If you haven't heard, Python loops can be slow--especially when working with large datasets. If you're trying to make calculations across millions of data points, execution time can quickly become a bottleneck. Luckily for us, Numba has a Just-in-Time (JIT) compiler that we can use to help speed up our numerical computations and loops in Python.

The other day, I found myself in need of a simple exponential smoothing function in Python. This function needed to take in array and return an array of the same length with the smoothed values. Typically, I try and avoid loops where possible in Python (especially when dealing with Pandas DataFrames). At my current level of capability, I didn't see how to avoid using a loop to exponentially smooth an array of values.

I am going to walk through the process of creating this exponential smoothing function and testing it with and without the JIT compilation. I'll briefly touch on JIT and how I made sure to code the the loop in a manner that worked with the nopython mode.

What is JIT?

JIT compilers are particularly useful with higher-level languages like Python, JavaScript, and Java. These languages are known for their flexibility and ease of use, but they can suffer from slower execution speeds compared to lower-level languages like C or C++. JIT compilation helps bridge this gap by optimizing the execution of code at runtime, making it faster without sacrificing the advantages of these higher-level languages.

When using the nopython=True mode in the Numba JIT compiler, the Python interpreter is bypassed entirely, forcing Numba to compile everything down to machine code. This results in even faster execution by eliminating the overhead associated with Python's dynamic typing and other interpreter-related operations.

Building the fast exponential smoothing function

Exponential smoothing is a technique used to smooth out data by applying a weighted average over past observations. The formula for exponential smoothing is:

where:

• $S_t$ : Represents the smoothed value at time $t$ .
• $V_t$ : Represents the original value at time $t$ from the values array.
• $\alpha$ : The smoothing factor, which determines the weight of the current value $V_t$ in the smoothing process.
• $S_{t-1}$ : Represents the smoothed value at time $t-1$ , i.e., the previous smoothed value.

The formula applies exponential smoothing, where:

• The new smoothed value $S_t$ is a weighted average of the current value $V_t$ and the previous smoothed value $S_{t-1}$ .
• The factor $\alpha$ determines how much influence the current value $V_t$ has on the smoothed value compared to the previous smoothed value $S_{t-1}$ .

To implement this in Python, and stick to functionality that works with nopython=True mode, we will pass in an array of data values and the alpha float. I default the alpha to 0.33333333 because that fits my current use case. We will initialize an empty array to store the smoothed values in, loop and calculate, and return smoothed values. This is what it looks like:

@jit(nopython=True) 
def fast_exponential_smoothing(values, alpha=0.33333333): 

    smoothed_values = np.zeros_like(values) # Array of zeros the same length as values
    smoothed_values[0] = values[0] # Initialize the first value 

    for i in range(1, len(values)): 
        smoothed_values[i] = alpha * values[i] + (1 - alpha) * smoothed_values[i - 1]
    return smoothed_values

Simple, right? Let's see if JIT is doing anything now. First, we need to create a large array of integers. Then, we call the function, time how long it took to compute, and print the results.

# Generate a large random array of a million integers
large_array = np.random.randint(1, 100, size=1_000_000)

# Test the speed of fast_exponential_smoothing
start_time = time.time()
smoothed_result = fast_exponential_smoothing(large_array)
end_time = time.time()
print(f"Exponential Smoothing with JIT took {end_time - start_time:.6f} seconds with 1,000,000 sample array.")

This can be repeated and altered just a bit to test the function without the JIT decorator. Here are the results that I got:

Wait, what the f***?

I thought JIT was supposed to speed it up. It looks like the standard Python function beat the JIT version and a version that attempts to use no recursion. That's strange. I guess you can't just slap the JIT decorator on something and make it go faster? Perhaps simple array loops and NumPy operations are already pretty efficient? Perhaps I don't understand the use case for JIT as well as I should? Maybe we should try this on a more complex loop?

Here is the entire code python file I created for testing:

import numpy as np
from numba import jit
import time

@jit(nopython=True) 
def fast_exponential_smoothing(values, alpha=0.33333333): 

    smoothed_values = np.zeros_like(values) # Array of zeros the same length as values
    smoothed_values[0] = values[0] # Initialize the first value 

    for i in range(1, len(values)): 
        smoothed_values[i] = alpha * values[i] + (1 - alpha) * smoothed_values[i - 1]
        return smoothed_values

def fast_exponential_smoothing_nojit(values, alpha=0.33333333):

    smoothed_values = np.zeros_like(values) # Array of zeros the same length as values
    smoothed_values[0] = values[0] # Initialize the first value 

    for i in range(1, len(values)): 
        smoothed_values[i] = alpha * values[i] + (1 - alpha) * smoothed_values[i - 1]
        return smoothed_values

def non_recursive_exponential_smoothing(values, alpha=0.33333333):
    n = len(values)
    smoothed_values = np.zeros(n)

    # Initialize the first value
    smoothed_values[0] = values[0]

    # Calculate the rest of the smoothed values
    decay_factors = (1 - alpha) ** np.arange(1, n)
    cumulative_weights = alpha * decay_factors
    smoothed_values[1:] = np.cumsum(values[1:] * np.flip(cumulative_weights)) + (1 - alpha) ** np.arange(1, n) * values[0]

    return smoothed_values

# Generate a large random array of a million integers
large_array = np.random.randint(1, 1000, size=10_000_000)

# Test the speed of fast_exponential_smoothing
start_time = time.time()
smoothed_result = fast_exponential_smoothing_nojit(large_array)
end_time = time.time()
print(f"Exponential Smoothing without JIT took {end_time - start_time:.6f} seconds with 1,000,000 sample array.")

# Test the speed of fast_exponential_smoothing
start_time = time.time()
smoothed_result = fast_exponential_smoothing(large_array)
end_time = time.time()
print(f"Exponential Smoothing with JIT took {end_time - start_time:.6f} seconds with 1,000,000 sample array.")

# Test the speed of fast_exponential_smoothing
start_time = time.time()
smoothed_result = non_recursive_exponential_smoothing(large_array)
end_time = time.time()
print(f"Exponential Smoothing with no recursion or JIT took {end_time - start_time:.6f} seconds with 1,000,000 sample array.")

I attempted to create the non-recursive version to see if vectorized operations across arrays would make it go faster, but it seems to be pretty damn fast as it is. These results remained the same all the way up until I didn't have enough memory to make the array of random integers.

Let me know what you think about this in the comments. I am by no means a professional developer, so I am accepting all comments, criticisms, or educational opportunities.

Until next time.

Happy coding!