Propagation of Interval and Probabilistic ... - Semantic Scholar
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Propagation of Interval and Probabilistic Uncertainty in Cyberinfrastructure-Related Data Processing and Data Fusion Christian Servin Computational Science Program University of Texas at El Paso El Paso, Texas 79968, USA [email protected]
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1.
Need for Data Processing and Data Fusion
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• For many quantities y, it is not easy (or even impossible) to measure them directly. • Instead, we measure related quantities x1 , . . . , xn , and use the known relation y = f (x1 , . . . , xn ) to estimate y.
Propagation of . . . What We Do in This . . . Chapter 2: Towards . . . Chapter 3: Towards . . . Chapter 4: Towards . . .
• Such data processing is especially important for cyberinfrastructure-related heterogenous data. • Example of heterogenous data – geophysics: – first-arrival passive (from actual earthquakes) and active seismic data (from seismic experiments); – gravity data; – surface waves, etc. • Before we start processing data, we need to first fuse data points corresponding to the same quantity.
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2.
Need to Take Uncertainty into Consideration
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• The result x e of a measurement is usually somewhat different from the actual (unknown) value x.
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• Usually, the manufacturer of the measuring instrument (MI) gives us a bound ∆ on the measurement error:
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def
|∆x| ≤ ∆, where ∆x = x e−x • Once we know the measurement result x e, we can conclude that the actual value x is in [e x − ∆, x e + ∆]. • In some situations, we also know the probabilities of different values ∆x ∈ [−∆, ∆]. • In this case, we can use statistical techniques. • However, often, we do not know these probabilities; we def only know that x is in the interval x = [e x − ∆, x e + ∆]. • In this case, we need to process this interval data.
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3.
Measurement Uncertainty: Traditional Approach
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def
• Usually, a meas. error ∆x = x e − x is subdivided into random and systematic components ∆x = ∆xs + ∆xr : – the systematic error component ∆xs is usually defined as the expected value ∆xs = E[∆x], while – the random error component is usually defined as def the difference ∆xr = ∆x − ∆xs . • The random errors ∆xr corresponding to different measurements are usually assumed to be independent. • For ∆xs , we only know the upper bound ∆s s.t. |∆xs | ≤ ∆s , i.e., that ∆xs is in the interval [−∆s , ∆s ]. • Because of this fact, interval computations are used for processing the systematic errors. • ∆xr is usually characterized by the corr. probability distribution (usually Gaussian, with known σ).
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4.
Expert Estimates and Fuzzy Data
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• There is no guarantee of expert’s accuracy. • We can only provide bounds which are valid with some degree of certainty.
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• This degree of certainty is usually described by a number from the interval [0, 1]. • So, for each β ∈ [0, 1], we have an interval x(α) containing the actual value x with certainty α = 1 − β. • The larger certainty we want, the broader should the corresponding interval be. • So, we get a nested family of intervals corresponding to different values α. • Alternative: for each x, describe the largest α for which x is in x(α); this αlargest is a membership function µ(x).
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5.
How to Propagate Uncertainty in Data Processing
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• We know that y = f (x1 , . . . , xn ).
What We Do in This . . .
• We estimate y based on the approximate values x ei as ye = f (e x1 , . . . , x en ).
Chapter 2: Towards . . .
• Since x ei 6= xi , we get ye 6= y; it is desirable to estimate def the approximation error ∆y = ye − y. • Usually, measurements are reasonably accurate, i.e., def ei − xi are small. measurement errors ∆xi = x • Thus, we can keep only linear terms in Taylor expann P ∂f sion: ∆y = Ci · ∆xi , where Ci = . ∂x i i=1 n P • For systematic error, we get a bound |Ci | · ∆si . i=1
• For random error, we get σ 2 =
n P
Ci2 · σi2 .
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6.
How to Propagate Uncertainty in Data Fusion: Case of Probabilistic Uncertainty
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• Reminder: we have several estimates x e(1) , . . . , x e(n) of the same quantity x. • Data fusion: we combine these estimates into a single estimate x e. (i) def
(i)
• Case: each estimation error ∆x = x e −x is normally distributed with 0 mean and known st. dev. σ (i) . • How to combine: use Least Squares, i.e., find x e that (i) 2 n P (e x −x e) minimizes ; (i) )2 2 · (σ i=1 n P x e(i) · (σ (i) )−2 • Solution: x e = i=1 P . n (i) −2 (σ ) i=1
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7.
Data Fusion: Case of Interval Uncertainty
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• In some practical situations, the value x is known with interval uncertainty. • This happens, e.g., when we only know the upper bound ∆(i) on each estimation error ∆x(i) : |∆x(i) | ≤ ∆i .
Propagation of . . . What We Do in This . . . Chapter 2: Towards . . . Chapter 3: Towards . . . Chapter 4: Towards . . .
• In this case, we can conclude that |x − x e(i) | ≤ ∆(i) , i.e., def that x ∈ x(i) = [e x(i) − ∆(i) , x e(i) + ∆(i) ]. • Based on each estimate x e(i) , we know that the actual value x belongs to the interval x(i) . • Thus, we know that the (unknown) actual value x belongs to the intersection of these intervals: def
x=
n \ i=1
x(i) = [max(e x(i) − ∆(i) ), min(e x(i) + ∆(i) )].
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8.
Propagation of Uncertainty: Challenges
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• In the ideal world: – we should have an accurate description of data uncertainty;
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– based on this description, we should use well-justified and efficient algorithms to propagate uncertainty.
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• In practice, we are often not yet in this ideal situation:
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– the description of uncertainty is often only approximate,
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– the algorithms for uncertainty propagation are often heuristics, i.e., not well-justified, and
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– the algorithms for uncertainty propagation are often not very computationally efficient.
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9.
What We Do in This Thesis
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• In Chapter 2, we show that the traditional idea of random and systematic components is an approximation: – we also need periodic components;
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– this is important in environmental studies. • In Chapter 3, on the example of a fuzzy heuristic, we show how a heuristic can be formally justified. • In Ch. 4, we show how to process more efficiently; e.g.:
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– first, we process data type-by-type;
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– then, we fuse the resulting models.
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• All these results assume that we have a good description of the uncertainty of the original data.
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• In practice, we often need to extract this information from the data; these are our future plans (Ch. 5).
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10.
Chapter 2: Towards More Accurate Description of Uncertainty
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• Often, the differences r = ∆x − s corr. to nearby times are strongly correlated.
What We Do in This . . .
• For example, meteorological sensors may have daytime or nighttime biases, or winter and summer biases.
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• To capture this correlation, environmental scientists proposed a semi-heuristic 3-component model of ∆x.
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• In this model, the difference ∆x − ∆xs is represented as a combination of: – a “truly random” error ∆xr (which is independent from one measurement to another), and – a new “periodic” component ∆xp . • We provide a theoretical explanation for this heuristic three-component model.
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11.
Error Components: Analysis
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• We want to represent measurement error ∆x(t) as a linear combination of several components.
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• We consider the most detailed level of granularity, w/each component determined by finitely many parameters ci .
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• Each component is thus described by a finite-dimensional linear space L = {c1 · x1 (t) + . . . + cn · xn (t) : c1 , . . . , cn ∈ IR}. • In most applications, signals are smooth and bounded, so we assume that xi (t) is smooth and bounded. • Finally, for a long series of observations, we can choose a starting point arbitrarily: t → t + t0 .
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• It is reasonable to require that this change keeps us within the same component, i.e., x(t) ∈ L ⇒ x(t + t0 ) ∈ L.
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12.
Error Components: Main Result
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• A function x(t) of one variable is called bounded if
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∃M ∀t (|x(t)| ≤ M ). • We say that a class F of functions of one variable is shift-invariant if
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∀x(t) (x(t) ∈ F ⇒ ∀t0 (x(t + t0 ) ∈ F )). • By an error component we mean a shift-invariant finitedimensional linear space of functions L = {c1 · x1 (t) + . . . + cn · xn (t) : ci ∈ IR}.
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• Theorem: Every error component is a linear combination of the functions x(t) = sin(ω · t) and x(t) = cos(ω · t).
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13.
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Proof
Measurement . . .
• Shift-invariance means that, for some ci (t0 ), we have
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xi (t + t0 ) = ci1 (t0 ) · x1 (t) + . . . + cin (t0 ) · xn (t).
Chapter 2: Towards . . .
• For n different values t = t1 , . . . , t = tn , we get a system of n linear equations with n unknowns cij (t0 ).
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• The Cramer’s rule solution to linear equations is a smooth function of all the coeff. & right-hand sides.
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• Since all the right-hand sides xi (tj +t0 ) and coefficients xi (tj ) are smooth, cij (t0 ) are also smooth. def
• Differentiating w.r.t. t0 and taking t0 = 0, for cij = c˙ij (0), we get
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x˙ i (t) = ci1 · x1 (t) + . . . + cin · xn (t). Full Screen
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14.
Proof (cont-d)
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• Reminder: x˙ i (t) = ci1 · x1 (t) + . . . + cin · xn (t). • A general solution of such system of equations is a linear combination of functions
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k
t · exp(λ · t), w/k ∈ N, k ≥ 0, λ = a + i · ω ∈ C.
Chapter 4: Towards . . . Chapter 5: Towards . . .
• Here, exp(λ · t) = exp(a · t) · cos(ω · t) + i · exp(a · t) · sin(ω · t). • When a 6= 0, we get unbounded functions for t → ∞ or t → −∞. • So, a = 0. • For k > 0, we get unbounded tk ; so, k = 0. • Thus, we indeed have a linear combination of sinusoids.
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15.
Practical Conclusions
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• Let f be the measurements frequency (how many measurements we perform per unit time).
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• When ω f , the values cos(ω · t) and sin(ω · t) practically do not change with time.
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• Indeed, the change period is much larger than the usual observation period. • Thus, we can identify such low-frequency components with systematic error component. • When ω f , the phases of the values cos(ω · ti ) and cos(ω · ti+1 ) differ a lot. • For all practical purposes, the resulting values of cosine or sine functions are independent. • Thus, high-frequency components can be identified with random error component.
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Practical Conclusions (cont-d)
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• Result: every error component is a linear combination of cos(ω · t) and sin(ω · t). • Notation: let f be the measurements frequency (how many measurements we perform per unit time).
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• Reminder: – we can identify low-frequency components (ω f ) with systematic error component; – we can identify high-frequency ones (ω f ) with random error component. • Easy to see: all other error components cos(ω · t) and sin(ω · t) are periodic.
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• Conclusion: we have indeed justified to the semi-empirical 3-component model of measurement error.
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17.
Chapter 3: Towards Justification of Heuristic Techniques for Processing Uncertainty
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• As we have mentioned, some methods for processing uncertainty are heuristic.
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• Such methods lack justification and are, therefore, less reliable.
Chapter 3: Towards . . .
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• Usually, techniques for processing interval and probabilistic uncertainty are well-justified. • However, many techniques for processing expert (fuzzy) data are still heuristic.
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• In Chapter 3: Page 18 of 65
– we consider a practically efficient heuristic fuzzy technique for decision making under uncertainty; – we show how this heuristic can be formally justified.
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18.
Traditional Approach to Decision Making: Reminder
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• The quality of each possible alternative is characterized by the values of several quantities.
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• For example, when we buy a car, we are interested in its cost, its energy efficiency, its power, size, etc.
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• For each of these quantities, we usually have some desirable range of values. • Often, there are several different alternatives all of which satisfy all these requirements. • The traditional approach assumes that there is an objective function that describes the user’s preferences. • We then select an alternative with the largest possible value of this objective function.
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Traditional Approach to Decision Making: Limitations
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• The traditional approach to decision making assumes: – that the user knows exactly what he or she wants — i.e., knows the objective function – and
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– that the user also knows exactly what he or she will get as a result of each possible decision. • In practice, the user is often uncertain: – the user is often uncertain about his or her own preferences, and – the user is often uncertain about possible consequences of different decisions. • It is therefore desirable to take this uncertainty into account when we describe decision making.
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20.
Fuzzy Target Approach (Huynh-Nakamori)
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• For each numerical characteristic of a possible decision, we form two fuzzy sets: – µi (x) describing the users’ ideal value; – µa (x) describing the users’ impression of the actual value.
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• For example, a person wants a well done steak, and the steak comes out as medium well done. • In this case, µi (x) corresponds to “well done”, and µa (x) to “medium well done”. • The simplest “and”-operation (t-norm) is min(a, b); so, the degree to which x is both actual and desired is min(µa (x), µi (x)). • The degree to which there exists x which is both possible and desired is d = max min(µa (x), µi (x)).
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Fuzzy Target Approach: Successes and Remaining Problems
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• The above approach works well in many applications. • Example: it predicts how customers select a handcrafted souvenir when their ideal ones is not available.
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• Problem: this approach is heuristic, it is based on selecting: – the simplest possible membership function and – the simplest possible “and”- and “or”-operations. • Interestingly, we get better predictions than with more complex membership functions and “and”-operations. • In this section, we provide a justification for the above semi-heuristic target-based fuzzy decision procedure.
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22.
Chapter 4: Towards More Computationally Efficient Techniques for Processing Uncertainty
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• Fact: computations often take a lot of time. • One of the main reasons: we process a large amount of data.
What We Do in This . . . Chapter 2: Towards . . . Chapter 3: Towards . . . Chapter 4: Towards . . .
• So, a natural way to speed up data processing is:
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– to divide the data into smaller parts,
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– to process each smaller part separately, and then
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– to combine the results of data processing. • In particular, when we are processing huge amounts of heterogenous data, it makes sense: – to first process different types of data type-by-type, – and then to fuse the resulting models. • This idea is explored in the first sections of Chapter 4.
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23.
Data Fusion under Interval Uncertainty: Reminder
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• Frequent practical situation:
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– we are interested in a quantity u;
Chapter 3: Towards . . .
– we have several measurements and/or expert estimates u1 , . . . , un of u. • Objective: fuse these estimates into a single more accurate estimate. • Interval case: each ui is known with interval uncertainty. • Formal description: for each i, we know the interval ui = [ui − ∆i , ui + ∆i ] containing u. n def T • Solution: u belongs to the intersection u = ui of
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i=1
these intervals.
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24.
Data Fusion under Probabilistic Uncertainty: Reminder
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def
• Probabilistic uncertainty: each measurement error ∆ui = ui − u is normally distributed w/0 mean and known σi .
What We Do in This . . .
• Technique: the Least Squares Method (LSM) n X (u − ui )2 → min . 2 2σ i i=1
Chapter 3: Towards . . .
• Resulting estimate: is i=1
ui · σi−2
n P i=1
. σi−2
• Standard deviation: σ2 = P n i=1
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n P
u=
Chapter 2: Towards . . .
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1 σi−2
,
with σ 2 σi2 .
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25.
New Problem: Different Resolution
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• Traditional data fusion: fusing measurement results with different accuracy. • Additional problem: different measurements also have different resolution.
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• Case study – geosciences: estimating density u1 , . . . , un at different locations and depths. • Examples of different geophysical estimates: – Seismic data leads to higher-resolution estimates u e1 , . . . , u en of the density values. – Gravity data leads to lower-resolution estimates, i.e., estimates u e for the weighted average u=
n X i=1
wi · ui .
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26.
Why This Is Important
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• Reminder: there are many sources of data for Earth models: – first-arrival passive seismic data (from the actual earthquakes), – first-arrival active seismic data (from the seismic experiments), – gravity data, – surface waves, etc. • At present: each of these datasets is processed separately, resulting in several different Earth models. • Fact: these models often provide complimentary geophysical information. • Idea: all these models describe the properties of the same Earth, so it is desirable to combine them.
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27.
New Idea: Model Fusion
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• Objective: to combine the information contained in multiple complementary datasets. • Ideal approach: it is desirable to come up with techniques for joint inversion of these datasets.
Propagation of . . . What We Do in This . . . Chapter 2: Towards . . . Chapter 3: Towards . . . Chapter 4: Towards . . .
• Problem: designing such joint inversion techniques is an important theoretical and practical challenge. • Status: such joint inversion methods are being developed. • Practical question: what to do while these methods are being developed?
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• Our practical solution: fuse the Earth models coming from different datasets.
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28.
Need to Take . . .
Model Fusion: Statistical Case
Measurement . . .
• Objective: find the values u1 , . . . , un of the desired quantity in different spatial cells.
Propagation of . . . What We Do in This . . . Chapter 2: Towards . . .
• Geophysical example: ui is the density at different
Chapter 3: Towards . . .
1 km × 1 km × 1 km cells.
Chapter 4: Towards . . . Chapter 5: Towards . . .
• Input: we have
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– high-resolution measurements, i.e., values u ei ≈ ui with st. dev. σi ; (k)
– lower-resolution measurements, i.e., values u e responding to blocks of neighboring cells: X (k) u e(k) ≈ wi · ui , with st. dev. σ (k) .
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cor-
i
• Additional information: a lower-resolution measurement result is representative of values within the block.
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29.
Model Fusion: Statistical Case (cont-d)
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(k)
• Formal description: when wi with st. dev. δ (k) .
6= 0, we have u e(k) ≈ ui ,
• How to estimate δ (k) : as the empirical st. dev. within the block.
Propagation of . . . What We Do in This . . . Chapter 2: Towards . . . Chapter 3: Towards . . . Chapter 4: Towards . . .
• High-resolution values (reminder): u ei ≈ ui w/st. dev. σi . • Lower-resolution values (reminder): X (k) (k) u e ≈ wi · ui , with st. dev. σ (k) . i
• LSM Solution: minimize the sum P (k) (k) (e u − wi · ui )2 2 XX (k) 2 X X (ui − u ei ) (ui − u e ) i + + . 2 (k) 2 (k) )2 σ (δ ) (σ i i i k k • How: differentiating w.r.t. ui , we get a system of linear equations with unknowns u1 , . . . , un .
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30.
Need to Take . . .
Model Fusion: Interval Case
Measurement . . .
• Quantities of interest: values u1 , . . . , un of the desired quantity in different spatial cells.
Propagation of . . .
• Objective: find the ranges u1 , . . . , un of possible values of u1 , . . . , un .
Chapter 2: Towards . . .
What We Do in This . . .
Chapter 3: Towards . . . Chapter 4: Towards . . .
• High-resolution measurements: values u ei ≈ ui with bound ∆i : u ei − ∆i ≤ ui ≤ u ei + ∆i .
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(k)
• Lower-resolution measurements: values u e corresponding to blocks of neighboring cells: X (k) u e(k) ≈ wi · ui , with bound ∆(k) . i
• Resulting constraint: (k)
u e
(k)
−∆
≤
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X i
(k)
wi · ui ≤ u e(k) + ∆(k) .
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31.
Model Fusion: Interval Case (cont-d)
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• Additional information: a priori bounds on ui :
Propagation of . . . What We Do in This . . .
ui ≤ ui ≤ ui . • Additional information: a priori bounds on the changes between neighboring cells: −δij ≤ ui − uj ≤ δij . • High-resolution measurements (reminder): u ei − ∆i ≤ ui ≤ u ei + ∆i . • Lower-resolution measurements (reminder): X (k) u e(k) − ∆(k) ≤ wi · ui ≤ u e(k) + ∆(k) . i
• Objective: minimize and maximize each ui under these constraints.
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Model Fusion: Interval Solution
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• Problem. Minimize (Maximize) ui under the following constraints:
Propagation of . . . What We Do in This . . . Chapter 2: Towards . . .
• ui ≤ ui ≤ ui .
Chapter 3: Towards . . .
• −δij ≤ ui − uj ≤ δij . •u ei − ∆i ≤ ui ≤ u ei + ∆i . P (k) •u e(k) − ∆(k) ≤ wi · ui ≤ u e(k) + ∆(k) .
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i
• Current solution method: linear programming. • Objective: provide more efficient algorithms for specific geophysical cases. • Preliminary results: some such algorithms have been developed.
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33.
Preliminary Experiments
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• What we have done: preliminary proof-of-concept experiments. • Simplifications:
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– equal weights wi ;
Chapter 4: Towards . . .
– simplified datasets.
Chapter 5: Towards . . .
• Conclusion: the fused model improves accuracy and resolution of different Earth models.
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34.
Chapter 5: Towards Better Ways of Extracting Information About Uncertainty from Data
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• Previous methods assume that we have a good description of the uncertainty.
What We Do in This . . .
• In practice, often, we do not have this information.
Chapter 3: Towards . . .
Chapter 2: Towards . . .
Chapter 4: Towards . . .
• We need to extract uncertainty information from the data. • In Chapter 5, we propose ideas on how this uncertainty information can be extracted from the data. • These ideas constitute our future work.
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35.
Extracting Uncertainty from Data: What Is Known
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• Traditional approach: use a “standard” (more accurate) measuring instrument SMI.
What We Do in This . . .
• Idea: values x eS measured by SMI are accurate: x eS ≈ x, def so x e−x eS ≈ ∆x = x e − x.
Chapter 3: Towards . . .
• Limitation: for cutting-egde measurements, we do not have more accurate instruments, these are the best. • Example: the Eddy convariance tower provides the best estimates for Carbon flux. • Idea: if we have two similar measuring instruments, we can estimate ∆x(1) − ∆x(2) as x e(1) − x e(2) . • If both error are normally distributed with st. dev. σ, then ∆x(1) − ∆x(2) is also normal, with variance 2σ 2 . • So, we can determine σ from observations.
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36.
Extracting Uncertainty from Data: Remaining Problem
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• Problem: if distribution of ∆x(i) is skewed, we cannot distinguish between two distributions: – the distribution of ∆x(i) , and – its mirror image, the distribution of −∆x(i) . • Our preliminary results: – this is the only non-uniquness, and – modulo this non-uniqueness, we can effectively reconstruct the distribution of ∆x(i) .
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37.
Summary: Main Problem
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• In the ideal world: – we should have an accurate description of data uncertainty;
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– based on this description, we should use well-justified and efficient algorithms to propagate uncertainty.
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• In practice, we are often not yet in this ideal situation:
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– the description of uncertainty is often only approximate,
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– the algorithms for uncertainty propagation are often heuristics, i.e., not well-justified, and
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– the algorithms for uncertainty propagation are often not very computationally efficient.
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38.
Summary: Conclusions
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• In Ch. 2, we showed that the traditional idea of random and systematic components is an approximation: – we also need periodic components;
Propagation of . . . What We Do in This . . . Chapter 2: Towards . . . Chapter 3: Towards . . .
– this is important in environmental studies. • In Chapter 3, on the example of a fuzzy heuristic, we showed how a heuristic can be formally justified. • In Ch. 4, we showed how to be more efficient; e.g.:
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– first, we process data type-by-type;
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– then, we fuse the resulting models.
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• All these results assume that we have a good description of the uncertainty of the original data.
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• In practice, we often need to extract this information from the data; these are our future plans (Ch. 5).
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39.
Acknowledgments
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I would like to express my deep-felt gratitude: • to my mentor Dr. Vladik Kreinovich;
Propagation of . . . What We Do in This . . . Chapter 2: Towards . . .
• to members of my committee, Dr. Aaron Velasco, Dr. Scott Starks, and Dr. Luc Longpr´e; • to Dr. Benjamin C. Flores, and to the Alliance for Minority Participation Bridge to the Doctorate program; • to Dr. Craig Tweedie for his suggestions, comments, and guidance in this work; • to Dr. Aaron Velasco, Dr. Vanessa Lougheed, Dr. William Robertson, and to all the GK-12 program staff; and • last but not the least, to all the faculty and staff of the Computational Science Program.
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40.
Publications
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• L. Longpr´e, C. Servin, and V. Kreinovich, “Quantum computation techniques for gauging reliability of interval and fuzzy data”, International Journal of General Systems, 2011, Vol. 40, No. 1, pp. 99–109.
Propagation of . . .
• O. Ochoa, A. A. Velasco, V. Kreinovich, and C. Servin, “Model fusion: a fast, practical alternative towards joint inversion of multiple datasets”, Abstracts of the Annual Fall Meeting of the American Geophysical Union AGU’08, San Francisco, California, December 15–19, 2008.
Chapter 4: Towards . . .
• O. Ochoa, A. A. Velasco, C. Servin, and V. Kreinovich, “Model Fusion under Probabilistic and Interval Uncertainty, with Application to Earth Sciences”, International Journal of Reliability and Safety, 2012, Vol. 6, No. 1–3, pp. 167–187.
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Publications (cont-d)
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• C. Servin, O. Ochoa, and A. A. Velasco, “Probabilistic and interval uncertainty of the results of data fusion, with application to geosciences”, Abstracts of 13th International Symposium on Scientific Computing, Computer Arithmetic, and Verified Numerical Computations SCAN’2008, El Paso, Texas, September 29 – October 3, 2008, p. 128.
Propagation of . . .
• C. Servin, V.-N. Huyhn, and Y. Nakamori, “Semiheuristic target-based fuzzy decision procedures: towards a new interval justification”, Proceedings of the Annual Conference of the North American Fuzzy Information Processing Society NAFIPS’2012, Berkeley, California, August 6–8, 2012.
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42.
Publications (cont-d)
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• C. Servin, C. Tweedie, and A. Velasco, “Towards a more realistic treatment of uncertainty in Earth and environmental sciences: beyond a simplified subdivision into interval and random components”, Abstracts of the 15th GAMM-IMACS International Symposium on Scientific Computing, Computer Arithmetic, and Verified Numerical Computation SCAN’2012, Novosibirsk, Russia, September 23–29, 2012, pp. 164–165. • C. Servin, M. Ceberio, A. Jaimes, C. Tweedie, and V. Kreinovich, “How to describe and propagate uncertainty When Processing time series: metrological and computational challenges, with potential applications to environmental studies”, In: S.-M. Chen and W. Pedrycz (eds.), Time Series Analysis, Modeling and Applications: A Computational Intelligence Perspective, Springer Verlag, to appear.
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43.
Appendix to Chapter 2: How to Propagate Uncertainty in the Three-Component Model
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• We are interested in the quantity
What We Do in This . . . Chapter 2: Towards . . .
y = f (x1 (t11 ), x1 (t12 ), . . . , x2 (t21 ), x2 (t22 ), . . . , xn (tn1 ), xn (tn2 ), . . .).Chapter 3: Towards . . . • Instead of the actual values xi (tij ), we only know the measurement results x ei (tij ) = xi (tij ) + ∆xi (tij ). • Measurement errors are usually small, so terms quadratic (and higher) in ∆xi (tij ) can be safely ignored. • For example, if the measurement error is 10%, its square is 1% which is much much smaller than 10%. • If the measurement error is 1%, its square is 0.01% which is much much smaller than 1%. • Thus, we can safely linearize the dependence of ∆y on ∆xi (tij ).
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How to Propagate Uncertainty (cont-d)
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• Reminder: we can safely linearize the dependence of ∆y on ∆xi (tij ), so XX ∂y def ∆y = Cij · ∆xi (tij ), with Cij = . ∂x (t ) i ij i j P • In general, ∆xi (tij ) = si +rij + ` A`i ·cos(ω` ·tij +ϕ`i ). P • Due to linearity, we have ∆y = ∆ys + ∆yr + ` ∆yp` , where XX XX ∆ys = Cij · si ; ∆yr = Cij · rij ; i
∆yp` =
j
i
XX i
j
Cij · A`i · cos(ω` · tij + ϕ`i ).
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j
• We know: how to compute ∆ys and ∆yr . • What is needed: propagation of the periodic component.
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45.
Propagating Periodic Component: Analysis
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• Reminder: for each component, we have XX ∆yp` = Cij · A`i · cos(ω` · tij + ϕ`i ). i
j
Propagation of . . . What We Do in This . . . Chapter 2: Towards . . . Chapter 3: Towards . . .
• It is reasonable to assume that different phrases ϕ`i are independent (and uniformly distributed). • Thus, by the Central Limit Theorem, the distribution of ∆yp` is close to normal, with 0 mean. 1 X 2 • The variance of ∆yp` is · A`i · (Kci2 + Ksi2 ). 2 i • Each amplitude A`i can take any value from 0 to the known bound P`i . 1 X 2 • Thus, the variance is bounded by · P`i ·(Kci2 +Ksi2 ). 2 i • So, we arrive at the following algorithm.
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46.
Propagating Periodic-Induced Component: Algorithm
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• First, we apply the algorithm f to the measurement results x ei (tij ) and get the estimate ye.
What We Do in This . . .
• Then, we select a small value δ and for each sensor i, we do the following:
Chapter 3: Towards . . .
(ci)
– take xi (tij ) = x ei (tij ) + δ · cos(ω` · tij ) for all moments j; (ci) – for other sensors i0 6= i, take xi0 (ti0 j ) = x ei (ti0 j ); (ci)
– substitute the resulting values xi0 (ti0 j ) into the data processing algorithm f and get the result y (ci) ; (si) – then, take xi (tij ) = x ei (tij ) + δ · sin(ω` · tij ) for all moments j; (si) – for all other i0 6= i, take xi0 (ti0 j ) = x ei (ti0 j ); (si)
– substitute the resulting values xi0 (ti0 j ) into the data processing algorithm f and get the result y (si) .
Chapter 2: Towards . . .
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47.
Algorithm (cont-d)
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• Reminder:
Propagation of . . . What We Do in This . . .
– First, we apply the algorithm f to the measurement results x ei (tij ) and get the estimate ye.
Chapter 2: Towards . . .
– Then, for each sensor i, we simulate cosine terms and get the results y (ci) .
Chapter 4: Towards . . .
– Third, for each sensor i, we simulate sine terms and get the results y (si) . • Finally, we estimate the desired bound σp` on the standard deviation of ∆yp` as v u (ci) 2 (si) 2 ! u1 X y − ye y − ye σp` = t · P`i2 · + . 2 i δ δ
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48.
Appendix to Chapter 3
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• We know: – a fuzzy set µi (x) describing the users’ ideal value; – the fuzzy set µa (x) describing the users’ impression of the actual value.
Propagation of . . . What We Do in This . . . Chapter 2: Towards . . . Chapter 3: Towards . . . Chapter 4: Towards . . .
• For crisp sets, the solution is possibly satisfactory if some of the possibly actual values is also desired. • In the fuzzy case, we can only talk about the degree to which the proposed solution can be desired. • A possible decision is satisfactory if either: – the actual value is x1 , and this value is desired, – or the actual value is x2 , and this value is desired, – ... • Here x1 , x2 , . . . , go over all possible values of the desired quantity.
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Derivation of the d-Formula (cont-d)
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• For each value xk , we know: – the degree µa (xk ) with which this value is actual, and – the degree µi (xk ) to which this value is desired. • Let us use min(a, b) to describe “and” – the simplest possible choice of an “and”-operation. • Then we can estimate the degree to which the value xk is both actual and desired as min(µa (xk ), µi (xk )). • Let us use max(a, b) to describe “or” – the simplest possible choice of an “or”-operation.
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• Then, we can estimate the degree d to which the two fuzzy sets match as d = max min(µa (x), µi (x)).
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Fuzzy Target Approach: How Are Membership Functions Elicited?
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• In many applications (e.g., in fuzzy control), the shape of a membership function does not affect the result.
What We Do in This . . .
• Thus, it is reasonable to use the simplest possible membership functions – symmetric triangular ones.
Chapter 3: Towards . . .
Chapter 2: Towards . . .
Chapter 4: Towards . . . Chapter 5: Towards . . .
• To describe a symmetric triangular function, it is sufficient to know the support [x, x] of this function.
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• So, e.g., to get the membership function µi (x) describing the desired situation: – we ask the user for all the values a1 , . . . , an which, in their opinion, satisfy the requirement; – we then take the smallest of these values as a and the largest of these values as a; – finally, we form symmetric triangular µi (x) whose support is [a, a].
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Analyzing the Problem
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• Reminder: all we elicit from the experts is two intervals: – an interval [a, a] = [e a − ∆a , e a + ∆a ] describing the set of all desired values, and – an interval [b, b] = [eb − ∆b , eb + ∆b ] describing the set of all the values which are possible. • Based on these intervals, we build triangular membership functions µi (x) and µa (x) centered in e a and eb. • For these membership functions, |eb − e a| d = max min(µa (x), µi (x)) = 1 − . x ∆a + ∆b • This is the formula that we need to justify.
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52.
Our Main Idea
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• If we knew the exact values of a and b, then we would conclude a = b, a < b, or b < a. • In reality, we know the values a and b with uncertainty.
Propagation of . . . What We Do in This . . . Chapter 2: Towards . . . Chapter 3: Towards . . .
• Even if the actual values a and b are the same, we may get approximate values which are different. • It is reasonable to assume that if the actual values are the same, then Prob(a > b) = Prob(b > a), i.e., Prob(a > b) = 1/2. • If the probabilities that a > b and that a < b differ, this is an indication that the actual value differ. • Thus, it’s reasonable to use |Prob(a > b)−Prob(b > a)| as the degree to which a and b may be different.
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53.
How To Estimate Prob(a > b) and Prob(b > a)
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• If we knew the exact values of a and b, then we could def check a > b by comparing r = a − b with 0. • In real life, we only know a and b with interval uncertainty, i.e., we only know that a ∈ [e a − ∆a , e a + ∆a ] and b ∈ [eb − ∆b , eb + ∆b ]. • In this case, we only know the range r of possible values of r = a − b; interval arithmetic leads to r = [(e a − eb) − (∆a + ∆b ), (e a − eb) + (∆a + ∆b )]. • We do not have any reason to assume that some values from r are more probable and some are less probable. • It is thus reasonable to assume that all the values from r are equally probable, i.e., r is uniformly distributed. • This argument is widely used in data processing; it is called Laplace Principle of Indifference.
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How To Estimate Probabilities (cont-d)
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• We estimate Prob(a > b) as Prob(a − b > 0). • We estimate Prob(a < b) as Prob(a − b < 0).
Propagation of . . . What We Do in This . . . Chapter 2: Towards . . .
• We assumed that r = a − b is uniformly distributed on [(e a − eb) − (∆a + ∆b ), (e a − eb) + (∆a + ∆b )]. • We can compute Prob(a − b > 0), Prob(a − b < 0), and |e a − eb| . |Prob(a > b) − Prob(b > a)| = ∆a + ∆b |eb − e a| • Since d = 1 − , we get ∆a + ∆b d = 1 − |Prob(a > b) − Prob(b > a)|.
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• We have produced a new justification for the d-formula.
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• This justification that does not use any simplifying assumptions about memb. f-s and/or “and”-operations.
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55.
Appendix to Chapter 4: Towards More Efficient Uncertainty Processing
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• Even after all algorithmic speed-ups are implemented, the computation time is still often too long.
What We Do in This . . .
• In such situations, the only remaining way to speed up computations is to speed up hardware.
Chapter 3: Towards . . .
Chapter 2: Towards . . .
Chapter 4: Towards . . . Chapter 5: Towards . . .
• Such ideas range from available (e.g., parallelization) to futuristic (e.g., quantum computing). • Parallelization has been largely well-researched. • The use of futuristic techniques in uncertainty estimation is still largely an open problem.
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• In the last section of Ch. 4, we show how quantum computing can be used to speed up computations.
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56.
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Reliability of Interval Data
Measurement . . .
• Usual assumption: all measuring instruments (MI) functioned correctly.
Propagation of . . .
• Conclusion: the resulting intervals [e x − ∆, x e + ∆] contain the actual value x.
Chapter 2: Towards . . .
What We Do in This . . .
Chapter 3: Towards . . . Chapter 4: Towards . . .
• In practice: a MI can malfunction, producing way-off values (outliers). • Problem: outliers can ruin data processing. • Example: average temperature in El Paso 95 + 100 + 105 – based on measurements, = 100. 3 95 + 100 + 105 + 0 – with outlier, = 75. 4 • Natural idea: describe the probability p of outliers.
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def
• Solution: out of n results, dismiss k = p · n largest values and k smallest.
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57.
Need to Gauge the Reliability of Interval Data
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• Ideal case: all measurements of the same quantity are correct.
Propagation of . . .
• Fact: resulting intervals x(1) , . . . , x(n) contain the same (actual) value x. n T • Conclusion: x(i) 6= ∅.
Chapter 2: Towards . . .
i=1
• Reality: we have outliers far from x, so
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n T
x(i) = ∅.
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i=1
• Expectation: out of n given intervals, ≥ n − k are correct – and hence have a non-empty intersection. • Conclusion: – to check whether our estimate p for reliability is correct, – we must check whether out of n given intervals, n − k have a non-empty intersection.
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58.
Need to Gauge Reliability of Interval Data: Multi-D Case
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• In practice, a measuring instrument often measure several different quantities x1 , . . . , xd .
What We Do in This . . .
• Due to uncertainty, after the measurement, for each quantity xi , we have an interval xi of possible values.
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• So, the set of all possible values of the tuple x = (x1 , . . . , xd ) is a box X = x1 ×. . .×xd = {(x1 , . . . , xd ) : x1 ∈ x1 , . . . , xd ∈ xd }. • Thus: – to check whether our estimate p for reliability is correct, – we must be able to check whether out of n given boxes, n − k have a non-empty intersection.
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59.
Resulting Computational Problem: Box Intersection Problem
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Thus, both in the interval and in the fuzzy cases, we need to solve the following computational problem:
What We Do in This . . . Chapter 2: Towards . . . Chapter 3: Towards . . .
• Given:
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• integers d, n, and k; and
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• n d-dimensional boxes (j)
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(j)
(j) X (j) = [x1 , x1 ] × . . . × [x(j) n , xn ], (j)
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(j)
j = 1, . . . , n, with rational bounds xi and xi .
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– we can select n − k of these n boxes – in such a way that the selected boxes have a nonempty intersection.
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60.
Results
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• First result: in general, the above computational problem is NP-hard. • Meaning: no algorithm is possible that solves all particular cases of this problem in reasonable time.
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• In practice: the number of d of quantities measured by a sensor is small: e.g., – a GPS sensor measures 3 spatial coordinates; – a weather sensor measures (at most) 5: ∗ temperature, ∗ atmospheric pressure, and ∗ the 3 dimensions of the wind vector.
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• Second result: for a fixed dimension d, we can solve the above problem in polynomial time O(nd ).
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61.
Algorithm: Description and Need for Speed Up
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• Lemma: if a set of boxes has a common point, then there is another common vector whose all components are endpoints.
What We Do in This . . .
• Proof: move to an endpoint in each direction.
Chapter 4: Towards . . .
Chapter 2: Towards . . . Chapter 3: Towards . . .
Chapter 5: Towards . . .
• Number of endpoints: n intervals have ≤ 2n endpoints. • Bounds on computation time: we have ≤ (2n)d combinations of endpoints, i.e., polynomial time. • Remaining problem: nd is too slow; – for n = 100 and d = 5, we need 1010 computational steps – very long but doable; 4
20
– for n = 10 and d = 5, we need 10 steps – which is unrealistic.
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62.
Use of Quantum Computing
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• Idea: use Grover’s algorithm for quantum search.
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• Problem: search for a desired element in an unsorted list of size N .
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• Without using quantum effects: we need – in the worst case – at least N computational steps. • A quantum computing √ algorithm can find this element much faster – in O( N ) time. • Our case: we must search N = O(nd ) endpoint vectors. √ • Quantum speedup: we need time N = O(nd/2 ). • Example: for of n = 104 and d = 5, – the non-quantum algorithm requires a currently impossible amount of 1020 computational steps, – while the quantum algorithm requires only a reasonable amount of 1010 steps.
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63.
Quantum Computing: Conclusion
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• In traditional interval computations, we assume that – the interval data corresponds to guaranteed interval bounds, and
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– that fuzzy estimates provided by experts are correct. • In practice, measuring instruments are not 100% reliable, and experts are not 100% reliable. • We may have estimates which are “way off”, intervals which do not contain the actual values at all. • Usually, we know the percentage of such outlier unreliable measurements.
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• It is desirable to check that the reliability of the actual data is indeed within the given percentage.
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64.
Quantum Computing: Conclusions (cont-d)
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In this section, we have shown that: • in general, the problem of checking (gauging) this reliability is computationally intractable (NP-hard);
Measurement . . . Propagation of . . . What We Do in This . . . Chapter 2: Towards . . . Chapter 3: Towards . . . Chapter 4: Towards . . .
• in the reasonable case – when each sensor measures a small number of different quantities, – it is possible to solve this problem in polynomial time; • quantum computations can drastically reduce the required computation time.
Chapter 5: Towards . . .
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Propagation of Interval and Probabilistic Uncertainty in Cyberinfrastructure-Related Data Processing and Data Fusion Christian Servin Computational Science Program University of Texas at El Paso El Paso, Texas 79968, USA [email protected]
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1.
Need for Data Processing and Data Fusion
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• For many quantities y, it is not easy (or even impossible) to measure them directly. • Instead, we measure related quantities x1 , . . . , xn , and use the known relation y = f (x1 , . . . , xn ) to estimate y.
Propagation of . . . What We Do in This . . . Chapter 2: Towards . . . Chapter 3: Towards . . . Chapter 4: Towards . . .
• Such data processing is especially important for cyberinfrastructure-related heterogenous data. • Example of heterogenous data – geophysics: – first-arrival passive (from actual earthquakes) and active seismic data (from seismic experiments); – gravity data; – surface waves, etc. • Before we start processing data, we need to first fuse data points corresponding to the same quantity.
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2.
Need to Take Uncertainty into Consideration
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• The result x e of a measurement is usually somewhat different from the actual (unknown) value x.
Propagation of . . .
• Usually, the manufacturer of the measuring instrument (MI) gives us a bound ∆ on the measurement error:
Chapter 2: Towards . . .
What We Do in This . . .
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def
|∆x| ≤ ∆, where ∆x = x e−x • Once we know the measurement result x e, we can conclude that the actual value x is in [e x − ∆, x e + ∆]. • In some situations, we also know the probabilities of different values ∆x ∈ [−∆, ∆]. • In this case, we can use statistical techniques. • However, often, we do not know these probabilities; we def only know that x is in the interval x = [e x − ∆, x e + ∆]. • In this case, we need to process this interval data.
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3.
Measurement Uncertainty: Traditional Approach
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def
• Usually, a meas. error ∆x = x e − x is subdivided into random and systematic components ∆x = ∆xs + ∆xr : – the systematic error component ∆xs is usually defined as the expected value ∆xs = E[∆x], while – the random error component is usually defined as def the difference ∆xr = ∆x − ∆xs . • The random errors ∆xr corresponding to different measurements are usually assumed to be independent. • For ∆xs , we only know the upper bound ∆s s.t. |∆xs | ≤ ∆s , i.e., that ∆xs is in the interval [−∆s , ∆s ]. • Because of this fact, interval computations are used for processing the systematic errors. • ∆xr is usually characterized by the corr. probability distribution (usually Gaussian, with known σ).
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4.
Expert Estimates and Fuzzy Data
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• There is no guarantee of expert’s accuracy. • We can only provide bounds which are valid with some degree of certainty.
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• This degree of certainty is usually described by a number from the interval [0, 1]. • So, for each β ∈ [0, 1], we have an interval x(α) containing the actual value x with certainty α = 1 − β. • The larger certainty we want, the broader should the corresponding interval be. • So, we get a nested family of intervals corresponding to different values α. • Alternative: for each x, describe the largest α for which x is in x(α); this αlargest is a membership function µ(x).
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5.
How to Propagate Uncertainty in Data Processing
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• We know that y = f (x1 , . . . , xn ).
What We Do in This . . .
• We estimate y based on the approximate values x ei as ye = f (e x1 , . . . , x en ).
Chapter 2: Towards . . .
• Since x ei 6= xi , we get ye 6= y; it is desirable to estimate def the approximation error ∆y = ye − y. • Usually, measurements are reasonably accurate, i.e., def ei − xi are small. measurement errors ∆xi = x • Thus, we can keep only linear terms in Taylor expann P ∂f sion: ∆y = Ci · ∆xi , where Ci = . ∂x i i=1 n P • For systematic error, we get a bound |Ci | · ∆si . i=1
• For random error, we get σ 2 =
n P
Ci2 · σi2 .
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6.
How to Propagate Uncertainty in Data Fusion: Case of Probabilistic Uncertainty
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• Reminder: we have several estimates x e(1) , . . . , x e(n) of the same quantity x. • Data fusion: we combine these estimates into a single estimate x e. (i) def
(i)
• Case: each estimation error ∆x = x e −x is normally distributed with 0 mean and known st. dev. σ (i) . • How to combine: use Least Squares, i.e., find x e that (i) 2 n P (e x −x e) minimizes ; (i) )2 2 · (σ i=1 n P x e(i) · (σ (i) )−2 • Solution: x e = i=1 P . n (i) −2 (σ ) i=1
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Data Fusion: Case of Interval Uncertainty
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• In some practical situations, the value x is known with interval uncertainty. • This happens, e.g., when we only know the upper bound ∆(i) on each estimation error ∆x(i) : |∆x(i) | ≤ ∆i .
Propagation of . . . What We Do in This . . . Chapter 2: Towards . . . Chapter 3: Towards . . . Chapter 4: Towards . . .
• In this case, we can conclude that |x − x e(i) | ≤ ∆(i) , i.e., def that x ∈ x(i) = [e x(i) − ∆(i) , x e(i) + ∆(i) ]. • Based on each estimate x e(i) , we know that the actual value x belongs to the interval x(i) . • Thus, we know that the (unknown) actual value x belongs to the intersection of these intervals: def
x=
n \ i=1
x(i) = [max(e x(i) − ∆(i) ), min(e x(i) + ∆(i) )].
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8.
Propagation of Uncertainty: Challenges
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• In the ideal world: – we should have an accurate description of data uncertainty;
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– based on this description, we should use well-justified and efficient algorithms to propagate uncertainty.
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• In practice, we are often not yet in this ideal situation:
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– the description of uncertainty is often only approximate,
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– the algorithms for uncertainty propagation are often heuristics, i.e., not well-justified, and
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– the algorithms for uncertainty propagation are often not very computationally efficient.
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What We Do in This Thesis
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• In Chapter 2, we show that the traditional idea of random and systematic components is an approximation: – we also need periodic components;
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– this is important in environmental studies. • In Chapter 3, on the example of a fuzzy heuristic, we show how a heuristic can be formally justified. • In Ch. 4, we show how to process more efficiently; e.g.:
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– first, we process data type-by-type;
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– then, we fuse the resulting models.
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• All these results assume that we have a good description of the uncertainty of the original data.
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• In practice, we often need to extract this information from the data; these are our future plans (Ch. 5).
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Chapter 2: Towards More Accurate Description of Uncertainty
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• Often, the differences r = ∆x − s corr. to nearby times are strongly correlated.
What We Do in This . . .
• For example, meteorological sensors may have daytime or nighttime biases, or winter and summer biases.
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• To capture this correlation, environmental scientists proposed a semi-heuristic 3-component model of ∆x.
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• In this model, the difference ∆x − ∆xs is represented as a combination of: – a “truly random” error ∆xr (which is independent from one measurement to another), and – a new “periodic” component ∆xp . • We provide a theoretical explanation for this heuristic three-component model.
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11.
Error Components: Analysis
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• We want to represent measurement error ∆x(t) as a linear combination of several components.
Propagation of . . .
• We consider the most detailed level of granularity, w/each component determined by finitely many parameters ci .
Chapter 2: Towards . . .
What We Do in This . . .
Chapter 3: Towards . . . Chapter 4: Towards . . .
• Each component is thus described by a finite-dimensional linear space L = {c1 · x1 (t) + . . . + cn · xn (t) : c1 , . . . , cn ∈ IR}. • In most applications, signals are smooth and bounded, so we assume that xi (t) is smooth and bounded. • Finally, for a long series of observations, we can choose a starting point arbitrarily: t → t + t0 .
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• It is reasonable to require that this change keeps us within the same component, i.e., x(t) ∈ L ⇒ x(t + t0 ) ∈ L.
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Error Components: Main Result
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• A function x(t) of one variable is called bounded if
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∃M ∀t (|x(t)| ≤ M ). • We say that a class F of functions of one variable is shift-invariant if
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∀x(t) (x(t) ∈ F ⇒ ∀t0 (x(t + t0 ) ∈ F )). • By an error component we mean a shift-invariant finitedimensional linear space of functions L = {c1 · x1 (t) + . . . + cn · xn (t) : ci ∈ IR}.
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• Theorem: Every error component is a linear combination of the functions x(t) = sin(ω · t) and x(t) = cos(ω · t).
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13.
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Proof
Measurement . . .
• Shift-invariance means that, for some ci (t0 ), we have
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xi (t + t0 ) = ci1 (t0 ) · x1 (t) + . . . + cin (t0 ) · xn (t).
Chapter 2: Towards . . .
• For n different values t = t1 , . . . , t = tn , we get a system of n linear equations with n unknowns cij (t0 ).
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• The Cramer’s rule solution to linear equations is a smooth function of all the coeff. & right-hand sides.
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• Since all the right-hand sides xi (tj +t0 ) and coefficients xi (tj ) are smooth, cij (t0 ) are also smooth. def
• Differentiating w.r.t. t0 and taking t0 = 0, for cij = c˙ij (0), we get
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14.
Proof (cont-d)
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• Reminder: x˙ i (t) = ci1 · x1 (t) + . . . + cin · xn (t). • A general solution of such system of equations is a linear combination of functions
Propagation of . . . What We Do in This . . . Chapter 2: Towards . . . Chapter 3: Towards . . .
k
t · exp(λ · t), w/k ∈ N, k ≥ 0, λ = a + i · ω ∈ C.
Chapter 4: Towards . . . Chapter 5: Towards . . .
• Here, exp(λ · t) = exp(a · t) · cos(ω · t) + i · exp(a · t) · sin(ω · t). • When a 6= 0, we get unbounded functions for t → ∞ or t → −∞. • So, a = 0. • For k > 0, we get unbounded tk ; so, k = 0. • Thus, we indeed have a linear combination of sinusoids.
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Practical Conclusions
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• Let f be the measurements frequency (how many measurements we perform per unit time).
Propagation of . . .
• When ω f , the values cos(ω · t) and sin(ω · t) practically do not change with time.
Chapter 2: Towards . . .
What We Do in This . . .
Chapter 3: Towards . . . Chapter 4: Towards . . .
• Indeed, the change period is much larger than the usual observation period. • Thus, we can identify such low-frequency components with systematic error component. • When ω f , the phases of the values cos(ω · ti ) and cos(ω · ti+1 ) differ a lot. • For all practical purposes, the resulting values of cosine or sine functions are independent. • Thus, high-frequency components can be identified with random error component.
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Practical Conclusions (cont-d)
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• Result: every error component is a linear combination of cos(ω · t) and sin(ω · t). • Notation: let f be the measurements frequency (how many measurements we perform per unit time).
Propagation of . . . What We Do in This . . . Chapter 2: Towards . . . Chapter 3: Towards . . . Chapter 4: Towards . . .
• Reminder: – we can identify low-frequency components (ω f ) with systematic error component; – we can identify high-frequency ones (ω f ) with random error component. • Easy to see: all other error components cos(ω · t) and sin(ω · t) are periodic.
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• Conclusion: we have indeed justified to the semi-empirical 3-component model of measurement error.
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17.
Chapter 3: Towards Justification of Heuristic Techniques for Processing Uncertainty
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• As we have mentioned, some methods for processing uncertainty are heuristic.
What We Do in This . . .
• Such methods lack justification and are, therefore, less reliable.
Chapter 3: Towards . . .
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Chapter 4: Towards . . . Chapter 5: Towards . . .
• Usually, techniques for processing interval and probabilistic uncertainty are well-justified. • However, many techniques for processing expert (fuzzy) data are still heuristic.
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– we consider a practically efficient heuristic fuzzy technique for decision making under uncertainty; – we show how this heuristic can be formally justified.
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18.
Traditional Approach to Decision Making: Reminder
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• The quality of each possible alternative is characterized by the values of several quantities.
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• For example, when we buy a car, we are interested in its cost, its energy efficiency, its power, size, etc.
Chapter 3: Towards . . .
Chapter 2: Towards . . .
Chapter 4: Towards . . . Chapter 5: Towards . . .
• For each of these quantities, we usually have some desirable range of values. • Often, there are several different alternatives all of which satisfy all these requirements. • The traditional approach assumes that there is an objective function that describes the user’s preferences. • We then select an alternative with the largest possible value of this objective function.
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Traditional Approach to Decision Making: Limitations
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• The traditional approach to decision making assumes: – that the user knows exactly what he or she wants — i.e., knows the objective function – and
What We Do in This . . . Chapter 2: Towards . . . Chapter 3: Towards . . . Chapter 4: Towards . . .
– that the user also knows exactly what he or she will get as a result of each possible decision. • In practice, the user is often uncertain: – the user is often uncertain about his or her own preferences, and – the user is often uncertain about possible consequences of different decisions. • It is therefore desirable to take this uncertainty into account when we describe decision making.
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20.
Fuzzy Target Approach (Huynh-Nakamori)
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• For each numerical characteristic of a possible decision, we form two fuzzy sets: – µi (x) describing the users’ ideal value; – µa (x) describing the users’ impression of the actual value.
Propagation of . . . What We Do in This . . . Chapter 2: Towards . . . Chapter 3: Towards . . . Chapter 4: Towards . . . Chapter 5: Towards . . .
• For example, a person wants a well done steak, and the steak comes out as medium well done. • In this case, µi (x) corresponds to “well done”, and µa (x) to “medium well done”. • The simplest “and”-operation (t-norm) is min(a, b); so, the degree to which x is both actual and desired is min(µa (x), µi (x)). • The degree to which there exists x which is both possible and desired is d = max min(µa (x), µi (x)).
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21.
Fuzzy Target Approach: Successes and Remaining Problems
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• The above approach works well in many applications. • Example: it predicts how customers select a handcrafted souvenir when their ideal ones is not available.
What We Do in This . . . Chapter 2: Towards . . . Chapter 3: Towards . . . Chapter 4: Towards . . .
• Problem: this approach is heuristic, it is based on selecting: – the simplest possible membership function and – the simplest possible “and”- and “or”-operations. • Interestingly, we get better predictions than with more complex membership functions and “and”-operations. • In this section, we provide a justification for the above semi-heuristic target-based fuzzy decision procedure.
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22.
Chapter 4: Towards More Computationally Efficient Techniques for Processing Uncertainty
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• Fact: computations often take a lot of time. • One of the main reasons: we process a large amount of data.
What We Do in This . . . Chapter 2: Towards . . . Chapter 3: Towards . . . Chapter 4: Towards . . .
• So, a natural way to speed up data processing is:
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– to divide the data into smaller parts,
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– to process each smaller part separately, and then
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– to combine the results of data processing. • In particular, when we are processing huge amounts of heterogenous data, it makes sense: – to first process different types of data type-by-type, – and then to fuse the resulting models. • This idea is explored in the first sections of Chapter 4.
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23.
Data Fusion under Interval Uncertainty: Reminder
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• Frequent practical situation:
What We Do in This . . . Chapter 2: Towards . . .
– we are interested in a quantity u;
Chapter 3: Towards . . .
– we have several measurements and/or expert estimates u1 , . . . , un of u. • Objective: fuse these estimates into a single more accurate estimate. • Interval case: each ui is known with interval uncertainty. • Formal description: for each i, we know the interval ui = [ui − ∆i , ui + ∆i ] containing u. n def T • Solution: u belongs to the intersection u = ui of
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i=1
these intervals.
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24.
Data Fusion under Probabilistic Uncertainty: Reminder
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def
• Probabilistic uncertainty: each measurement error ∆ui = ui − u is normally distributed w/0 mean and known σi .
What We Do in This . . .
• Technique: the Least Squares Method (LSM) n X (u − ui )2 → min . 2 2σ i i=1
Chapter 3: Towards . . .
• Resulting estimate: is i=1
ui · σi−2
n P i=1
. σi−2
• Standard deviation: σ2 = P n i=1
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n P
u=
Chapter 2: Towards . . .
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1 σi−2
,
with σ 2 σi2 .
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25.
New Problem: Different Resolution
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• Traditional data fusion: fusing measurement results with different accuracy. • Additional problem: different measurements also have different resolution.
Propagation of . . . What We Do in This . . . Chapter 2: Towards . . . Chapter 3: Towards . . . Chapter 4: Towards . . .
• Case study – geosciences: estimating density u1 , . . . , un at different locations and depths. • Examples of different geophysical estimates: – Seismic data leads to higher-resolution estimates u e1 , . . . , u en of the density values. – Gravity data leads to lower-resolution estimates, i.e., estimates u e for the weighted average u=
n X i=1
wi · ui .
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Why This Is Important
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• Reminder: there are many sources of data for Earth models: – first-arrival passive seismic data (from the actual earthquakes), – first-arrival active seismic data (from the seismic experiments), – gravity data, – surface waves, etc. • At present: each of these datasets is processed separately, resulting in several different Earth models. • Fact: these models often provide complimentary geophysical information. • Idea: all these models describe the properties of the same Earth, so it is desirable to combine them.
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New Idea: Model Fusion
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• Objective: to combine the information contained in multiple complementary datasets. • Ideal approach: it is desirable to come up with techniques for joint inversion of these datasets.
Propagation of . . . What We Do in This . . . Chapter 2: Towards . . . Chapter 3: Towards . . . Chapter 4: Towards . . .
• Problem: designing such joint inversion techniques is an important theoretical and practical challenge. • Status: such joint inversion methods are being developed. • Practical question: what to do while these methods are being developed?
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• Our practical solution: fuse the Earth models coming from different datasets.
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28.
Need to Take . . .
Model Fusion: Statistical Case
Measurement . . .
• Objective: find the values u1 , . . . , un of the desired quantity in different spatial cells.
Propagation of . . . What We Do in This . . . Chapter 2: Towards . . .
• Geophysical example: ui is the density at different
Chapter 3: Towards . . .
1 km × 1 km × 1 km cells.
Chapter 4: Towards . . . Chapter 5: Towards . . .
• Input: we have
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– high-resolution measurements, i.e., values u ei ≈ ui with st. dev. σi ; (k)
– lower-resolution measurements, i.e., values u e responding to blocks of neighboring cells: X (k) u e(k) ≈ wi · ui , with st. dev. σ (k) .
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cor-
i
• Additional information: a lower-resolution measurement result is representative of values within the block.
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29.
Model Fusion: Statistical Case (cont-d)
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(k)
• Formal description: when wi with st. dev. δ (k) .
6= 0, we have u e(k) ≈ ui ,
• How to estimate δ (k) : as the empirical st. dev. within the block.
Propagation of . . . What We Do in This . . . Chapter 2: Towards . . . Chapter 3: Towards . . . Chapter 4: Towards . . .
• High-resolution values (reminder): u ei ≈ ui w/st. dev. σi . • Lower-resolution values (reminder): X (k) (k) u e ≈ wi · ui , with st. dev. σ (k) . i
• LSM Solution: minimize the sum P (k) (k) (e u − wi · ui )2 2 XX (k) 2 X X (ui − u ei ) (ui − u e ) i + + . 2 (k) 2 (k) )2 σ (δ ) (σ i i i k k • How: differentiating w.r.t. ui , we get a system of linear equations with unknowns u1 , . . . , un .
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30.
Need to Take . . .
Model Fusion: Interval Case
Measurement . . .
• Quantities of interest: values u1 , . . . , un of the desired quantity in different spatial cells.
Propagation of . . .
• Objective: find the ranges u1 , . . . , un of possible values of u1 , . . . , un .
Chapter 2: Towards . . .
What We Do in This . . .
Chapter 3: Towards . . . Chapter 4: Towards . . .
• High-resolution measurements: values u ei ≈ ui with bound ∆i : u ei − ∆i ≤ ui ≤ u ei + ∆i .
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(k)
• Lower-resolution measurements: values u e corresponding to blocks of neighboring cells: X (k) u e(k) ≈ wi · ui , with bound ∆(k) . i
• Resulting constraint: (k)
u e
(k)
−∆
≤
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X i
(k)
wi · ui ≤ u e(k) + ∆(k) .
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31.
Model Fusion: Interval Case (cont-d)
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• Additional information: a priori bounds on ui :
Propagation of . . . What We Do in This . . .
ui ≤ ui ≤ ui . • Additional information: a priori bounds on the changes between neighboring cells: −δij ≤ ui − uj ≤ δij . • High-resolution measurements (reminder): u ei − ∆i ≤ ui ≤ u ei + ∆i . • Lower-resolution measurements (reminder): X (k) u e(k) − ∆(k) ≤ wi · ui ≤ u e(k) + ∆(k) . i
• Objective: minimize and maximize each ui under these constraints.
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32.
Model Fusion: Interval Solution
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• Problem. Minimize (Maximize) ui under the following constraints:
Propagation of . . . What We Do in This . . . Chapter 2: Towards . . .
• ui ≤ ui ≤ ui .
Chapter 3: Towards . . .
• −δij ≤ ui − uj ≤ δij . •u ei − ∆i ≤ ui ≤ u ei + ∆i . P (k) •u e(k) − ∆(k) ≤ wi · ui ≤ u e(k) + ∆(k) .
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i
• Current solution method: linear programming. • Objective: provide more efficient algorithms for specific geophysical cases. • Preliminary results: some such algorithms have been developed.
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33.
Preliminary Experiments
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• What we have done: preliminary proof-of-concept experiments. • Simplifications:
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– equal weights wi ;
Chapter 4: Towards . . .
– simplified datasets.
Chapter 5: Towards . . .
• Conclusion: the fused model improves accuracy and resolution of different Earth models.
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34.
Chapter 5: Towards Better Ways of Extracting Information About Uncertainty from Data
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• Previous methods assume that we have a good description of the uncertainty.
What We Do in This . . .
• In practice, often, we do not have this information.
Chapter 3: Towards . . .
Chapter 2: Towards . . .
Chapter 4: Towards . . .
• We need to extract uncertainty information from the data. • In Chapter 5, we propose ideas on how this uncertainty information can be extracted from the data. • These ideas constitute our future work.
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35.
Extracting Uncertainty from Data: What Is Known
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• Traditional approach: use a “standard” (more accurate) measuring instrument SMI.
What We Do in This . . .
• Idea: values x eS measured by SMI are accurate: x eS ≈ x, def so x e−x eS ≈ ∆x = x e − x.
Chapter 3: Towards . . .
• Limitation: for cutting-egde measurements, we do not have more accurate instruments, these are the best. • Example: the Eddy convariance tower provides the best estimates for Carbon flux. • Idea: if we have two similar measuring instruments, we can estimate ∆x(1) − ∆x(2) as x e(1) − x e(2) . • If both error are normally distributed with st. dev. σ, then ∆x(1) − ∆x(2) is also normal, with variance 2σ 2 . • So, we can determine σ from observations.
Chapter 2: Towards . . .
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Extracting Uncertainty from Data: Remaining Problem
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• Problem: if distribution of ∆x(i) is skewed, we cannot distinguish between two distributions: – the distribution of ∆x(i) , and – its mirror image, the distribution of −∆x(i) . • Our preliminary results: – this is the only non-uniquness, and – modulo this non-uniqueness, we can effectively reconstruct the distribution of ∆x(i) .
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37.
Summary: Main Problem
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• In the ideal world: – we should have an accurate description of data uncertainty;
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– based on this description, we should use well-justified and efficient algorithms to propagate uncertainty.
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• In practice, we are often not yet in this ideal situation:
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– the description of uncertainty is often only approximate,
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– the algorithms for uncertainty propagation are often heuristics, i.e., not well-justified, and
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– the algorithms for uncertainty propagation are often not very computationally efficient.
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38.
Summary: Conclusions
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• In Ch. 2, we showed that the traditional idea of random and systematic components is an approximation: – we also need periodic components;
Propagation of . . . What We Do in This . . . Chapter 2: Towards . . . Chapter 3: Towards . . .
– this is important in environmental studies. • In Chapter 3, on the example of a fuzzy heuristic, we showed how a heuristic can be formally justified. • In Ch. 4, we showed how to be more efficient; e.g.:
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– first, we process data type-by-type;
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– then, we fuse the resulting models.
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• All these results assume that we have a good description of the uncertainty of the original data.
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• In practice, we often need to extract this information from the data; these are our future plans (Ch. 5).
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39.
Acknowledgments
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I would like to express my deep-felt gratitude: • to my mentor Dr. Vladik Kreinovich;
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• to members of my committee, Dr. Aaron Velasco, Dr. Scott Starks, and Dr. Luc Longpr´e; • to Dr. Benjamin C. Flores, and to the Alliance for Minority Participation Bridge to the Doctorate program; • to Dr. Craig Tweedie for his suggestions, comments, and guidance in this work; • to Dr. Aaron Velasco, Dr. Vanessa Lougheed, Dr. William Robertson, and to all the GK-12 program staff; and • last but not the least, to all the faculty and staff of the Computational Science Program.
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40.
Publications
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• L. Longpr´e, C. Servin, and V. Kreinovich, “Quantum computation techniques for gauging reliability of interval and fuzzy data”, International Journal of General Systems, 2011, Vol. 40, No. 1, pp. 99–109.
Propagation of . . .
• O. Ochoa, A. A. Velasco, V. Kreinovich, and C. Servin, “Model fusion: a fast, practical alternative towards joint inversion of multiple datasets”, Abstracts of the Annual Fall Meeting of the American Geophysical Union AGU’08, San Francisco, California, December 15–19, 2008.
Chapter 4: Towards . . .
• O. Ochoa, A. A. Velasco, C. Servin, and V. Kreinovich, “Model Fusion under Probabilistic and Interval Uncertainty, with Application to Earth Sciences”, International Journal of Reliability and Safety, 2012, Vol. 6, No. 1–3, pp. 167–187.
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Publications (cont-d)
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• C. Servin, O. Ochoa, and A. A. Velasco, “Probabilistic and interval uncertainty of the results of data fusion, with application to geosciences”, Abstracts of 13th International Symposium on Scientific Computing, Computer Arithmetic, and Verified Numerical Computations SCAN’2008, El Paso, Texas, September 29 – October 3, 2008, p. 128.
Propagation of . . .
• C. Servin, V.-N. Huyhn, and Y. Nakamori, “Semiheuristic target-based fuzzy decision procedures: towards a new interval justification”, Proceedings of the Annual Conference of the North American Fuzzy Information Processing Society NAFIPS’2012, Berkeley, California, August 6–8, 2012.
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42.
Publications (cont-d)
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• C. Servin, C. Tweedie, and A. Velasco, “Towards a more realistic treatment of uncertainty in Earth and environmental sciences: beyond a simplified subdivision into interval and random components”, Abstracts of the 15th GAMM-IMACS International Symposium on Scientific Computing, Computer Arithmetic, and Verified Numerical Computation SCAN’2012, Novosibirsk, Russia, September 23–29, 2012, pp. 164–165. • C. Servin, M. Ceberio, A. Jaimes, C. Tweedie, and V. Kreinovich, “How to describe and propagate uncertainty When Processing time series: metrological and computational challenges, with potential applications to environmental studies”, In: S.-M. Chen and W. Pedrycz (eds.), Time Series Analysis, Modeling and Applications: A Computational Intelligence Perspective, Springer Verlag, to appear.
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43.
Appendix to Chapter 2: How to Propagate Uncertainty in the Three-Component Model
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• We are interested in the quantity
What We Do in This . . . Chapter 2: Towards . . .
y = f (x1 (t11 ), x1 (t12 ), . . . , x2 (t21 ), x2 (t22 ), . . . , xn (tn1 ), xn (tn2 ), . . .).Chapter 3: Towards . . . • Instead of the actual values xi (tij ), we only know the measurement results x ei (tij ) = xi (tij ) + ∆xi (tij ). • Measurement errors are usually small, so terms quadratic (and higher) in ∆xi (tij ) can be safely ignored. • For example, if the measurement error is 10%, its square is 1% which is much much smaller than 10%. • If the measurement error is 1%, its square is 0.01% which is much much smaller than 1%. • Thus, we can safely linearize the dependence of ∆y on ∆xi (tij ).
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How to Propagate Uncertainty (cont-d)
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• Reminder: we can safely linearize the dependence of ∆y on ∆xi (tij ), so XX ∂y def ∆y = Cij · ∆xi (tij ), with Cij = . ∂x (t ) i ij i j P • In general, ∆xi (tij ) = si +rij + ` A`i ·cos(ω` ·tij +ϕ`i ). P • Due to linearity, we have ∆y = ∆ys + ∆yr + ` ∆yp` , where XX XX ∆ys = Cij · si ; ∆yr = Cij · rij ; i
∆yp` =
j
i
XX i
j
Cij · A`i · cos(ω` · tij + ϕ`i ).
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j
• We know: how to compute ∆ys and ∆yr . • What is needed: propagation of the periodic component.
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45.
Propagating Periodic Component: Analysis
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• Reminder: for each component, we have XX ∆yp` = Cij · A`i · cos(ω` · tij + ϕ`i ). i
j
Propagation of . . . What We Do in This . . . Chapter 2: Towards . . . Chapter 3: Towards . . .
• It is reasonable to assume that different phrases ϕ`i are independent (and uniformly distributed). • Thus, by the Central Limit Theorem, the distribution of ∆yp` is close to normal, with 0 mean. 1 X 2 • The variance of ∆yp` is · A`i · (Kci2 + Ksi2 ). 2 i • Each amplitude A`i can take any value from 0 to the known bound P`i . 1 X 2 • Thus, the variance is bounded by · P`i ·(Kci2 +Ksi2 ). 2 i • So, we arrive at the following algorithm.
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46.
Propagating Periodic-Induced Component: Algorithm
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• First, we apply the algorithm f to the measurement results x ei (tij ) and get the estimate ye.
What We Do in This . . .
• Then, we select a small value δ and for each sensor i, we do the following:
Chapter 3: Towards . . .
(ci)
– take xi (tij ) = x ei (tij ) + δ · cos(ω` · tij ) for all moments j; (ci) – for other sensors i0 6= i, take xi0 (ti0 j ) = x ei (ti0 j ); (ci)
– substitute the resulting values xi0 (ti0 j ) into the data processing algorithm f and get the result y (ci) ; (si) – then, take xi (tij ) = x ei (tij ) + δ · sin(ω` · tij ) for all moments j; (si) – for all other i0 6= i, take xi0 (ti0 j ) = x ei (ti0 j ); (si)
– substitute the resulting values xi0 (ti0 j ) into the data processing algorithm f and get the result y (si) .
Chapter 2: Towards . . .
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47.
Algorithm (cont-d)
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• Reminder:
Propagation of . . . What We Do in This . . .
– First, we apply the algorithm f to the measurement results x ei (tij ) and get the estimate ye.
Chapter 2: Towards . . .
– Then, for each sensor i, we simulate cosine terms and get the results y (ci) .
Chapter 4: Towards . . .
– Third, for each sensor i, we simulate sine terms and get the results y (si) . • Finally, we estimate the desired bound σp` on the standard deviation of ∆yp` as v u (ci) 2 (si) 2 ! u1 X y − ye y − ye σp` = t · P`i2 · + . 2 i δ δ
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48.
Appendix to Chapter 3
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• We know: – a fuzzy set µi (x) describing the users’ ideal value; – the fuzzy set µa (x) describing the users’ impression of the actual value.
Propagation of . . . What We Do in This . . . Chapter 2: Towards . . . Chapter 3: Towards . . . Chapter 4: Towards . . .
• For crisp sets, the solution is possibly satisfactory if some of the possibly actual values is also desired. • In the fuzzy case, we can only talk about the degree to which the proposed solution can be desired. • A possible decision is satisfactory if either: – the actual value is x1 , and this value is desired, – or the actual value is x2 , and this value is desired, – ... • Here x1 , x2 , . . . , go over all possible values of the desired quantity.
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Derivation of the d-Formula (cont-d)
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• For each value xk , we know: – the degree µa (xk ) with which this value is actual, and – the degree µi (xk ) to which this value is desired. • Let us use min(a, b) to describe “and” – the simplest possible choice of an “and”-operation. • Then we can estimate the degree to which the value xk is both actual and desired as min(µa (xk ), µi (xk )). • Let us use max(a, b) to describe “or” – the simplest possible choice of an “or”-operation.
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• Then, we can estimate the degree d to which the two fuzzy sets match as d = max min(µa (x), µi (x)).
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Fuzzy Target Approach: How Are Membership Functions Elicited?
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• In many applications (e.g., in fuzzy control), the shape of a membership function does not affect the result.
What We Do in This . . .
• Thus, it is reasonable to use the simplest possible membership functions – symmetric triangular ones.
Chapter 3: Towards . . .
Chapter 2: Towards . . .
Chapter 4: Towards . . . Chapter 5: Towards . . .
• To describe a symmetric triangular function, it is sufficient to know the support [x, x] of this function.
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• So, e.g., to get the membership function µi (x) describing the desired situation: – we ask the user for all the values a1 , . . . , an which, in their opinion, satisfy the requirement; – we then take the smallest of these values as a and the largest of these values as a; – finally, we form symmetric triangular µi (x) whose support is [a, a].
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51.
Analyzing the Problem
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• Reminder: all we elicit from the experts is two intervals: – an interval [a, a] = [e a − ∆a , e a + ∆a ] describing the set of all desired values, and – an interval [b, b] = [eb − ∆b , eb + ∆b ] describing the set of all the values which are possible. • Based on these intervals, we build triangular membership functions µi (x) and µa (x) centered in e a and eb. • For these membership functions, |eb − e a| d = max min(µa (x), µi (x)) = 1 − . x ∆a + ∆b • This is the formula that we need to justify.
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52.
Our Main Idea
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• If we knew the exact values of a and b, then we would conclude a = b, a < b, or b < a. • In reality, we know the values a and b with uncertainty.
Propagation of . . . What We Do in This . . . Chapter 2: Towards . . . Chapter 3: Towards . . .
• Even if the actual values a and b are the same, we may get approximate values which are different. • It is reasonable to assume that if the actual values are the same, then Prob(a > b) = Prob(b > a), i.e., Prob(a > b) = 1/2. • If the probabilities that a > b and that a < b differ, this is an indication that the actual value differ. • Thus, it’s reasonable to use |Prob(a > b)−Prob(b > a)| as the degree to which a and b may be different.
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53.
How To Estimate Prob(a > b) and Prob(b > a)
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• If we knew the exact values of a and b, then we could def check a > b by comparing r = a − b with 0. • In real life, we only know a and b with interval uncertainty, i.e., we only know that a ∈ [e a − ∆a , e a + ∆a ] and b ∈ [eb − ∆b , eb + ∆b ]. • In this case, we only know the range r of possible values of r = a − b; interval arithmetic leads to r = [(e a − eb) − (∆a + ∆b ), (e a − eb) + (∆a + ∆b )]. • We do not have any reason to assume that some values from r are more probable and some are less probable. • It is thus reasonable to assume that all the values from r are equally probable, i.e., r is uniformly distributed. • This argument is widely used in data processing; it is called Laplace Principle of Indifference.
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54.
How To Estimate Probabilities (cont-d)
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• We estimate Prob(a > b) as Prob(a − b > 0). • We estimate Prob(a < b) as Prob(a − b < 0).
Propagation of . . . What We Do in This . . . Chapter 2: Towards . . .
• We assumed that r = a − b is uniformly distributed on [(e a − eb) − (∆a + ∆b ), (e a − eb) + (∆a + ∆b )]. • We can compute Prob(a − b > 0), Prob(a − b < 0), and |e a − eb| . |Prob(a > b) − Prob(b > a)| = ∆a + ∆b |eb − e a| • Since d = 1 − , we get ∆a + ∆b d = 1 − |Prob(a > b) − Prob(b > a)|.
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• We have produced a new justification for the d-formula.
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• This justification that does not use any simplifying assumptions about memb. f-s and/or “and”-operations.
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55.
Appendix to Chapter 4: Towards More Efficient Uncertainty Processing
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• Even after all algorithmic speed-ups are implemented, the computation time is still often too long.
What We Do in This . . .
• In such situations, the only remaining way to speed up computations is to speed up hardware.
Chapter 3: Towards . . .
Chapter 2: Towards . . .
Chapter 4: Towards . . . Chapter 5: Towards . . .
• Such ideas range from available (e.g., parallelization) to futuristic (e.g., quantum computing). • Parallelization has been largely well-researched. • The use of futuristic techniques in uncertainty estimation is still largely an open problem.
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• In the last section of Ch. 4, we show how quantum computing can be used to speed up computations.
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56.
Need to Take . . .
Reliability of Interval Data
Measurement . . .
• Usual assumption: all measuring instruments (MI) functioned correctly.
Propagation of . . .
• Conclusion: the resulting intervals [e x − ∆, x e + ∆] contain the actual value x.
Chapter 2: Towards . . .
What We Do in This . . .
Chapter 3: Towards . . . Chapter 4: Towards . . .
• In practice: a MI can malfunction, producing way-off values (outliers). • Problem: outliers can ruin data processing. • Example: average temperature in El Paso 95 + 100 + 105 – based on measurements, = 100. 3 95 + 100 + 105 + 0 – with outlier, = 75. 4 • Natural idea: describe the probability p of outliers.
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• Solution: out of n results, dismiss k = p · n largest values and k smallest.
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57.
Need to Gauge the Reliability of Interval Data
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• Ideal case: all measurements of the same quantity are correct.
Propagation of . . .
• Fact: resulting intervals x(1) , . . . , x(n) contain the same (actual) value x. n T • Conclusion: x(i) 6= ∅.
Chapter 2: Towards . . .
i=1
• Reality: we have outliers far from x, so
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n T
x(i) = ∅.
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• Expectation: out of n given intervals, ≥ n − k are correct – and hence have a non-empty intersection. • Conclusion: – to check whether our estimate p for reliability is correct, – we must check whether out of n given intervals, n − k have a non-empty intersection.
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Need to Gauge Reliability of Interval Data: Multi-D Case
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• In practice, a measuring instrument often measure several different quantities x1 , . . . , xd .
What We Do in This . . .
• Due to uncertainty, after the measurement, for each quantity xi , we have an interval xi of possible values.
Chapter 3: Towards . . .
Chapter 2: Towards . . .
Chapter 4: Towards . . . Chapter 5: Towards . . .
• So, the set of all possible values of the tuple x = (x1 , . . . , xd ) is a box X = x1 ×. . .×xd = {(x1 , . . . , xd ) : x1 ∈ x1 , . . . , xd ∈ xd }. • Thus: – to check whether our estimate p for reliability is correct, – we must be able to check whether out of n given boxes, n − k have a non-empty intersection.
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59.
Resulting Computational Problem: Box Intersection Problem
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Thus, both in the interval and in the fuzzy cases, we need to solve the following computational problem:
What We Do in This . . . Chapter 2: Towards . . . Chapter 3: Towards . . .
• Given:
Chapter 4: Towards . . .
• integers d, n, and k; and
Chapter 5: Towards . . .
• n d-dimensional boxes (j)
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(j) X (j) = [x1 , x1 ] × . . . × [x(j) n , xn ], (j)
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(j)
j = 1, . . . , n, with rational bounds xi and xi .
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– we can select n − k of these n boxes – in such a way that the selected boxes have a nonempty intersection.
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60.
Results
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• First result: in general, the above computational problem is NP-hard. • Meaning: no algorithm is possible that solves all particular cases of this problem in reasonable time.
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• In practice: the number of d of quantities measured by a sensor is small: e.g., – a GPS sensor measures 3 spatial coordinates; – a weather sensor measures (at most) 5: ∗ temperature, ∗ atmospheric pressure, and ∗ the 3 dimensions of the wind vector.
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• Second result: for a fixed dimension d, we can solve the above problem in polynomial time O(nd ).
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Algorithm: Description and Need for Speed Up
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• Lemma: if a set of boxes has a common point, then there is another common vector whose all components are endpoints.
What We Do in This . . .
• Proof: move to an endpoint in each direction.
Chapter 4: Towards . . .
Chapter 2: Towards . . . Chapter 3: Towards . . .
Chapter 5: Towards . . .
• Number of endpoints: n intervals have ≤ 2n endpoints. • Bounds on computation time: we have ≤ (2n)d combinations of endpoints, i.e., polynomial time. • Remaining problem: nd is too slow; – for n = 100 and d = 5, we need 1010 computational steps – very long but doable; 4
20
– for n = 10 and d = 5, we need 10 steps – which is unrealistic.
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Use of Quantum Computing
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• Idea: use Grover’s algorithm for quantum search.
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• Problem: search for a desired element in an unsorted list of size N .
What We Do in This . . . Chapter 2: Towards . . . Chapter 3: Towards . . .
• Without using quantum effects: we need – in the worst case – at least N computational steps. • A quantum computing √ algorithm can find this element much faster – in O( N ) time. • Our case: we must search N = O(nd ) endpoint vectors. √ • Quantum speedup: we need time N = O(nd/2 ). • Example: for of n = 104 and d = 5, – the non-quantum algorithm requires a currently impossible amount of 1020 computational steps, – while the quantum algorithm requires only a reasonable amount of 1010 steps.
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63.
Quantum Computing: Conclusion
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• In traditional interval computations, we assume that – the interval data corresponds to guaranteed interval bounds, and
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– that fuzzy estimates provided by experts are correct. • In practice, measuring instruments are not 100% reliable, and experts are not 100% reliable. • We may have estimates which are “way off”, intervals which do not contain the actual values at all. • Usually, we know the percentage of such outlier unreliable measurements.
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• It is desirable to check that the reliability of the actual data is indeed within the given percentage.
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64.
Quantum Computing: Conclusions (cont-d)
Need for Data . . . Need to Take . . .
In this section, we have shown that: • in general, the problem of checking (gauging) this reliability is computationally intractable (NP-hard);
Measurement . . . Propagation of . . . What We Do in This . . . Chapter 2: Towards . . . Chapter 3: Towards . . . Chapter 4: Towards . . .
• in the reasonable case – when each sensor measures a small number of different quantities, – it is possible to solve this problem in polynomial time; • quantum computations can drastically reduce the required computation time.
Chapter 5: Towards . . .
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