a cross-impact simulation and analysis of events through time with a micro-computer

North-Holland Microprocessing and Microprogramming 13 (1984) 189 198

189

A Cross-Impact Simulation and Analysis of Events Through Time with a Micro-Computer*

Eduardo Rivera Chilpancingo No. 8, 06100 Mexico, D.F., Mexico

and

Ma. Luisa Revilla

Fundaci6n Javier Barros Sierra, Apartado Postal 20 061, 01000 M#xico, D.F., Mexico

It is of the utmost importance, in strategic planning and the making of projections, both in private business as well as in governmental agencies, and for decisionmakers in general, to possess the ability to evaluate rationally their conjectures as to the possibility for the occurrence of events in the future. This kind of planning can hardly be left to one's subordinates, but learning to do it requires experimentation and practice. With the advent of the mini-computers it is now possible to have an instrument which is portable, interactive and easy to use. In this article, we will present the use of a method called 'SECITE' which has been developed by the authors and is illustrated with an example from the educational field.

Keywords: Cross-impact analysis, Monte-Carlo simulation model, planning applications, BASIC, interactive system, scenario, conditional probability of events.

I. Introduction

By experimenting with forecasting techniques, decisionmakers and other potential users of these techniques can improve their understanding of the possible future consequences of decisions being made in the present.

The long-term planning process cannot be carried out under conditions of a blind acceptance of the goals to be achieved without first having in- vestigated different options, the probabili ty of their realization, and the consequences of these decisions as they affect other events. Carried out in

* This paper is a modified version of a lecture presented at the 15th International Symposium on Mini and Micro Computers, April 13-15, 1981, Mexico City, Mexico.

this fashion, one moves f rom the sphere of deterministic predictions to that of probability forecasting and the rational examination of pro- jected alternatives.

The method which has been developed here permits one to create experimental models of the total problem (even when it is perceived subjectively) by means of the interaction of events and the probability of their occurrence in order to configure and evaluate alternative options. The method described here forms part of the larger concept called 'Cross- Impact Analysis ' .

S. Enzer [6] has defined cross-impact analysis as being 'a generic term for a family of techniques which attempts to evaluate changes in the probability of the occurrence of certain events - among a total group of possible future events - in order to demonstrate the limits to change in the probabili ty for some points in the entire group ' .

This type of cross-impact analysis of models necessitates large computers and long periods of time to carry out and evaluate. For this reason, its use has remained restricted to large corporations with planning departments. S. Alter [1] and others discovered a mathematical formula which permitted the reduction of the number of cross-impact evaluations when there was non-occurrence. Nevertheless, Alter 's formula left intact the long Monte Carlo simulations or the scenarios were designed in an interactive form. Due to a modular reprogramming in which an interactive method was designed for the testing of the convergence of the Monte Carlo simulation, the authors found a way to implement the method on a micro-computer on which the results can be obtained in very short time periods. We developed a program, SECITE, which is both portable and interactive, to be used in the design and evaluation of the most probable scenarios through time. The program makes use of cross-impact principles.

190 E. Rivera, M.L. Revilla / Cross-Impact Simulation and Analysis of Events

For the implementation of the program, we used a commercial micro-computer (personal computer) and the most common language used by these computers (BASIC).

2. Antecedents

Prior to the cross-impact method, other forecasting evaluation techniques had also been developed, e.g., the Delphi method - in conjunction with expert consultation - and those employed by the Rand Corporation, such as the in- terdependency matrices.

The cross-impact method, as a technique, was first conceived and put into operation by T.J . Gor- don and H. Haywood in 1968, using a computer to aid them in the synthesis of the interaction of events. The first operational analysis using the cross-impact method was done by Selwyn Enzer, T.J . Gordon, R. Rochberg and R. Buchele in 1970.

Subsequently, the method was criticized in diverse ways and various solutions were proposed. In this manner the method developed and evolved, empirically as well as theoretically.

Dalkey [3] proposed the rule of the triangle theorem with the object of verifying or correcting the estimates for conditional probabilities. Later, Turof f demonstrated that such probabilities through time are not conditional probabilities. More recently, Alter has demonstrated that the Bayes theorem cannot be applied. Almost simultaneously, Turoff , Kane and others developed a cross-impact method based on an interactive program, KSIM, which made use of the additive principle of impacts.

S. Alter [1] improved the 'multiplicative method ' , initially proposed by S. Enzer in 1970. Rosove and others, in 1973, used the cross-impact matrix in the evaluation of inter-relation tendencies.

Duperrin and Godet [4] developed the SMIC in order to find the most probable scenarios. This method is based on the optimal adjustment of the probabilities in order to seek their convergence with classical probability theory.

M. Bloom [2] proposed to relate the cross- impact technique with the Dynamic Systems

methodology for the crossing of tendencies. O. Helmer [10] proposed the space method -

another multiplicative transformation. R. Eymard, in the same year, worked with the application of Markovian techniques to the cross-impact model.

S. Enzer [8] proposed a different interpretation which in the use of static conditional probabilities is congruent with the Bayes theorem, while in the case of dynamic conditional probabilities requires an adaptation of the formulas.

Novaky and Lorant [14] developed an interactive variant to delimit the probabilities for the occurrence of chains of events.

Mitchell and Tyderman, at the University of Canberra (Australia), following the line developed by Godet, proposed a static method for the solu- tion to the problem of conditional probabilities. Similarly, Kaya and others, in 1979, applied the linear programming method with the object of inter-relating the minimum and maximum probabilities for each possible scenario.

O. Helmer [11] proposed the use of a cross- impact model with an interactive focus, using a simulation in the analysis of tendencies.

The most elaborate model and use of the cross- impact is the INTERAX, developed by Enzer, Alter, et al. [8]. The model permits the evaluation of the economic climate for big industry in the United States. The model is based on the multiplicative model and makes use of a Monte Carlo simulation in which each scenario can be interactively modified.

All the models can be included in the scheme of Fig. 1.

3. Analysis of the Cross-Impact

This technique requires the conceptualization of a series of events which are representative of tendencies that can be considered as a system. The series of events must be inter-related in a causal fashion to form a model which can be described in the following way: ' I f such an event occurs it will modify the probability of the occurrence of other events, and if the event does not occur it will also affect the probability of the occurrence of subsequent events'. This model can be represented in

E. Rivera, M.L. Revilla / Cross-Impact Simulation and Analysis of Events 191

J Additive

Cross-Impact

Treatment and Concruencv with Correction of °robahil5 t" Probabilities

~'ultiplicative St~ace P ~djustpent Ot~tiplzat~<.'n

Dynamic Sta t i c

/ ~4onte Carlo

\ I n t e r a c t i v e

J

Conjunction with other technioues:

bna]ysis of tendencies

Simulation

Distances and clusters

Delphi and expert consultation

etc .

Fig. 1.

describe the way the occurrence or non-occurrence o f an event (horizontal line) affect the others (ver- tical line) (Fig. 2). Obviously, the diagonal is com- posed o f neutral values since it is supposed that the occurrence o f an event has no influence upont itself. The construction of the matrix rests on the fo l lowing suppositions:

[::wN~ t i

Event j

I ...... Value of the impact

1. Events can occur only once (therefore, there can be no feedback on a single event).

2. The interaction between events take place, principally, on a one-to-one basis, i.e., the occurrence (or non-occurrence) o f two or more events does not significantly the one-to-one interactions.

3. There are no s imultaneously incompatible or exclusive events.

4. There is an interdependence between the series o f events.

5. The series o f events is sufficient, i.e., there is no relevant external event which is not already included.

4. The Mechanics of Secite

Cross-Impact Matrix

Fig. 2. Cross-Impact Matrix.

The program which was developed is called SECITE (in Spanish: Simulaci6n para la Evalua- ci6n Cruzada de Impactos en el T iempo de Even- tos; in English: Cross-Impact Simulation for the

192 E. Rivera, M,L. Revilla / Cross-Impact Simulation and Analysis of Events

Analysis of Events through Time). The mechanics used in this program was that of the multiplicative impact method, i.e., for a given simulation time interval an occurrence probabili ty is associated in- dependently with each event. This probabil i ty will be affected by an impact which multiplies it measuredly, in accord with the law of the equilibrium of probabilities through time [3]. Therefore, the impacts can have values between <0, + oo>. I f the impact has a value between <0,1 > the probabil i ty for the occurrence of the impacted event will be diminished in a multiplicative form, while if the impact is between < 1, + oo > its pro-

bability will be increased in a multiplicative form. I f the impact has a value of 1, the impact will be null or will have no effect.

By means of the random generation of numbers between 0 and 1, the occurrence or non-occurrence of each event in a given time interval is determined. This is determined by means of the comparison of the probabil i ty interval of the event with the random number. I f this random number is less, the event occurred, at which point the modification in

the probabilities for the rest of the events is effectuated in accord with the cross-impact matrix in a multiplicative form. In the case of a non- occurrence, the modification is effectuated through the non-occurrence impact matrix. Upon the completion of the series of time intervals, a compound scenario is obtained by the occurrence of some events at certain time intervals (Fig. 3). By simulating this process several times (Monte Carlo simulation) one can observe, on the average, which event has the most probabili ty of occurring and at which interval. Thus - beginning with a matrix which presents the occurrence of all the events - by means of the comparison of the average total occurrence of events with the number of occurrences of each event in each of the intervals, one obtains

the scenario or configuration of that which is most probable. I f the number obtained in this comparison is less than the average, one can suppose that this event has no probabili ty of occurring. In the contrary case, the interval in which one finds the greatest number of occurrences is considered to be the scene in which said event occurred. Should

New Scenario

I C a l c u l a t i o n of the C o n d i t i o n a l P r o b a b i l i t y / (PC) means of the d i s t r i b u t i o n (DP) fo r event~

byGenerati°nchance of numbers I

Yes )

& Modification of PC by the I ,Non-Occurrence Matrix(NE 1

ICorrection of the OE I and NE Matrices I

4

Modification of PC I by the Occurrence Matrix I(OE) I

Fig. 3. Simulation of an Interval.

E. Rivera, M.L. Revilla / Cross-Impact Simulation and Analysis of Events 193

it happen that there are two intervals with an equal bable is determined according to the greater value number of occurrences, the one which is most pro- of the probability interval.

Reading of the data and simulation conditions ]

[ "New Scenario beoins time J

IN 1 Simulation in one probability interval and one impact interval

o.@ Yes

Yes

No I No

Generation of most probable confimuration

"4

I Calculation and visualization of results

Fig. 4. Scheme of SECITE.

f

INPUT

Cross-Impact Matrix

Probability Distribution of

Simulation Conditions

Interactive ~ I Chanaes of the~ C o t ~ /

Sumary ~ost Probable Matrix Configuration of Impacts ~atrix (CMP)

S E C I T E I

Visualization Histoaram of Most of Probable Impacts Scenario

OUTPUT

Probability Distribution for the CMP

Fig. 5. Input and Output for SECITE.


Principal Formulas Employed

Calculation of the Probability Interval DP(I, t ) - DP(I , t - 1)

PC(I, t ) = 1 - D P ( I , t - 1)

Calculation of the Probability Interval of an event given the occurrence of another PC(l, t )OE(K, I)

PC(l , t / k ) = 1 - PC(l , t) + PC(I, t )OE(K, I)

Probability Interval of an event given the non-occurrence of another PC(l, t ) l - PC(l, t) + PC(l, t )OE(K, I) - DP(I , t - 1)OE(K, I)

PC(I, t / k ) = 1 - DP(I , t - 1 )1 - PC(1, t) + PC(l, t )OM(K , I)

Calculation of the New Cross-Impact Matrix for the occurrence of the event PC(l, t / k ) PC(I, t)

OE(K, I) = + 1 - PC(l, t / k ) 1 - PC(l, t)

New Non-Occurrence Matrix PC(I, t/[~)

NE(K, I) = + 1 - PC(l, t /E)

PC(l, t)

1 - PC(l, t)

Modification of the Probability Interval by the Occurrence Matrix PC(l, t )OE(K, I)

PC(I, t) = 1 - PC(l, t) + PC(l, t )OE(K, I)

Modification of the Probability Interval by the Non-Occurrence Matrix PC(l, t )NE(K, I)

PC(I, t) = 1 - PC(l, t) + PC(l, t )NE(K, I)

With these probability intervals, the calculations for the subsequent time interval are made.

The number of repetitions necessary to complete the simulation will be at the moment in which the conditions of convergency are met - i.e., the number of most probable equal consecutive con- figurations - determined by the user (see Fig. 4).

The principal formulas that were employed are shown in the box below. They were taken and adapted from Enzer [8]. Our use of the formulas differs from that of Enzer's insofar as he used them only interactively to generate scenarios in which, on the basis of chance, the scenarios are analyzed one by one but not as a group.

5. Results

The results obtained (Fig. 5) permit one to observe

the total accumulated occurrences of the events for

all the scenarios of the Monte Carlo simulation. This permits the deduction of other scenarios with a high probability as well as the most probable configuration. The histogram of accumulated impacts for each event allows one to distinguish the most important events in the group - those events being the ones which have been most affected by other events and, consequently, those which have had the most impact on others•

Finally, the distribution probability is recal-

E. Rivera, M.L. Revilla / Cross-Impact Simulation and Analysis o f Events 195

culated for the most probable configuration, such that comparing it with the initial probabil i ty

distribution, one is permitted to see in which intervals the greatest changes take place and, therefore, which intervals are the most critical.

5.1. Example

With the objective of demonstrat ing how the SECITE system operates, we have taken the following example which forms part of a larger study and model. The events or circumstances are the following: 1. The participation of women in the Economical-

ly Active Populat ion has risen f rom 23°7o (1977)

to 35%. 2. The annual rate of economic growth has not re-

mained constant at 7 % . 3. The participation of women in high school has

risen to 45°7o of the total enrollment. 4. The university preparatory education has come

to represent 20°7o of the graduate university education.

5. The graduation rate f rom high school decreased to 7°7o annual rate of growth.

6. The participation of women in university education has risen to 40°7o of the total enrollment.

The cross-impact matrix is given in Fig. 6,

80 85 90 95 00

E1 .22 .27 .3 .46 .6

E2 .18 .31 .47 .6 .69

E3 .17 .36 .5 .69 .79

E4 .17 .28 .3] .37 .42

E5 .16 .24 .31 .37 .42

E6 .17 .34 .48 .58 .68

Event /T ime

Fig. 7. The Initial Probability Distribution.

6 2 0 2 3 4 I 2 2 2 4 5 4 2 0 2 2 ] ] 3 4 2 0 I 0 1 5 2 6 1

Fig. 8. Summary Matrix of Impacts.

1 1 1 1 1 1

1 1 2.5 1.5 2 2.2

1.75 1 1 1 1 3

1 1 i 1 0.2 1

1 1 1 1 1 2

2 1 1 2.5 1 1

Event /T ime

I 0 0 0 0 I 0 0 0 0 0 1 0 0 0 0 0 0 0 i I 0 0 0 0 0 0 0 I 0

Fig. 9. Most Probable Configuration Matrix.

Fig. 6. The Cross-Impact Matrix.

whereas the initial probabili ty distribution is presented in Fig. 7.

The algorithm for the example reached the convergence at the 20th Scenario (i.e. it has 3 consecutive equal most possible scenarios). Some of the printed results are presented in the next figures:

A summary of matrix impacts (Fig. 8), the most probable configuration matrix (Fig. 9), the most probable scenario (Fig. 10), the histogram of impacts (Fig. 11), the interval probabili ty distribution for the CMP and the accumulated probabili ty distribution (Fig. 12) and finally the first three graphs of the accumulated probabili ty distribution (Fig. 13).


TIME=I

TIME=2

TIME=3

TIME=4

T IM_E= 5

*******

1

* * * * * * *

*******

. 3 ,

*******

*******

* 6 *

*******

* * * * * * *

* 4 *

*******

******* *******

* 2 * * 5 * * * * * * * * * * * * * * *

+

*************************** E 1

*************************** 13

+

+

******************************** 15

+

******************* 9

+

************** E 5

************** 7

+

******************************** E 6 ********************************

******************************** 15

-F

Fig. 10. Most Probable Scenario. Fig. 11. Histogram of Impacts.

Event I Event 2 Event 3 Event 4 Event 5 Event 6

22 18 17 17 16 17

193370166 .013017335 .0448285173 5.1]424226E-05 150536585 .231884058 .245283019 .225 185651278 .411764706 .]80389972 4.75963829E-04 093579562 9.90031403E-03 5.62082561E-05 2.73862784E-05 0404040404 .0761801736 .0307758671 ].31232715E-04 )1207458 .038235514 3.99435931E-05 .804878049

Event I .22 .270828729 .379018863 .406856526 .406886961 Event 2 .18 .31 .47 .6 .69 Event 3 .17 .324090561 .602406212 .674126952 .674282056 Event 4 .17 .247671037 .25511933 .255161198 .255181596 Event 5 .16 .193939394 .255345231 .278262627 .278357343 Event 6 .17 .263021902 .29]288638 .29122895 .86170321

Fig. 12. The Interval Probability Distribution for the CMP (top) and the Accumulated Probability Distribution (bottom).

6. Rivera, M.L. Revilla / Cross-Impact Simulation and Analysis of Events 197

. 22 .4

+

+

+

"k +

+

Time

Probability

Event 1

.17 .69

Time

• Probability

Event

.17 .67

Time

• Probability

Event

Fig. 13. Some Graphs of the ACC Probability Distribution.

6. Conclusions

The method presented here has the advantage of being simple to use in strategy planning, specially when placed at the disposition of decision-makers in a micro-computer using standard BASIC, which makes it transferable• Given the facilities possessed by the A P P L E micro-computer, the method is being reprogrammed in PASCAL, thus permitting the running of long examples in a question of

seconds• The technique presented here takes into account the uncertainty and probability in the design and understanding of models which will permit the configuration and evaluation of scenarios. Such models explain the hypothesis and also help in testing internal coherency•

Nevertheless, the model has its limitations since it can only deal with the interaction of events on a two-by-two basis while the impact is taken to be constant• For this reason the model should be used in planning as one technique among others• The principal use of the method is for micro-planning in business and institutions in the evaluation of future alternatives, expressed in terms of the occurrence of events, for specific problems•

References

[1] Alter Steve, The computational mathematics of time- dependant cross-impact modeling, Center for Futures Research, report M-26, University of Southern California (Oct., 1976).

[2] F. Mitchell, Bloom, Deterministic trend cross-impact forecasting, Technological Forecasting and Social Change 8, Nr. 1, (1975) 35-74.

[3] N. Dalkey, An elementary cross-impact model, Technological Forecasting and Social Change 3, Nr. 3, (1972) 341-351.

[4] J.C. Dupperrin and M. Godet, SMIC 74. A method for construction and ranking scenarios, Futures 7, Nr. 4, (1979) 302-312.

[5] Enzer Selwyn, A case study using forecasting as a decision-making aid, Futures 2, Nr. 4 (1970) 341-362.

[6] Enzer Selwyn, Cross-impact technique in technology assessment, Futures 4, Nr. 1 (1971)30-51.

[7] Enzer Selwyn, Alter Steve, INTERAX. An interactive en- vironmental simulator for corporate planning, Center for Futures Research, report M-29, University of Southern California (June, 1977).

[8] Enzer Setwyn, Alter Steve, Verification of unsistency between cross-impact analysis and classical probability, Center for Futures Research, report M-32, University of Southern California (Feb., 1978).

[9] T. Gordon and A. Hayword, Initial experiments with the cross-impact matrix method of forecasting, Futures 1, Nr. 2 (1968) 100-116.

[10] Helmer Olaf, Problems of futures research, Futures 9, Nr. 1, (1977) 17-31.

[11] Helmer Olaf, Cross-impact gaming applied to global planning, Simulation 5, Nr. 2 (1979) 73-86.

[12] J. Kane, A primer for a new cross-impact language - KSIM, Technological Forecasting and Social Change 4, Nr. 2 (1972) 129-142.


[13] R.B. Mitchell, Tyderman J., Subjective conditional probability modelling, Technological Forecasting and Social Change 11, Nr. 2 (1978) 133-152.

[14] Nov&ky Erzs~bet, L6r&nt K~roly, A method for analysis of interrelationships between mutually connected events: a cross-impact model, Technological Forecasting and Social Change 12, (1978) 201-212.

[15] Perry E. Rosove, A trend impact matrix for societal impact assessment, Center for Futures Research, University of Southern California (Apr., 1973).

[16] Murray Turroff, An alternative approach to cross-impact analysis, Technological Forecasting and Social Change 3, (1972) 309-339.

Eduardo Rivera holds a B.S. in Physics, M.A. and Doctorate in Informatics; Doctorate studies in future studies and science policy, he has teached at National University of Mexico and Polytechnic Institute. Presently he is professor of Technology and Production Department of Universidad Aut6noma Metropolitana. He was a researcher and Director of Research at the future studies center 'Fundaci6n Javier Barros Sierra' in Mexico City. He is now a consultant at National Computer Policy Direction in Mexico.

Maria Luisa Revilla G., holds a B.S. in Physics, B.A. in Philosophy. Being a system analyst, she was a researcher at Fundaci6n J. Barros Sierra. She is presently a consultant at Na- tional Computer Policy Direction in Mexico.

a cross-impact simulation and analysis of events through time with a micro-computer

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