Need for Data Processing Need to Estimate . . . Traditional Statistical . . . Heavy-Tailed . . . Processing Quantities with Result for Addition . . . Heavy-Tailed Distribution of Case of a General . . . Case of the Product . . . Measurement Uncertainty: General Asymptotics . . . What If We Only Have . . . How to Estimate Home Page the Tails of the Results Title Page of Data Processing ◭◭ ◮◮ ◭ ◮ capek 1 and Vladik Kreinovich 2 Michal Holˇ Page 1 of 16 1 Centre of Excellence IT4Innovations, University of Ostrava, Go Back Institute for Research and Applications of Fuzzy Modeling, Ostrava, Czech Republic, michal.holcapek@osu.cz Full Screen 2 University of Texas at El Paso El Paso, Texas 79968, USA, vladik@utep.edu Close Quit
Need for Data Processing Need to Estimate . . . 1. Need for Data Processing Traditional Statistical . . . • We are often interested in the values of a quantity y Heavy-Tailed . . . which is not easy to measure directly, e.g.: Result for Addition . . . Case of a General . . . – tomorrow’s weather, Case of the Product . . . – distance to a faraway planet, General Asymptotics . . . – amount of oil in an oil well. What If We Only Have . . . • In such situations in which we cannot measure y di- Home Page rectly , we can often measure y indirectly , i.e.: Title Page – measure auxiliary quantities x 1 , . . . , x n related to ◭◭ ◮◮ the desired quantity y by a known relation ◭ ◮ y = f ( x 1 , . . . , x n ); Page 2 of 16 – use the results � x 1 , . . . , � x n of measuring the quanti- Go Back ties x i to compute the estimate � y = f ( � x n ). x 1 , . . . , � Full Screen • The process of computing � y = f ( � x 1 , . . . , � x n ) is known as data processing. Close Quit
Need for Data Processing Need to Estimate . . . 2. Need to Estimate Uncertainty of the Result of Traditional Statistical . . . Data Processing Heavy-Tailed . . . • Measurements are never 100% accurate. Result for Addition . . . Case of a General . . . • In general, the measurement results � x i are somewhat Case of the Product . . . def different from the actual values x i : ∆ x i = � x i − x i � = 0. General Asymptotics . . . • Since � x i � = x i , the estimate � y = f ( � x 1 , . . . , � x n ) is, in gen- What If We Only Have . . . eral, different from the actual value y = f ( x 1 , . . . , x n ). Home Page • Often, there is additional difference since the depen- Title Page dence between y and x i is only approximately known. ◭◭ ◮◮ • It is therefore important to gauge how much the actual ◭ ◮ value y can differ from this estimate. Page 3 of 16 • In other words, we need to gauge the uncertainty of Go Back the result of data processing. Full Screen Close Quit
Need for Data Processing Need to Estimate . . . 3. Traditional Statistical Approach Traditional Statistical . . . • Usually, there are many different (and independent) Heavy-Tailed . . . factors which contribute to the measurement error. Result for Addition . . . Case of a General . . . • Due to Central Limit Theorem, the distr. of the joint Case of the Product . . . effect of numerous independent factors is ≈ normal. General Asymptotics . . . • To describe a normal distribution, it is sufficient to What If We Only Have . . . know the mean µ and the standard deviation σ . Home Page • If µ � = 0, we can compensate for this bias, so each ∆ x i Title Page is normally distributed with mean 0 and st. dev. σ i . ◭◭ ◮◮ • The measurement errors ∆ x i are usually small, so ◭ ◮ n � ∂f ∆ y = f ( � x n ) − f ( x 1 , . . . , x n ) ≈ · ∆ x i , x 1 , . . . , � Page 4 of 16 ∂x i i =1 Go Back • Thus, ∆ y is normally distributed with 0 mean and vari- � ∂f � 2 � Full Screen n ance σ 2 = · σ 2 i . ∂x i Close i =1 Quit
Need for Data Processing Need to Estimate . . . 4. Heavy-Tailed Distributions Traditional Statistical . . . • In practice, the probability distribution of the mea- Heavy-Tailed . . . surement error is often different from normal. Result for Addition . . . Case of a General . . . • In many such situations, the variance is infinite. Case of the Product . . . • Such distributions are called heavy-tailed . General Asymptotics . . . • Mandelbrot (of fractal fame) found that price fluctua- What If We Only Have . . . Home Page tions follows the Pareto power-law Title Page ρ ( x ) = A · x − α , α ≈ 2 . 7 . ◭◭ ◮◮ • For this empirical value α , variance is infinite. ◭ ◮ • We need to estimate ∆ y for the case when distributions Page 5 of 16 for ∆ x i are heavy-tailed. Go Back Full Screen Close Quit
Need for Data Processing Need to Estimate . . . 5. Result for Addition y = f ( x 1 , x 2 ) = x 1 + x 2 Traditional Statistical . . . • For addition, ∆ y = ∆ x 1 + ∆ x 2 . Heavy-Tailed . . . Result for Addition . . . • Let us assume that: Case of a General . . . – the measurement error ∆ x 1 of the first input has a Case of the Product . . . tail with asymptotics ρ 1 (∆ x 1 ) ∼ A 1 · | ∆ x 1 | − α 1 ; General Asymptotics . . . – the measurement error ∆ x 2 of the second input has What If We Only Have . . . a tail with asymptotics ρ 2 (∆ x 2 ) ∼ A 2 · | ∆ x 2 | − α 2 , Home Page • Then the tail for ∆ y has the asymptotics Title Page ◭◭ ◮◮ ρ (∆ y ) ∼ A · | ∆ y | − α , with α = min( α 1 , α 2 ) . ◭ ◮ Page 6 of 16 Go Back Full Screen Close Quit
Need for Data Processing Need to Estimate . . . 6. Case of a General Linear Combination Traditional Statistical . . . • Let us assume that: Heavy-Tailed . . . � m Result for Addition . . . • y = a 0 + a i · x i ; and Case of a General . . . i =1 • the measurement error ∆ x i of the i -th input has a Case of the Product . . . tail with asymptotics General Asymptotics . . . What If We Only Have . . . ρ i (∆ x i ) ∼ A i · | ∆ x i | − α i . Home Page • Then the tail for ∆ y has the asymptotics Title Page ρ (∆ y ) ∼ A · | ∆ y | − α with α = min( α 1 , . . . , α m ) . ◭◭ ◮◮ ◭ ◮ Page 7 of 16 Go Back Full Screen Close Quit
Need for Data Processing Need to Estimate . . . 7. Case of the Product y = f ( x 1 , x 2 ) = x 1 · x 2 : Result Traditional Statistical . . . • Let us assume that: Heavy-Tailed . . . Result for Addition . . . – the measurement error ∆ x 1 of the first input has a Case of a General . . . tail with asymptotics ρ 1 (∆ x 1 ) ∼ A 1 · | ∆ x 1 | − α 1 ; Case of the Product . . . – the measurement error ∆ x 2 of the second input has General Asymptotics . . . a tail with asymptotics ρ 2 (∆ x 2 ) ∼ A 2 · | ∆ x 2 | − α 2 . What If We Only Have . . . • Then, ρ (∆ y ) ∼ A · | ∆ y | − α with α = min( α 1 , α 2 ). Home Page • Similar formulas hold for an arbitrary combination Title Page � m x a i y = a 0 · i : ◭◭ ◮◮ i =1 ◭ ◮ – when the meas. error ∆ x i of the the i -th input has a tail with asymptotics ρ i (∆ x i ) ∼ A i · | ∆ x i | − α i , Page 8 of 16 – then the tail for ∆ y has the asymptotics Go Back ρ (∆ y ) ∼ A · | ∆ y | − α with α = min( α 1 , . . . , α m ). Full Screen Close Quit
Need for Data Processing Need to Estimate . . . 8. Epistemic vs Aleatory Uncertainty Traditional Statistical . . . • The main objective of this paper is to deal with mea- Heavy-Tailed . . . surement (epistemic) uncertainty. Result for Addition . . . Case of a General . . . • However, the same formula can be used if we have Case of the Product . . . aleatory uncertainty. General Asymptotics . . . • For example, we can use these formulas to analyze what What If We Only Have . . . happens if: Home Page – we have a population of two-job individuals; Title Page – we know the distribution ρ 1 ( x 1 ) of first salaries; ◭◭ ◮◮ – we know the distribution ρ 2 ( x 2 ) of second salaries; ◭ ◮ – we know that these distributions are independent, Page 9 of 16 and Go Back – we want to find the distribution of the total salary Full Screen y = x 1 + x 2 . Close Quit
Need for Data Processing Need to Estimate . . . 9. General Asymptotics Remains a Challenge Traditional Statistical . . . • For a normal distribution, Prob( | ∆ x i | > 6 σ i ) ≈ 10 − 6 %. Heavy-Tailed . . . Result for Addition . . . • Such large deviations can be safely ignored, so mea- Case of a General . . . surement errors ∆ x i are small. Case of the Product . . . • So, we can approximate the dependence y = f ( x 1 , . . . , x n ) General Asymptotics . . . by linear terms in its Taylor expansion. What If We Only Have . . . • For ρ (∆ x ) ≈ A · | ∆ x | − α with α = 2, the probability of Home Page ∆ x exceeding 6 σ is ≈ 6 − 2 ≈ 3%: quite probable. Title Page • Even deviations of size 100 σ are possible: they occur ◭◭ ◮◮ once every 10,000 trials. ◭ ◮ • For such large deviations, we can no longer ignore quadratic Page 10 of 16 or higher order terms. Go Back • So, we can no longer reduce any smooth function to its linear approximation. Full Screen • Each smooth function has to be treated separately. Close Quit
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