Understanding Semantic Code and Factor Analysis in Multivariate Statistics: Insights from Nonlinear Dynamic System Theor

1. APPENDIX 1

2. If we have questions about • Semantic code • Factor analysis and • More generally about multivariate statistics • Using Methods of Nonlinear Dynamic System Theory And time to answer its

3. Semantic Code • Low integrative model n+ – number of subjects who scaled the concept as positive n- – number of subjects who scaled the concept as negative N – total number of subjects: N= n+ + n- 12 – chi-square distribution with 1 degree of freedom • If SC > 12then this scale is significant positive (ifn+ > n-) or negative (ifn+ < n-) for this concept.

4. Factor analysis is the most frequently used method of multivariate statistics • Reconstruction of geometrical model of Semantic space • Scales form factors (according their factor loadings) • Factor scores are used as coordinates of concepts in geometrical space of factors • Measurement of cognitive complexity (number and significance of extracted factors) • Classification of additional concepts not included in factor analysis (according their factor scores)

5. Comparing FA with other multivariate methods frequently used in Psychosemantics • Cluster analysis gives classification based on the ONE integrated feature • Multidimensional scaling provides less objective interpretation of the factors – there are no scales just concepts.

6. Clusters: Cluster structer of Russian Political Liders (1997)

7. Vladimir Zhirinovsky Boris Eltsin Unpopular Leaders Gennady Zuganov Victor Chernomyrdin Government Old orthodox opposition Anatoly Chubais Boris Berezovsky Egor Gaidar Ex-Government Michail Gorbachev

8. Boris Nemtsov Popular Leaders Free market’s ideas, but opposition to government way Grigory Yavlinsky Yurij Luzhkov Alexander Lebed Neo-Communistic and Socialistic ideas Ivan Rybkin Genadij Seleznev Egor Stroev Stanislav Govorukhin Neo-National-Patriotic ideas Aman Tuleev Alexander Rutskoy

9. Semantic space of images of Russian political parties in social consciousness before Parliament’s election (December 1995) F2+ Social-oriented economy F1- West-oriented F1+ Totalitarian isolationism F2- Criminal economy

10. The cloud of distribution of Kazakh subjects in the political semantic space (1991) F1- Support the Separated national Republic F1+ Support the Union of the Republics of the Former USRR Zone of possible electorate

11. The cloud of distribution of Russian subjects in the political semantic space (1991) F1- Support the Separated national Republic F1+ Support the Union of the Republics of the Former USRR

12. Compare different forms of geometrical representations Psychosemantic space of geopolitical representations (1999). From: Petrenko, Mitina, Bertnikov 2003

13. Traditional way using factor space

14. Semantic space of images Europe countries By factor: Friendliness to Russia

15. Determinacy analysis (DA) • As usual mostly methods of data analysis in Humanitarian and Social sciences got birth in North America and West Europe. DA is an exclusion. The author of this method is Russian mathematician S.Chesnokov.

16. DA algorithms to the maximum extent approach the principles of logical reasoning in natural language. • The basic concept of DA is “determination” which establishes correspondence between two statements, objects, events, etc. according to the rule “if a, then b”. a and b are fixed answers to different questions of the survey or any other features, properties of the investigated objects. • Here b is the object’s property the appearing of which is being explained, and a is the property of the object by the influence of which b is explained. • For the analysis of the determinations the conditional frequencies P(b/a), P(a/b) of the appearance of a and b features are used.

17. To analyze determination following indices are used: exactness (I), completeness (C) and essentiality (E). I characterizes the degree of correctness of the choice of the explanative feature. C is used in order to determine how often the explanative feature a chosen induces the presence of the feature b which is being explained. E shows in what degree the portion of objects possessing both a and b features among objects with the feature a is less or greater than the portion of objects with the featureb in the whole sample. • The advantage of DA is its orientation towards the work with nominative, non-parametric data.

18. From Pictures to Movies How to use semantic spaces to build Dynamical Cognitive Models

19. Model build-up • The model offered here involves the construction of trajectories in phase space. • In phase space time is represented in the trajectories rather than with its own axis. • The phase space allows to present graphically the evolution of system’s condition in time, i.e. consecutive evolution of its state, with the help of curves (trajectories)–a geometrical set of points appropriate to the system’s changing position in the phase space, which this system occupied at the consecutive moments of time. • These trajectories allow to see all set of movements that can appear at every possible initial conditions.

20. For building the phase trajectories use • Differential equations (used when we speak about the systems who's variables can be considered continuous) • Difference equations (used when we speak about the systems who's variables can be considered discrete) • These equations allow to describe the dynamics of a process as functional dependence of different initial states of the system.

21. The solution of a difference equation can be obtained • With the help of calculus of finite differences. • Analytically with the help of the transition to the limit to the continuous differential equation, if there is an algorithm of direct analytical integration of this differential equation.

22. The basic idea of the methodology of difference equations • If the law of evolution on an interval between two moments of time is known, it is possible to connect the points on a trajectory at the moment of time Tn and Tn+1 with the help of functional dependence. • The mathematical model of a dynamic system S, set up with the help of a difference equation, is based on the condition of the system Sn, which is understood as the description of this system at the moment of time Tn, and on the operator F, determining the transformation of system S in time. The operator F describes an iterative process: F[S], F[F[S]] . . . and also specifies transformation of the dynamical system Sn at the moment of time Tn to its condition Sn+1 at the moment of time Tn+1: Sn+1=F [Sn] (1) • The set of all possible states of a system S forms a phase space of states Ф(S). This space together with the operator F, form the mathematical model of the dynamical system defined by the difference equation (1).

23. Stationary state • The fulfillment of the condition Sn=F[Sn](2) means, that the system is in such condition that all its objects in each moment of time following Tn move into themselves, i.e. remain in the same place (are motionless). Thus, the state of the system Sn, satisfying the condition (2) is stationary at a fixed point attractor. • There are stable and unstable stationary states.

24. The hypothesis about the ergodicity of dissipative systems allows: • In case of absence of large sequence of data on the state of process at each moment of time S1, S2, S3, . . . Sn . . ., where n is a large enough number from the point of view of statistical and computing procedures, there is no opportunity to write and solve a difference equation. • It is possible to build a pseudo-phase space and extrapolate the function Sn+1=F(Sn) on a multiple set of values {St, St+T}, obtained as a result of observation and measurement of values of the whole ensemble of points, representing the process at two different moments of time t and t+T. • The synergetic approach allocating general laws of functioning of natural and social systems proves the acceptance of the ergodicity in our case. However, the strict proof of the ergodicity hypothesis is very difficult and more often still remains an unsolved task. • It allows avoiding difficulties arising at "development in time" of this or that process, and to replace it by "development in space", i.e. with the data on a large number of objects in the system from information received at any moment of time. Using this method it is possible to predict the behavior of the system at other stages of its development, even in the area of human sciences, when carrying out a number of measurements in longitudinal research frequently appears inconvenient.

25. The system S is defined by the position of the included concepts. • The location of concepts is defined by a set of its’ coordinates in the semantic space. • Each concept O with coordinates Oij, where i=1 if we are speaking about the coordinates of concept O, when the system is in a condition S1, and i=2, if we are speaking about the coordinates of concept O, when the system is in a condition S2, and j changes from 1 up to N. • S1 and S2 are the two states of the same system.

26. Let A be the regression operator constructed on the basis of the statistical analysis of empirical values, specifying the coordinates of each of concepts for condition S1 and S2 in a space of dimension N, so that from the point of view of statistical criteria S2=A(S1) is the best theoretical approximation of the experimental data. Then it is possible to write down the following simultaneous equations with the help of the regression operator: O2j=A(O1j), j=1... N • The choice of a type of the regression operator A is probably one of the most difficult methodical questions and should be resolved on the basis of additional reasons concerning the laws and properties of the dependence under study.

27. In our work we restricted ourselves to a linear operator, proceeding from the assumption that the majority of the nonlinear operators in the limited vicinity can be approximated by linear operators. • The following model regression formulas were used: Xn+1=a0+a1 Xn+a2 Yn Yn+1=b0+b1 Xn+b2 Yn • These equations can be transformed in the following way: Xn+1 –Xn= a0+(a1–1) Xn +a2 Yn Yn+1 – Yn= b0+b1 Xn+(b2–1) Yn • The latter are replaced by the following simultaneous linear differential equations: dx/dt= a0+(a1–1) x+ a2 y dy/dt= b0+b1 x+(b2–1) y

28. Results of application the model to the data getting from psychosemantic research Dynamical Cognitive Models of Political-Social Issues in Russia (1994-1998) (Mitina, Petrenko 2001)

30. List of the concepts

31. Neural networks are considering as algorithms of data analysis, not as models for constructions and functioning the Image of the world. • The possible domains of applications: • Classification • Patterns recognition