# Numerical Solutions to ODEs

In this post I’ll present some theory and Python code for solving ordinary differential equations numerically. I’ll discuss Euler’s Method first, because it is the most intuitive, and then I’ll present Taylor’s Method, and several Runge-Kutta Methods. Obviously, there is top notch software out there that does this stuff in its sleep, but it’s fun to do math and write programs. This material is adapted from the excellent textbook by Burden and Faires, Numerical Analysis 8th Ed., which is easily worth whatever they’re asking for it these days.

# Linear Regression with Python

In this post I will use Python to explore more measures of fit for linear regression. I will consider the coefficient of determination (R2), hypothesis tests (, , Omnibus), AIC, BIC, and other measures. This will be an expansion of a previous post where I discussed how to assess linear models in R, via the IPython notebook, by looking at the residual, and several measures involving the leverage.

# Smoothing with Exponentially Weighted Moving Averages

A moving average takes a noisy time series and replaces each value with the average value of a neighborhood about the given value. This neighborhood may consist of purely historical data, or it may be centered about the given value. Furthermore, the values in the neighborhood may be weighted using different sets of weights. Here is an example of an equally weighted three point moving average, using historical data,