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Unformatted text preview: CS 70 Discrete Mathematics and Probability Theory Spring 2010 Alistair Sinclair Lecture 20 Inference Example 3: 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 is yet another example of an inference problem. In Note 18 we considered examples where the unknown and the observations are discrete (multi-armed bandit, communication over binary sym- metric channels); 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 (in a very simple special case). Modeling The situation is shown in Figure 1. The underlying signal is modeled by a sequence of random variables X , 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 signals X i are described by: X ∼ N ( μ , σ 2 ) (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 and of the Z i ’s....
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This note was uploaded on 09/21/2010 for the course CS 70 taught by Professor Papadimitrou during the Spring '08 term at Berkeley.
- Spring '08