**Name** (please print):
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1. *Data processing: interval uncertainty.*

Let us assume that data
processing is performed by using a function y = f(x_{1}, x_{2})
= x_{1}^{2} + x_{2}, the measurement results are
X_{1} = 1.0 and X_{2} = 2.0, and we only know the upper bounds
on the measurement errors: Δ_{1} = 0.1 and
Δ_{2} = 0.2.
Find the upper bound Δ on the approximation error Δy of the result
of data processing. Since the function is monotonic, you can compute the exact
range. Compare this range with the results of linearized computations.

2. *Interval uncertainty in data processing: computational aspects.*

For the formula for computing upper bound on the error the result of data
processing, explain how the computational complexity (= number of computational
steps) depends on the choice of the parameters h_{i} used in numerical
differentiation, and what is the choice for which the computational complexity
is the smallest.

3. *Interval uncertainty in data processing: Monte-Carlo method.*

Explain
why Monte-Carlo method is useful in computing interval uncertainty of the
result of data processing, and for what number of inputs it is useful.

4. *Bisection method.* Describe a bisection method for finding the value x
on a given interval [l,u] for which f(x) = 0. Follow the first two steps of
this method for finding the square root of 2, i.e., the value x from the
interval [0,2] for which f(x) = x^{2} - 2 = 0. Explain where the
bisection method is used in Monte-Carlo estimation of interval uncertainty.

5. *Interval uncertainty in data fusion.*

Let us assume that we have made
three measurements of the same quantity x.

- In the first measurement, we
have a measurement result X
_{1}= 1.0, and the known upper bound on the measurement error is Δ_{1}= 0.1. - In the second measurement, we
have a measurement result X
_{2}= 0.97, and the known upper bound on the measurement error is Δ_{2}= 0.03. - In the third measurement, we
have a measurement result X
_{3}= 1.15, and the known upper bound on the measurement error is Δ_{3}= 0.2.

6. *Effect of reliability: case of independence.*

Let us assume that we are producing a 1D map,
and that to approximate a value on a grid point, we use the 1-Nearest Neighbor
(1NN) algorithm, i.e., we take the values of the nearest point at which the
measurement was performed. We are interested in estimating the value at a point
x = 1. We have made three measurements:

- The first measurement was performed at location x
_{1}= 0.7, the measured value is v_{1}= 1.0. - The second measurement was performed at location x
_{2}= 1.4, the measured value is v_{2}= 0.9. - The third measurement was performed at location x
_{3}= 0.5, the measured value is v_{3}= 1.1.

- The first value v
_{1}is reliable with probability p_{1}= 0.9. - The second value v
_{2}is reliable with probability p_{2}= 0.8. - The third value v
_{3}is reliable with probability p_{3}= 0.9.

7. *Monte-Carlo approach: need for re-scaling.*

Explain why for very
reliable components, we cannot directly use the Monte-Carlo method, we need a
re-scaling. Provide a numerical example of the number of iterations that are
needed to achieve a given accuracy. Describe the main idea of the re-scaling and
how it helps.

8. *Possibility of re-scaling: case of independence.*

On the example of two cases:

- case when f trusts t with probability p
_{1}= 1 - Δp_{1}and t trusts s with probability p_{2}= 1 - Δp_{2}; - case when f has two reasons for trusting s: with probability
p
_{1}= 1 - Δp_{1}and with probability p_{2}= 1 - Δp_{2};

9. *Probabilities: case of possible dependence.*

Assume that the probability
p(A) of the event A is 0.8, and the probability p(B) of the event B is 0.8.

- What is the probability p(A & B) that both A and B hold if these events are independent?
- What is the range of possible values of p(A & B) when we do not have any information about the dependence between A and B? Draw examples illustrating the possibilities of the smallest and the largest values from this range.
- What is the probability p(A \/ B) that one of the events A or B holds if these events are independent?
- What is the range of possible values of p(A \/ B) when we do not have any information about the dependence between A and B? Draw examples illustrating the possibilities of the smallest and the largest values from this range.

10. *Reliability: case of possible dependence.*

On the example of two cases:

- case when f trusts t with probability p
_{1}= 1 - Δp_{1}and t trusts s with probability p_{2}= 1 - Δp_{2}; - case when f has two reasons for trusting s: with probability
p
_{1}= 1 - Δp_{1}and with probability p_{2}= 1 - Δp_{2};