## Test 1, CS 5353, Fall 2008

Date: Tuesday, September 30, 2008.

1. What is cyberinfrastructure.
Explain what is cyberinfrastructure, why it is useful, and why it did not appear many years ago, when computer communications were slower.

2. Statistical foundations.
Explain, in detail, why for two independent events A and B, the probability of A & B is equal to the product of the probabilities of A and B: P(A & B) = P(A) * P(B).

3. Statistical foundations.
What is the Maximum Likelihood Method? Explain, in detail, how for the Gaussian distribution, with the probability density d(x)= const * exp(-(1/2)*(x-a)2/s2), this method leads to an estimate of the mean a as the arithmetic average of the measurement results x1, ..., xn: a = (1/n) * (x1 + ... + xn).

4. Linearization.
Illustrate, on the example of the function y = x12 + x2, with the nominal values x1 = 1.0 and x2 = 2.0 and measurement errors dx1 = 0.1 and dx2 = -0.1, what will be the error dy as estimated by the linearization method, and how this estimated error compares with the actual value of this error. Compare the analytical expressions for the corresponding partial derivatives with the results of numerical differentiation.

5. Data fusion.
Let us assume that we have measured the same quantity with two different measurement instruments. Both measuring instrument have 0 mean; the first instrument has standard deviation 0.1, the second has standard deviation 0.2. The result of the first measurement is 1.1, the result of the second measurement is 0.9. Combine these two results into a single "fused" value. What is the accuracy (i.e., in this case, what is the standard deviation) of this fused value? Where do the resulting formulas come from (no need for detailed derivations, just explain the main ideas.)

6. Need for data processing.
Explain why we cannot directly measure all physical quantities of interest to us, why we need data processing. Feel free top use either a radar example that we discussed in class or any other example.

7. Data processing: uncertainty.
Let us assume that data processing is performed by using a function y = f(x1, x2) = x12 + x2, the measurement results are X1 = 1.0 and X2 = 2.0, and the measurement errors are independent and normally distributed, with 0 mean and standard deviations s1=0.1 and s1=0.2. Find the standard deviation s of the result of data processing.

8. Uncertainty in data processing: computational aspects.
For the formula for computing uncertainty (i.e., standard deviation) of the result of data processing, explain how the computational complexity (= number of computational steps) depends on the choice of the parameters hi used in numerical differentiation, and what is the choice for which the computational complexity is the smallest.