AP08_5.vp 1 Introduction The problem of complex probability functions is gaining in importance in quantum mechanics [1, 9], where the phase functions has been recognized as necessary for information processing and quantum system modeling. The contextual interpretation of phase functions was presented in [7], and the wave probabilistic models were introduced as necessary part of probabilistic multi-models. A more rigorous introduc- tion to wave probabilistic models was presented in [14], where phase parameters are interpreted as dependency functions between events. The link between wave probabilistic functions and the complementarity principle was first introduced in [10]. The quantization principle as the consequence of phase parameters was defined in [6]. The goal of this paper is to continue in this way of think- ing and to provide a more rigorous definition of wave proba- bilistic models together with their basic features. Chapter 2 presents the mathematical theory of wave functions, together with their geometric interpretation. Chapter 3 covers the link between wave probabilities and entanglement. Chapter 4 de- scribes the general estimation algorithm of wave probabilities. Chapter 5 presents the methodology for modeling non-er- godic binary time series. Chapter 6 offers an illustrative example of binary time series, and Chapter 7 concludes the paper. 2 Mathematical theory of wave probabilistic functions A probability space consists of a sample space S and a probability function P(.), mapping the events of S to real numbers in [0,1], such that P(S) � 1, and if A1, A2, … is a se- quence of disjoint events, then the sum rule is fulfilled: P A P(Ai i N i i N� � � � � � � � � � ) . (1) If the events A1, A2, … are not disjoint the following (product and inclusion-exclusion) rules can be defined: P(A A A P(A P(A A P(A A A P(A A A 1 2 1 2 1 3 1 2 1 � � � � � � � � � � � � N N N ) ) ) ) 1), (2) P(A A A P(A P(A A P(A A A 1 2 1 � � � � � � � � � � � N i i N i j i j N i j k ) ) ) ) i j k N N N � � � � � � � � � �( ) ).1 1 1 2P(A A A (3) Taking into consideration the basic laws of probability de- fined above, we can rewrite them with the help of the complex representations (wave functions) summarized in Theorem 1. Theorem 1: Let us define N events Ai, i N�{ , , , }1 2 � of a sample space S, with defined probability functions P A( )i , i N�{ , , , }1 2 � and let us define the N complex functions � � � �( ( ) , { , , , }A ) e P A ei i j i ji i i N� � � � � 1 2 � (4) together with their superposition state � � as a quantum ob- ject at its measurement place (or at time series position) �: � � � � � � � � � � � � � � � ( ) ( ) ( ) A A A A A A 1 1 2 2 � N N (5) with modules P A( )i and phases �i where the reference phase assigned to event A1 is chosen as �1 0� , then the inclusion-ex- clusion rule given in (3) is represented for measurements on quantum objects � �� �1, ,� N (or at a time series window) by: P A A A A( ) ( )1 1 2 2 1 2 � �� � � � � � � � � � � N N i i N , (6) where the phases �i are given: �i i i i i a� � � � � � � � � cos ( ) ) ( ) ( 1 2 1 2 1 1 2 1 P( A A A A P A A A P A � � ) , , , � � � � � � � �1 2 1� i (7) and �1 2 1, , ,� i is computed: � � � � � ( ) ( ) ( ) , , , , A A A A A e 1 2 1 1 1 1 2 1 1 2 � � � � � � � � � � � � i i i j , ,� i 1 (8) Proof: The proof is presented in [15]. 42 © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ Acta Polytechnica Vol. 48 No. 5/2008 Wave Probabilistic Model of Binary Time Series M. Svítek This paper presents the theory of wave probabilistic models, together with important features, such as the inclusion-exclusion rule, the product rule, the complementary principle and entanglement. These features are mathematically described, and an illustrative example of binary time series is shown to demonstrate possible applications of the theory. Keywords: quantum models, wave probabilistic functions, binary time series. In the next part of this paper, the set of complex functions (4) plus superposition state (5) together with the inclusion-ex- clusion rule (6, 7, 8) is called the wave probabilistic model. Theorem 2: Let us define N events Ai, i N�{ , , , }1 2 � of a sample space S, with defined probability functions P(A i), i N�{ , , , }1 2 � , and the N complex functions �(A i) with mod- ules P A( )i and phases �i defined in (4–8), where the refer- ence phase assigned to event A1 is chosen as �1 0� , then the inclusion-exclusion rule given for the subset of events A r, r k k km�{ , , , }1 2 � is given as: P P ( ) lim ( ( ) , , A A A A A k k k m k k k i m k m 1 2 1 1 2 0 � �� � � � � � � � � � � � � � ) i N � 1 2 (9) Proof: The proof of Theorem 2 arises directly from Theorem 1, which was proven for all probabilistic values P A( )i , A i, i N�{ , , , }1 2 � including zeros. The zero probabilistic values have an impact on phase parameters �i and change them in such a way that equation (9) is fulfilled. In this paper we assume that quantum objects � are distin- guishable. When two identical particles interact (there is a sig- nificant overlap of their wave functions), we can not distin- guish between them. These overlapping quantum objects are in general bosons (plus sign corresponding to a symmetric wave function under exchange of quantum objects) or fer- mions (minus sign corresponding to an anti-symmetric wave function under exchange of quantum objects). 3 Wave probabilities and entanglement Quantum entanglement (the definition for describing the entanglement principle from Wikipedia, available on: http://en.wikipedia.org/wiki/Quantum_entanglement ) is a quantum mechanical phenomenon in which the quantum states of two or more quantum objects have to be described with reference to each other, even though the individual objects may be spatially separated. This leads to correlations between observable physical properties of the systems. For example, it is possible to prepare two particles in a single quantum state such that when one is observed to be spin-up, the other one will always be observed to be spin-down and vice versa, despite the fact that it is impossible to predict, ac- cording to quantum mechanics, which set of measurements will be observed. As a result, measurements performed on one system seem to be instantaneously influencing other systems entangled with it. Theorem 3: Let us define N events Ai, i N�{ , , , }1 2 � of a sample space S, with defined probability functions P(A i), i N�{ , , , }1 2 � , and the N complex functions �(A i) with mod- ules P A( )i and phases �i defined in (4–8), where the refer- ence phase assigned to event A1 is chosen as �1 0� , then all events A r, r r r rn�{ , , , }1 2 � with only two possible states A r ii �� A r ii �� are entangled if the following form holds: P P ( ) lim ( ) ( ( ) , , , A A A A r r n k r r r n k n 1 1 2 1 0 1 � � � � � � � � � � � � � � � A A2 2 0) ( )� � �� � N (10) which yields into form: P A A( )r r nn1 1 1 � �� � � � �� (11) where A r ii �� means the inversion state in comparison with A r ii �� on quantum object ( )� � i . Proof: Theorem 3 emerges directly from the inclusion-exclu- sion rules and from the definition of wave probabilities. Phase parameters assigned into wave functions can be either posi- tive or negative. Form (10) modifies the phases for the selected subset of events { , , , }A A Ar r rn1 2 � to comply with the inclusion-exclu- sion rule (9). For special cases, the inclusion-exclusion rule can yield into zero due to wave resonances within the phases of events. This special case can occur for a special set up of phases. Zero probability (10) directly yields into equation (11), which defines that the state characterized by A Ar rn1 � �� will surely occur. This state is not random but fully determinis- tic and so spatially spread within events { , , , }A A Ar r rn1 2 � . It can be stated that entanglement is the logical result of probabilistic wave functions, and represents something like the resonance wave functions yielding into deterministic states. 4 Estimation algorithm of the wave probabilistic model of time series We assume the time series composed of many quan- tum objects, each of which is described by wave function covering the superposition principle of all possible events { , , , }A A A1 2 � N : � � � � � � � � � � �1 1A AN N� , (12) where � means the �-th quantum object. If we take into consideration the window of i quantum ob- jects, the corresponding wave function ~� � is given by the Kronecker Product [14]: ~ � � � � � � � � � � � � � � �1 2 � i, (13) where ~� � 2 give us probabilities that measurements on the set of i quantum objects { , , }� �� �1 � i will yield into a series of predefined events. Now we assume that we have the time series, and by re- turn, we estimate the wave functions (4). The algorithm for es- timating the parameters of the wave functions can be decom- posed into following steps: 1. Let us start with two events { , }A A1 2 and assume ��1 0� . © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ 43 Acta Polytechnica Vol. 48 No. 5/2008 2. We can estimate the phase ��2 from the following equation: P A A A A( ) ( ) ( ) � � � � cos 1 1 2 2 1 2 2 1 2 2 2 1 22 � � � � � � � � � � � � � � � � � ( � � ) � ,� � �2 1 1 2 2 � (14) with the help of occurrence rates � , � , � , �,� � � �1 2 1 2 1 estimated from time series. In principle we estimate the probabilities: P A P A P A A( ), ( ), ( )1 2 1 2 1� � � �� � and compute the parameters � , � , � ,� � �1 2 1 2. 3. We continue and extend the algorithm for three events { , , }A A A1 2 3 P A A A A A A ( ) ( ) ( ) ( ) � � 1 1 2 2 3 3 1 2 3 2 1 2 2 2 � � � � � � � � � � � � � � � � � � � � � � cos( � � ) � � cos( � � ) � � � � � � � � � 3 2 1 2 2 1 2 3 3 2 2 2 2 � � � � � � cos( � � ),� � � �1 3 3 1� � (15) where the estimate of angle ��3 can be computed from the following equation: � cos( � � ) � cos( � � ) � cos( � , ,� � � � � � � � 1 2 3 1 2 2 3 2 1 3 � � � � � � )�1 (16) under knowledge of the estimated parameters � , � , � ,,� � �1 2 1 2 � , � , � ,� � �1 2 1 2. This step must be made numerically because equation (16) is non-linear. 4. The above described procedure can be extended to a gen- eral N-step, where the unknown angle ��N is assumed to be numerically computed from the following equation: � cos( � � ) � cos( � � , , , , , ,� � � � � � 12 1 12 1 1 � �N N N N N N � � � � � � � � � � 1 2 2 2 2 1 ) � cos( � � ) � cos( � � ) � co � � � � � � � N N N N� s( � � )� �N 1 (17) under knowledge of the estimated parameters � , ,�1 � � , � , � , � , , � , �, , , , , ,� � � � � �N N N N 1 1 2 1 1 1 1 1 2 1� �� . Fortunately, for binary time series we need only step 1 and 2, and the result can be given analytically. 5 Wave probabilistic model of non-ergodic binary time series We start our discussion with probabilistic binary time se- ries, and we will show models of the occurrence of zero or one (probability and structure) in the form of wave probabilistic functions. Let us define the binary quantum object in “bra-ket” form [17]: � � � � � � � � � �1 0 1p p je . (18) Parameter p defines the probability of occurrence of state 1 � of time series position �. The probability of occurrence of a state 0 � must be (1 p). Phase � plays the role of “struc- tural” parameter that expresses the rate of time series ran- domness [16]. The ergodic theorem allows the time average of a con- forming process to equal the ensemble average. In practice, this means that statistical sampling can be performed at one instant across a group of identical processes or sampled over time on a single process with no change in the measured result. Quantum objects defined in (18) fulfill the ergodic theo- rem, because the time average of a series along the time trajectories exists almost everywhere and is related to the space (set of realizations) average: E d� � � � � � � � � � � � � ��� , ( ) lim , , t t t nn n 1 � , (19) where E� is the mean value of a stochastic process and �(�) is a probability measure. The first part of (19) is the �-ensemble average, which does not depend on time t and so the process is stationary. The second part of (19) is the time-average of quantum objects � � � � , ,t tn1 � �� with respect to selected process realization �. If the complex parameters of the “bra-ket” model are time dependent we speak of a non-ergodic probabilistic bi- nary process. Since non-ergodic processes are very difficult to model we will ease our requirements only to a special class of quasi-non-ergodic processes. Quasi-non-ergodic processes are characterized by linear time invariant (LTI) evolution of complex wave functions �( , )A i t assigned into states A i � in time interval t and position �. If we add the time varying phase parameter into a binary process we can rewrite “bra-ket” objects (18) as follows: � � � � � � � , ( ) t j j t j t � � � � � � � � � � �� 1 0 1 1 0 1 A A e e A A e � , (20) where parameter A defines the probability of occurrence of state 1 � . Parameter � expresses the initial structure of the studied set of �-positioned states. Parameter represents the frequency of continual structuring and randomizing of � positioned states. Due to the time evolution of complex pa- rameters (20) the ergodic condition is not fulfilled. The parameters for state 0 � can easily be computed from the parameters assigned to state 1 � because A must be maxi- mally equal to one to be a probability function and the phase assigned to state 0 � is assumed to be the normalized ref- erence phase presented in Theorem 1 and equal to zero. Frequency can be interpreted as the energy spent to “struc- ture” or “randomize” the set of �-positioned states with respect to chosen frequency . We use the notation A , � as modulus and initial phase parameters assigned to frequency . They represent the fre- quency decomposition in the same way as the Fourier transform. Equation (20) can be used as one frequency com- 44 © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ Acta Polytechnica Vol. 48 No. 5/2008 ponent (modulus and initial phase) of a non-ergodic binary quantum object. The general periodic non-ergodic behavior can be ex- pressed as the sum of different frequency components: � � � � � � � , ( , ) ( , ) ( ) t t t i i ij t i N � � � � � � � � � � � � 0 0 1 1 1 1 A e � � � � � � � � � � � �� � 2 1 0 1 � � � A e i i ij t i N ( ) , (21) where the final modulus and phase assigned to state 1 � are: A e A e � �� i i ij t i N j tt� � �� � � ( ) ~ )~( ) 1 , (22) where ~ ( )A t is the time evolution of the probability of state 1 � , and ~( )� t is the evolution of the link among different � posi- tioned quantum objects (expressing structuring and random- izing). The complex parameter assigned to 0 � is computed from the normalization and reference condition: � � � �� � , ~ ~ ~ )t t t j t� � � � ��1 0 1A( ) A( ) e . (23) Equation (23) can be understood as the “bra-ket” repre- sentation of general non-ergodic binary quantum objects. We can see in (21) that for every state 0 � , 1 � the discrete modules and phase spectrum can be defined. This means that the time-evolution is modeled by a periodic function. The dis- crete spectrum can be replaced by a continuous spectrum. In this case, the sums in (21) are replaced by integrals. This replacement means the transition from a Fourier series into the Fourier transform. 6 Illustrative example-binary time series Let us take two values { , }A A0 10 1� � time series rep- resented by two complex wave functions �( )A 0 and �( )A1 . In the next part we will use for simplicity the notation � �( )A 0 0� and � �( )A1 1� . A. Complementarity principle We demonstrate the complementarity principle of binary time series. Discrete Fourier Transform (DFT) is defined for k, i N� { , , , }0 1 1� : X e X e k i j i k N i N i k j i k N k N x x N � � � � � � � � � � � � 2 0 1 2 0 1 1 � � , . Probability in x-representation is given from (4): P A P A ( ) ( ) * * 0 0 0 1 1 1 0 1 � � � � � � � � � � (24) First we compute DFT of wave functions �0, �1: � � � � � � �� ( ) , ( ) k k i i i i j i � � � � � � � � � � � � � 0 1 0 1 0 0 1 0 1 0 e e �1, (25) where �(.) are DFT transformed functions. The probability function in k-representation can be given [10]: ~ ( ) ~( ) ~ ( ) cos( ), P k k k� � � � � � � � � � 0 0 0 20 2 1 2 0 1 � � � � � � � (26) ~ ( ) ~( ) ~ ( ) cos( ), P k k k� � � � � � � � � 1 1 1 20 2 1 2 0 1 � � � � � � � (27) where � means the phase difference between complex num- bers �0 and �1. The Inverse DFT of probabilities (26) and (27) yields into convolution in x-representation and describes the links be- tween two successive quantum objects. If the phase parameter is zero � � 0 the values are strong- ly independent and the time series is fully random. Phase � �� 2 explains that the probability of changes {0, 1} or {1, 0} in the time series is very low. A binary time series looks like { , , , , , , , , , , , }0 0 0 0 0 0 0 1 1 1 1 1 1 1� � . On the other hand if the phase parameter is � �� the probability of finding a pair {1, 1} or {0, 0} limits to zero. The corresponding time se- ries looks like {0, 1, 0, 1, …, 0, 1, 0, 1} and seems to be fully deterministic. B. Entanglement principle Let us take two quantum objects ~� � of a time series with wave functions �0 and �1 defined in (13) and (18). Let us de- fine the probability: P A A(( ) ( ) cos( ), � � � � � � � � � � � � � � � � �0 1 2 1 0 2 1 2 0 1 (28) where � is the phase difference between wave functions �0 and �1. Suppose now with respect to Theorem 3 that: P A A(( ) ( )� �� � � ��0 1 01 . (29) This case can occur for the following values of �: � � � � � � � � � � � � � � a cos 1 2 0 2 1 2 0 1 . (30) If, for example, � �0 1 1 2 � � then � �� represents the entanglement. As the result of entanglement we can write that the follow- ing events will surely happen (there are no random values): P A A(( ) ( )� �� � � ��1 0 11 . (31) We can also start with the following probability, instead of (29): © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ 45 Acta Polytechnica Vol. 48 No. 5/2008 P A A(( ) ( )� �� � � ��1 0 01 . (32) Then the entanglement yields into: P A A(( ) ( )� �� � � ��0 1 11 . (33) Equations (32) and (33) can both be written to quantum “bra-ket” representation: ~ ( ).� � � � 1 2 01 10 (34) Measuring the first quantum object from representation ~ � � (the probability of measuring event 0 is 1/2 and the prob- ability of measuring 1 is also 1/2) fully determines the value which will be measured on the second object. Equation (34) is the well-known Bell state, which is used in many applications, e.g., in quantum teleportation, quantum cryptography, etc. C. Wave function estimation Let us define the time series: {0, 0, 1, 1, 0, 1, 0, 0, 1, 1, 1, 1, 0, 0, 0} with the estimated occurrence rates: �( )P A � � �1 7 15 , �( )P A � � �0 8 15 (35) �( )P A A� �� � � ��0 1 3 141 . (36) By using the inclusion-exclusion rule we can apply the es- timated probabilities and write the following equation: � ( ) ( )) � ) � ) � ) P( A A P(A P(A P(A � � � � � � � � � � � � � � � � 0 1 0 1 2 0 1 1 � ) cos( ) � ) � ) �( P(A P(A P(A P A A � � � � � �� � � � � � � � � � � 1 1 1 1 0 1 0 � 1) (37) from which the estimated phase parameter can be computed: � arccos �( ) � ) � ) � � � � � � � � � � � � � � � � � � � � P A A P(A P(A 0 1 2 0 1 1 1 � 13543. . (38) If we use the estimated angle we can compute the inclu- sion-exclusion probability as: � ( ) ( )) � ) � ) � ) P( A A P(A P(A P(A � � � � � � � � � � � � � � � � 0 1 0 1 2 0 1 1 � ) cos( � ) ,P(A � �� � � �1 1 11 14 (39) where the estimate (39) corresponds to the occurrence rate directly estimated from the time series: � ( ) ( ))P( A A� �� � � ��0 1 11 141 . (40) We can see the compliance between (39) and (40). 7 Conclusion Wave probabilistic models have been introduced and a mathematical comparison between usually used probabilistic models and wave probabilistic models has been presented. Mathematical theory points to the applicability of wave probabilistic models and their special features. Quantum en- tanglement is explained as the consequence of phase para- meters and it can be interpreted as the resonance principle of wave functions. The results of wave function resonance are fully deterministic, spatially distributed states with many properties. The complementarity principle presents the studied time series in both x- and k-representation, where x-representation provides us with probabilities of occurrence of different events and k-representation caries information about the links be- tween events and how the time series is structured. The general estimation algorithm for phase parameters of wave probabilities was introduced and shown on an illustra- tive example – a binary time series. We can understand that this methodology yields into new models taking into account the structure of time series. The application of the methodology presented here for non-ergodic time series modelling is also described and shown on binary time series. This opens new ways for model- ling non-ergodic or quasi-non-ergodic processes of this kind. The inspiration for the problem defined here came from quantum physics [1, 4, 6, 7]. The analogy with quantum me- chanics seems very interesting and will inspire future work in statistical modelling area and wave probabilistic models. 8 Acknowledgments This project was supported by MŠMT COST 356, OC194 and AV ČR, IAA201240701. References [1] Gold, J. F.: Knocking on the Devil’s Door – A Naive Introduc- tion to Quantum Mechanics, Tachyon Publishing Com- pany, 1995. [2] Raymer, M. G.: Measuring the Quantum Mechanical Wave Function. Contemporary Physics, Vol. 38 (1997), No. 5, p. 343–355. [3] Radon, J.: Über die Bestimmung von Funktionen durch ihre Interalwerte längs gewisser Mannigfaltigkeiten. Berichte Sachsiche Akademie der Wissenschaften, Leipzich, Mathematisch-Physikalische Klasse, Vol. 69 (1917), p. 262–277. [4] Bogdanov, Yu. I.: The Sixth Hilbert’s Problem and the Prin- ciples of Quantum Informatics. Institute of Physics and Technology, Russian Academy of Sciences [http://arxiv.org/ftp/quant-ph/papers/0612/0612025.pdf] [5] Clifton, R.: Complementarity between Position and Momen- tum as a Consequence of Kochen-Specker Arguments [http://eprintweb.org/S/article/quant-ph/9912108] [6] Svítek, M.: Dynamical Systems with Reduced Dimensionality. Monograph NNW No. 6, Czech Academy of Science, 2006, 161 p., ISBN 80-903298-6-1. [7] Svítek, M.: Probabilistic Multi-Modeling by Quantum Calculus. In ICTTA’06, IEEE Conference Damascus, Syria, 2006. [8] Deutsch, D.: Quantum Theory, the Church-Turing Prin- ciple and the Universal Quantum Computer, Proceedings of the Royal Society of London Series A400, 1985, p. 97–117. 46 © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ Acta Polytechnica Vol. 48 No. 5/2008 [9] Lawden, D. F.: The Mathematical Principles of Quantum Mechanics. Dover Publication, Inc., Mineola, New York, 1995, ISBN 0-486-44223-3. [10] Svítek, M.: Complementary Variables and its Applica- tion in Statistics. Neural Network World, 2007, No. 3, Prague, p. 237–253. [11] Svítek, M., Novovičová, J.: Performance Parameters Def- inition and Processing. Neural Network World, 2005, No. 6, p. 567–577. [12] Svítek, M.: Theory and Algorithm for Time Series Com- pression, Neural Network World, 2005, No. 1, p. 53–67. [13] Clarke, C. J. S. The role of Quantum Physics in the The- ory of Subjective Consciousness. Mind and Matter, Vol. 5 (2007), No. 1, p. 45–81. [14] Vedral, V.: Introduction to Quantum Information Science, Oxford University Press, 2006. [15] Svítek M.: Wave probabilistic models. Neural Network World, 2007, No. 5, p. 469–481. [16] Svítek, M.: Quantum System Modelling. International Journal on General Systems, Vol. 37 (2008), No. 5, p. 603–626, . Prof. Dr. Ing. Miroslav Svítek phone: +420 224 359 631 Fax: +420 224 359 545 e-mail: svitek@fd.cvut.cz Department of Control and Telematics Czech Technical University in Prague Faculty of Transportation Sciences Konviktská 20 110 00 Prague 1, Czech Republic © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ 47 Acta Polytechnica Vol. 48 No. 5/2008