Towards Chemical Traditional . . . Applications From Continuous to - PowerPoint PPT Presentation

Formulation of the . . . Rota’s Dempster- . . . Relation to the . . . Practical Applications . . . Towards Chemical Traditional . . . Applications From Continuous to . . . Discrete Taylor . . . of Dempster-Shafer-Type Comparing Poset and . . . Proof that The . . . Approach: Home Page Case of Variant Ligands Title Page ◭◭ ◮◮ Jaime Nava ◭ ◮ Department of Computer Science Page 1 of 17 University of Texas at El Paso El Paso, TX 79968 Go Back jenava@miners.utep.edu Full Screen Close Quit

Formulation of the . . . Rota’s Dempster- . . . 1. Formulation of the Problem: Extrapolation Is Relation to the . . . Needed Practical Applications . . . • In many practical situations, molecules can be obtained Traditional . . . from a “template” molecule like benzene C 6 H 6 . From Continuous to . . . Discrete Taylor . . . • How: by replacing some of its hydrogen atoms with Comparing Poset and . . . ligands (other atoms or atom groups). Proof that The . . . • Fact: there can be many possible replacements of this Home Page type. Title Page • Testing of all possible replacements would be time- ◭◭ ◮◮ consuming. ◭ ◮ • It is desirable: to test some of the replacements and Page 2 of 17 then extrapolate to others. Go Back • Thus: only the promising molecules will have to be synthesized and tested. Full Screen Close Quit

Formulation of the . . . Rota’s Dempster- . . . 2. Formulation of the Problem: Extrapolation Is Relation to the . . . Needed (cont-d) Practical Applications . . . • D. J. Klein and co-authors proposed to use a poset Traditional . . . extrapolation technique developed by G.-C. Rota. From Continuous to . . . Discrete Taylor . . . • In many practical situations, this technique indeed leads Comparing Poset and . . . to accurate predictions of many important quantities. Proof that The . . . • Limitation: this technique has been originally proposed Home Page on a heuristic basis. Title Page • There is no convincing justification of its applicability ◭◭ ◮◮ to chemical (or other) problems. ◭ ◮ • In this presentation: we show that this equivalence can Page 3 of 17 be extended to the case when we have variant ligands. Go Back Full Screen Close Quit

Formulation of the . . . Rota’s Dempster- . . . 3. Rota’s Dempster-Shafer-Type Poset Approach Relation to the . . . to Extrapolation: Reminder Practical Applications . . . • Rota considererd the situation in which there is Traditional . . . From Continuous to . . . – a natural partial order relation ≤ on some set of Discrete Taylor . . . objects, and Comparing Poset and . . . – a numerical value v ( a ) associated to each object a Proof that The . . . from this partially ordered set (poset). Home Page • Main idea: is that we can represent an arbitrary de- Title Page pendence v ( a ) as � ◭◭ ◮◮ v ( a ) = V ( b ) ◭ ◮ b : b ≤ a for some values V ( b ). Page 4 of 17 • To find values V ( b ), solve a system of linear equations Go Back with as many unknowns V ( b ) as the number of objects. Full Screen • It was proven that the poset-related system always has Close a solution. Quit

Formulation of the . . . Rota’s Dempster- . . . 4. Relation to the Dempster-Shafer Approach Relation to the . . . • The poset formula is identical to one of the main for- Practical Applications . . . mulas of the Dempster-Shafer approach. Traditional . . . From Continuous to . . . • Specifically, in this approach: Discrete Taylor . . . – in contrast to a probability distribution when prob- Comparing Poset and . . . abilities are assigned to different elements x ∈ X , Proof that The . . . – we have “masses” (in effect, probabilities) assigned Home Page to subsets A ⊆ X of the set X . Title Page • For each expert: ◭◭ ◮◮ – B is the set of alternatives that is possible according ◭ ◮ to this expert, and Page 5 of 17 – m ( B ) is the probability that this expert is correct Go Back based on his or her previous performance. Full Screen Close Quit

Formulation of the . . . Rota’s Dempster- . . . 5. Relation to the Dempster-Shafer Approach (cont- Relation to the . . . d) Practical Applications . . . • For every set A ⊆ X and for every expert, the expert’s Traditional . . . set B of possible alternatives can be contained in A . From Continuous to . . . Discrete Taylor . . . • This means that this expert is sure that all possible Comparing Poset and . . . alternatives are contained in the set A . Proof that The . . . • Thus: our overall belief bel( A ) that the actual alterna- Home Page tive is contained in A can be computed as Title Page � bel( A ) = m ( B ) . ◭◭ ◮◮ B ⊆ A ◭ ◮ • This is the exact analog of the above formula, with Page 6 of 17 – v ( a ) instead of belief, Go Back – V ( b ) instead of masses, and Full Screen – B ⊆ A as the ordering relation b ≤ a . Close Quit

Formulation of the . . . Rota’s Dempster- . . . 6. Practical Applications of the Poset Approach Relation to the . . . • In practice: many values V ( b ) turn out to be negligible Practical Applications . . . and thus, can be safely taken as 0s. Traditional . . . From Continuous to . . . • If we know which values V ( b 1 ) , . . . , V ( b m ) are non-zeros, Discrete Taylor . . . we can then: Comparing Poset and . . . – measure the value v ( a 1 ) , . . . , v ( a p ) of the desired Proof that The . . . quantity v for p ≪ n different objects a 1 , . . . , a p ; Home Page – use the Least Squares techniques to estimate the Title Page values V ( b j ) from the system � ◭◭ ◮◮ v ( a i ) = V ( b j ) , i = 1 , . . . , p ; ◭ ◮ j : b j ≤ a i Page 7 of 17 – use the resulting estimates V ( b j ) to predict all the remaining values v ( a ) ( a � = a 1 , . . . , a m ), as Go Back � v ( a ) = V ( b j ) . Full Screen j : b j ≤ a Close Quit

Formulation of the . . . Rota’s Dempster- . . . 7. Traditional (Continuous) and Discrete Taylor Relation to the . . . Series Practical Applications . . . • In physical and engineering applications, most param- Traditional . . . eters x 1 , . . . , x n are continuous . From Continuous to . . . Discrete Taylor . . . • The dependence y = f ( x 1 , . . . , x n ) is also usually con- Comparing Poset and . . . tinuous and smooth (differentiable). Proof that The . . . • Smooth functions can be usually expanded into Taylor Home Page series around some point � x = ( � x n ): x 1 , . . . , � Title Page n � ∂f ◭◭ ◮◮ f ( x 1 , . . . , x n ) = f ( � x 1 , . . . , � x n ) + · ∆ x i + ∂x i i =1 ◭ ◮ n n � � ∂ 2 f 1 Page 8 of 17 2 · ∂x i ∂x i ′ · ∆ x i · ∆ x i ′ + . . . , Go Back i =1 i ′ =1 def Full Screen where ∆ x i = x i − � x i . Close Quit

Formulation of the . . . Rota’s Dempster- . . . 8. Traditional (Continuous) and Discrete Taylor Relation to the . . . Series (cont-d) Practical Applications . . . • In practice, we can ignore higher-order terms. Traditional . . . From Continuous to . . . • Example: if linear approximation is not accurate enough, Discrete Taylor . . . we can use quadratic approximation. Comparing Poset and . . . • If we do not know the exact expression for f ( x 1 , . . . , x n ), Proof that The . . . we do not know the values of its derivatives. Home Page • All we know is that we approximate a general function Title Page by a general linear or quadratic formula ◭◭ ◮◮ n n n � � � ◭ ◮ c ii ′ · ∆ x i · ∆ x i ′ . f ( x 1 , . . . , x n ) ≈ c 0 + c i · ∆ x i + i =1 i =1 j =1 Page 9 of 17 Go Back • The values of the coefficients c 0 , c i , and (if needed) c ii ′ can then be determined experimentally. Full Screen Close Quit

Formulation of the . . . Rota’s Dempster- . . . 9. From Continuous to Discrete Taylor Series Relation to the . . . • General case: y = f ( x 11 , . . . , x 1 N , . . . , x n 1 , . . . , x nN ), Practical Applications . . . so Traditional . . . n N n N n N � � � � � � From Continuous to . . . y = y 0 + y ij · ∆ x ij + y ij,i ′ j ′ · ∆ x ij · ∆ x i ′ j ′ , Discrete Taylor . . . i =1 j =1 i =1 j =1 i ′ =1 j ′ =1 Comparing Poset and . . . def where ∆ x ij = x ij − d i 0 j . Proof that The . . . Home Page • Let ε ik denote the discrete variable that describes the presence of a ligand of type k at the location i : Title Page – when there is no ligand of type k at the location i , ◭◭ ◮◮ we take ε ik = 0, and ◭ ◮ – when there is a ligand of type k at the location i , Page 10 of 17 we take ε ik = 1. Go Back • If no ligand, x ij = d i 0 j . Thus ∆ x ij = d i 0 j − d i 0 j = 0 . Full Screen • If ligand, x ij = d ikj . Thus ∆ x ij = d ikj − d i 0 j is equal Close def to ∆ ikj = d ikj − d i 0 j . Quit