Formulation of the . . . Formulation of the . . . How to Best Process Data If Formulation of the . . . Recommendation We Have Both Absolute and Algorithm Relative Measurement Home Page Errors: A Pedagogical Title Page Comment ◭◭ ◮◮ ◭ ◮ Ana Maria Hernandez Posada 1 , Maria Isabel Olivarez 1 , Page 1 of 6 Christian Servin 2 , and Vladik Kreinovich 1 1 Department of Computer Science Go Back University of Texas at El Paso, El Paso, TX 79968, USA Computer Science and IT Program, El Paso Community College Full Screen amhernandezegias@miners.utep.edu, miolivares@miners.utep.edu Close servin1@epcc.edu, vladik@utep.edu Quit
1. Formulation of the Problem Formulation of the . . . Formulation of the . . . • In many practical situations, we need to find the de- Formulation of the . . . pendence of a quantity y on quantities x = ( x 1 , . . . , x n ). Recommendation • Usually, we know the type of the dependence, i.e., we Algorithm know that f = f ( p, x ) for some parameters Home Page p = ( p 1 , . . . , p m ) . Title Page • We just need to find p . ◭◭ ◮◮ • For example, the dependence may be linear, then ◭ ◮ n Page 2 of 6 � f ( x, p ) = p i · x i + p n +1 . Go Back i =1 Full Screen • To find this dependence, we measure x i and y in several situations k . Close • Then, we find p for which f ( p, x ( k ) ) ≈ y ( k ) for all k . Quit
2. Formulation of the Problem (cont-d) Formulation of the . . . Formulation of the . . . • The measurement error is often caused by a large num- Formulation of the . . . ber of independent factors of about the same size, Recommendation • In this case the Central Limit Theorem implies that it Algorithm is normally distributed. Home Page • Usually, it is assumed that the bias is 0, so we only have standard deviation σ . Title Page ◭◭ ◮◮ • Sometimes, we have absolute error σ = const, in which case we use the usual Least Squares method ◭ ◮ ( y ( k ) − f ( p, x ( k ) )) 2 → min . � Page 3 of 6 k Go Back • In other cases, we have relative error, in which case we Full Screen ( y ( k ) − f ( p, x ( k ) )) 2 find p for which � → min. Close ( y ( k ) ) 2 k Quit
3. Formulation of the Problem (cont-d) Formulation of the . . . Formulation of the . . . • In practice, we usually have both absolute and relative Formulation of the . . . error components. Recommendation • Namely, ∆ y = ∆ y abs + ∆ y rel , with σ abs = σ 0 and σ rel = Algorithm σ 1 · | y | for some σ i . Home Page • How should we then process data? Title Page ◭◭ ◮◮ ◭ ◮ Page 4 of 6 Go Back Full Screen Close Quit
4. Recommendation Formulation of the . . . Formulation of the . . . • In this case, the variance of the measurement error if σ 2 = σ 2 Formulation of the . . . 0 + σ 2 1 · y 2 . Recommendation • So, we use Maximum Likelihood method and maximize Algorithm the expression − ( y ( k ) − f ( p, x ( k ) )) 2 Home Page 1 � � � √ 1 · ( y ( k ) ) 2 · exp . 2( σ 2 0 + σ 2 1 · ( y ( k ) ) 2 ) Title Page � σ 2 0 + σ 2 2 π · k ◭◭ ◮◮ • In this talk, we present an iterative algorithm for find- ◭ ◮ ing p . Page 5 of 6 Go Back Full Screen Close Quit
5. Algorithm Formulation of the . . . • The above problem is complex, so what we can do is Formulation of the . . . Formulation of the . . . solve it iteratively. Recommendation Algorithm • First, we assume that σ 1 = 0. • Then, we compute ( σ ( k ) ) 2 = σ 2 0 + σ 2 1 · ( y ( k ) ) 2 . Home Page • After that, we use the Least Squares and find p that Title Page ( y ( k ) − f ( p, x ( k ) )) 2 minimizes � . ◭◭ ◮◮ ( y ( k ) ) 2 k ◭ ◮ • Once we find these values p , we again use the Least Squares to find the values σ 2 0 and σ 2 1 for which Page 6 of 6 ( y ( k ) − f ( p, x ( k ) )) 2 ≈ σ 2 0 + σ 2 1 · ( y ( k ) ) 2 . Go Back Full Screen • Then, we again compute ( σ ( k ) ) 2 , find p , etc., until the Close process converges. Quit
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