Need to Fuse Models: . . . Fusion: General Problem Motivation for Using . . . Statistical Fusion: . . . Towards Model Fusion in Traditional Methods . . . Geophysics: How to How to Estimate . . . How to Estimate . . . Estimate Accuracy of An Idea of How to . . . Resulting Algorithm Different Models Home Page Omar Ochoa, Aaron Velasco, Title Page Christian Servin, and Vladik Kreinovich ◭◭ ◮◮ Cyber-ShARE Center ◭ ◮ University of Texas at El Paso El Paso, TX 79968, USA Page 1 of 15 Go Back omar@miners.utep.edu, aavelasco@utep.edu christians@utep.edu, vladik@utep.edu Full Screen Close Quit
Need to Fuse Models: . . . Fusion: General Problem 1. Need to Fuse Models: Geophysics Motivation for Using . . . • One of the main objectives of geophysics: find the den- Statistical Fusion: . . . sity ρ ( x, y, z ) at different depths z and locations ( x, y ). Traditional Methods . . . How to Estimate . . . • There exist several methods for estimating the density: How to Estimate . . . – we can use seismic data, An Idea of How to . . . – we can use gravity measurements. Resulting Algorithm Home Page • Each of the techniques for estimating ρ has its own advantages and limitations. Title Page ◭◭ ◮◮ • Example: seismic measurements often lead to a more accurate value of ρ than gravity measurements. ◭ ◮ • However, seismic measurements mostly provide infor- Page 2 of 15 mation about the areas above the Moho surface. Go Back • It is desirable to combine (“fuse”) the models obtained Full Screen from different types of measurements into a single model. Close Quit
Need to Fuse Models: . . . Fusion: General Problem 2. Fusion: General Problem Motivation for Using . . . • Similar situations are frequent in practice: Statistical Fusion: . . . Traditional Methods . . . – we are interested in the value of a quantity, and How to Estimate . . . – we have reached the limit of the accuracy achievable How to Estimate . . . by using a single measuring instrument. An Idea of How to . . . • Objective: to further increase the estimation accuracy. Resulting Algorithm Home Page • Idea: perform several measurements of the desired quan- tity x i . Title Page ◭◭ ◮◮ • Comment: we may use the same measuring instrument or different measuring instruments. ◭ ◮ • Then, we combine the results x i 1 , x i 2 , . . . , x im of these Page 3 of 15 measurement into a single more accurate estimate � x i . Go Back Full Screen Close Quit
Need to Fuse Models: . . . Fusion: General Problem 3. Motivation for Using Normal Distributions Motivation for Using . . . • The need for fusion comes when we have extracted all Statistical Fusion: . . . possible accuracy from each measurements. Traditional Methods . . . How to Estimate . . . • This means, in particular, that we have found and elim- How to Estimate . . . inated the systematic errors. An Idea of How to . . . • Thus, the resulting measurement error has 0 mean. Resulting Algorithm Home Page • It also means that that we have found and eliminated the major sources of the random error. Title Page • Since all big error components are eliminated, what is ◭◭ ◮◮ left is the large number of small error components. ◭ ◮ • According to the Central Limit Theorem, the distribu- Page 4 of 15 tion is approximately normal. Go Back • Thus, it is natural to assume that each measurement Full Screen def error ∆ x ij = x ij − x i is normally distributed. Close Quit
Need to Fuse Models: . . . Fusion: General Problem 4. Statistical Fusion: Formulas Motivation for Using . . . • Each measurement error is normally distributed: Statistical Fusion: . . . � � Traditional Methods . . . − ( x ij − x i ) 2 1 ρ ij ( x ij ) = √ · exp . How to Estimate . . . 2 σ 2 2 π · σ j j How to Estimate . . . • It is reasonable to assume that measurement errors An Idea of How to . . . corr. to different measurements are independent, so Resulting Algorithm � � Home Page m � − ( x ij − x i ) 2 1 L = √ · exp . Title Page 2 σ 2 2 π · σ j j j =1 ◭◭ ◮◮ • According to the Maximum Likelihood Principle, we ◭ ◮ select most probable value x i s.t. L → max: Page 5 of 15 � m σ − 2 · x ij j Go Back j =1 x i = ; so, we must know the accuracies σ j . � m Full Screen σ − 2 j j =1 Close Quit
Need to Fuse Models: . . . Fusion: General Problem 5. Traditional Methods of Estimating Accuracy Motivation for Using . . . Cannot Be Directly Used in Geophysics Statistical Fusion: . . . • Calibration is possible when we have a “standard” (sev- Traditional Methods . . . eral times more accurate) measuring instrument (MI). How to Estimate . . . How to Estimate . . . • In geophysics, seismic (and other) methods are state- An Idea of How to . . . of-the-art. Resulting Algorithm • No method leads to more accurate determination of Home Page the densities. Title Page • In some practical situations, we can use two similar ◭◭ ◮◮ MIs to measure the same quantities x i . ◭ ◮ • In geophysics, we want to estimate the accuracy of a Page 6 of 15 model, e.g., a seismic model, a gravity-based model. Go Back • In this situation, we do not have two similar applica- tions of the same model. Full Screen Close Quit
Need to Fuse Models: . . . Fusion: General Problem 6. Maximum Likelihood (ML) Approach Cannot Motivation for Using . . . Be Applied to Estimate Model Accuracy Statistical Fusion: . . . • We have several quantities with (unknown) actual val- Traditional Methods . . . ues x 1 , . . . , x i , . . . , x n . How to Estimate . . . How to Estimate . . . • We have several measuring instruments (or geophysical An Idea of How to . . . methods) with (unknown) accuracies σ 1 , . . . , σ m . Resulting Algorithm • We know the results x ij of measuring the i -th quantity Home Page x i by using the j -th measuring instrument. Title Page • At first glance, a reasonable idea is to find all the un- ◭◭ ◮◮ known quantities x i and σ j from ML: � � ◭ ◮ n m � � − ( x ij − x i ) 2 1 L = √ · exp → max . Page 7 of 15 2 σ 2 2 π · σ j j i =1 j =1 Go Back • Fact: the largest value L = ∞ is attained when, for Full Screen some j 0 , we have σ j 0 = 0 and x i = x ij 0 for all i . Close • Problem: this is not physically reasonable. Quit
Need to Fuse Models: . . . Fusion: General Problem 7. How to Estimate Model Accuracy: Idea Motivation for Using . . . • For every two models, the difference x ij − x ik = Statistical Fusion: . . . ∆ x ij − ∆ x ik is normally distributed, w/variance σ 2 j + σ 2 k . Traditional Methods . . . How to Estimate . . . • We can thus estimate σ 2 j + σ 2 k as How to Estimate . . . n � = 1 An Idea of How to . . . def σ 2 j + σ 2 ( x ij − x ik ) 2 . k ≈ A jk n · Resulting Algorithm i =1 Home Page • So, σ 2 1 + σ 2 2 ≈ A 12 , σ 2 1 + σ 2 3 ≈ A 13 , and σ 2 2 + σ 2 3 ≈ A 23 . Title Page • By adding all three equalities and dividing the result ◭◭ ◮◮ 3 = A 12 + A 13 + A 23 by two, we get σ 2 1 + σ 2 2 + σ 2 . ◭ ◮ 2 • Subtracting, from this formula, the expression for Page 8 of 15 1 ≈ A 12 + A 13 − A 23 σ 2 2 + σ 2 3 , we get σ 2 . Go Back 2 Full Screen 2 ≈ A 12 + A 23 − A 13 3 ≈ A 13 + A 23 − A 12 • Similarly, σ 2 and σ 2 . 2 2 Close Quit
Need to Fuse Models: . . . Fusion: General Problem 8. How to Estimate Model Accuracy: General Case Motivation for Using . . . and Challenge Statistical Fusion: . . . • General case: we may have M ≥ 3 different models. Traditional Methods . . . • Then, we have M · ( M − 1) How to Estimate . . . different equations 2 How to Estimate . . . σ 2 j + σ 2 k ≈ A jk to determine M unknowns σ 2 j . An Idea of How to . . . • When M > 3, we have more equations than unknowns, Resulting Algorithm Home Page • So, we can use the Least Squares method to estimate the desired values σ 2 Title Page j . ◭◭ ◮◮ = A 12 + A 13 − A 23 1 ≈ � def • Challenge: the formulas σ 2 V 1 2 ◭ ◮ are approximate. Page 9 of 15 • Sometimes, these formulas lead to physically meaning- less negative values � Go Back V 1 . Full Screen • It is therefore necessary to modify the above formulas, to avoid negative values. Close Quit
Need to Fuse Models: . . . Fusion: General Problem 9. An Idea of How to Deal With This Challenge Motivation for Using . . . • The negativity challenge is caused by the fact that the Statistical Fusion: . . . estimates � V j for σ 2 j are approximate. Traditional Methods . . . How to Estimate . . . def = � V j − σ 2 • For large n , the difference ∆ V j j is asymptot- How to Estimate . . . ically normally distributed, with asympt. 0 mean. An Idea of How to . . . • We can estimate the standard deviation ∆ j for this Resulting Algorithm difference. Home Page j = � • Thus, σ 2 V j − ∆ V j is normally distributed with mean Title Page � V j and standard deviation ∆ j . ◭◭ ◮◮ • We also know that σ 2 j ≥ 0. ◭ ◮ • As an estimate for σ 2 j , it is therefore reasonable to use a Page 10 of 15 � � � � � � � conditional expected value E V j − ∆ V j V j − ∆ V j ≥ 0 . Go Back • This new estimate is an expected value of a non-negative Full Screen number and thus, cannot be negative. Close Quit
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