**π‘ Problem Formulation:** When working with time series data, calculating the rolling mean is a common task for smoothing the data and identifying trends. Suppose you have a Pandas DataFrame, with a column of numerical data, and you desire to compute the rolling mean with a specific window size. The goal of this article is to demonstrate how to find the rolling mean in Python using Pandas, transforming the input data into a new series where each element is the calculated mean of the preceding elements defined by the window size.

## Method 1: Using `rolling()`

Function with `mean()`

This method involves the `rolling()`

function provided by Pandas, which creates a Rolling object. Upon this object, you can then call the `mean()`

method to compute the rolling mean. It’s a flexible way to specify the number of periods to use for calculating the mean, and it’s particularly effective for time-series data.

Here’s an example:

import pandas as pd # Creating a sample DataFrame data = pd.DataFrame({'values': [2, 4, 6, 8, 10]}) # Calculating the rolling mean with a window of 2 periods rolling_means = data['values'].rolling(window=2).mean() print(rolling_means)

Output:

0 NaN 1 3.0 2 5.0 3 7.0 4 9.0 Name: values, dtype: float64

This code snippet creates a DataFrame with a single column and calculates the rolling mean over a window of two data points. The output shows the rolling mean, with the first value being NaN because there’s no prior data point to form a pair for the first value.

## Method 2: Applying `lambda`

Function with `rolling()`

The `lambda`

function can be utilized in conjunction with the `rolling()`

method to perform more complex rolling calculations, not just the mean. However, for the rolling mean, a lambda function serves to provide explicitness or to chain additional operations

Here’s an example:

import pandas as pd # Creating a sample DataFrame data = pd.DataFrame({'values': range(10)}) # Using a lambda function to calculate the rolling mean rolling_means = data['values'].rolling(window=3).apply(lambda x: x.mean()) print(rolling_means)

Output:

0 NaN 1 NaN 2 1.0 3 2.0 4 3.0 5 4.0 6 5.0 7 6.0 8 7.0 9 8.0 Name: values, dtype: float64

In this code, a lambda function is passed to `apply()`

to calculate the rolling mean over a window of three data points. The lambda function simply calls `mean()`

on the window elements. This can be useful for chaining complex operations within the rolling window.

## Method 3: Expanding Windows

Expanding window calculations start with the first element and increase the window size until it encompasses the entire data set. The `expanding()`

method in conjunction with `mean()`

can be used to calculate a cumulative mean.

Here’s an example:

import pandas as pd # Creating a sample DataFrame data = pd.DataFrame({'values': [1, 3, 5, 7, 9]}) # Calculating the expanding mean expanding_mean = data['values'].expanding().mean() print(expanding_mean)

Output:

0 1.0 1 2.0 2 3.0 3 4.0 4 5.0 Name: values, dtype: float64

This snippet illustrates the use of the expanding mean, which is the mean of the data up to the current point. It differs from a rolling mean in that the window size grows with each new data point.

## Method 4: Weighted Rolling Mean

The weighted rolling mean assigns different weights to the observations in the window, rather than treating them equally. This can be done using the `rolling()`

method combined with the `apply()`

method where a custom function calculates the weighted mean.

Here’s an example:

import pandas as pd import numpy as np # Creating a sample DataFrame data = pd.DataFrame({'values': range(5)}) # Define a custom function for a weighted mean def weighted_mean(x): weights = np.array([0.2, 0.8]) return np.dot(x, weights) / weights.sum() # Applying the weighted mean function over a rolling window weighted_means = data['values'].rolling(window=2).apply(weighted_mean, raw=True) print(weighted_means)

Output:

0 NaN 1 0.8 2 1.8 3 2.8 4 3.8 Name: values, dtype: float64

In this example, a custom weighted mean function is defined which applies higher weight to the more recent value in the window of two. The `apply()`

method uses this function to compute the weighted mean for each window position.

## Bonus One-Liner Method 5: Using `ewm()`

for Exponential Weighted Moving Mean

The `ewm()`

method in Pandas computes the exponential weighted moving average (EWMA), which gives more weight to recent observations. This method can be considered a form of weighted rolling mean where the weights decrease exponentially.

Here’s an example:

import pandas as pd # Creating a sample DataFrame data = pd.DataFrame({'values': range(5)}) # Calculating the exponential weighted moving mean with a span of 2 exp_weighted_mean = data['values'].ewm(span=2).mean() print(exp_weighted_mean)

Output:

0 0.000000 1 0.750000 2 1.615385 3 2.550000 4 3.520661 Name: values, dtype: float64

This one-liner demonstrates the use of the `ewm()`

method to calculate the EWMA with a span of 2, which provides a smoother series that reacts more to recent values in the time series.

## Summary/Discussion

**Method 1: Rolling Mean with**Standard approach for moving averages. Simple to use. Does not handle NaN values by default, and it requires a sufficient number of observations to fill the window.`mean()`

.**Method 2: Rolling Mean with**Offers customizability for complex operations. Slightly more verbose. Suitable for chaining multiple operations.`lambda`

.**Method 3: Expanding Mean.**Useful for cumulative mean over time. Only requires a single data point to begin. Can be less representative of recent trends due to cumulative nature.**Method 4: Weighted Rolling Mean.**Beneficial when different weights are needed. Requires custom function for weights. More complex but allows for customization of weights.**Method 5: Exponential Weighted Moving Mean.**Great for emphasizing recent data. Reacts quickly to changes. May be too reactive for some applications depending on span.