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Visualizing Company Fitness in High-Dimensional Parameter Space

Brendan Kitts 10 Rockwell Street, Cambridge, MA. 02139. USA Email: bj@datasage.com, Web: http://www.cs.brandeis.edu/~brendy

Tord Beding Hagatornet AB, Haga Nygata 28, S-411 22 Gothenburg, SWEDEN Ph: (S) 708-668738 Fax: (S) 31-139060 Email: hagatornet@goteborg.mail.telia.com

Catherine Harris Department of Psychology Boston University, Boston, MA. 02215. USA Ph: (US) 617 353-2956, Email: charris@bu.edu, Web: http://www.bu.edu/psych/faculty/charris

Preprint submitted to Decision Sciences, September 4th, 1998

Abstract

This paper explores the use of Data Mining algorithms to reduce and project internal company parameter data onto a two dimensional landscape which can show the course and trajectory of the company. We show how the resulting landscape can be used for analysis, planning, prediction, and ultimately in some ways, navigation.

Keywords: Business Process Reengineering, Scientific Visualization, Artificial Intelligence, Organizational Learning, Computer Science, What-if analysis

1.0 Introduction

We will define a company’s “state” as aspects of its internal processes which are changeable. The objective of this document is to visualize the state of these internal parameters, and the company’s surrounding state space.

The question of how to formalize a company’s internal processes is nontrivial. For this paper we have chosen to use Edvinsson and Malone’s (1997) Intellectual Capital measurement system to define a company’s state position. There are several advantages to using Intellectual Capital (IC): (a) IC covers five very different company domains including human, structural, process, renewal, and financial variables. This encourages

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diverse aspects of the company’s operations to be captured. (b) IC metrics are chosen with great care, and are already correlated with fitness. Thus they tend to be a good set of variables for predicting that company’s fitness, which is a requirement for functions described later. (c) There is the possibility that some form of IC measurement will become standard analogous to the American Consumer Satisfaction Index (Bryant, et. al., 1997), so that different companies can be compared.

Using proven mathematical techniques for visualizing high dimensional data, this document investigates techniques for reducing the inherent dimensionality in IC measurements, and projecting those measures into a more manageable, two-dimensional space. This projection, which we refer to as a “map”, shows the company’s position in IC space, its past trajectory, and its current heading.

The present document reviews the techniques used in reducing high dimensional data, and examines maps of five different companies in the American, Swedish, and UK branches of the multinational Skandia Investment firm.

2.0 Overview of mapping

The mapping process consists of three stages. The first task is to project the high dimensional company data into two dimensions. The second is to add a fitness variable to the 2D vector as a third dimension. The final task is to interpolate between those points to predict the shape of the surface connecting these regions.

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Multidimensional scaling

The objective of MDS is to represent high dimensional data in 1-3 dimensions so that a human being can visually understand the data. Because its not possible to directly visualize n-dimensions, the method focuses instead on preserving the inter-point distances of the high-dimensional data. MDS therefore finds a set of points in two dimensions which have distances as close as possible to the inter-point distances in the high dimensional space. If the method succeeds, then a human looking at the points will be able to correctly judge that their present position is “very different from” some other point on the map, and that judgement will also hold in the difference between the two high-dimensional vectors. (Young, 1998; Young and Hamer, 1987; Norusis, 1997)

It should be noted that the axes of the new, low-dimensional space are not intuitively interpretable. Each 2D dimension is constructed out of convenience to retain the distance relationships, and could represent several high dimensional axes that have been collapsed. The only features which make sense in the new landscape are the concepts of “more similar to-“ or “more different from-“ x, where x is a known case. Thus, this is somewhat analogous to a paper topographic map, where landmarks are shown (the other datapoints), and the user can see how distant these landmarks are to his or her position.

Fitness as the Third dimension

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The most important feature of the landscape we want to plot is fitness. Fitness refers to the health of the company, and examples can include “total operating income”, or “net profit after tax”.

Because knowing the fitness of the company at a given position is so important, we propose to carry fitness into the 2D projection unaltered, and have it plotted as the third dimension, giving rise to a “topographic fitness landscape”. As a result, fitness data is perfectly accurate, however, the underlying 2D point may be subject to some error.

Descriptive labels on the original cases can be added also. For example, in 1994 we might add a label, “competitor x came onto the market”. These labels can provide some descriptive information, analogous to labeled terrain, and are attached to the topographic map at the location of the respective datapoints which are labeled.

Interpolating the surrounding surface

After the above step we have a set of 3D points. Unfortunately, these 3D scatterpoints are also difficult to interpret. To infer the appearance of a lanscape, we need a model which predicts fitness at any point on the 2D map, and so infers the topographic surface around the known points. To generate this, we can use function estimation techniques such as neural networks, splines, and regression to fill in these regions.

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Estimating a surface can be difficult, as points must be estimated outside of regions of known points. Therefore, we measure the error of the surface model by inducing the surface on a subset of the points, and then testing the model on the remaining points, a technique known as cross-validation (described in more detail later). The resulting map will resemble figure 1.

Applications

Mapping can be used for a range of applications. The types of applications are summarized below.

a) Viewing Company Trajectory across landscape:

By connecting the company’s positions in chronological order, we can view the historical movement of the company. This can be especially useful in interpreting past performance of the company.

b) Predicting Company position and state

By extrapolating the trajectory of the company, (for instance, drawing a line through the last few points), we can estimate where the company will be in the next period of time. At this new point, we can also read off the fitness.

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Current position of the company

Region of very poor “fitness”

Extrapolated position of company +6months

Policy change comes into effect Position of the company 24 months ago

Figure 1: Viewing a company’s weight timeseries allows us to plot its speed and direction of movement, and allows us to anticipate problems it will encounter in the future.

c) What-if analysis

We can look for desirable regions, and find out their underlying IC vector. What-if analysis involves building an inverse model mapping 2D surface position back to highdimensional IC vector.

The remainder of this paper will describe the specific MDS algorithms used, the model estimation methods, results with Skandia group companies, and what-if models.

3.0 Mapping in Detail

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The Skandia Insurance Company Limited

Skandia is an international corporation specializing in insurance and financial services, with a home market in Nordic countries. The parent company is Skandia Insurance Company Ltd. established in 1855. The corporation markets insurance and savings products including Life, Health, Commercial, Industrial, Marine and Offshore, Motor, Personal accident, Personal property, Aviation, and Space insurance; Banking and Investment management.

Skandia is the parent company for six subsidiary branch companies. These are American Skandia, Skandia Real Estate, Skandia Life UK Group, Dial, SkandiaLink, and SkandiaBanken.

SkandiaBanken has 11 IC dimensions and four points of data corresponding to the years 1994-97. (see table 19). Our data is thus a 4x11 matrix. We have placed Total Operating Income on the height axis as our “fitness” variable. The examples that follow in figures 6 to 15 all use SkandiaBanken’s landscape.

High dimensional reduction

Algorithms for Multi-dimensional Scaling

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To find a set of coordinates in two dimensions which respect the distance relationships between the same points in the n-dimensional space, one typically has to minimize some measure of error. The standard “error” is known as Stress, and was introduced by Kruskal and Shepard (Kruskal, 1964; Shepard, 1962):

Stress(d, ? ) = ∑ ∑

i j

n

n

(d ij ? ? ij ) 2 ? ij

2

(1)

Many different algorithms can be used to perform stress minimization. We will review three of them, Nelder-Mead simplex algorithm, Torgerson’s classic MDS algorithm, and the Kohonen Self-organizing map.

Nelder-Mead simplex algorithm

The Nelder-Mead algorithm (1965) is a non-gradient search algorithm that is used in many numerical packages. The algorithm starts off with a “simplex” of initial guesses.

Figure 2: Reflection is used to choose a new point in Nelder-Mead optimization. Each vertex consists of a complete set of coordinates in two dimensions. The fitness of each vertex is equal to the (negation of) stress.

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Simplexes are geometric objects each edge connected to each other edge by a vertex, and the property that if there are d dimensions, then there are d+1 vertices and d+1 edges. After its initial simplex of guesses, the Nelder-Mead algorithm then removes the poorest point (the poorest guess), and replaces it with another guess. In this way, the simplex “moves” across the space, and hopefully finds better points (coordinate sets). The precise manner in which it selects new points for the simplex are as follows:

?

Reject the worst point, and initialize a new point to be the reflection of the worst point

?

If the new point is returning to an old point, then reject this movement, and use the second-worst as the rejection point.

?

If a point has been unchanged in the simplex for more than a certain number of trials, try rejecting it.

?

If error is improving, increase the scale of the reflections by adding to the scale coefficient

?

If error is getting worse, decrease the scale of the reflections by decreasing the scale coefficient

In previous work one of the authors has used the Nelder-Mead algorithm for 2D object recognition using elastic prototypes (Kitts, Cooke, et. al., 1996), and the algorithm proved to be reliable in finding good solutions.

Torgerson’s Classic Metric MDS algorithm

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Torgerson showed in the 1950s that if a distance matrix was double centered such that, given an original distance matrix B,

dij = -0.5 (bij2 – bi.2 – b.j2 + b..2)

(2)

then the following relationship held:

D = XXT

(3)

This meant that any distance matrix could be changed into a coordinate matrix by taking a matrix square root of D. To derive a set of coordinates with less than the original number of dimensions, singular value decomposition could be used to identify the two largest eigenvalues, and then reconstruct coordinates with only the two largest eigenvalues. This will result in coordinates which capture as much of the variance as possible in the selection of coordinates.

UΣUT = D

(4)

X = UΣ1/2

(5)

Self-organizing map

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Self-organizing maps are a new technique originally formulated as a model of human visual cortex (von der Marsburg, 1973). The self-organizing map consists of an array of neurons which each store a prototype of the input they most prefer (their tuning curve). As input comes in, the cell with the prototype vector most similar to the input “wins”, and adjusts its prototype to be more similar to the input. Each cell also has a “neighborhood” of other cells. This can be seen in the s below (figure 3 and 8) and as a mesh connecting the cells. When all the cells start off, they are in a perfect grid. Over time this grid will deform to “map” the input.

Whenever one neuron adjusts its vector, the other cells in the neighborhood also have their prototypes adjusted in exactly the same direction, but to a smaller degree. The result is that neighboring cells represent similar inputs. If we then “read off” what each cell represents, we will find that input vectors have migrated to different regions of the selforganizing map. These different regions are the low-dimensional manifestation of the high-dimensional data. Other details of the Self-organizing map can be found in Kohonen (1997).

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1.6

1.55

1.5

1.45

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1.35

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1.25 70

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100

105

Figure 3: Top-view of the Kohonen self-organizing map, as it attempts to map a higher dimensional object. Areas of “compression” in the map represent regions in the high dimensional space where high concentrations of datapoints exist. Because we have so few datapoints, the “high concentrations” will consist of areas with one or more points. High concentrations of points encourage centroids in the Kohonen map to “crowd in” try to to map those denser regions of points. Also shown are the values for the first two dimensions of the 11-dimensional SkandiaBanken datapoints. Because we are only seeing the first twodimensions, the areas of compression on the map and the datapoints aren’t aligning properly. However, we should conjecture that the map is picking up some of the hidden dimensions.

The equations for a Kohonen net are as follows:

Given a learning constant 1>α>0, a c×v matrix of centroids which is initially random, the centroids are changed as follows, upon presentation of x to the network:

?c n∈N(min) dt

= F( c n ? c min )(x ? c n )

(6)

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(7)

c min = c i ? min c i ? x

?? i ? ? i ?? ? grid(i) = ? ? ? gsize ? , i ? gsize ? gsize ? ? ?? ? ? ??

(8)

d ? ? 1? F(d) = ? nsize , d < nsize ? ?0, otherwise

(9)

(10)

j ∈ N(i) ? grid(i) ? grid(j) < nsize

where nsize is neighborhood size, gsize is gridsize, The first equation states that a neighbor cn of the closest centroid cmin, is moved towards the input x at a rate equal to a function of its distance on the mesh to cmin. This speed function, F, is linear with decreasing activation from distance 0 from cmin to 0 at distance nsize. Thus, the closest centroid moves fastest towards the input.

N(cmin) is a set of neighbors of cmin. In this implementation neighbors form a square grid around cmin, where their grid coordinates are given by grid(.). In addition to the learning algorithm above, the SOM also undergoes annealing of the neighborhood size nsize, and learning rate α. These details can be found in Kohonen (1997). Typical parameters used in our implementation were α=0.005, gsize=10, nsizeinit=gsize/2.

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Comparison of map quality for each method

Duch and Naud (1998a, 1998b) compared self-organizing maps to classical MDS methods. These authors generated n-dimensional equilateral simplexes and mapped them to two dimensions. Duch and Naud reported that MDS generated symmetric 2D geometric shapes which they conjectured were global stress optima, while SOMs did not generally approach these low stress figures.

We have replicated Duch and Naud’s results using our own implementation of MDS, SOM, and Torgerson’s projection. Our results corroborate Duch and Naud’s findings. MDS minimization results in low-stress geometric figures of the type seen in figure 4.

Figure 4: 7-dimensional simplex, 8-dimensional simplex, stress=3.610143. 12-dimensional simplex, stress 5.3911, 10,000 iterations

Figure 5 compares the three methods used in this paper on Duch and Naud’s simplex benchhmark. Torgerson’s method (figure 5C) was the poorest on this test data. Torgerson’s method works by recovering coordinates from the distance matrix, and then finding the principal components of the coordinates, and projecting the coordinates onto

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these principal components. The resulting 2 dimensional coordinates capture as much of the variance as possible. However, all of the distances between each vertex are equal to 1. Therefore, in this particular application all of the dimensions are equally important, and taking the principal components just results in loosing d-2 important dimensions.

Self-organizing maps (figure 5B) fair better, but are still twice the stress of MDS (figure 5A). MDS generates the best results by far (figure 5A).

Figure 5: 6D simplexes represented in two-dimensions using three different algorithms. (from left) MDS stress minimization, Iterations = 10,000, Stress = 2.786404, SOM 10x10 grid, Stress = 5.078828, Torgerson, stress = 20.9641

Li (1993 and 1995) has also compared the self-organizing map with Kruskal’s monotonic MDS method, and Sammon’s method on five widely-used machine learning benchmarks (Metz and Murphy, 1998). He found that MDS and Sammon’s mapping were consistently better than SOM in terms of distance-distance correlation and Procrustes error. Therefore, SOMs are a fascinating model which could be being used by the brain, but as techniques

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for reproducing distances as precisely as possible (the objective of this application), MDS methods are currently a better method. This actually stands to reason. SOMs do not explicitly minimize a stress function. Thus, it is natural to expect that SOM stresses aren’t as low as methods which explicitly minimize this measure.

High-dimensional Mapping Results for SkandiaBanken

Z-scores

Before translating our high-dimensional company data into low-dimensional projection, we first applied a Z-score transformation to the data. This transform re-scales all columns of the data (each IC metric) to have a mean of zero, and a variance of 1, and can be computed as follows:

x ij =

(x ij ? x .j )

.j

(11)

This prevents any one dimension from dominating the distances simply because it happened to be measured with a larger number, eg. millions of Kroner, versus % of GDP. By executing this transform, we are making the assumption that each dimension has equal importance in fixing the company’s position in 2D space.

Results

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Results of using each of the methods above to project SkandiaBanken’s IC vectors (table 19) into two dimensions are shown over the following pages (figures 6, 7, and 9). The most important thing to note about these results, is that the projection for each different method is very similar. This indicates that the projections we have come up with are good representations of the high dimensional data.

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400

Total operating income

350

300

250

200 2 1 0 -1 -2 -4 -3 -2 -1 0 1 2 3 4

Figure 6: Skandiabanken. Change in total operating income for years 1994-2997, ploted in 2D space representing 11 IC variables. Kuskal stress optimization using Nelder-Mead, Stress = 2.001988

400

350

300

250

200 2 1 0 -1 -2 -10 -5 0 5

Figure 7: Torgerson metric MDS Stress = 42.18

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26

24

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12 1.6 1.5 1.4 100 105

1.3

1.2 70

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Figure 8: Side view of the kohonen net, showing how it tries to “reach” out to the different points, constrained by its 2D mesh topology.

400

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200 20 15 8 10 4 5 2 6 12 10

Figure 9: Self-organizing map 20x20 square grid, lc = 0.005, stress = 50178. Using a 40x40 grid, stress = 3.59

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Interpolating the surrounding surface

After we have projected our data we are left with a series of points in 3D space. This is a skeleton of what the actual space looks like. We would next like to visualize the surface connecting these regions. For example, the shape of the surface could be such that the four points lead up a hill, or they could represent a series of plateaus and ravines.

In the following figures (10 to 15) we have used polynomial and splines to estimate the surface which the data lies on. Many other techniques can be used to estimate this surface, including backpropagation, k-nearest neighbor, and hyper-basis functions.

Polynomials

Polynomial regression is used widely in statistics and system modeling, and consists of finding a polynomial equation of a given degree which fits the data in the least squares sense. The polynomial model is as follows:

X ? W = ∑ ∑ xd vd w vd

v =1 d =1

V

D

(12)

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Where X is a r × (v.d)+1 design matrix and W is a (v.d)+1×t weight matrix. r is the number of rows (observations), v is the number of variables (dimensions), d is the highest degree of the polynomial, and t is the number of target or dependent dimensions.

The work reported here shows that polynomial models aren’t well suited to this particular domain. The reason is that polynomials attempt to globally fit the data, much as ordinary linear regression attempts to draw a straight line through a cluster of points. This apporach is thus unable to fit local aspects of the data, such as a sinkhole or peak.

Splines

Splines are a non-parametric regression technique which approximate a function by (a) finding a set of high-density “centroids” in the function (b) projecting all data onto a new coordinate system where each axis is a centroid, and the value on the axis is the distance from this data point to that centroid, (c) performing a least squares mapping from the new points to the target.

The spline model is defined as follows: Let C be a c×2 basis matrix of centroid vectors, sometimes called knot points. Let S be a r×c basis-transformed input matrix, and let W be a c×1 matrix of weights. Given an r×2 data matrix X, a spline approximates a function by applying the following formula:

S?W

(13)

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where S = G(D XC )

(14)

S is a representation of the input which has been transformed by the basis function given by G(.). G is usually one of the radial functions given below (Karur and Ramachandran, 1995). The Gaussian spline has come to be known as a “Radial Basis Function network” in the neural networks literature (Girosi, Jones, and Poggio, 1993; Orr, 1996), however all of the functions below are actually radial functions, so ‘RBF’ is a slight misnomer.

Radial Basis net (Gaussian spline)

G(D XC ) = e

?D 2 ? ? XC ? ?

? ? ? ?

(15)

Thin-plate spline

(16)

G(D XC ) = D XC log(D XC + 1)

2

Cubic

(17)

G(D XC ) = 1 + D XC

3

Multi-quadratic G(D XC ) = β 2 + D XC

(

n 2 2

)

(18)

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Linear spline G(D XC ) = 1 + D XC

(19)

2

, log and e operate over the individual elements of the matrix. The DXC term is the

distance between each row of X and the centroid points, C. Therefore, S can be thought of as a dot product between the input and the bases, normalized by the size of both X and C, and then put through a non-linear transform.

D XC = ?2C ? X T + C T C + X T X

We also ensured that all activations were normalized such that

V

(20)

?r ∑ S rv = 1

v =1

(21)

Where V is the number of centroids. Normalization meant that extrapolation outside the range of datapoints did not result in overly large or small values. Instead, the surface far from the points converges to the average of the datapoint values. Given the small number of datapoints, we also set the rows of C to be equal to the existing datapoints, so each centroid was a datapoint.

Accuracy of methods

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The results of polynomial and spline interpolation are shown on the following pages. A set of six statistics is reported with each figure. These indicate the accuracy of the induced surface. The statistics are:

Stddev or standard deviation, is the average difference between a point and the mean. This is the error that we would incur if we just fitted a model which was equal to the mean. The model errors (reported below) should be lower than this value.

nd modelerror is the average error for the model predicting outcomes in the Ndimensional space. Thus, a high nd modelerror relative to stdev means that the domain might not have a lot of predictability to start with, or it could indicate that the model isn’t doing a very good job predicting in this domain.

2d stress indicates how well distances were replicated in the 2-dimensional space. A high stress indicates that the underlying high dimensional projection was difficult, and that the vertices of the surface aren’t accurate representations of the high dimensional data..

2d modelerror is the prediction error from fitting the model to the 2D datapoints. This tells us the overall amount of error on our 2D reconstruction of the surface.

To calculate model error, a k-fold cross-validation estimate was performed. This involves dividing the data into k equal sets, inducing the model with (k-1) subsets of the data, and then finding out the error on the remaining holdout portion of the data. This is done k

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times, holding out a different set in turn. The errors are then averaged. This procedure therefore finds a relatively good picture of the error.

When looking at these results, the comparison between stddev and 2d modelerror is probably the most important statistic to look at. Ideally models should have an error which is less than half the standard deviation.

In our experiments polynomial models gave the highest error (figures 12 and 13). The problem seems to be that with only four points, polynomials both overfit the data, and are constrained to a global model of fixed form (eg. linear, quadratic, etc). Increasing the degree of the polynomial actually makes the situation worse, since the higher the degree, the more “steep” the polynomials become, and the more extreme the overfit. As a result, hold-out error got worse with higher order models.

Splines on the other hand are based on fitting local surfaces around the known datapoints, and as such do a much better job of reconstructing the surface (figures 10 and 11). This is also reflected in their cross-validation errors, which are much better. The surfaces also look more reasonable given what we know about the data and its skeleton projection in two-dimensions. As a result, for this initial four-point analysis, splines are a better choice.

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Low model error

Figure 10 and 11: Linear spline and cubic spline approximation to SkandiaBanken’s surface

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High model error

Figure 12 and 13: Linear and quadratic polynomials

High model error

Note that a parabolic surface is inappropriate given what we know about the data (one side is lower than the other). This shows a deficiency of using a global polynomial model.

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thin plate spline | stddev 56 | nd modelerror 172 | 2d stress 2.001988 | 2d modelerror 125 20

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Figure 14 and 15: Different methods for showing the landscape.

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20

4.0 Company maps

On the next pages we show maps of companies listed under the Skandia Annual Report.

Figure 16 shows American Skandia with operating result on the height dimension. The central depression is interesting, and could be an artifact of the cubic spline.

Figure 17 shows the Skandia Life UK Group plotted against assets under management. This plot shows that assets increased over the years 1994-1997, and may have done so by zig-zagging across the face of a cliff. It is not clear whether the fitness in year 1997 has reached a plateau or will continue to rise.

Figure 18 plots Skandia Real Estate’s market value for the past four years. This is the only Skandia Group company which has reported a downward trend in the time period measured. The shape of the surface suggests that the dramatic sink hole in year 97 might have been avoided if the path implied in years 94-95 had been continued. This would have involved maintaining the IC variable change delta from 94-95, into 95-96. The way this trajectory curves back on itself is also fascinating. This implies that by 1997 some of the parameters have apparently returned to a level near where they were in 1994. However, the variables which are “causing” the company to be in the sink hole are still at their aberrant levels.

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Figure 19 shows the landscape of SkandiaBanken, looking “down” from the top of the peak. SkandiaBanken was used as the main example throughout this paper, and other graphs of SkandiaBanken can be found in figures 6-7, 9-15. It looks as though if the company continues on its present course, its fitness will continue to increase. Note that on the very top right hand side, the landscape starts to curve down. This is a side-effect of there being very few points out that far. Because spline contributions are proportional to distance, far from a set of known points, the contribution of each to the surface will become roughly equal, and the estimation will approach the mean of the points. Therefore, it is a little risky to draw conclusions about the shape of the landscape too far from known datapoints. If we are able to obtain data from other companies, it may be possible to have datapoints scattered richly throughout the landscape, resulting in an interesting landscape.

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American Skandia

Plotted against operating result

Figure 16: American Skandia IC Focus Financial Financial Customer Customer Customer Customer Human Human Human Human Process Process Process Renewal Renewal Renewal Financial IC Variable Return on capital employed (%) Value added per employee Number of contracts Savings per contract (SEK 000s) Surrender ratio (%) Points of sale Number of employees, full time Number of managers Of whom women Training expense per employee (SEK 000s) Number of contracts per employee Adm exp per gross premiums written (%) IT expense per admin expense (%) Increase in net premiums written (%) Development expense / admin expense (%) Share of staff under 40 years (%) Operating result (MSEK) 1994 12.2 1666 59089 333 4.2 11573 220 62 13 9.8 269 2.9 8.8 17.8 11.6 72 115 1995 28.7 1904 87836 360 4.1 18012 300 81 28 2.5 293 3.3 13.1 29.9 10.1 81 355 1996 27.1 2206 133641 396 4.4 33287 418 86 27 15.4 320 2.9 12.5 113.7 9.9 78 579 1997 21.9 2616 189104 499 4.4 45881 599 88 50 2.7 316 3.5 8.1 31.9 9.8 76 1027

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Skandia Life UK Group

Plotting against Assets under management

Figure 17: Skandia Life UK Group IC Focus Financial Financial Customer Customer Customer Human Process Renewal Renewal Renewal Financial IC Variable Return on capital employed (%) Operating result (MSEK) Number of contracts Savings per contract (SEK 000s) Service awards (max value=5) Number of employees, full time Number of contracts per employee Increase in net premium, new sales (%) Pension products, share of new sales (%) Increase in assets under management (%) Assets under management (MSEK) 1994 33.2 483 228397 149 5 741 308 95 13 31 34274 1995 25.5 518 250807 159 5 814 308 -35 23 25 37859 1996 26.1 619 304425 171 5 993 307 51 28 24 52400 1997 19.3 633 361664 192 4 1242 291 10 31 22 70018

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Skandia Real Estate

Plotted against Market value

Figure 18: Skandia Real Estate IC Focus Financial Financial Financial Customer Human Human Human Process Process Process Process Renewal Renewal Renewal Renewal Renewal Financial IC Variable Direct yield (%) Net operating income (MSEK) Total yield (%) Average rent (SEK per square meter) Employee turnover (%) Average years of service with company College graduates / total number of office staff (%) Occupancy rate measured by area (%) Financial occupancy rate (%) Net operating income per square meter (SEK) Costs per square meter, Sweden (SEK) Property turnover: purchases (%) Property turnover: sales (%) Change and development of existing holdings (MSEK) Training expense / administrative expense (%) Share of staff under 40 years (%) Market value (MSEK) 1994 6.6 1399 4.44 1041 7.7 10.2 31 89.3 91.2 657 272 0.8 0.4 313 1.0 72 21504 1995 6.16 1258 5.06 970 7.9 10.1 31 89.7 93.0 590 276 3.2 6.1 333 1.5 81 20702 1996 5.93 1215 -0.62 960 10.1 10.0 32 91.8 94.9 569 274 3.1 1.1 311 1.0 78 20092 1997 5.96 1130 7.73 951 10.0 12.0 36 93.7 96.2 553 304 0.2 8.1 235 0.8 76 19206

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Skandiabanken

Plotted against Total operating income

Figure 19: SkandiaBanken IC Focus Financial Financial Financial Customer Human Human Process Renewal Renewal Renewal Renewal Financial IC Variable Operating income (MSEK) Income/expense ratio after load losses Capital ratio Number of customers Average number of employees Of whom, women (%) Payroll costs/administrative expenses (%) Total assets (MSEK) Deposits and borrowing, general public (MSEK) Lending and leasing (MSEK) Net asset value of funds (MSEK) Total operating income 1994 75 1.49 25 38000 130 42 38 3600 1300 3200 4700 226 1995 85 1.32 24.48 126000 163 45 42 5600 4300 3700 6300 351 1996 86 1.30 14.95 157000 200 49 46 8100 6200 7600 7400 373 1997 104 1.35 12.90 197000 218 56 49 9100 7600 8500 9900 398

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5.0 What-if analysis

The most compelling use for maps is in what-if analysis. Using the map, a user can look around, and seek good or bad regions on the landscape. He or she can then bring up the IC vectors which underlie those “geological formations”.

After finding a point’s IC vector, the vector difference between the current point and this desired point can then be computed. The difference vector between the two prescribes a list of adjustments that need to be made to the organization, to allow it to move towards that new state.

For example, a user might find that the most significant difference between the current state and a desired location, is that employee turnover is 6% lower at the desired location. Armed with this knowledge, a manager can take practical steps to address the employee turnover problem.

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Figure 20: What-if is the practice of trying to chart a course to reach a desired point on the map, for example, moving from a sink area to a higher region.

Inverse model method

Determining what IC vector is associated with a given 2D map location is a difficult problem. Because a function estimator constructed much of the surface, there is often no known IC vector for many of the points on the map.

N-dimensional company state (IC) vectors Inverse model MDS

Figure 21: An inverse model attempts to infer a mapping from 2D map location to IC vector. The difficulty is reverse mapping from a novel point on the 2D map.

A solution is to construct an inverse model predicting given 2D location, the IC vector (figure 21). Because of the risk of getting the reverse model wrong, we have elected to use a very stable method, which otherwise may not give as smooth surface as some of the other methods.

The method we used for the inverse model has been named by Masters (1994) a Generalized Regression Neural Network. We will refer to it as “inverse-distance interpolation”. Let C equal the set of known cases in 2D space. These are the points

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which were projected from their high-dimensional representation. Let X equal the position we wish to test in 2D space. We first calculate the distance to each of the known points in 2D space using equation 20. Then let dij equal the ijth element of DXC. We then invert these distances, so that a low distance to a point becomes a very large value.

D XC = 1 D XC +

(22)

Then we normalize so that all distances become percentages of the total distance.

d ij =

d ij

∑d

k =1

n

(23)

ik

Finally, we know that the points corresponding to those known 2D locations C, is the matrix of n-dimensional coordinates, Y. We now linearly combine these highdimensional IC vectors, weighted by their distance coefficient calculated in the 2D space.

Y * = D XC ? Y

(24)

Figure 22 shows what surfaces built using inverse-distance interpolation look like. In this figure inverse-distance interpolation was used to estimate the 2D position-fitness surface, and the resulting surface can be compared against those in figures 10-11. The method causes very sharp sink holes and peaks, surrounded by areas which are fairly uniform. Although the landscape is less smooth than the previous figures, the method has the

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advantage of always choosing values which are interpolations of the true (known) values. This minimizes the chance of an artifact in the approximation scheme causing unusual values to be returned from a what-if query. To be even more conservative, we could have instituted a nearest neighbor scheme that would have caused the result of a what-if query to be the value of the nearest input example. However in our experiments the inversedistance method worked well.

Figure 22: Inverse-distance interpolation

Results

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A prototype what-if analysis was performed on the Skandia Real Estate map. As described in figure x, Skandia Real Estate is the only member of the Skandia Group companies which has experienced a decline over the past four years. Thus, a manager may be especially interested in determining which variables are causing this problem. After selecting a new point (on the highest region, towards the top left of the map), the most significant differences are between the current and new IC location were then reported. These differences are reported in table 24.

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Skandia Real Estate IC Map

What-if location

Current location

Figure 23: Top-view of the Skandia Real Estate map, which has experienced a decline for the past four consecutive years. Current position (1997) and queried position are shown.

The number at right in table 24 is the number of standard deviations which the current location and the what-if location differ by. Thus, Property turnover: sales differs by 1.5 standard deviations at the new location.

Property turnover: sales (%)

1.549655

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Net operating income (MSEK) Financial occupancy rate (%)

1.514948 1.336784

Table 24: Most significant differences between present and queried point on Skandia Real Estate Map

This finding is consistent with the data. Property turnover increased from 0.4% in 1994 to 8.1% in 1997, and was the largest percentage change of any variable. Compared to the other variables and their value ranges over the period, this variable does indeed stand out as being a worrisome indicator.

6.0 Conclusion

This paper has presented a method for strategic business analysis, visualization and whatif analysis. Despite the many sources of error with the method – error caused by highdimensional reduction, surface estimation, and inverse surface estimation, the models produced are surprisingly accurate. The three MDS methods tested all converged to a very similar low-dimensional representation of the data. The topographic aspects of the fitness landscape made intuitive when compared to the actual progress of these companies over time. Finally, the inverse model also presented reasonable results, which was another welcome surprise. As a result of these checks, we are led to conclude that this method could be used in a highly practical manner for depicting a company’s state space.

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There are many enhancements we would like to investigate. Error bars could be added to form a smoothly varying “atmosphere” over the landscape. This could help users to discount topographic formations far from training cases, which are likely to be artifacts of the extrapolation/estimation technique.

Not all variables should be included in the multi-dimensional projection. Specifically we should drop dimensions which don’t predict fitness well. Therefore, before forcing the map to take into account many extraneous dimensions with the resulting high stress, we should perform a variable selection process to remove variables which only contribute noise to the model.

The present paper only deals with a company’s own trajectory through space, and tries to fill in the region surrounding this path. However, it would be desirable to add in the IC vectors of other equivalent companies in other markets. Whereas a map of the company’s past trajectory can show where it has been, a map which uses data from other sources can offer predictions as to new regions which the company has yet to explore.

Building these composite maps will introduce more sources of error to the landscape, since companies in different markets may be qualitatively different. The procedure for handling these effects will could involve a form of weighted MDS and prediction, where companies are weighted inversely to their similarity to the original company.

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