BAKER ( ) - Entropy - Measure of Diversity

7/30/2019 BAKER ( ) - Entropy - Measure of Diversity

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Entropy: Measure ofDiversity?

David Lee Baker

David E. Booth

William Acar

Management & Information Systems Working Paper MIS2007-08:1(Do not cite without permission)


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EntropyIn Management Strategy I

Managers are interested in knowing howdiversified are the firms lines of business

It has been considered (and aptlydiscussed and hotly debated) that morediverse businesses are more profitable

May not always be the case though


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EntropyIn Management Strategy II

In diversification we need to consider related(similar to the firms core business) versusunrelated (dissimilar) diversification

In related diversification the firms several linesof business, even though distinct, still possesssome kind of fit

In unrelated diversification there is no common

linkage or element of fit among the firms lines ofbusiness In this sense unrelated diversification may be

considered as pure diversification


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DiversityAn Example

If we take a beaker of water (H2O) and a veryconcentrated solution of red food coloring andwe then add a drop of the coloring to the water

we will see the red color diffusing throughout thewater and thus, we go from concentration todiversification

Economists, as well as chemists and physicists,

want to define concentration vs. diversification

Concentration and diversification are two endsof the spectrum


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EntropyBackground & History I

Introduction of a mysterious entity calledthe H-function, or statistical entropy, by

Ludwig Boltzmann (1896) Defined as the mean value of the logarithm ofa probability density function (p.d.f.)

Measured the amount of uncertainty about thepossible states of a physical system

Because there was still disagreement about theexistence of atoms his statistical entropygenerated much debate


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EntropyBackground & History II

Claude E. Shannon (1948) generalized Boltzmannsentropy to information theory and proved that it hadthe properties that allow it to be taken as theaverage amount of information conveyed by a

discrete random variable about another Mathematicians have further refined the Shannonentropy, and new tools, such as the relative orconditional entropies have been developed

Norbert Wiener and Claude E. Shannon along with

others extended Boltzmanns earlier theories tomore general cases Shannon had studied under Wiener at MIT in the late 1930s,

graduating with both a masters and doctorate in mathematics


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EntropyBackground & History III

Mathematically this reduces to the amountof uncertainty contained in a probabilistic

experimentA as measured by thefunction:

Hm (p1, . . . , pm) = i = 1 m pi log pi,


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EntropyBackground & History IV

The Entropy (inverse) measure of industryconcentration weights each pi by the

logarithm (log) of 1/pi, e.g.:

E =i = 1 n pi log 1/pi,

Notice that we have replaced H by E andused the fact that log(A)=log(1/A)


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Herfindhahls Measure

Herfindahls contribution to diversificationmeasures was the suggestion that the

share of each firm be weighted by itself,i.e..: (using H for Herfindahl)

H = i = 1 n pi pi


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Decomposability of Entropy

Entropy is a decomposable measure(Khinchins Decomposition Theorem)

Herfindahl is decomposable because in fact,

Herfindahl is an approximation to Entropy

2 & 4 digit SIC code is compatible withthese decompositions but is NAICS?

Further research is needed


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Diversification-Score AnomaliesBased on Entropy Decomposition proposed in this paper.

Probably violates the Decomposition Theorem

Note that columns 17 & 18 do not sum to column 16.

Source: Ragunathan (1995), Journal of Management, 21(5), J une, excerpts of p. 992.


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Corporate DiversificationCorrect Totals

Note that columns 8 & 9 add up to column 7, as they should.

Source: J acquemin & Berry (1979),The Journal of Industrial Economics, 27(4), J une, p. 362.


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Triangular Numbers I

Source: http://www.mathematische-basteleien.de/triangularnumber.htm

These are the first 100 triangular numbers:


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Triangular Numbers II

You can illustrate the name triangular number by thefollowing drawing:



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Sample Triangular Load

DistributionGraphTriangular Load1; 0.3333

2; 0.2667

4; 0.1333

5; 0.0667

3; 0.2000

0.0000

0.0500

0.1000

0.1500

0.2000

0.2500

0.3000

0.3500

0 1 2 3 4 5 6


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Triangular DistributionsExamples

Position in Pascal's Triangle top

... ...

Pascal's triangle makes a contribution tomany fields of the number theory.

The red numbers are triangle numbers.You even can find the sum of thetriangular numbers easily.Example: 1+3+6+10+15=35

You can express the triangular numbers as binomial coefficients



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Triangular DistributionsAnalyses I

Triangular

Samples,

Figure # n

3.1 5/15, 4/15, 3/15, 2/15, 1/15 5

3.2 10/55, 9/55, 8/55, 7/55, . . . 1/55 10

Range of Values


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Triangular DistributionsAnalyses II

TRIANGULAR

SAMPLES

Figure # n Uncalibrated

Herfinda

hl

Calibrated

Herfinda

hl

Uncalibrated

Entropy

Calibrated

Entropy

Calibrated

A1,

Acar-Bhatnagar

Calibrated

A2,

Acar-Troutt

Single-S

umF

ormula

3.1 5/15, 4/15, 3/15, 2/15, 1/15 5 0.75556 0.94444 1.48975 0.92563 0.50000 0.66667

3.2 10/55, 9/55. 8/55, 7/55, . . . 1/55 10 0.87273 0.96970 2.15128 0.93429 0.51279 0.66667

Range of Values


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Concluding Remarks

and Future Directions II In their seminal article, J acquemin and

Berry (1979) have specified how the

decomposition of the Entropy measurecan be related to SIC codes by breakingdown the diversity measurement betweenthe 2-digit and the 4-digit codes. We now

need to see if that is still true for NAICS We will be following up on their work and

further examining statistical properties


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Triangular Number Theory

A triangular numberis the sum of the n natural numbers from 1 to n.Triangular numbers are so called because they describe numbers ofballs that can be arranged in a triangle. The nth triangular number isgiven by the following formula:

Tn =k=1n k = 1+2+3+ . . . +(n-2)+(n-1)+n = n(n+1) = n2+1 = (n+1)2 2 ( 2 )

As shown in the rightmost term of this formula, every triangular numberis a binomial coefficient: the nth triangular is the number of distinctpairs to be selected fromn + 1 objects. In this form it solves the'handshake problem' of counting the number of handshakes if eachperson in a room shakes hands once with each other person.

The sequence of triangular numbers (sequenceA000217 in OEIS) forn =1, 2, 3... is:

1, 3, 6, 10, 15, 21, 28, 36, 45, 55, ...


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Thermodynamics

The first law of Thermodynamics which statesthat energy is neither created or destroyeddirects us to a world where energy is lost

The second law says that entropy always tendsto increase in a closed system, forecasting auniverse that is constantly winding down

The tension between the first and second lawsruns like a recurring theme between turn-of-thecentury cultural formations


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NAICS vs. SIC Codes

The North American Industry ClassificationSystem (NAICS) has replaced the U.S.Standard Industrial Classification (SIC)

system. NAICS will reshape the way weview our changing economy.

NAICS was developed jointly by the U.S.,Canada, and Mexico to provide newcomparability in statistics about businessactivity across North America.


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NAICS

The official 2007 US NAICS Manual NorthAmerican Industry Classification System--UnitedStates, 2007 includes definitions for eachindustry, tables showing correspondence

between 2007 NAICS and 2002 NAICS forcodes that changed, and a comprehensiveindex--features also available on this web site.

To order the 1400-page 2007 Manual, in print,call NTIS at (800) 553-6847 or (703) 605-6000,

or check the NTIS web site. The 2002 Manual,showing correspondence between 2002 NAICSand 1997 NAICS, and the 1997 Manual,showing correspondence between 1997 NAICSand 1987 SIC, are also available.

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BAKER ( ) - Entropy - Measure of Diversity