gianluca Posted September 24 Share Posted September 24 Hello, An overdetermined system of linear equation y = A x + z with y vector of known real numbers of dimension m; x vector of unknown real numbers of dimension n; z vector of Gaussian noise of dimension m and A the known coefficient matrix. it is characterized by 3 aspects: 1) The unknown x exhibits elements with order of magnitude difference among them. example: x is 4 elements and I know in advance that two of them will be around 10^4 and 2 around 10^0 2) The vector z is a noise and each of its element is a Gaussian number with zero mean and known variance. Basically those are measurements coming from sensors of different "quality", i.e., different variance 3) Eventually z is composed by elements with a predominant variance. Example, 80% of the elements of z comes from the same sensor with the same variance and 20% from others Question: can someone please link me to a textbook where such numerical aspects are elaborated? I'm not an expert but I guess that a simple pseudoinverse is not the "best" solution Thanks in advance, g. Link to comment Share on other sites More sharing options...

studiot Posted September 24 Share Posted September 24 Hello, You have an over determined system that is linear. There are no unique solutions to this situation however The branch of mathematics dealing with this is called Linear Programming ( There is also non linear programming for non linear equations). The method identifies a 'convex hull' in the variable space, bounded in the linear case by lines or flat surfaces / hypersurfaces depending upon the number of variables. The over determined system is turned into a determined system by introducing additional constraints to identify the optimum maximum or minimum solution. Probabilities are dealt with by adding weighting coefficients to the equation matrix. https://en.wikipedia.org/wiki/Linear_programming Link to comment Share on other sites More sharing options...

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