Constructive decision theory (1982) by Shafer G.

By Shafer G.

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Additional info for Constructive decision theory (1982)

Example text

Thus the problem of learning the source simplifies to estimating x and I . Additionally, it is also known that the warehouse is large, the initial temperature is a constant throughout the warehouse, and the thermal conductivity is large and uniform throughout the medium, and known. To map the model set for the source to a model set for the sensor measurements we will use the diffusion equation. Let C(s, t; x, I ) denote the temperature at a point s at time t when the source is at x and intensity is I .

Theorem 3 Let the Assumptions of Theorem 1 hold. Additionally, assume that X is bounded and a constant stepsize α is used. Then, with basic quantization the iterates {xk } in (8) satisfy √ nΔ maxx,y∈X x − y α + liminf E[ f (xk )] ≤ f + k→0 2α 2 m ∗ i=1 √ nΔ Ci + νi + 2α 2 , and with dither quantization the iterates satisfy liminf E[ f (xk )] ≤ f ∗ + k→0 √ nΔ maxx,y∈X x − y α + 2α 2 m Ci + νi + i=1 Furthermore, the bounds hold with probability 1 for infk≥0 f (xk ). √ nΔ √ α 6 2 . S. Ram et al. Observe that with dither quantization it is possible to obtain a better bound.

Distributed and Recursive Parameter Estimation 31 xk−1 = z m,k−1 = z 0,k , ψi,k 0 Fi (z i,k−1 ) = ( ) ∇ ( ) Fi (z i,k−1 ) Fi (z i,k−1 ) χi,k (12) + ψi,k−1 K i (z i,k−1 ) r + ( ) ∇ ( ) K i (z i,k−1 ) i,k−1 χi,k−1 G i (z i,k−1 ) u ∇ ( ) G i (z i,k−1 ) i,k ζi,k Hi (z i−1,k ) 0 ( ) = ξi,k ∇ Hi(l) (z i−1,k ) Hi (z i−1,k ) εi,k = ri,k − ζi,k , ( ) ( ) = z i−1,k − αk z i,k ( ) ξi,k (1) (d) . . z i,k z i,k = z i,k T T ψi,k ( ) , χi,k εi,k , . ( ) The initial values for the recursion are fixed at x0 = xs , ψi,0 = ψi,s and χi,0 = ( ) χi,s .

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