Taking Into Account Interval Case of Fuzzy Uncertainty (and Fuzzy) - PowerPoint PPT Presentation

Data Processing: . . . Usually Linearization . . . Least Squares . . . Case When We Do . . . Taking Into Account Interval Case of Fuzzy Uncertainty (and Fuzzy) Uncertainty Can Case When We Need . . . Combining . . . Lead to More Adequate From Formulas to . . . But Where Do We Get . . . Statistical Estimates Home Page Title Page Ligang Sun, Hani Dbouk, Ingo Neumann, Steffen Schoen Leibniz University of Hannover, Germany ◭◭ ◮◮ ligang.sun@gih.uni-hannover.de, dbouk@mbox.ife.uni-hannover.de neumann@gih.uni-hannover.de, schoen@ife.uni-hannover.de ◭ ◮ Vladik Kreinovich Page 1 of 31 University of Texas at El Paso, El Paso, TX 79968, USA Go Back vladik@utep.edu Full Screen Close Quit

Data Processing: . . . Usually Linearization . . . 1. Data Processing: General Introduction Least Squares . . . • Some quantities, we can directly measure. Case When We Do . . . Case of Fuzzy Uncertainty • For example, we can directly measure the distance be- Case When We Need . . . tween two points. Combining . . . • However, many other quantities we cannot measure di- From Formulas to . . . rectly. But Where Do We Get . . . • For example, we cannot directly measure the spatial Home Page coordinates. Title Page • To estimate such quantities X i , we measure them in- ◭◭ ◮◮ directly : ◭ ◮ – we measure easier-to-measure quantities Y 1 , . . . , Y m Page 2 of 31 – which are connected to X j in a known way: Go Back Y i = f i ( X 1 , . . . , X n ) for known functions f i . Full Screen Close Quit

Data Processing: . . . Usually Linearization . . . 2. Sometimes, Measurement Results Also Depend Least Squares . . . on Additional Factors of No Interest to Us Case When We Do . . . • Sometimes, the measurement results also depend on Case of Fuzzy Uncertainty auxiliary factors of no direct interest to us. Case When We Need . . . Combining . . . • For example, the time delays used to measure distances From Formulas to . . . depend: But Where Do We Get . . . – not only on the distance, Home Page – but also on the amount of H 2 0 in the troposphere. Title Page • In such situations, we can add these auxiliary quanti- ◭◭ ◮◮ ties to the list X j of the unknowns. ◭ ◮ • We may also use the result Y i of additional measure- Page 3 of 31 ments of these auxiliary quantities. Go Back Full Screen Close Quit

Data Processing: . . . Usually Linearization . . . 3. Data Processing (cont-d) Least Squares . . . • Example: Case When We Do . . . Case of Fuzzy Uncertainty – we want to measure coordinates X j of an object; Case When We Need . . . – we measure the distance Y i between this object and Combining . . . objects with accurately known coordinates X ( i ) j : From Formulas to . . . � � But Where Do We Get . . . 3 � � � ( X j − X ( i ) j ) 2 . Home Page Y i = j =1 Title Page • General case: ◭◭ ◮◮ – we know the results � Y i of measuring Y i ; ◭ ◮ – we want to estimate the desired quantities X j . Page 4 of 31 Go Back Full Screen Close Quit

Data Processing: . . . Usually Linearization . . . 4. Usually Linearization Is Possible Least Squares . . . • In most practical situations, we know the approximate Case When We Do . . . values X (0) of the desired quantities X j . Case of Fuzzy Uncertainty j Case When We Need . . . • These approximation are usually reasonably good, in def = X j − X (0) Combining . . . the sense that the difference x j are small. j From Formulas to . . . • In terms of x j , we have But Where Do We Get . . . Y i = f ( X (0) + x 1 , . . . , X (0) Home Page + x n ) . 1 n Title Page • We can safely ignore terms quadratic in x j . ◭◭ ◮◮ • Indeed, even if the estimation accuracy is 10% (0.1), its square is 1% ≪ 10%. ◭ ◮ Page 5 of 31 • We can thus expand the dependence of Y i on x j in Taylor series and keep only linear terms: Go Back n � = ∂f i Full Screen Y i = Y (0) a ij · x j , Y (0) def = f i ( X (0) def 1 , . . . , X (0) + n ) , a ij . i i ∂X j Close j =1 Quit

Data Processing: . . . Usually Linearization . . . 5. Least Squares Least Squares . . . • Thus, to find the unknowns x j , we need to solve a Case When We Do . . . system of approximate linear equations Case of Fuzzy Uncertainty Case When We Need . . . � n def Y i − Y (0) = � a ij · x i ≈ y i , where y i . Combining . . . i j =1 From Formulas to . . . But Where Do We Get . . . • Usually, it is assumed that each measurement error is: Home Page – normally distributed Title Page – with 0 mean (and known st. dev. σ i ). ◭◭ ◮◮ • The distribution is indeed often normal: ◭ ◮ – the measurement error is a joint result of many in- Page 6 of 31 dependent factors, Go Back – and the distribution of the sum of many small in- dependent errors is close to Gaussian; Full Screen – this is known as the Central Limit Theorem. Close Quit

Data Processing: . . . Usually Linearization . . . 6. Least Squares (cont-d) Least Squares . . . • 0 mean also makes sense: Case When We Do . . . Case of Fuzzy Uncertainty – we calibrate the measuring instrument by compar- Case When We Need . . . ing it with a more accurate, Combining . . . – so if there was a bias (non-zero mean), we delete it From Formulas to . . . by re-calibrating the scale. But Where Do We Get . . . • It is also assumed that measurement errors of different Home Page measurements are independent. Title Page • In this case, for each possible combination x = ◭◭ ◮◮ ( x 1 , . . . , x n ), the probability of observing y 1 , . . . , y m is:     � � 2 ◭ ◮ � n     y i − a ij · x j Page 7 of 31     m � 1 j =1     √ · exp −     . Go Back 2 σ 2     2 π · σ i i     i =1 Full Screen Close Quit

Data Processing: . . . Usually Linearization . . . 7. Least Squares (final) Least Squares . . . • It is reasonable to select x j for which this probability Case When We Do . . . is the largest, i.e., equivalently, for which Case of Fuzzy Uncertainty � � 2 Case When We Need . . . � n y i − a ij · x j Combining . . . n � j =1 From Formulas to . . . → min . σ 2 But Where Do We Get . . . i i =1 Home Page • The set S γ of all possible combinations x is: Title Page   � � 2   �  n  ◭◭ ◮◮    y i − a ij · x j      n � j =1 ◭ ◮ ≤ χ 2 S γ = x : . m − n,γ σ 2     i Page 8 of 31   i =1       Go Back Full Screen • If S = ∅ , this means that some measurements are out- liers. Close Quit

Data Processing: . . . Usually Linearization . . . 8. Simple Example Least Squares . . . • Suppose that we have m measurements y 1 , . . . , y m of Case When We Do . . . the same quantity x 1 , with 0 mean and st. dev. σ i . Case of Fuzzy Uncertainty Case When We Need . . . • Then, the least squares estimate for x 1 is Combining . . . � m σ − 2 · y i From Formulas to . . . i i =1 x 1 = ˆ . But Where Do We Get . . . � m σ − 2 Home Page i i =1 Title Page 1 • The accuracy of this estimate is σ 2 [ x 1 ] = ◭◭ ◮◮ . � m σ − 2 ◭ ◮ i i =1 Page 9 of 31 • In particular, for σ 1 = . . . = σ m = σ , we get Go Back x 1 = y 1 + . . . + y m σ √ m. ˆ , with σ [ x 1 ] = m Full Screen Close Quit

Data Processing: . . . Usually Linearization . . . 9. Least Squares Approach Is Not Always Appli- Least Squares . . . cable Case When We Do . . . • There are cases when this Least Squares approach is Case of Fuzzy Uncertainty not applicable. Case When We Need . . . Combining . . . • The first case is when we use the most accurate mea- suring instruments. From Formulas to . . . But Where Do We Get . . . • In this case, we have no more accurate instrument to Home Page calibrate. Title Page • So, we do no know the mean, we do not know the ◭◭ ◮◮ distribution. ◭ ◮ • The second case is when we have many measurements. Page 10 of 31 • If we simply measure the same quantity m times, we σ get an estimate (average) with accuracy √ m. Go Back Full Screen • So, if we use GPS with 1 m accuracy million times, we can 1 mm accuracy, then microns etc. Close Quit

Data Processing: . . . Usually Linearization . . . 10. Least Squares Approach Is Not Always Appli- Least Squares . . . cable (cont-d) Case When We Do . . . • This makes no physical sense. Case of Fuzzy Uncertainty Case When We Need . . . • When we calibrate, we guarantee that the systematic Combining . . . error (mean) is much smaller than the random error. From Formulas to . . . • However: But Where Do We Get . . . Home Page – when we repeat measurements – and take the av- erage – we decrease random error, Title Page – however, the systematic error does not decrease, ◭◭ ◮◮ – so, systematic error becomes larger than the re- ◭ ◮ maining random error. Page 11 of 31 • Let us consider these two cases one by one. Go Back Full Screen Close Quit