! " " ### &' " " ( ) ( * ( & * + # $ % 1 ( - " ) . # # # & / 0 ( $ ( # , 2 $ # Recorded Temperature 30 Current Temperature x1 y0 20 )3 4 3!) 5 4 ) 5! 6 7 10 x0 y1 0 F $ x y 1 2 ( # 9 2 $ # &' Recorded Temperature Bound for Current Temperature 30 y0 20 ) x0 10 0 x F y $ ( # 9 8 2 $ # &' Recorded Temperature ' Bound for Current Temperature 30 ) 20 y0 10 6 x0 0 ; F $ x y : 3 $ = # ! # 5 # % 9 7 # )!<3> !,3> 7 0 / # # $ < 6 @# # 6 D 7 A6 BBC * # A 3,C 7 7 * ' D 7 6 * AD63%C # 7 $ ? 4 D ( 7 9 ( 7 7 ) $ B 2 - ( * / - * - /0 / 0 0 - / 0 $ 55 5 E 2 fi(x,t) – uncertainty pdf Ti.a(t) [li(t) 9) ui(t)] Uncertainty Interval Ui(t) ! ) * A6 BBC $ 5% $ 5 F # ! 7 /A " ! /7 /G % C 0 * 0 H3I 0 ( # 7 $ !# * 7 # ! * 7 $ 5, 6 $ & F & # 9 $ # 9$ 6 ); 6 # ( & ( $ $ 2 $ 2 )$ 6 ,3@; 9FF$ 92 $ 92 )$ 6 ; 6 53@ ; 7 7 $ 92 $ + 0 9 51 5!= * * - 9 /- 0 # / & + 0 $ 58 7 92 $ %! = T4 T3 T2 T1 0 0 J K # -5 -% -, -1 * # $ 5: 9 -% T4 T3 T2 0 l2 l1 l3 l4 l5 x x+dx =-% l3 l2 % ∈A % ,C -% ! f 2 ( x) • (1 − P1 ( x))dx + l3 # l4 l3 l2 f 2 ( x) • ∏ (1 − Pk ( x))dx + k =1, 3 $ l5 l4 f 2 ( x) • (1 − P1 ( x))dx f 2 ( x) • ∏ (1 − P ( x))dx k =1, 3, 4 k 5< 8 $ F # L 7 L "Which sensor, among 4, has the minimum reading?" 100% (assuming only 1 answer exists, if values are known precisely) 100% T2 T3 Sensors T4 25% 25% 0% T1 50% 25% 0% 0% 0% 25% 75% 25% Probability Probability 75% 50% 25% 100% T1 T2 T3 Sensors T4 0% 7 $ 5? ( # $ $ "Is reading of sensor i in range [l,u] ?" 7 L L L L# 7 0 0 0 # # # l 533> 9$ b) M B8>! % M 8>! %/B8> M 83>! %/ Score = u a) c) A 0 C0 | pi − 0.5 | 0.5 Score _ of _ an _ ERQ = $ 1 | pi − 0.5 | | R | i∈R 0.5 5B 9 $ 9& ( $ /5" %0 "Which sensor, among n, has the minimum reading?" MG /- 0I G /-5 ,3>0 /-% 13>0 /-, ,3>0I ' # D ' M A53 %3C 7 2 2 ) FF * 9$ / 0 # " $ $ 9& ( N/O0 %3 $ O /O5 P O # n H (X ) = i =1 / / D p ( X i ) log 2 /%" %0 /O50 P /O 00 1 p( X i ) ∃ ! /O 0 M 5 / 00 OQ % 30 7 ! Score _ of _ Entity _ Aggr _ Query = − H ( R )× | B | / # 0 #/ ' 0 $ %5 10 &( $ "What is the minimum value among n sensors?" ! G /)0 !) ∈ A A53 %3C CI /)0 R A53 %3C u H ( X ) = − p ( x ) log2 p ( x )dx l 2 # O# Score _ of _ Value _ Aggr _ Query = − H ( X ) $ = ( # $ E - %% 7 # 7 7 $ %, 11 E / E S 0 T . / . S T 2 2 T 2) T 0 - # # ) $ 9) # %1 ! 5 5333 # # L 2 L7 $ %8 12 9) 2 ! 3 3 3 # 3 $ R /λ0 533 $ 9 ' # %: 92 $ 0 -0.5 -1 Score -1.5 Glb_RR Loc_RR MinMin MaxUnc -2 -2.5 -3 -3.5 -4 -4.5 200 250 300 350 Bandwidth $ 400 450 500 %< 13 9 ' # - 0.9 0.8 Glb_RR Loc_RR MinMin MaxUnc 0.7 Time (sec) 0.6 0.5 0.4 0.3 0.2 0.1 0 250 300 350 400 450 500 Bandwidth $ 6 %? ! 0 7 0 0 0 ) 7 7 0 7 7 7 0 # # 7 $ %B 14 = &" '( ### " " U )% ( * ### U " R (* ### U " " $ 92 $ ,3 ,!9 T4 T3 T2 T1 x 0 x x+dx . %/)0 -% =-% ∈ A) )V )C -% # %/)0 W/5& 5/)00 ) /)0 $ -5 H -% ,5 15 +) 0) , - ../D 6 $ ( - 2 = </,0 5BBB , '% / $ * 5B = 999 = = 9= ,- D X6 # ' & = (2 = E2D %33% $ 9 ' ,% # 1.4 1.2 1 Uncertainty ) %33, / D + Glb_RR Loc_RR MinMin MaxUnc MinExpEntr 0.8 0.6 0.4 0.2 0 200 250 300 350 400 450 500 Bandwidth $ ,, 16
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