Hypothesis Testing with An Important New . . . Interval Data: Case - PowerPoint PPT Presentation

Hypothesis Testing: A . . . Traditional Statistical . . . How to Describe . . . Types of Uncertainty: . . . Hypothesis Testing with An Important New . . . Interval Data: Case of Case of Probabilistic . . . Example: Car Testing . . . Regulatory Constraints Case of Interval . . . Case of Fuzzy Uncertainty Sa-aat Niwitpong 1 , Hung T. Nguyen 2 , Home Page Vladik Kreinovich 3 , and Ingo Neumann 4 Title Page 1 Department of Applied Statistics ◭◭ ◮◮ King Mongkut’s University of Technology North Bangkok ◭ ◮ Bangkok 10800, Thailand, snw@kmitnb.ac.th 2 Mathematics, New Mexico State University Page 1 of 18 Las Cruces, NM 88003, USA, hunguyen@nmsu.edu 3 Computer Science, University of Texas at El Paso Go Back El Paso, TX 79968, USA, vladik@utep.edu Full Screen 4 Geodetic Institute, Leibniz University of Hannover D-30167 Hannover, Germany, neumann@gih.uni-hannover.de Close Quit

Hypothesis Testing: A . . . Traditional Statistical . . . 1. Hypothesis Testing: A General Problem How to Describe . . . • It is often desirable to check whether a given object (or Types of Uncertainty: . . . situation) satisfies a given property. An Important New . . . Case of Probabilistic . . . • Examples: Example: Car Testing . . . – whether a patient has flu, Case of Interval . . . – whether a building or a bridge is structurally stable. Case of Fuzzy Uncertainty • In statistics, this problem is called hypothesis testing: Home Page – we have a hypothesis – that a patient is healthy, Title Page that a building is structurally stable – ◭◭ ◮◮ – and we want to test this hypothesis based on the ◭ ◮ available data. Page 2 of 18 • This hypothesis H 0 is usually called a null hypothesis : Go Back – if H 0 is satisfied, no (“null”) action is required, Full Screen – if H 0 is not satisfied, action is needed: cure a pa- tient, reinforce the building, etc. Close Quit

Hypothesis Testing: A . . . Traditional Statistical . . . 2. Hypothesis Testing: Ideal Case of Complete How to Describe . . . Knowledge Types of Uncertainty: . . . • In the ideal case, we know the exact values of all the An Important New . . . quantities x 1 , . . . , x n that characterize the object o . Case of Probabilistic . . . Example: Car Testing . . . • Since x i are all the quantities characterize the object, Case of Interval . . . they determine whether o satisfies the property. Case of Fuzzy Uncertainty • Thus, the set X of all possible values of the tuple x = Home Page ( x 1 , . . . , x n ) can be divided into: Title Page – the acceptance region A of all the tuples that satisfy ◭◭ ◮◮ the desired property; and ◭ ◮ – the rejection region R of all the tuples that do not satisfy the desired property. Page 3 of 18 • Thus, once we know the tuple x characterizing o , we: Go Back – accept the hypothesis if x ∈ A , and Full Screen – reject the hypothesis if x ∈ R (i.e., if x �∈ A ). Close Quit

Hypothesis Testing: A . . . Traditional Statistical . . . 3. Hypothesis Testing: Realistic Case of Incom- How to Describe . . . plete Knowledge Types of Uncertainty: . . . • In practice, we usually only have an incomplete knowl- An Important New . . . edge about an object. Case of Probabilistic . . . Example: Car Testing . . . • Based on this partial information, we cannot always Case of Interval . . . tell whether an object satisfies the given property. Case of Fuzzy Uncertainty • Example: H 0 is x 1 + x 2 ≤ x 0 , and we only know x 1 : Home Page – for some x 2 (when x 2 ≤ x 0 − x 1 ) we have x 1 + x 2 ≤ Title Page x 0 and thus, the hypothesis H 0 is satisfied; ◭◭ ◮◮ – for some x 2 (when x 2 > x 0 − x 1 ) H 0 is not satisfied. ◭ ◮ • In such situations, the decision may be erroneous: Page 4 of 18 – false positive (Type I error): the object o satisfies Go Back H 0 , but we classify it as not satisfying H 0 ; Full Screen – false negative (Type II error): the object o does not satisfy H 0 , but we conclude that it does. Close Quit

Hypothesis Testing: A . . . Traditional Statistical . . . 4. Traditional Statistical Approach to Hypothesis How to Describe . . . Testing Types of Uncertainty: . . . • We assume that we know the probability distribution An Important New . . . of objects that satisfy the given hypothesis H 0 . Case of Probabilistic . . . Example: Car Testing . . . • We are given the allowed probability p 0 of Type I error. Case of Interval . . . • Idea: we select the accept and reject regions A and R Case of Fuzzy Uncertainty so as to minimize the probability p II of Type II error. Home Page • Example: in 1-D case, the distribution is usually Gaus- Title Page sian, with known mean a and standard deviation σ . ◭◭ ◮◮ • Usually, situations are anomalous when the quantity ◭ ◮ (e.g., blood pressure or cholesterol level) is too high. Page 5 of 18 • In this case, we take A = { x 1 : x 1 ≤ x 0 } for some x 0 : x 0 = a + 2 σ for p 0 = 5%, x 0 = a + 3 σ for p 0 = 0 . 05%. Go Back Full Screen • To find p II , we also need to know probability distribu- tion for all objects (not necessarily satisfying H 0 ). Close Quit

Hypothesis Testing: A . . . Traditional Statistical . . . 5. Limitations of the Traditional Statistical Ap- How to Describe . . . proach to Decision Making Types of Uncertainty: . . . • Main problem: how to determine Type I error p 0 . An Important New . . . Case of Probabilistic . . . • Fact: decreasing p 0 increases Type II probability p II . Example: Car Testing . . . • Example: mass screening for breast cancer; when the Case of Interval . . . result is suspicious, we apply a more complex text. Case of Fuzzy Uncertainty • Consequences: Type I error means missing cancer, Type Home Page II error means re-testing. Title Page • If p 0 is too low , we apply the more complex test to too ◭◭ ◮◮ many people – so expenses are unrealistic. ◭ ◮ • If p 0 is too high , we miss many cancers. Page 6 of 18 • To find desirable p 0 , we must know the society’s limi- Go Back tations and preferences. Full Screen • To determine p 0 from preferences, we must learn how to describe these preferences. Close Quit

Hypothesis Testing: A . . . Traditional Statistical . . . 6. How to Describe Preferences: the Notion of How to Describe . . . Utility Types of Uncertainty: . . . • To get a scale, we select two alternatives: a very nega- An Important New . . . tive alternative A 0 and a very positive alternative A 1 . Case of Probabilistic . . . Example: Car Testing . . . • For every p ∈ [0 , 1], we consider an event L ( p ) in which Case of Interval . . . we get A 1 w/prob. p and A 0 w/prob. 1 − p . Case of Fuzzy Uncertainty • The larger p , the better L ( p ): L (0) < L ( p ) < L (1). Home Page • ∀ event E , there exists a p for which E is equivalent to Title Page L ( p ): E ∼ L ( p ); this p is called the utility u ( E ) of E . ◭◭ ◮◮ • Let an action A lead to alternatives a 1 , . . . , a m with ◭ ◮ utilities u i and probabilities p i . Page 7 of 18 • Since a i ∼ L ( u i ), A is equivalent to having L ( u i ) w/prob. p i , i.e., to having A 1 w/prob. p = p 1 · u 1 + . . . + p n · u n . Go Back Full Screen • Thus, the utility u ( A ) of an action is equal to the ex- pected value E [ u ] = � p i · u i of the utilities u i . Close Quit

Hypothesis Testing: A . . . Traditional Statistical . . . 7. Utility Is Defined Modulo Linear Transforma- How to Describe . . . tions Types of Uncertainty: . . . • By definition u ( E ) is the value for which E is equivalent An Important New . . . to L ( u ), i.e., to A 1 w/prob. u and A 0 w/prob. 1 − u . Case of Probabilistic . . . Example: Car Testing . . . • The numerical value of u ( E ) depends on the choice of Case of Interval . . . A 0 and A 1 : Case of Fuzzy Uncertainty 1 , and let u ′ be utility based on • Let A ′ 0 < A 0 < A 1 < A ′ Home Page A ′ 0 and A ′ 1 . Title Page • By definition, A 0 ∼ L ′ ( u ′ ( A 0 )) and A 1 ∼ L ′ ( u ′ ( A 1 )). ◭◭ ◮◮ • Thus, E is equivalent to a composite event: ◭ ◮ L ′ ( u ′ ( A 0 )) w/prob. u and L ′ ( u ′ ( A 1 )) w/prob. 1 − u . Page 8 of 18 • In this composite event, we get A ′ 1 with probability u · u ′ ( A 1 ) + (1 − u ) · u ′ ( A 0 ) . Go Back • Thus, in the new scale, u ′ = u · u ′ ( A 1 )+(1 − u ) · u ′ ( A 0 ), Full Screen i.e., u ′ = a · u + b for a > 0. Close Quit

Hypothesis Testing: A . . . Traditional Statistical . . . 8. Types of Uncertainty: Probabilistic, Interval, How to Describe . . . Fuzzy Types of Uncertainty: . . . • Uncertainty means that our estimate � x differs from the An Important New . . . def actual (unknown) value x : ∆ x = � x − x � = 0. Case of Probabilistic . . . Example: Car Testing . . . • Ideal case: we know the probabilities of different pos- Case of Interval . . . sible values of approximation error ∆ x . Case of Fuzzy Uncertainty • Interval case: often, we only know the upper bound ∆ Home Page on the the approximation error: | ∆ x | ≤ ∆. Title Page • Based on � x , we conclude that the actual value of x is ◭◭ ◮◮ def in the interval x = [ � x − ∆ , � x + ∆]. ◭ ◮ • In addition to ∆, experts can provide us with smaller Page 9 of 18 bounds corr. to different degrees of uncertainty α . Go Back • Fuzzy uncertainty: the resulting intervals can be viewed Full Screen as α -cuts of a fuzzy set. Close Quit