How Can Generalized Inverse Help Analyze Non-Uniform Tidal Data?

[henrybrent]

Homework Statement

A magnetic data set is believed to be dominated by a strong periodic tidal signal of known tidal period$$\Omega$$ The field strength $$F(t)$$ is assumed to follow the relation:

$$F=a+b\cos\Omega t + c\sin\Omega t$$

If the data were evenly spaced in time, then Fourier analysis would enable simple determination of the three parameters {a, b, c}. For non-uniform data, one technique to obtain the parameters is to calculate a generalized matrix inverse.

a) Define the model vector m for this problem.
b) Assume we have three measurements $${F_1, F_2, F_3}$$ at times $${t_1, t_2,t_3}$$. Write down the data vector $$\gamma$$ and matrix A you would derive for these three measurements.

c) Hence, calculate the normal equations Matrix $$A^T A$$ and right-hand side vector $$A^T \gamma$$.

d) By generalizing your argument to N data, write down the normal equations matrix.

f) Imagine you now have many evenly spaced data over one full period of the oscillation. Explain why the off leading-diagonal terms of the matrix are now 0. What are the diagonal terms?

g) when the data are evenly spaced, explain why the estimates of the parameters {a,b,c} are independent.

h) What physical properties of the tidal signal could be derived from the values for b and c?

(20 marks)

Homework Equations

Given a vector of model parameters m, a data vector $$\gamma$$ and a matrix A to connect the two vectors, such that $$\gamma = Am$$

a solution for the model parameters can be obtained by solving (inverting) the equation $$(A^T A)m = A^T \gamma$$

The Attempt at a Solution

[/B]
Starting with a), I'm trying to define my model vector.

$$m = 1/(A^T A) * A^T \gamma$$ ??

 

[Mark44]

Homework Statement

A magnetic data set is believed to be dominated by a strong periodic tidal signal of known tidal period$$\Omega$$ The field strength $$F(t)$$ is assumed to follow the relation:

$$F=a+b\cos\Omega t + c\sin\Omega t$$

If the data were evenly spaced in time, then Fourier analysis would enable simple determination of the three parameters {a, b, c}. For non-uniform data, one technique to obtain the parameters is to calculate a generalized matrix inverse.

a) Define the model vector m for this problem.
b) Assume we have three measurements $${F_1, F_2, F_3}$$ at times $${t_1, t_2,t_3}$$. Write down the data vector $$\gamma$$ and matrix A you would derive for these three measurements.

c) Hence, calculate the normal equations Matrix $$A^T A$$ and right-hand side vector $$A^T \gamma$$.

d) By generalizing your argument to N data, write down the normal equations matrix.

f) Imagine you now have many evenly spaced data over one full period of the oscillation. Explain why the off leading-diagonal terms of the matrix are now 0. What are the diagonal terms?

g) when the data are evenly spaced, explain why the estimates of the parameters {a,b,c} are independent.

h) What physical properties of the tidal signal could be derived from the values for b and c?

(20 marks)

Homework Equations

Given a vector of model parameters m, a data vector $$\gamma$$ and a matrix A to connect the two vectors, such that $$\gamma = Am$$

a solution for the model parameters can be obtained by solving (inverting) the equation $$(A^T A)m = A^T \gamma$$

The Attempt at a Solution

[/B]
Starting with a), I'm trying to define my model vector.

$$m = 1/(A^T A) * A^T \gamma$$ ??

What do you know about the matrix A? Is it a square matrix? If so, is it invertible?

If A is invertible, then so is A^T, so solving the equation $A^TAm = A^T\nu$ involves nothing more than left-multiplying both sides of the equation by $(A^T)^{-1}$, and then left-multiplying both sides by $A^{-1}$. There is no division operation for matrices.