Understanding the Statistics of Kalman Filters I

Kalman Filters uses the concept of correlation and regression. I use the following example.

Let use predict the position of car at time k on the basis of position of car at time  k-1 and velocity of the car at time k. 

So, we apply the concepts of simple linear regression. Here A is the change in the position of car at time k with a unit change in the position of car at time k-1. B is the change in the position of car at time k with a unit change in the velocity of car at time k. Further, 

As these two linear models mentioned above approximate a real life scenario some amount of error is always involved. This error is represented by wk and vk. These are independently and identically normally distributed with a mean 0.These errors are called measurements errors or white noise. This discussion will be continued in the next entries of this blog.

Why do we need to "Test the Hypothesis"?

Testing ESP

The  following example (metaphorically and physically) illustrates the power of the "concept  of hypothesis testing" in introducing objectivity  in any type of research. This could range from social science to medicine. This concept is explained by an example on Extra Sensory Perception (ESP).  A  person  with  ESP is normally is looked at with suspicion as ESP cannot be validated with exact science with measurements and readings. But with "Testing of Hypothesis" ESP or Sixth Sense can be tested or validated objectively. 

Suppose there are 50 red and black cards and a person guesses the color of 32 cards out of 50 cards correctly , in an experiment. In this experiment the person with ESP is supposed to guess the color of cards correctly. The cards are displayed in one room and the person with ESP is in next room. He/she has no prior knowledge of the color arrangement of the cards.

Under normal situations if a person guesses the color of 50 cards out of 50 cards correctly, he is supposed to have some unsual power of foresight called ESP. Also if a person guesses 0 out of 50 cards correctly, he is supposed to have no foresight or no ESP.  Now we try to answer the following  important question statistically.

OUT OF 50 CARDS HOW MANY CARDS SHOULD BE GUESSED CORRECTLY IN ORDER TO BE 99% SURE THAT THE PERSON HAS ESP?

Statistical Illustration of the Example

Statistical Solution to ESP Example