16.322 Stochastic Estimation and Control, Fall 2004 Prof vander Velde Differential equations relating dynamic parameters Physical characteristics of environment Suppose you are given a set of data and asked to smooth it. Unless you know the physics of the situation you cannot opt for one scheme over another(e. g elevation data for a satellite overpass) To determine where to point your telescope, you can use prior data to calculate the constants that describe the satellite' s orbit, but it may be easier to use the satellite' s current position and velocity to estimate its future position Direct measurements are convenient, but not always possible e. g. temperature of a star. Inferential measurements depend on physical constraints relating measured to estimated quantities. There may be uncertainty in these relationships, which should be modeled somehow A basic principle: should formulate the complete estimate problem at once especially if any non-linear relations would be involved in deriving the desired quantities Example estimation problems Example: Estimate x scalar constant A deterministic quantity, x, is being observed directly-these observations are not necessarily each of equal quality e observations Ek =x+nk The noises are independent, unbiased, normal random variables. They may have different variances ok. The conditional probability density function for Fk, conditioned on a given value of x, is just the density function for n, Page 4 of 816.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde Page 4 of 8 Differential equations relating dynamic parameters Physical characteristics of environment Suppose you are given a set of data and asked to smooth it. Unless you know the physics of the situation you cannot opt for one scheme over another (e.g., elevation data for a satellite overpass). To determine where to point your telescope, you can use prior data to calculate the constants that describe the satellite’s orbit, but it may be easier to use the satellite’s current position and velocity to estimate its future position. Direct measurements are convenient, but not always possible e.g., temperature of a star. Inferential measurements depend on physical constraints relating measured to estimated quantities. There may be uncertainty in these relationships, which should be modeled somehow. A basic principle: should formulate the complete estimate problem at once – especially if any non-linear relations would be involved in deriving the desired quantities. Example estimation problems Example: Estimate x scalar constant A deterministic quantity, x , is being observed directly – these observations are not necessarily each of equal quality. The observations: k k z xn = + The noises are independent, unbiased, normal random variables. They may have different variances 2 σ k . The conditional probability density function for k z , conditioned on a given value of x , is just the density function for k n centered around x