Need to Take into . . . Need for Guaranteed . . . Locating Local . . . Locating Local Extrema Definitions: Locations . . . Definitions: . . . under Interval Uncertainty: Definitions: Locating . . . Definitions: Locating . . . Multi-D Case First Result Karen Villaverde 1 and Second Result Vladik Kreinovich 2 Home Page Title Page 1 Department of Computer Science New Mexico State University ◭◭ ◮◮ Las Cruces, NM 88003, USA ◭ ◮ kvillave@cs.nmsu.edu Page 1 of 16 2 Department of Computer Science University of Texas at El Paso Go Back El Paso, TX 79968, USA vladik@utep.edu Full Screen Close Quit
Need to Take into . . . 1. The Problem of Locating Local Extrema Is Im- Need for Guaranteed . . . portant Locating Local . . . Definitions: Locations . . . • In spectral analysis , chemical species are identified by Definitions: . . . locating local maxima of the spectra. Definitions: Locating . . . • In radioastronomy , radiosources are identified as local Definitions: Locating . . . maxima of the measured brightness. First Result • Elementary particles are identified as local maxima of Second Result Home Page scattering y as a function of energy t . Title Page • Different clusters correspond to local maxima of the probability density function. ◭◭ ◮◮ ◭ ◮ Page 2 of 16 Go Back Full Screen Close Quit
Need to Take into . . . 2. The Problem of Locating Local Extrema: Pre- Need for Guaranteed . . . cise Formulation Locating Local . . . Definitions: Locations . . . In each of these applications, the following problem arises: Definitions: . . . • we know that a physical quantity y is a function of one Definitions: Locating . . . or several ( m ≥ 1) other physical quantities t 1 , . . . , t m : Definitions: Locating . . . First Result y = f ( t 1 , . . . , t m ); Second Result • we have n situations, i = 1 , . . . , n , in each of which we Home Page know the values of all m quantities: v i = ( t i 1 , . . . , t im ); Title Page • in each of these n situations, we have measured the ◭◭ ◮◮ values y 1 = f ( v 1 ) , . . . , y n = f ( v n ) of the quantity y ; ◭ ◮ • based on this information, we want to locate the local Page 3 of 16 maxima and/or the local minima of the function f . Go Back Full Screen Close Quit
Need to Take into . . . 3. Need to Take into Account Interval Uncertainty Need for Guaranteed . . . Locating Local . . . • Observed values y i = f ( v i ) come from measurements, Definitions: Locations . . . and measurements are never absolutely accurate. Definitions: . . . • The measurement results � y i are, in general, different Definitions: Locating . . . from the actual (unknown) values y i . Definitions: Locating . . . First Result • In some cases, we know the probabilities of different def Second Result values of the measurement error ∆ y i = � y i − y i . Home Page • In many practical cases, however, we only know the Title Page upper bound ε > 0 on the measurement error: ◭◭ ◮◮ | ∆ y i | < ε. ◭ ◮ • Then, the only information that we have about y i is Page 4 of 16 that y i belongs to the interval ( � y i − ε, � y i + ε ). Go Back • We thus need to locate the local maxima and local Full Screen minima of a function under such interval uncertainty. Close Quit
Need to Take into . . . 4. Need for Guaranteed Results Need for Guaranteed . . . Locating Local . . . • Due to measurement uncertainty, the actual observed Definitions: Locations . . . values fluctuate. Definitions: . . . • Hence, the f-n corresponding to the actual measure- Definitions: Locating . . . ment results usually has many local maxima (minima). Definitions: Locating . . . • Most of these local maxima and minima: First Result Second Result – are caused by the measurement errors and Home Page – do not have any physical significance. Title Page • We only want to keep those local maxima and minima ◭◭ ◮◮ which reflect the actual dependence, ◭ ◮ • In other words, we want to keep only those local ex- trema that guaranteed to correspond to: Page 5 of 16 – source components, Go Back – chemical substances, Full Screen – etc. Close Quit
Need to Take into . . . 5. Case of Fuzzy Uncertainty Need for Guaranteed . . . Locating Local . . . • Often: Definitions: Locations . . . – in addition (or instead) the guaranteed bound ε for Definitions: . . . the measurement error ∆ y i , Definitions: Locating . . . – an expert can provide bounds that contain ∆ y i with Definitions: Locating . . . a certain degree of confidence. First Result Second Result • Usually, we know several such bounding intervals cor- Home Page responding to different degrees of confidence. Title Page • Such a nested family of intervals is equivalent to a fuzzy ◭◭ ◮◮ set (to be more precise, to its α -cuts). ◭ ◮ • From the algorithmic viewpoint, fuzzy uncertainty can be thus reduced to interval uncertainty. Page 6 of 16 • Because of this reduction, we will be concentrating on Go Back the algorithms for solving the interval problem. Full Screen Close Quit
Need to Take into . . . 6. Locating Local Extrema Under Interval Uncer- Need for Guaranteed . . . tainty: What Is Known Locating Local . . . Definitions: Locations . . . • A feasible (polynomial-time) algorithm is known for Definitions: . . . m = 1, when there is only one input variable t 1 . Definitions: Locating . . . • In many practical applications, we need to solve a sim- Definitions: Locating . . . ilar problem in a situation when we have several inputs First Result Second Result t 1 , . . . , t m , m > 1 . Home Page • For example, in locating components of a radioastro- Title Page nomical source, we start with a 2-D intensity function. ◭◭ ◮◮ • In clustering, we also need to consider local maxima of ◭ ◮ functions of several variables, etc. Page 7 of 16 • In this talk, we describe a polynomial-time algorithm Go Back that solves the problem for case of several variables. Full Screen Close Quit
Need to Take into . . . 7. Definitions: Locations and Connectedness Need for Guaranteed . . . Locating Local . . . • Let G be a finite undirected graph; its vertices will be Definitions: Locations . . . called locations . Definitions: . . . • If the vertices x, y ∈ G are connected by an edge, we Definitions: Locating . . . will call them neighbors and denote it by x ∼ y . Definitions: Locating . . . First Result • We say that a function f : G → I R has a local minimum at location x if f ( x ) ≤ f ( y ) for all neighbors y of x . Second Result Home Page • We say that a function f : G → I R has a local maximum Title Page at location x if f ( x ) ≤ f ( y ) for all neighbors y of x . ◭◭ ◮◮ • Let S ⊆ G . We say that x, y ∈ S are S -connected if there exists a S -connecting sequence ◭ ◮ Page 8 of 16 x 0 = x ∈ S, x 1 ∈ S, . . . , x m − 1 ∈ S, x m = y ∈ S s.t. ∀ i ( x i ∼ x i +1 ) . Go Back • We say that a subset S ⊆ G is connected if every two locations x, y ∈ S are S -connected. Full Screen Close Quit
Need to Take into . . . 8. Definitions: Measurement Results Need for Guaranteed . . . Locating Local . . . • Let G be a graph. By a measurement result , we mean Definitions: Locations . . . a pair f = � f 0 , ε � , where: Definitions: . . . • f 0 : G → I R is a rational-valued function whose Definitions: Locating . . . values f 0 ( x ) are called measured values ; Definitions: Locating . . . • ε > 0 is a rational number called measurement ac- First Result curacy . Second Result Home Page • A measurement result will also be called an interval- valued function and denoted by Title Page ◭◭ ◮◮ f ( x ) = ( f 0 ( x ) − ε, f 0 ( x ) + ε ) . ◭ ◮ • We say that a function f : G → I R is consistent with Page 9 of 16 f ( x ) = ( f 0 ( x ) − ε, f 0 ( x )+ ε ) if f ( x ) ∈ ( f 0 ( x ) − ε, f 0 ( x )+ ε ) for every location x . Go Back Full Screen • We will denote this consistency by f ∈ f . Close Quit
Need to Take into . . . 9. Definitions: Locating Local Minimum Need for Guaranteed . . . Locating Local . . . • Let G be a graph, and let f be an interval-valued func- Definitions: Locations . . . tion on this graph. Definitions: . . . • We say that a connected set S is a local minimum set Definitions: Locating . . . of f if the following properties are satisfied: Definitions: Locating . . . First Result – every function f ∈ f attains a local minimum at some location x ∈ S ; Second Result Home Page – each location x m ∈ S at which f ∈ f attains its smallest value on S is a local minimum of f on G ; Title Page – for S ′ ⊂ S , S ′ � = S , there is a function f ∈ f that ◭◭ ◮◮ does not have any local minimum on S ′ . ◭ ◮ • For S = { x 0 } , for every f ∈ f , the value f ( x 0 ) is smaller Page 10 of 16 than or equal to the value at all neighbors y ∼ x 0 . Go Back • When S = { x 1 , x 2 , . . . } , different f ∈ f may attain Full Screen local minimum at different locations x i ∈ S . Close Quit
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