Quantum Models Are . . . Ubiquity of Multi-D . . . Let Us Simplify the . . . For Complex Systems, . . . Why Quantum (Wave Probability) Need for a Geometric . . . Models Are a Good Description of Many Geometric Description . . . Non-Quantum Complex Systems, and Using Geometric . . . How to Go Beyond Quantum Models For k = 2 , the Above . . . What Can We Do to . . . ıtek 1 , Olga Kosheleva 2 , Miroslav Sv´ Home Page Vladik Kreinovich 2 , and Thach Ngoc Nguyen 3 , Title Page and Thach Ngoc Nguyen 3 ◭◭ ◮◮ 1 Czech Technical University in Prague, svitek@fd.cvut.cz ◭ ◮ 2 University of Texas at El Paso, El Paso, Texas 79968, USA olgak@utep.edu, vladik@utep.edu Page 1 of 21 3 Banking University of Ho Chi Minh City, Vietnam, Thachnn@buh.edu.vn Go Back Full Screen Close Quit
Quantum Models Are . . . Ubiquity of Multi-D . . . 1. Quantum Models Are Often a Good Descrip- Let Us Simplify the . . . tion of Non-Quantum Systems For Complex Systems, . . . • Quantum physics has been designed to describe quan- Need for a Geometric . . . tum objects. Geometric Description . . . Using Geometric . . . • These are objects – mostly microscopic but sometimes For k = 2 , the Above . . . macroscopic as well – that exhibit quantum behavior. What Can We Do to . . . • Somewhat surprisingly, however, it turns out that quantum- Home Page type techniques can also be useful in describing non- Title Page quantum complex systems. ◭◭ ◮◮ • For example, they describe economic systems and other ◭ ◮ systems involving human behavior. Page 2 of 21 • Why quantum techniques can help in non-quantum sit- uations is largely a mystery. Go Back Full Screen Close Quit
Quantum Models Are . . . Ubiquity of Multi-D . . . 2. Quantum Models (cont-d) Let Us Simplify the . . . • The next natural question is related to the fact that: For Complex Systems, . . . Need for a Geometric . . . – while quantum models provide a good description Geometric Description . . . of non-quantum systems, Using Geometric . . . – this description is not perfect. For k = 2 , the Above . . . • So, a natural question: how to get a better approxima- What Can We Do to . . . tion? Home Page • In this talk, we provide answers to the above two ques- Title Page tions. ◭◭ ◮◮ ◭ ◮ Page 3 of 21 Go Back Full Screen Close Quit
Quantum Models Are . . . Ubiquity of Multi-D . . . 3. Ubiquity of Multi-D Normal Distributions Let Us Simplify the . . . • To describe the state of a complex system, we need to For Complex Systems, . . . describe the values of some quantities x 1 , . . . , x n . Need for a Geometric . . . Geometric Description . . . • In many cases, the system consists of a large number Using Geometric . . . of reasonably independent parts; in this case: For k = 2 , the Above . . . – each of the quantities x i describing the system What Can We Do to . . . – is approximately equal to the sum of the values that Home Page describes these parts. Title Page • E.g., the country’s trade volume is the sum of the ◭◭ ◮◮ trades performed by all its companies. ◭ ◮ • The number of country’s unemployed people is the sum Page 4 of 21 of numbers from different regions, etc. Go Back Full Screen Close Quit
Quantum Models Are . . . Ubiquity of Multi-D . . . 4. Multi-D Normal Distributions (cont-d) Let Us Simplify the . . . • It is known that: For Complex Systems, . . . Need for a Geometric . . . – the distribution of the sum of a large number of Geometric Description . . . independent random variables Using Geometric . . . – is – under certain reasonable conditions – close to For k = 2 , the Above . . . Gaussian (normal). What Can We Do to . . . • This result is known as the Central Limit Theorem . Home Page • Thus, with reasonable accuracy, we can assume that: Title Page ◭◭ ◮◮ – the vectors x = ( x 1 , . . . , x n ) formed by all the quan- tities that characterize the system as a whole ◭ ◮ – are normally distributed. Page 5 of 21 Go Back Full Screen Close Quit
Quantum Models Are . . . Ubiquity of Multi-D . . . 5. Let Us Simplify the Description of the Multi-D Let Us Simplify the . . . Normal Distribution For Complex Systems, . . . • A multi-D normal distr. is uniquely characterized: Need for a Geometric . . . Geometric Description . . . def – by its means µ = ( µ 1 , . . . , µ n ), µ i = E [ x i ], and Using Geometric . . . def – by its covariance matrix σ ij = E [( x i − µ i ) · ( x j − µ j )]. For k = 2 , the Above . . . • By observing the values x i corresponding to different What Can We Do to . . . Home Page systems, we can estimate the mean values µ i . Title Page • Instead of the original values x i , we can consider devi- def ◭◭ ◮◮ ations δ i = x i − µ i ; then, E [ δ i ] = 0, so: ◭ ◮ – to fully describe the distribution of the correspond- ing vector δ = ( δ 1 , . . . , δ n ), Page 6 of 21 – it is sufficient to know the covariance matrix σ ij . Go Back • Since E [ δ i ] = 0, we have σ ij = E [ δ i · δ j ]. Full Screen Close Quit
Quantum Models Are . . . Ubiquity of Multi-D . . . 6. For Complex Systems, with a Large Number of Let Us Simplify the . . . Parameters, a Further Simplification Is Needed For Complex Systems, . . . • To fully describe the distribution, we need to describe Need for a Geometric . . . all the values of the n × n covariance matrix σ ij . Geometric Description . . . • In general, an n × n matrix contains n 2 elements. Using Geometric . . . For k = 2 , the Above . . . • Ssince the covariance matrix is symmetric, we only What Can We Do to . . . = n 2 need to describe n · ( n + 1) 2 + n 2 parameters. Home Page 2 • Can we determine all these parameters from the obser- Title Page vations? In general in statistics: ◭◭ ◮◮ – if we want to find a reasonable estimate for a pa- ◭ ◮ rameter, Page 7 of 21 – we need to have a certain number of observations. Go Back • Based on N observations, we can find the value of each 1 Full Screen √ quantity with accuracy ≈ . N Close Quit
Quantum Models Are . . . Ubiquity of Multi-D . . . 7. Simplification Is Needed (cont-d) Let Us Simplify the . . . • Thus, to be able to determine a parameter with a rea- For Complex Systems, . . . sonable accuracy of 20%, we need to select N for which Need for a Geometric . . . Geometric Description . . . 1 √ ≈ 20% = 0 . 2 , i.e., N = 25 . Using Geometric . . . N For k = 2 , the Above . . . • So, to find the value of one parameter, we need approx- What Can We Do to . . . imately 25 observations. Home Page • By the same logic, for any integer k , to find the values Title Page of k parameters, we need to have 25 k observations. ◭◭ ◮◮ ≈ n 2 • In particular, to determine n · ( n + 1) ◭ ◮ 2 parame- 2 ters, we need to have 25 · n 2 Page 8 of 21 2 observations. Go Back • Each fully detailed observation of a system leads to n Full Screen numbers x 1 , . . . , x n and thus, to n numbers δ 1 , . . . , δ n . Close Quit
Quantum Models Are . . . Ubiquity of Multi-D . . . 8. Simplification Is Needed (cont-d) Let Us Simplify the . . . • So, to estimate 25 · n 2 For Complex Systems, . . . 2 = 12 . 5 · n 2 parameters, we need Need for a Geometric . . . to have 12 . 5 · n different systems. Geometric Description . . . • And we often do not have that many system to observe. Using Geometric . . . For k = 2 , the Above . . . • For example, for a detailed analysis of a country’s econ- What Can We Do to . . . omy, we need to have n ≥ 30 parameters. Home Page • To fully describe the joint distribution of all these pa- Title Page rameters, we need ≥ 12 . 5 · 30 ≈ 375 countries. ◭◭ ◮◮ • We do not have that many countries. ◭ ◮ • This problem occurs not only in econometrics, it is even Page 9 of 21 more serious, e.g., in medical bioinformatics. Go Back • There are thousands of genes, and not enough data to Full Screen be able to determine all the correlations between them. Close Quit
Quantum Models Are . . . Ubiquity of Multi-D . . . 9. Simplification Is Needed (cont-d) Let Us Simplify the . . . • So we cannot determine the covariance matrix σ ij ex- For Complex Systems, . . . actly. Need for a Geometric . . . Geometric Description . . . • Thus, we need to come up with an approximate de- Using Geometric . . . scription, with fewer parameters. For k = 2 , the Above . . . What Can We Do to . . . Home Page Title Page ◭◭ ◮◮ ◭ ◮ Page 10 of 21 Go Back Full Screen Close Quit
Quantum Models Are . . . Ubiquity of Multi-D . . . 10. Need for a Geometric Description Let Us Simplify the . . . • What does it means to have a good approximation? For Complex Systems, . . . Need for a Geometric . . . • Intuitively, approximations means having a model which Geometric Description . . . is, in some reasonable sense, close to the original one. Using Geometric . . . • In other words, we need model whose distance from the For k = 2 , the Above . . . original model is small. What Can We Do to . . . Home Page • So, we need to represent objects by points in a metric space. Title Page • So, it is desirable to use an appropriate geometric rep- ◭◭ ◮◮ resentation of multi-D normal distributions. ◭ ◮ • Such a representation is well known. Page 11 of 21 Go Back Full Screen Close Quit
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