Calculating Running Averages with NumPy and SciPy

Understanding Running Averages

A running average (also known as a moving average or rolling mean) is a technique used to analyze data points by creating a series of averages of different subsets of the complete data set. It’s commonly used in finance to smooth out price data, in signal processing to reduce noise, and in various other applications where trend analysis is important.

In essence, you take a "window" of a specific size (N) and slide it across your data. At each step, you calculate the average of the data points within that window. This provides a smoothed representation of the underlying trend, filtering out short-term fluctuations.

Implementing Running Averages in Python

Python, with its powerful numerical libraries like NumPy and SciPy, makes calculating running averages straightforward. Let’s explore a few approaches.

1. Using NumPy’s convolve Function

The numpy.convolve function performs convolution, a mathematical operation that can be easily adapted to calculate running averages. The key idea is that a running average is a discrete convolution with a uniform kernel (a set of equal weights).

import numpy as np

def running_mean_convolve(x, N):
  """
  Calculates the running mean of a 1D array using convolution.

  Args:
    x: The input 1D NumPy array.
    N: The window size for the running mean.

  Returns:
    A 1D NumPy array containing the running mean.
  """
  return np.convolve(x, np.ones(N) / N, 'valid')

# Example Usage
data = np.random.rand(100)
window_size = 10
running_mean = running_mean_convolve(data, window_size)
print(running_mean)

In this code:

• np.ones(N) / N creates the uniform kernel – an array of N ones divided by N, giving each element a weight of 1/N.
• np.convolve(x, kernel, 'valid') performs the convolution. The 'valid' mode ensures that the output only includes points where the entire window fits within the input array. This results in an output array that is shorter than the input array by N-1 elements.

2. Using NumPy’s Cumulative Sum (cumsum)

An efficient alternative is to use NumPy’s cumsum function to calculate cumulative sums. This approach avoids explicit looping and leverages optimized NumPy operations.

import numpy as np

def running_mean_cumsum(x, N):
  """
  Calculates the running mean of a 1D array using cumulative sum.

  Args:
    x: The input 1D NumPy array.
    N: The window size for the running mean.

  Returns:
    A 1D NumPy array containing the running mean.
  """
  cumsum = np.cumsum(np.insert(x, 0, 0)) 
  return (cumsum[N:] - cumsum[:-N]) / float(N)

# Example Usage
data = np.random.rand(100)
window_size = 10
running_mean = running_mean_cumsum(data, window_size)
print(running_mean)

Here’s how it works:

• np.insert(x, 0, 0) inserts a zero at the beginning of the array. This simplifies the calculation of the running sum.
• np.cumsum(...) calculates the cumulative sum of the modified array.
• (cumsum[N:] - cumsum[:-N]) / float(N) calculates the running mean by subtracting the cumulative sum N elements ago from the current cumulative sum, and then dividing by N.

Important Note: While this cumsum method is often faster, be aware of potential floating-point precision issues when dealing with large datasets. Repeated addition and subtraction can lead to accumulated errors.

3. Using SciPy’s uniform_filter1d

SciPy’s ndimage module provides uniform_filter1d, which is specifically designed for calculating uniform filters (including running averages). This is often the fastest and most accurate option.

import numpy as np
from scipy.ndimage import uniform_filter1d

def running_mean_uniform_filter1d(x, N):
  """
  Calculates the running mean of a 1D array using SciPy's uniform_filter1d.

  Args:
    x: The input 1D NumPy array.
    N: The window size for the running mean.

  Returns:
    A 1D NumPy array containing the running mean.
  """
  return uniform_filter1d(x, size=N, mode='constant', origin=-(N//2))[:-(N-1)]

# Example Usage
data = np.random.rand(100)
window_size = 10
running_mean = running_mean_uniform_filter1d(data, window_size)
print(running_mean)

In this code:

• uniform_filter1d(x, size=N, mode='constant', origin=-(N//2)) applies a uniform filter to the input array x with a window size of N. The mode='constant' handles boundary conditions by extending the array with constant values. origin=-(N//2) centers the window.
• [:-(N-1)] slices the result to match the output length of the other methods.

Performance Considerations

The performance of these methods can vary depending on the size of your dataset and the window size. Generally:

• uniform_filter1d is the fastest and most accurate option, especially for large datasets.
• cumsum can be faster than convolve for smaller datasets, but it’s prone to floating-point errors.
• convolve is a versatile option but can be slower than the other methods.

It’s always a good idea to profile your code with different methods to determine the best approach for your specific use case.