# Linear Parameter Estimation with Singular Value Decomposition

Topic 5.6 will introduce the principles involved in performing parameters estimation with model linearization in terms of the required parameters. This procedures assumes that noise corrupting the data is white, therefore we will also provide an introduction to data whitening.

The complete procedure can be summarised as:

1. Model linearization. The model linearized in terms of the fit parameters. Which correspond to substitute the model with its Taylor series expansion at the first order.
2. Data whitening. The noise on our data is non-white. Since least-squares methods can be rigorously applied only if the noise corrupting the data is white and uncorrelated, we need a noise whitening step before the fitting operation.
3. Generate whitened templates. In order to correctly perform the fit, both the nominal response and the fit basis need to be whitened. The nominal response is the model response when parameters are set to their nominal values. Fit basis is the response of the first order term of the Taylor series expansion of the model.
4. Basis change. It often happens that the fit basis is composed of linearly dependent elements since several parameters are physically indistinguishable. The system cannot be solved in such conditions, therefore we perform a change of basis with the singular value decomposition (SVD) algorithm. The SVD ensures that we deal with a linearly independent fit basis. The procedure is suited to perform the estimation of parameters value from the combination of the knowledge provided by different experiments on the same system. This ensure, in principle, that the value of each parameter can be obtained with proper accuracy.
5. Fit. The system of equations for the t are nally solved in order to get the best estimation for the parameters.
6. Check convergence and goto 3. The process is iterated until the convergence of the parameters is reached. At each step the nominal values of the parameters are updated with the current knowledge.
7. Change basis back to physical parameters. At this stage we revert the change of the fit basis and recover the values for the Physical parameters with their corresponding errors.

Next sections cover:

1. Building whitening filters
2. Linear Parameter Estimation