Practical Problem: . . . F-Transform Approach . . . Triangular Functions: . . . What Is a Trend: . . . Why Triangular Membership We Want to Select the . . . Functions Are Often Efficient What Noises n ( t ) . . . Case of Interval . . . in F-Transform Applications: Relation to Haar Wavelets Case of Probabilistic . . . Relation to Probabilistic and Home Page Interval Uncertainty and to Title Page Haar Wavelets ◭◭ ◮◮ ◭ ◮ Olga Kosheleva and Vladik Kreinovich Page 1 of 30 University of Texas at El Paso, El Paso, TX 79968, USA olgak@utep.edu, vladik@utep.edu Go Back Full Screen Close Quit
Practical Problem: . . . F-Transform Approach . . . 1. Practical Problem: Need to Find Trends Triangular Functions: . . . • In many practical situations, we analyze how a certain What Is a Trend: . . . quantity x changes with time t . We Want to Select the . . . What Noises n ( t ) . . . • For example, we may want to analyze how an economic Case of Interval . . . characteristic changes with time: Relation to Haar Wavelets – we want to analyze the trends, Case of Probabilistic . . . – we want to know what caused these trends, and Home Page – we want to make predictions and recommendations Title Page based on this analysis. ◭◭ ◮◮ • To perform this analysis, we observe the values x ( t ) of ◭ ◮ the desired quantity at different moments of time t . Page 2 of 30 • Often, however, the observed values themselves do not provide a good picture of the corresponding trends. Go Back Full Screen • Indeed, the observed values contain some random fac- tors that prevent us from clearly seeing the trends. Close Quit
Practical Problem: . . . F-Transform Approach . . . 2. Need to Find Trends (cont-d) Triangular Functions: . . . • For economic characteristics such as the stock market: What Is a Trend: . . . We Want to Select the . . . – on top of the trend – in which we are interested, What Noises n ( t ) . . . – there are always day-by-day and even hour-by-hour Case of Interval . . . fluctuations. Relation to Haar Wavelets • For physical measurements, a similar effect can be caused Case of Probabilistic . . . by measurement uncertainty. Home Page • As a result, the measured values x ( t ) differ from the Title Page clear trend by a random measurement error. ◭◭ ◮◮ • This error differs from one measurement to another. ◭ ◮ • How can we detect the desired trend in the presence of Page 3 of 30 such random noise? Go Back Full Screen Close Quit
Practical Problem: . . . F-Transform Approach . . . 3. F-Transform Approach to Solving this Prob- Triangular Functions: . . . lem: a Brief Reminder What Is a Trend: . . . • One of the successful approach for solving the above We Want to Select the . . . trend-finding problem comes from the F-transform idea. What Noises n ( t ) . . . Case of Interval . . . • We want not only a quantitative mathematical model. Relation to Haar Wavelets • We want a good qualitative understanding of the cor- Case of Probabilistic . . . responding trend – and of how it changes with time. Home Page • For example, we want to be able to say that the stock Title Page market first somewhat decreases, then rapidly increases. ◭◭ ◮◮ • In other words, we want these trends to be described ◭ ◮ in terms of time-localized natural-language properties. Page 4 of 30 • First, we select these properties. Go Back • Then, we can use fuzzy logic techniques to describe Full Screen these properties in computer-understandable terms. Close Quit
Practical Problem: . . . F-Transform Approach . . . 4. F-Transform Approach (cont-d) Triangular Functions: . . . • So, we get time-localized membership functions What Is a Trend: . . . We Want to Select the . . . x 1 ( t ) , . . . , x n ( t ) . What Noises n ( t ) . . . Case of Interval . . . • Time-localized means that when we analyze the pro- Relation to Haar Wavelets cess x ( t ) on a wide time interval [ T, T ]: Case of Probabilistic . . . – the 1st membership function x 1 ( t ) is different from Home Page 0 only on a narrow interval [ T 1 , T 1 ], where T 1 = T ; Title Page – the 2nd membership function x 2 ( t ) is � = 0 only on ◭◭ ◮◮ a narrow interval [ T 2 , T 2 ], where T 2 ≤ T 1 , etc. ◭ ◮ • The whole range [ T, T ] is covered by the corresponding ranges [ T i , T i ]. Page 5 of 30 Go Back Full Screen Close Quit
Practical Problem: . . . F-Transform Approach . . . 5. F-Transform Approach (cont-d) Triangular Functions: . . . • Once we have these functions x i ( t ), then: What Is a Trend: . . . We Want to Select the . . . – as a good representation of the original signal’s What Noises n ( t ) . . . trend, Case of Interval . . . – it is reasonable to consider, e.g., linear combina- Relation to Haar Wavelets n � tions x a ( t ) = c i · x i ( t ) of these functions; Case of Probabilistic . . . i =1 Home Page – this will be the desired reconstruction for the no- noise signal. Title Page ◭◭ ◮◮ • This approach has indeed led to many successful ap- plications. ◭ ◮ Page 6 of 30 Go Back Full Screen Close Quit
Practical Problem: . . . F-Transform Approach . . . 6. In Many Practical Applications, Triangular Mem- Triangular Functions: . . . bership Functions Work Well What Is a Trend: . . . • Which membership functions should we use in this ap- We Want to Select the . . . proach? What Noises n ( t ) . . . Case of Interval . . . • The objective of a membership function is to capture the expert reasoning. Relation to Haar Wavelets Case of Probabilistic . . . • So, we may expect that: Home Page – the more adequately these functions capture the Title Page expert reasoning, ◭◭ ◮◮ – the more adequate will be our result. ◭ ◮ • From this viewpoint, we expect complex membership functions to work the best. Page 7 of 30 • However, in many practical applications, the simplest Go Back possible triangular membership functions work the best: Full Screen � 1 − | x − c | � x i ( t ) = max , 0 . Close w Quit
Practical Problem: . . . F-Transform Approach . . . 7. Triangular Functions: Why? Triangular Functions: . . . � 1 − | x − c | � What Is a Trend: . . . x i ( t ) = max , 0 . w We Want to Select the . . . • These functions: What Noises n ( t ) . . . Case of Interval . . . – linearly rise from 0 to 1 on the interval [ c − w, c ], Relation to Haar Wavelets and then Case of Probabilistic . . . – linearly decrease from 1 to 0 on [ c, c + w ]. Home Page • The above empirical fact needs explanation: why tri- Title Page angular membership functions work so well? ◭◭ ◮◮ • In this talk, we provide a possible explanation for this ◭ ◮ empirical phenomenon. Page 8 of 30 Go Back Full Screen Close Quit
Practical Problem: . . . F-Transform Approach . . . 8. What Is a Trend: Discussion Triangular Functions: . . . • A trend may mean increasing or decreasing, decreasing What Is a Trend: . . . fast vs. decreasing slow, etc.; We Want to Select the . . . What Noises n ( t ) . . . – in the ideal situation with no random fluctuations, Case of Interval . . . – all these properties can be easily described in terms = dx Relation to Haar Wavelets def of the time derivative x ′ ( t ) dt . Case of Probabilistic . . . • From this viewpoint, understanding the trend means Home Page reconstructing the derivative x ′ ( t ); so: Title Page – once we have applied the F-transform technique ◭◭ ◮◮ and obtained the desired no-noise expression ◭ ◮ n � x a ( t ) = c i · x i ( t ) , Page 9 of 30 i =1 Go Back – what we really want is to use its derivative Full Screen n � x ′ c i · x ′ a ( t ) = i ( t ) . Close i =1 Quit
Practical Problem: . . . F-Transform Approach . . . 9. What Is a Trend: Discussion (cont-d) Triangular Functions: . . . • So, we must: What Is a Trend: . . . We Want to Select the . . . def = x ′ ( t ) of the orig- – approximate the derivative e ( t ) What Noises n ( t ) . . . inal signal Case of Interval . . . def – by a linear combination of the derivatives e i ( t ) = Relation to Haar Wavelets x ′ i ( t ): Case of Probabilistic . . . n Home Page � e ( t ) ≈ e a ( t ) = c i · e i ( t ) . i =1 Title Page • In these terms, we approximate the original derivative ◭◭ ◮◮ by a function from a linear space spanned by e i ( t ). ◭ ◮ • In this sense, selecting the functions x i ( t ) means select- Page 10 of 30 ing the proper linear space – i.e., the functions e i ( t ). Go Back Full Screen Close Quit
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