# Natural Image Statistics

I’m doing some simple exploration of image statistics on a large database of natural images. The first thing that I tried was computing the histogram of neighboring image pixel intensity differences. Here is the graph for that using a log y axis, for a few pixel separations.

It is clear that large differences occur much more rarely and that the most probable pixel to pixel spatial change in intensity is zero. However the tails are heavy, so it is nothing like a Gaussian distribution. The adjacent pixel intensity difference log probabilities were fairly well fitted by a function that goes like $-|kx|^{0.5}$, and the pixels further apart require a smaller exponent.

# Analyzing the Market – Part 2

This is part 2 of my series of posts on the statistics of financial markets. Part 1 is here.

I have established that a double exponential distribution fits price movements when they are converted to log prices, at least for bitcoin, Apple, and Dell. (Actually I have checked it on a few other NASDAQ stocks too.)

Once we have a statistical model, we can generate some data to see if it produces results that look like the actual price graph. Below you can see the real 2 month bitcoin price graph, together with two graphs that were obtained by using a model based on the Continue reading

# Analyzing the Market – Part 1

This series of blog posts is intended to document some mathematical analysis that I have been doing on the bitcoin price graph and on price histories of securities in the stock market. The purpose is to understand something about the statistics of these price movements, and to learn about the behavior of the stock market in general.

One thing that is useful about bitcoin is that trading is never stopped. Because everything runs 24 hours 7 days per week, there are no artifacts to do with starting and stopping trading on specific exchanges and transitioning between financial Continue reading