Estimation of Markov Chains transition probabilities by means of Conjoint Analysis approach Danilo Leone

Marilena Fucili

[email protected] [email protected] Dipartimento di Matematica e Statistica Università degli Studi di Napoli “Federico II” – Napoli -Italia Keywords: Markov Chain, Transition probabilities, Conjoint Analysis, Design of experiments, Temporal-spatial data, modeling and simulation.

1. Introduction The Markov Chains framework is widely used to model dynamic systems: examples come from engineering and social applications and from the class of problems solved by markovian decision processes. Recently these tools have been greatly used in Machine Learning studies where the transition probabilities are estimated using training data sets and techniques that take in account the sequences of states and the number of occurences of transitions. We refer to maximum likelihood estimations or logistic regression estimations. If the model’s parameters are not known a priori, they must be learned during the exploration and exploitation activities; no matter if the system is completely or partially observable: in this latter case the parameters of the Hidden Markov Model may infact be estimated by using the Baum-Welch algorithm. The proposed methodology has the purpose to estimate transition probabilities on the base of a request of judgements by experts instead of resorting to a sequence of observational data. The judges are asked to identify a subset of destination states for each state included in a fractional ortogonal design of starting states. They are also asked to assign transition probabilities to the identified scenarios. By means of Conjoint Analysis approach we finally reconstruct and analyze the complete markovian chain providing an original way to visualize the results. The advantages of this methodology are manifold: • it may be useful in very frequent situations of data absence (for example in the case of mapping states and actions), • the fractional factorial design is used to cope with the course of dimensionality, • the decompositional approach gives the opportunity to operate a generalization by reconstructing the transition probabilities not included in the orthogonal design definition, • the opportunity of a direct visualization of associations among starting states, destination states and judges on graphical displays.

2. Motivation for resorting to experts Dynamic systems can be classically modeled by using more components: • a set of states S, • a set of actions A, • a reward function SxA → R , • a state transition matrix P. An action a∈A can be interpreted as one of all the actions an agent can perform for regulate the system and guide it towards preferred states. Different actions, or a complete absence of actions, have clearly different effects on the system dynamics and the transitions among states S.

The reward is the result (outside feedback) that the environment gives to the agent as consequence of his action. Following this notation it’s possible to write P(i, a, j ) , or equivalently Pij (a ) , for the probability of making a transition from state i to state j using action a. The Markov Chains may be different for the different actions (Figure 1)

P00 (a1 ) P01 (a1 ) P02 (a1 ) L L P10 (a1 ) P11 (a1 ) P12 (a1 ) L L Pa1 = M Pi 0 (a1 ) Pi1 (a1 ) Pi 2 (a1 ) L M M M M

P00 (a 2 ) P01 (a 2 ) P02 (a 2 ) L L P10 (a 2 ) P11 (a 2 ) P12 (a 2 ) L L Pa2 = M Pi 0 (a 2 ) Pi1 (a 2 ) Pi 2 (a 2 ) L M M M M

…

Fig. 1 In order to simplify the notation, hereafter, we’ll use the symbol Pij instead of Pij (a ) for a transition matrix assuming its association to a specific action a. You can find databases with different kind of mappings: S → R (value function), SxS → [0,1] (transition probabilities); here it possible to identify dependent and indipendent variables or, as well, symmetrical relations among variables. With these data available, the problem of estimating transition probabilities has just been solved and in these situation we don’t need any further examination: it’s possible to derive the maximumlikelihood estimates for transition probabilities of a Markov Chain directly using: Pˆij =

nij ni

where: • nij equals the number of times that the process has been observed to go from state i directly to state j; • Pˆij is the “intuitive” estimate of Pij ; namely it equals the proportion of time that the process enter state j, in one step, when leaving state i. Otherwise the so called “policy mappings”, S → A (policies), SxA → P , are seldom available in the form of training sets of input-outupt pairs linking actions and states transition (i.e. strategies and results). That’s why it became difficult to make a query of the transitions specifically linked to predetermined actions, especially in systems not progammed to be under strict monitoring activities. One example can be borrowed from company environment: here it’s possible to find up huge database of market basket data, kind of products sold, revenues and expenditures, prices of the products, selling agents, selling zones. On the other hand, you can difficulty gather data about sequences of actions and strategies the managers pursued in order to drive the company towards desidered level of performance indicators or scenarios.

More often you have to ask for this informations directly to the experts involved with the system and, even in this situation, it is not easy to pair states and actions. In this situation, in fact, the outside feedback recorded cannot be considered a supervised feedback (such as in regression models, supervised NN,…) with correct output for each input instance. In our situation, knowledge about an outcome is useful for “evaluating” the total system’s performance but it says nothing about which actions were instrumental for the ultimate win or loss. The data absence, commonly frequent, impede predictions and modelization for the dynamic system when requested for different sequences of actions. The difficulties even more grows when the aim is providing scenarios analysis involving future states and actions perhaps never performed before. In this situation we move from “learning” to “planning” and we find a first similarity with Conjoint Analysis framework: we need information gathered from respondents (experts, judges) and we cannot (or deliberately don’t want) resort to past data.

3. The transition probabilities model We assume that the starting states (and the destination states) of the system are defined by combinations of “n” key attributes each with Lk (k=1…n)1 levels. The identification of the attributes just now mentioned can be the result of a more or less rigorous process conducted by experts as well as the result of feature selection or feature creation tasks. To simplify the reading we define l ik the level value of attribute “k” in the starting states and l jk the level value of attribute “k” in the destination states; according to this notation, if we use 3 attributes (n=3), a generic transition probability can be defined as Pij = P(l j1 , l j 2 , l j 3 | l i1 , l i 2 , l i 3 ) where (Figure 2) the state i is given by the combination of attributes l i1 , l i 2 , li 3 and the state j is given by the combination of attributes l j1 , l j 2 , l j 3 .

L L L l i1 , l i 2 , li 3 = i M Pij M L L L

Fig. 2 The total number of states is given by the product I = L1 ⋅ L2 ⋅ ...Ln ; moreover the total number of transition probabilities Pij (i,j=1…I) is equal to I2. If we further assume, in the exemplification with 3 attributes, L1 = 3, L2 = 3, L3 = 2 then the number of states I = 3 ⋅ 3 ⋅ 2 = 18 and the transition probabilities are 182=324. The probability of observing the destination level l jk of a specific attribute “k” is conditioned by the starting state “i” specified by the levels of all the “n” attributes jointly considered. This probability may be expressed as: 1

The case of continuous variables can be handled by discretizing them.

P(l jk | l i1 , l i 2 , K, l in ) . For the case in exemplification we can for example, refer to l jk = l11 and we can arrive there by arriving in one of the 6 destination states identified by the bracket (Figure 3), when leaving a specific starting state i = l 21 , l12 , l 23 .

l11 , l12 , l13 l11 , l12 , l 23 l11 , l 22 , l13 l11 , l 22 , l 23 l11 , l 32 , l13 l11 , l 32 , l 23 l 21 , l12 , l13 l 21 , l12 , l 23 l 21 , l 22 , l13 l 21 , l 22 , l 23 l 21 , l32 , l13 l 21 , l32 , l 23 K K Fig.3 The probability is given by the sum of the probabilities of arriving in destination states having l11 as level of the first attribute (Figure 4).

Pij Σ

Fig. 4

The transition probability from a specific starting state “i” to a specific destination state “j” of the Markov Chain is given by the joint probability of the simultaneous (independent) occurrence of specific levels of the attributes in the destination state given the starting state. n

Pij = ∏ P (l jk | l i1 , li 2 ,K , l in ) k =1

For example if i = l21 , l12 , l23 and j = l11 , l22 , l23 then: Pij = P(l11 | l 21 , l12 , l 23 ) ⋅ P(l 22 | l 21 , l12 , l 23 ) ⋅ P(l 23 | l 21 , l12 , l 23 ) In accordance with the markovian framework, state transitions are dependent only on the starting states. Our initial purpose of estimating transition probabilities Pij may be seen as the purpose of estimating the single P(l jk | l i1 , l i 2 , K, l in ) . This formulation of the problem is the logistic regression one but it is complicated by two aspects: • the resorting to experts; • the greatly potential growth of the problem. The complete number of states, with the n attributes, is defined by the combination of all the states. If the number of attributes and levels is high, the problem enters the course of dimensionality that’s why it become unfeaseble to ask even a little judgment to experts for an high number (more than 16) of states or scenarios. Even in the little proposed exemplification with 324 transition probabilities, the figures are greater than acceptable.

4. Organization of the Fractional Factorial Design It’s seems possible to make use of fractional factorial designs (FFD) to partially overcome the problem of dimensionality. A decompositional approach can, later, give the opportunity to operate a generalization by reconstructing the transition probabilities not included in the orthogonal design definition. This solution is another common point with the Conjoint Analysis framework. We want the experts to express their own opinions about transition and dynamics of the system, it is seemingly compulsory to organize the scenarios (alias treatments, experiments…) as a reduced number of transitions. However the organization of the FFD is not immediate because it must be in possession of a specific set of properties. The FFD: • must not be finalized in giving the importance and “utility” of a single attribute but to deal with transitions; • must be balanced such as to give the same weights to starting states and destination states, • the less the attributes in the FFD the better is the reconstruction; • the less the attributes in the FFD the less invasive is the data survay. Still, even in the little proposed exemplification, the number of 324 transition cannot be greatly reduced: common statistical package (SPSS) reduce a complete 324 treatments plan to an 80 treatments plan; it is a good result but not effective for the aim.

We propose to exploit the previously quoted Pij decomposition in order to built a FFD only on the starting states I. By using this solution we can start from I instead of I2 and can still profit of the fractional factorial design. This choice will require peculiar modalities of managing the data survay in comparison with Conjoint Analysis: these modalities are dealt in the next section.

5. The data survay The typical conjoint data survay is given by the provision of a set ot treatments to a panel of g judges asked to express a ranking or rating judgement for each treatment. In our case the experts build and select contestually the reduced number of transition expressing their opinion about the foreseeable destination states leaving each of the starting states selected in the FFD. They’ll be asked about a reduced number of starting states (the FFD ones) and the results will be generalized to the others non included in the FFD under the assumptions of independence of the attributes at an individual level for the respondents. The data survays for each starting state of the FFD is organized in multiple steps in order to submit all scenarios in a rational and not invasive sequence. The scenario firstly submitted to the judges is preferred to be the current state of the dynamic system object of analysis (the FFD may be adjusted to include this state). Starting for the best know current state: • the experts are asked to disclose their opinions about the reasonable destination states; • the experts identify the presumable destination states whose number is variable from 1 to I (but it is expected to be a short list because of the reduced number of eligible destination scenarios leaving from one specific state); • the experts are asked to quantify the probability of occurrence of the transitions towards each destination proposed scenarios. The remaining destination states not mentioned by the judges are assumed to be reached with very low values (close to zero) uniformely distributed. If one or more destination states mentioned by the expert are included in the FDD the operator will repeat the survay for this states in order to follow a logical path of evolution for the system simplifing enough the problem of processing multiple transitions. Otherwise he will go to the next FFD treatment. This procedure is replicated for all the starting states of the FFD; the interview to the first iudge ends when the analyst have the “evolution” of the system from each of the state of the FFD. At the end of this data collection procedure the operator’ll have at his disposal a partially (because of the partial fractional design) compiled markovian chain (Figure 5) for each of the respondents. Results of the interview for a single judge. Proposed Destination States from the Complete Factorial Design

Starting States of the Fractional Factorial Design

Disclosed probabilities

Fig. 5

The result of the phase of Data Survay may be represented, for an easier understanding, in the sequence of their obtaining from the experts (Figure 6). The starting state is supposed to be i = l 21 , l12 , l 23 and it’s supposed to belong to the FFD. Example of firsts phases of data survay Starting states of the FFD

Current state

l21 , l22 , l23

Proposed Proposed Destination Probabilities states

l 21 , l12 , l 23

l 21 , l12 , l 23 75% l 21 , l 22 , l 23 15% l21 , l22 , l23 10%

…

…

≈0%

Proposed Proposed Destination Probabilities states

… …

…% …%

Fig. 6 The next step consists of running a number of logistic regressions equal to the number of the attributes used in the model; the use of logistic regression is due to the presence of a probability measure associated to the categorical response variables.

6. The use of logistic regressions As above mentioned we want to estimate the single P(l jk | l i1 , l i 2 , K, l in ) by means of logistic regression techniques. It’s possible to compute all the needed informations by using the data survay and by adding the probabilities of arriving in states with the specified level of the attribute of interest. In order to built the inputs data of the logistic regression from the simple exemplification of Figure 6 it’s possible to write: … P(l 21 | l 21 , l12 , l 23 ) = 75% + 15% + 10% = 100% … P(l12 | l 21 , l12 , l 23 ) = 75% … P(l 22 | l 21 , l12 , l 23 ) = 15% + 10% = 25% … P(l 33 | l 21 , l12 , l 23 ) = 10% … These data being available, it is possible to use, in accordance with the situations at hand, the following forms of logistic regressions (LR):

•

LR with dichotomous dependent variable (such as in the case of the attribute 3 for the exemplification) and categorical independent variables; • LR with polytomous dependent variable (such as in the cases of attributes 1 and 2 for the exemplification) and categorical independent variables. The LR model examines the relationships between the indipendent variable and the log-odds of the outcome variable; log-odds leads to the simplest description consistent with the rules for probability. Let’s first center on the dichotomous case. The formula for conversion odds into probabilities and vice versa are: odds =

prob ; (1 − prob)

prob =

odds . (1 + odds )

The model on log-odds (logit) scale is linear so it’s possible to process data on log-odds scale in order to transform later the results in terms of probabilities. The trasfomed model is: Logit = log

P = Xβ 1− P

It becames possible to use a decompositional approach in order to estimate the model’s parameters and, consequently, the transition probabilities for each attribute. Most common statistical software package give the estimate of the parameters of the linear models, the probabilities, the confidence intervals and goodness of fit measures. The probabilities follow the standard S-shape curve that is characteristic of all logistic regression models. The curve levels off at 0 on one side and at 1 on the other assuring, in that, that the estimated probabilities are always in correct ranges. The logistic regression theory provide for crude models (looking at how a single attribute affects the resulting value) and adjusted models that incorporates covariates. These adjustments may be interesting for in-deep analysis of interactions and confounders. We can briefly illustrate the procedure for the dichotomous case with ipothetical figures (Table 1) for a single respondent. Numerical FFD

FFD

l31 l 21

l12 l 32

l13 l13

l13

l 32

l13

l31 l31

l 22 l 32

l13 l 23

l 21 l 22

l13

l 21 l12 l11 l 22

l 23 l 23

l11

l13

l12

3 2 1 3 3 2 2 1 1

1 3 3 2 3 2 1 2 1

1 1 1 1 2 1 2 2 1

Attribute 1 Attribute 2 in Dummy in Dummy Variables Variables D11 D21 D31 D12 D22 D32

0 0 1 0 0 0 0 1 1

0 1 0 0 0 1 1 0 0

1 0 0 1 1 0 0 0 0

1 0 0 0 0 0 1 0 1

0 0 0 1 0 1 0 1 0

0 1 1 0 1 0 0 0 0

Table 1

Attribute 3 in Dummy Prob. of success Variables (success=l22) D13 D23

1 1 1 1 0 1 0 0 1

0 0 0 0 1 0 1 1 0

odds

logodds

l 22

60%

1,5

0,4055

l 22 l 22

80% 80%

4

1,3863

4

1,3863

l 22

60%

1,5

0,4055

l 22

99,99%

l 22

60%

l 22 l 22

99,99% 99,99%

l 22

60%

9999 9,2102 1,5

0,4055

9999 9,2102 9999 9,2102 1,5

0,4055

The specific set up for the dummy explanatory variables corresponding to the logit model refer to: • the last two columns of dummy variables for attribute 1 and 2 (D21, D31, D22, D32); • the last column of dummy variables for attribute 3 (D23). The OLS estimates of the model: Logit = β1 + β 2 D21 + β 3 D31 + β 4 D22 + β 5 D32 + β 6 D23 + U are: Logit = 0,528+ 0,286 D21 − 0,04 D31 − 0,245 D22 + 0,613 D32 + 8,191 D23 + U ( 0 , 204 )

( 0 , 229 )

( 0 , 229 )

( 0 , 258 )

( 0 , 204 )

( 0 , 204 )

where the estimated coefficient standard errors are in parentheses. Different methods of estimation, such us generalized least squares or maximum likelihood estimates, can be computed and used for comparisons. The cases of attributes 1 and 2 in the exemplification (with more then two levels) need the different structure of polytomous logit model whose parameters can be estimated (either by GLS or ML methods) using Newton Raphson or Fisher iterative algorithms. The estimated parameters are used to reconstruct the probability of arriving in lja also for starting states not directly included in the FFD. Finally the probabilities of destination states for the single respondent are given by the product: n

Pij = ∏ P (l jk | l i1 , li 2 ,K , l in ) . k =1

According to the terminology proposed we remember how the transition probabilities are linked to a specific action, that’s to say Pij = Pij (a ) . If the analyst needs a recostruction of a complete transition probabilities matrix on the basis of all the respondents it’s possible to operate a mean of the coefficients associated to each of them. The probabilities of destination states can be computed by using the previous formula on the bases of the new computed mean coefficients.

7. Application on real data We propose an application on real data to illustrate the use of our methology. The application is intentionally simple and refers to political scenario analysis. The selected attributes are all dichotomous: • PRIV {1=yes, 0=no}: trend towards privatization (less pubblic channels, less pubblic health, to put up for sale of pubblic properties, more funds to private schools…) • FLEX {1=yes, 0=no}: trend towards flexibility on the labour market (easier dismissals, atypical employment…) • IMM_EMB {1=yes, 0=no}: trend towards immigration embitterment; • FREE_DR {1=yes, 0=no}: trend towards drugs free trade; • MIN_P {1=yes, 0=no}: trends towards higher minimal pensions; • LIB {1=yes, 0=no}: liberals government (in opposition to labour government). Thus, a sample survay has been carried out by interviewing a sample of 10 judges; the main goal is to understand the likelihood of different sequences of political scenarios.

The FFD is the one proposed in the following table 2.

FFD State Number 1 2 3 4 5 6 7 8

PRIV

FELX

0 1 1 0 1 0 0 1

1 1 0 0 0 0 1 1

IMM_ FREE_ MIN_P EMB DR 0 1 1 1 0 0 1 0

0 0 0 1 1 0 1 1

LIB

1 0 1 1 0 0 0 1

1 1 0 1 1 0 0 0

Table 2

100101

6

000000

7

011100

8

110110

20 % 50 %

50 %

50 %

50 %

50 %

…

5

40 %

000110

001111

30 %

000100

4

70 %

001100

101010

110011

3

110111

111001

80 %

111111

2

20 % 50 %

011111

010011

001010

1

000010

FFD State 111000

FFD State Num.

011011

Resp. 1

111011

The disclosed destination states and transition probabilities2 from the firsts two respondents are given in the following tables 3 and 4.

50 %

40 %

100

%

100

%

Table 3

2

The tables refers in blanck spaces to transition probabilities equals to zero. The data must be changed in order to give a more correct little uniform probability to arrive in the different states.

5

100101

6

000000

7

011100

8

110110

80 %

20 %

60 % 40 % 50 %

40 % 10 %

111001

001111

011001

4

000000

101010

111001

3

001110

111001

111001

2

40 % 60 %

110011

010011

110111

1

000010

FFD State 011011

FFD State Num.

111011

Resp. 2

50 %

60 % 90 % 100

%

70 %

30 %

Table 4 The logistic regression coefficients for the firsts two respondents are given3 in the following table 5 and 6. Resp. 1

β0 D21 -9,21024 -9,21024 -9,31615 -9,26092 -8,10964 -8,05896

β1 D21

β2 D22

β3 D23

β4 D24

β5 D25

β6 D26

5,298267 0 0 0 -5,29827 13,12221 -8,9E-16 -8,9E-16 -8,9E-16 -8,9E-16 -8,9E-16 18,42048 -4,81694 4,816945 8,998416 -4,3933 4,393296 9,422065 -0,10137 4,503754 0,101366 13,91673 0,101366 -4,50375 7,009047 -2,20119 -2,20119 2,201194 11,41143 7,009047 -2,30256 -2,30256 -2,30256 2,30256 2,30256 16,11792

L11 L12 L13 L14 L15 L16

Table 5

Resp. 2

β0 D21 -9,33284 -9,21024 -8,21556 -9,10433 -9,53558 -9,21024

β1 D21

β2 D22

β3 D23

β4 D24

β5 D25

β6 D26

5,05306 8,965033 -5,05306 4,15718 0,245207 9,455448 0 9,21024 0 9,21024 0 9,21024 2,615751 2,413018 6,797222 7,220871 -2,1921 6,59449 4,393296 -0,21182 0,211824 -0,21182 4,393296 -4,3933 4,15718 0,650672 0,650672 -0,65067 14,2633 4,15718 0 9,21024 0 9,21024 0 9,21024

L11 L12 L13 L14 L15 L16

Table 6 These calculation are conducted for each respondents; if desidered it’s possible to opearte a mean of the coefficients for each respondents by matching every rows.

3

The complete data (cohomprensive of standard errors) are available c/o the authors.

8. Opportunity of visualizations In this section we signify how the methodology produce a rich set of multidimensional data concerning starting states, destination states, and respondents; these data give the opportunity of visualizations of the associations among respondents and transitions. We are thinking to cluster analysis and symmetrical factorial data analysis for the associations among respondents on the basis ot the estimated coefficients; furthermore we can refer to non symmetrical factorial data analysis for the transition probabilities of the Markov Chain. However, the visualization of this kind of data is beyond the scope of the present work and will be objects of further extensions. We’ll only briefly show a simple example displaying the associations among a subset of 5 respondents with reference to the application proposed in section 7 and the probability of success for the attribute PRIV. The political subject of the application can be misleading because the reader may expect associations among respondents in accordance with their political opinions and approval (for example a first axis that opposes liberals versus labourists); this is not the case because we are interested in likelihood of political scenarios. Two subjects can give high probability to the occorrence of a specific transition even if one of them wishes the transition and the other opposes it; these two subjects will be neighbour becouse both think to a similar future scenario. The factorial analysis we are going to use is the Nonsymmetrical Principal Components Analysis first proposed by D’Ambra and Lauro (1982). The input matrices are the matrix of FFD and the judges responses reproduced in the following table 7.

state _ 1 _ FFD state _ 2 _ FFD state _ 3 _ FFD state _ 4 _ FFD state _ 5 _ FFD state _ 6 _ FFD state _ 7 _ FFD state _ 8 _ FFD

G1 0,2 0,999 0,0001 0,2 0,999 0,0001 0,0001 0,0001

G2 0,999 0,999 0,001 0,370 0,999 0,001 0,220 0,999

G3 0,119 0,999 0,999 0,999 0,999 0,575 0,240 0,003

G4 G5 0,382 0,999 0,999 0,999 0,482 0,273 0,617 0,999 0,999 0,999 0,211 0,0002 0,482 0,273 0,517 0,914

Table 7 The results refer to the firsts two factorial axis respectively with eigenvalues λ1=3,745 (62,04%) and λ2=1,879 (31,13%). In the graphical display (figure 7) you can easily notice difference among respondents having convinciments about preferred destination states (G1,G2,G5) and respondents who give similar probabilities at each of the disclosed destination states (G3, G4). You can also notice what are the starting attributes judged weighty to the “success” of finding privatization in the next state. These attributes are: • LIB+ at the time t;

1

• PRIV+ at the time t; just like it happen in states 5 and 2 of the factorial design.

2

Fig. 7

9. Strengths, weaknesses and future extensions The strengths and weaknesses of the proposed methodology are linked to the different principles used: the Markov models, the conjoint analysis framework and logistic regression. The possible strengths of the methodology have just been listed in the introduction: • it may be useful in very frequent situations of data absence; • the fractional factorial design is used to cope with the course of dimensionality. You can work with a realistic number of attributes in order to performe scenario analysis; • the decompositional approach operate a generalization by reconstructing the transition probabilities not included in the orthogonal design definition; • it gives the opportunity of a direct visualization of associations among starting states, destination states and judges on graphical displays. The weaknesses are those properly of the proposed methodology and those properly of the different principles used. First of all we think to: • the difficulties the respondendts may meet with while analyzing the starting state and while they are asked to disclose the probably destination states. The difficulties may grow if the number of attributes grows; • Psychological critics may be raised because of the the respondents aren’t asked according to the well procedure of rating or ranking the FFD treatments by comparing each one of them. We ask for a quantified measure of probability and we may also expect a tendency to give repeatedly same groups of probabilities (such as 80%, 20%; 70%, 20%, 10%;50,50);

•

the computational effort due to the necessity of performing a number of logistic regression that equals the number of the attributes; and furthermore: • the selection of the attributes is prearranged. The respondents cannot propose new features without requiring a return to the starting line and a re-formulation of the FFD; • the number of observations is given by the dimension of the FFD; this number is usually small when compared to the number of parameters to be estimated; • the additive model is not effective in cases of interation among factors; • the logistic regression techniques encounter problems when the number of variables and the number of levels for each variable grow. In this section we omit any reference to the goodness of graphical representations because of their foreingness to the core topics of reasonable dynamic system modelization and inferencial process. Some of these weakness belong to the specific techniques at hand; we suggest to refer to specific texts for in-depth studies. Some of the weakness can be partially overcomed for example: • by introducing additional interaction variables, • by adopting feature selection tasks, • by performing cross-validation tests. These topics, togheter with the graphical ones, will be object of future extensions.

9. Conclusions We have considered the problem of transition probabilities estimation when the data are absent or when you intend to perform scenario analysis and planning activities with reference to different actions. It is useful in dynamic systems and markovian decisional processes studies. We have used a fractional factorial design to select a reduced number of states; we have asked respondents for possible scenarios. We have used the data survay results to reconstruct the transition probabilities for the full factorial design that is the states transitions of the Markov Chain.

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