that fund managers generally tend tomove either toward or away from a single central clus-ter. Currently, our analysis suggests that this clus-ter contains portfolios that are weighted closer to themarket index. That is, more conservative portfoliosthat are weighted to hedge against significant volatil-ity. The fact that outliers tend to move either towardsor away from this central cluster (rather than aroundit) may suggest that when they are confident (possi-bly buoyed by a stronger market) they move towardsmore volatile, higher risk, stocks and when the marketis less stable they move back towards the safer “blue-chip” shares in the index. Of course it may also bean artifact of PCA. Work is continuing to investigatethis phenomenon in greater depth.

4 User Navigation

An essential part of any 3D (or 2 12D) visualisation

is providing the user with the ability to freely ro-tate, zoom-in or otherwise “fly” around the 3D model.When viewing a static projection of a 3D visualisationthe user has no sense of depth and the extra dimensionis wasted1. In our system this capability is providedin a fairly standard way with mouse interaction.

However, PCA gives us the ability to navigatearound the dataset in some fundamentally differentways. Using the two eigenvectors associated with the

1Indeed it is difficult to properly show the use of 3D visualisa-tion in a paper with only static black and white figures!

Figure 1: A worm-view visualisation showing the movements for the 391 largest fund managers over a 12 monthperiod.

two largest eigenvalues to define the projection planeby definition captures the greatest variance in thedata. However, one can just as easily use any pairof eigenvectors. Allowing the user to cycle throughthe largest eigenvectors to choose the projection planeprovides a high-dimensional rotation which may cap-ture different aspects of variation.

A zoomed view is also easily obtained by produc-ing another PCA projection of a subset of the visibledata. In Figure 1 the user is in the process perform-ing such a zooming operation. In the cross-sectionon the right hand side they have selected a subset ofportfolios by sweeping out a rectangular area with themouse. They can then select a subset of the availabletime samples by moving the water-level up or down,thus creating the transparent box seen in the 2 1

2Dview on the left-hand side of the figure. The selectedsubset of the data will then be re-projected as be-fore and the zoomed view shown in a new window.The zoomed window is shown in Figure 2. An exam-ple where a similar PCA based rotation and zoomingstrategy is used to view high dimensional graph em-beddings is given in (Harel & Koren 2002).

5 Detailed Graph Visualisation Based View

The “worm” visualisation described thus far providesus with an overview of our data set. In order to cap-ture broad movements across as much data as possiblethe PCA based dimensional scaling was a necessary,though severe, abstraction of the underlying detail.To visualise the detailed behaviour of an individualfund-manager as they re-balance their portfolio a dif-ferent approach is required.

Suppose an analyst selects an individual wormfrom the overview for closer inspection. Effectively,they have isolated a set of data for a single portfolioover a number of time periods. In our dataset thisincludes share price data and the count of shares of aparticular type held in the portfolio. That is, we havetwo matrices in which the columns are associated withmarket sectors (or any other aggregation of stocks, orindividual stocks) and the rows are associated with

each of the sample times (the examples shown herehave 12 monthly samples). In the first matrix P wehave share price data. When an aggregation of sharesis used this will be the average unit price across theaggregate. The second matrix Q contains the countsof shares held in the portfolio.

These matrices could be visualised by simple 3Darea charts. For example, Figure 3 shows the shareprice data P , in this case the average share price foreach of n = 50 market sectors for a year’s worth ofdata. Figure 4 shows the counts of shares held in aparticular fund manager’s portfolio across the sametime period. Figure 5 charts the total value of theportfolio across the time period, where the value x ateach month j is:

xj =n∑i

PijQij

The relatively flat curve in Figure 5 shows thatby re-weighting the portfolio the fund manager hasmanaged to more or less even out the volatility in theshare prices. In visualisations like that of Figures 3and 4 we can see a lot of activity taking place. How-ever, one must ask whether it is possible to producea visualisation which focuses an analyst’s attentionmore specifically on the fund manager’s movementbetween market sectors.

In (Dwyer & Eades 2002) I proposed a graph vi-sualisation based approach for visualising the move-ments of fund managers between different stocks ormarket sectors (in the sequel I’ll refer only to sectors).The graph for fund manager movement is defined asG = (V,E) consisting of a set V of vertices repre-senting sectors and a set E of edges where each edgee(u, v) ∈ E represents “movement” of one or morefund managers from sector u ∈ V to sector v ∈ V .

For the matrix Q of share counts in each sector wecan construct a graph in which a vertex representseach non-zero column. Beginning with the secondrow we compare the values in each column againstthat column’s value in the previous row. For eachcolumn (sector) showing a decreased holding we con-struct an edge to all the columns with an increased

Figure 2: A zoomed view of the highlighted portion of Figure 1. 8 months are shown.

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Figure 3: A 3D area chart showing the fluctuations in share price in a particular portfolio over 12 months.Each of the columns is a different stock, the depth axis is time (most recent at the front) and the vertical(labelled axis) shows share price in GBP (British Pounds Stirling).

0

10000

20000

30000

40000

50000

60000

Figure 4: A 3D area chart showing the changing count of shares in the portfolio over 12 months. The axes areas in Figure 3 except the vertical axis which shows count of shares

0

50000

100000

150000

200000

250000

1 2 3 4 5 6 7 8 9 10 11 12

Months

To

tal V

alu

e (G

BP

)

Figure 5: This chart shows the net effect on thetotal value of a portfolio of the share price fluctu-ations, shown in Figure 3, and the fund manager’sre-weighting, shown in Figure 4.

holding. To allow the user to focus on more signifi-cant movements they can adjust a threshold (τe) inincrease or decrease of stock price below which noedge is created. We continue comparing each pair ofrows to create layers of edges (ie, a set of edges foreach time period) until all rows have been examined.

To visualise this graph so that it is easy to see atwhat time different movements occurred it is possi-ble to use a 2 1

2D paradigm similar to that followedfor the worm view. That is, the graph drawing is ex-truded into the third dimension, see figures 8, 9 and10. The vertices become pillars or columns parallel tothe new third axis and the edges are placed perpen-dicular to the axis at a level dependent on the time(matrix row) at which the movement they representoccurred. The total value of a market sector at eachpoint in time (ie, PijQij) can be shown by setting theradius of the column vertex representing that marketsector. Clutter in the graph may be reduced by onlyincluding vertices for which the maximum value isgreater than a threshold τv. In other words, it is onlynecessary to include market sectors which make up asignificant proportion of the portfolio. The changingaverage share price information from P can be shownby colouring each segment of the column. In the ex-amples the columns are shaded towards green if thereis an increase in average share price from one periodto the next. The shading tends towards red if thereis a decrease. The default colour, light grey, meansthere is no change.

In the extruded 2 12D view the edges can be shown

as tubes or pipes between the columns. The totalvalue of a given movement is shown by adjusting theradius of the edges. That is for an edge showing a

Figure 10: The stratified graph from 8 viewed fromabove.

change in holding from time samples k and k + 1 be-tween vertices representing columns i and j in P andQ the radius of the edge is set to:

radius ∝ (Qi(k+1)−Qik)Pi(k+1)+(Qjk−Qj(k+1))Pj(k+1)

6 Stratified Graph Layout

I have chosen to refer to such graphs in which edgesappear in layers corresponding to a particular timeperiod as “Stratified” graphs2. Visualisations ofStratified graphs have appeared in the literature fromtime to time but the question of how to arrange themis yet to be thoroughly examined. (Koike 1993) waspossibly the earliest, using a stratified graph to showthe flow of control in parallel software execution, butthe author left placement of the columns representingvertices to the user. (Brandes & Corman 2002) choseto represent social group interactions as stratifiedgraphs using the well known force-directed methodsimulating physical attractive and repulsive forcesin an iterative algorithm that attempts to balancethe opposing forces typically creating an aestheticallypleasing arrangement. In (Dwyer & Eades 2002) weused a similar force-directed method but allowed thecolumns to bend in order to further reduce the forces.We showed that this last variation was useful in high-lighting clusters in the graph. A stratified portfo-lio graph arranged using the force-directed approach(without bendy columns) is shown in Figure 6.

In this paper another well known layout methodis introduced to the problem of stratified graphlayout: the Sugiyama (or Layered Graph Layout)Algorithm(Eades & Sugiyama 1990). The chief ad-vantage of the Sugiyama algorithm over the other lay-out methods described is that it clearly shows flow indirected graphs. This is because the first stage of thealgorithm creates a layering of the nodes such that netsources (nodes with mostly outgoing edges) are usu-ally placed closer to the top and net sinks (nodes withmostly incoming edges) are placed nearer the bottom.

2borrowing the geological term used to describe rock consistingof layers of sediment. I use this term to avoid confusion with theconventional usage of the term “layered graph drawing” associatedwith the Sugiyama algorithm

The final arrangement thus ensures that most of theedges are downward pointing and net flow can be saidto be from top to bottom.

This graph theoretic concept of flow is useful whenconsidering layout of our portfolio movement graphsin that it roughly corresponds to the flow of moneybetween different stocks. That is, source nodes rep-resent stocks that are mostly being sold and sinknodes represent stocks that are more commonly beingbought.

To generate the layout shown I use an implemen-tation of the Sugiyama algorithm based on the Dotprogram that comes with AT&T’s Graphviz package(http://www.graphviz.org). I have modified thisalgorithm to handle separate edge sets for each of thestrata. Specifically this involved the following modi-fications:

• the edge concentration3 method was changed sothat edges on different strata would not be con-centrated

• the crossing minimisation was changed such thatonly crossings between edges on the same strataare considered. The idea being that when edgeson different strata overlap the crossing can beresolved by rotating in 3D

7 Conclusion / Further Work

I have demonstrated two visualisations for fund man-ager movement data which may be coupled to providean holistic, overview and detail system. The first vi-sualisation, the PCA worm view, compresses a greatdeal of information about the entire dataset into asingle scene and takes advantage of the speed andflexibility of PCA to allow a user to quickly focus inon smaller regions of detail. The second visualisationparadigm, the graph based column view, brings to-gether the most important information from all threecharts shown in figures 3,4 and 5 into a single visuali-sation and draws an analysts attention to the featuresin which they are most interested. Particularly, it al-lows an analyst to directly see the correlation (if any)between stock price and a fund managers behaviourin re-weighting the portfolio.

In this paper a broad overview and definition ofthese two paradigms has been given. Work is pro-gressing on validating the techniques by using themin a field study with industry experts.

The problem of extending existing layout algo-rithms such as the Sugiyama algorithm, to producethe best possible results for stratified graphs is alsoongoing.

Finally, since the paradigms proposed in this papershould be applicable to any high-dimensional, multi-variate dataset I hope in future to test their utility inother domains.

8 Acknowledgements

This work was conducted with the support of theCapital Markets CRC Limited. The author wouldalso like to thank the members of the University ofSydney Information Visualisation Research Group fortheir comments and suggestions.

3Edge concentration involves condensing edges so that a singleedge can split and branch and a single branching edge may beshared by a number of nodes

Figure 6: A stratified graph representing the movements in the portfolio shown in figures 3, 4 and 5, laid outusing a basic force-directed algorithm. The thresholds are τe = ±5% and τv = £5, 000.

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Figure 7: The stratified graph from 6 arranged using the Sugiyama algorithm.

Figure 8: Another stratified graph showing the movements in the portfolio from Figure 7 in lower detail. Inthis figure fewer edges are visible since τe± has been raised to 8%.

Figure 9: The stratified graph from 8 viewed from the side to better show the strata.