R and Bigdata and wolfram mathematica

Big Data Analitycs - Spring 2016 - Homework #4 In this homework you are asked to write a program which would im- plement and demonstrate various aspects of optimal linear estimation. The underlying process imitates a simple signal measurement experi- ment.

1. Measurement Simulation.

(a) Choose some profile x or generate it randomly:

i. Generate randomly µ ∼ (0, I), i.e., mi are i.i.d. with Eµ = 0 and Var(µ) = 1.

ii. “Smooth” it with some matrix B. iii. Then x = Bµ ∼ (0, F), where F = BBT.

(b) Create a matrix A.

(c) Simulate a measurement y = Ax + ν.

2. Estimation: Construct an optimal linear estimate x̂ and the variance matrix Var(x̂ − x). Show on the same graph:

(a) The original signal x (a curve with components xi),

(b) Its estimate x̂ (a curve with components x̂i),

(c) Standard deviations for the estimates x̂i (= √

Var(x̂ − x)ii, can be illustrated by showing the corresponding “corridor” around x̂i).

3. Illustrate estimation (Phase 2) in different settings:

(a) Single measurement (y, A, S).

i. Transform (y, A, S) to canonical form (T, v). ii. Construct the estimate, based on the canonical information.

(b) Single measurement (y, A, S) with a priori information x ∼ (0, F): i. Transform a priori information to canonical form.

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ii. Transform the measurement to canonical form. iii. Combine pieces of canonical information. iv. Construct the estimate, based on the combined canonical infor-

mation.

(c) Many measurements (yj, Aj, Sj), no a priori information.

i. Simulate a sequence of measurements of the same signal x, but with differing matrices Aj (and, possibly, Sj).

ii. Extract canonical information from each measurement. iii. Combine pieces of canonical information. iv. Construct the estimate, based on the combined canonical infor-

mation.

(d) Many measurements (yj, Aj, Sj), now with a priori information x ∼ (0, F). Same steps as in item (c). i. Extract canonical information from a priori information.

ii. Simulate a sequence of measurements of the same signal x, but with differing matrices Aj (and, possibly, Sj).

iii. Extract canonical information from each measurement. iv. Combine pieces of canonical information. v. Construct the estimate, based on the combined canonical infor-

mation.

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