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Unformatted text preview: CS 70-2 Discrete Mathematics and Probability Theory Spring 2009 Alistair Sinclair, David Tse Lecture 20 Inference Example 4: The Kalman Filter Question: You would like to build an automatic system to land a spacecraft on the moon. To steer the spacecraft properly, the system needs to estimate the current location of the spacecraft relative to the moon surface. Unfortunately, the sensors are noisy. How can the system best estimate the current location given all the noisy measurements of the past trajectory of the spacecraft? Comment 1: This is an example of a very common problem in a diverse range of fields, such as control, signal and image processing, computer vision, finance etc. The general problem is to recover an underlying signal from noisy observations and perhaps to predict its future trajectory. The signal may be an image, an audio signal, trajectory of an aircraft, quality of a stock, etc. This class of problem is called filtering , denois- ing or prediction . The idea is to separate out the underlying signal from the noise. What distinguishes the signal from the noise is that the signal is often ”smooth”: location of the spacecraft from one measurement to the next does not change much, values of adjacent pixels of an image are likely to be similar. On the other hand, the noise is highly random and varies significantly from one measurement to the next. Comment 2: This problem is yet another example of inference problems. We have considered examples where both the unknown and the observations are discrete (communication over binary symmetric channels), the unknown is discrete and the observations are continuous (communication over the Gaussian channel), and now we are considering an example where both the unknown (the underlying signal) and the noisy observations are continuous. Comment 3: Historically, this problem first arose in the 1960’s in the Apollo space program to land Ameri- cans on the moon. The solution of this problem is the celebrated Kalman filter , which we will now describe (for a very simple model). Modeling The situation is shown in Figure 1. The underlying signal is modeled by a sequence of random variables X 1 , X 2 ,... . The noisy observations are Y 1 , Y 2 ,... , given by Y i = X i + Z i , i = 1 ,... The Z i ’s are i.i.d. N ( , σ 2 Z ) r.v.’s and independent of the X i ’s. The signal X i ’s are described by: X 1 ∼ N ( μ X , σ 2 X ) (1) X i + 1 = α X i + W i , i = 1 , 2 ,..., (2) where the W i ’s are i.i.d. N ( , σ 2 W ) r.v.’s, independent of X 1 and of the Z i ’s. Note that the observation noises Z i ’s are independent from measurement to measurement. On the other hand, the signal values at different times can be strongly dependent (think of the case when α is close to 1 and the perturbation W i has a small variance.). Thus the signal varies relatively smoothly compared to the CS 70-2, Spring 2009, Lecture 20 1 Figure 1: The system diagram for the filtering problem.Figure 1: The system diagram for the filtering problem....
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- Spring '08
- Conditional Probability, Probability theory, conditional density