(Sandipan Dey 5 Feb 2017)
In the edX Course DelftX: OT.1x Observation theory: Estimating the Unknown,a very systematic and conceptual approach in dealing with estimation problems was introduced. The approach involves a physical world, and a mathematical world. Conceptually, when we deal with a real-life problem, there are the following three stages: In the first stage our problem is defined in the real world, but we need to translate it to the mathematical world. In this mathematical world we can apply all our tricks and methods to get optimal numbers as a result. We call this ‘linking the mathematical model to the real world’. In the last stage, we need to go in the other direction: the results from the procedures in the mathematical world need to be transferred back to reality, where decisions need to be made. We call this the ‘unlinking’ or the disengagement of the ’mathematical model’and one particular type that is discussed in this course is the “Model of Observation Equations”.
The observations are denoted with the letter ‘y’. Consider observations in the case the measurements have not been taken yet, in which case, they are generally referred to as ‘observables, whose values should be interpreted as stochastic (or random) variables. The unknown parameter is indicated by the letter ’x’. For the functional model, the notation for this functional relation is denoted as ‘A’ and initially the linear relations are dealt with, in which case, ‘A’ becomes a matrix. Since all measurements will be affected by errors (either they are small or big), we need to include these errors also in the model. This is taken care of by the Stochastic model, which we need to describe the uncertainty in our observations due to the random errors.
With the above introduction from the course itself let’s consider solving the following problems that appeared as assignments in the same course and visualize the outputs from the models fitted. The following figure shows the summary of the theory to be used to solve the parameter estimation problems.Here the Least Square Estimator (LSE), Weighted Least Square Estimator (WLSE) with a given weight matrix and the Best Linear Unbiased Estimators (BLUE) will be considered.
Here are the equations that are going to be used.
Here is the observation dataset:
The octave output is shown below:
Next figure shows the R output of the 97% confidence interval for the BLUE.