What is the difference between Kalman filter and moving average? I am computing a very simple Kalman filter (random walk + noise model).
I find that the output of the filter is very similar to a moving average.
Is there an equivalence between the two?
If not, what is the difference?
 A: A random walk + noise model can be shown to be equivalent to a EWMA (exponentially weighted moving average). The kalman gain ends up being the same as the EWMA weighting.
This is shown to some details in Time Series Analysis by State Space, if you Google Kalman Filter and EWMA you will find a number of resources that discuss the equivalence. 
In fact you can use the state space equivalence to build confidence intervals for EWMA estimates, etc.
A: To Start: The equivalence of Kalman filter with EWMA is only for the case of a "random walk plus noise" and it is covered in the book, Forecast Structural Time Series Model and Kalman Filter by Andrew Harvey. The equivalence of EWMA with Kalman filter for random walk with noise is covered on page 175 of the text. There the author also mentions that the equivalence of the two was first shown in 1960 and gives the reference to it. Here is the link for that page of the text:
https://books.google.com/books?id=Kc6tnRHBwLcC&pg=PA175&lpg=PA175&dq=ewma+and+kalman+for+random+walk+with+noise&source=bl&ots=I3VOQsYZOC&sig=RdUCwgFE1s7zrPFylF3e3HxIUNY&hl=en&sa=X&ved=0ahUKEwiK5t2J84HMAhWINSYKHcmyAXkQ6AEINDAD#v=onepage&q=ewma%20and%20kalman%20for%20random%20walk%20with%20noise&f=false
Now here is reference which covers an ALETERNATIVE to the Kalman and Extended Kalman filters -- it yielded results that match the Kalman filter but the results are obtained much faster!  It is "Double Exponential Smoothing: An Alternative to Kalman Filter-Based Predictive Tracking."  In Abstract of the paper (see below) the authors state "...empirical results supporting the validity of our claims that these predictors are faster, easier to implement, and perform equivalently to the Kalman and extended Kalman filtering predictors..."
http://cs.brown.edu/~jjl/pubs/kfvsexp_final_laviola.pdf 
This is their Abstract "We present novel algorithms for predictive tracking of user position and orientation based on double exponential smoothing. These algorithms, when compared against Kalman and extended Kalman filter-based predictors with derivative free measurement models, run approximately 135 times faster with equivalent prediction performance and simpler implementations. This paper describes these algorithms in detail along with the Kalman and extended
Kalman Filter predictors tested against. In addition, we describe the details of a predictor experiment and present empirical results supporting the validity of our claims that these predictors are faster, easier to implement, and perform equivalently to the Kalman and extended Kalman filtering predictors."
