Fitting a Sine Function to Scatter Plot Data

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What will you learn?

In this comprehensive tutorial, you will master the art of fitting a sine function to scatter plot data using Python. This skill is invaluable for various applications such as signal processing and analyzing periodic phenomena.

Introduction to the Problem and Solution

Encountering datasets with periodic tendencies, like seasonal temperature variations or sound waves, is common. By modeling such data with a sine function, we can unveil underlying patterns. The challenge lies in determining the optimal parameters (amplitude, frequency, phase shift) for the sine wave that best fits our data.

To tackle this challenge, we will leverage curve fitting techniques available in Python libraries like NumPy and SciPy. These tools empower us to define a generic sine function and iteratively adjust its parameters until we achieve the ideal fit. This process not only ensures accurate modeling of our dataset but also reveals insightful characteristics within the data.

Code

import numpy as np
from scipy.optimize import curve_fit
import matplotlib.pyplot as plt

# Define our sine function: f(x) = A * sin(Bx + C)
def sine_function(x, A, B, C):
    return A * np.sin(B * x + C)

# Example data (replace these with your actual data points)
x_data = np.linspace(0, 4*np.pi, 100)
y_data = 3 * np.sin(2 * x_data + 1.5) + np.random.normal(size=len(x_data))

# Fit our sine function to the data
params, params_covariance = curve_fit(sine_function, x_data, y_data)

# Plotting original data vs fitted curve 
plt.figure(figsize=(10, 6))
plt.scatter(x_data,y_data,label='Data')
plt.plot(x_data,sine_function(x_data,params[0],params[1],params[2]),color='red',label='Fitted function')
plt.legend()
plt.show()

# Copyright PHD

Explanation

The solution commences by defining sine_function, representing our fitting model that computes (A \cdot \sin(Bx + C)), where (A), (B), and (C) denote amplitude, frequency (angular), and phase shift respectively.

Subsequently, synthetic sample data (x_data and y_data) mimicking real-world scenarios is generated; noise is introduced into y_data through np.random.normal. In practical scenarios, this step would involve loading your dataset.

The crux of our approach utilizes curve_fit from SciPy’s optimization module. This function takes our defined sine_function, along with initial parameter estimates (if needed), refining these values iteratively until they minimize the disparity between model predictions and observed values – effectively tailoring our model to fit observations accurately.

Finally plotting both original scattered datapoints alongside our fitted curve visually confirms how well our sine model approximates real behavior within given dataset.

    1. How do I choose initial parameter estimates for better convergence? One way is by visually inspecting your dataset�s pattern or utilizing domain knowledge about expected frequency/amplitude ranges which may inform better starting guesses.

    2. Can I fit functions other than a sine wave? Absolutely! The approach remains similar; just define your target function with unknown parameters you aim to optimize.

    3. What if my fitting doesn’t converge? Try different initial parameter estimates or increase max iterations allowed (maxfev parameter in curve_fit). Sometimes preprocessing steps like normalization might help too.

    4. How do I evaluate my fit’s quality? Common metrics include sum of squared errors (SSE) available via covariance matrix returned by curve_fit or R-squared statistic comparing fitted values against mean of observed ones.

    5. Can I apply weights to my datapoints during fitting? Yes! Use ‘sigma’ argument in curve_fit if some datapoints are more reliable than others so they have more influence on final fitted parameters.

Conclusion

Mastering the technique of fitting a sine wave onto scatter plot data unlocks powerful analysis capabilities across various domains where cyclic behaviors prevail. Python’s scientific stack simplifies intricate numerical computations ensuring accessible yet robust solutions suitable for beginners seeking fundamental comprehension and experts aiming at precision-engineered outputs alike.

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