Least squares optimization in SciPy

Introduction

Least squares optimization helps find the best fit line or curve to data by minimizing the total error. It makes predictions closer to actual points.

You want to fit a straight line to data points to see a trend.

You need to find parameters of a model that best explain your data.

You want to reduce errors between predicted and actual values in a dataset.

You want to smooth noisy data by fitting a curve.

You want to estimate unknown values from observed data.

Syntax

SciPy

from scipy.optimize import least_squares

result = least_squares(fun, x0, args=(), kwargs=None, method='trf')

fun is the function that calculates residuals (differences) between model and data.

x0 is the initial guess for the parameters to optimize.

Examples

Fit a line y = a*x + b to data by minimizing residuals.

SciPy

def residuals(params):
    a, b = params
    return a * x_data + b - y_data

result = least_squares(residuals, x0=[1, 0])

Fit a quadratic curve to data using extra arguments.

SciPy

def residuals(params, x, y):
    return params[0] * x ** 2 + params[1] * x + params[2] - y

result = least_squares(residuals, x0=[1, 1, 0], args=(x_data, y_data))

Sample Program

This code fits a quadratic curve to the sample data points by minimizing the difference between the curve and the data.

SciPy

import numpy as np
from scipy.optimize import least_squares

# Sample data points
x_data = np.array([0, 1, 2, 3, 4, 5])
y_data = np.array([1, 3, 7, 13, 21, 31])

# Define residuals function for model y = a*x^2 + b*x + c
def residuals(params, x, y):
    a, b, c = params
    return a * x**2 + b * x + c - y

# Initial guess for parameters a, b, c
initial_guess = [1, 1, 1]

# Run least squares optimization
result = least_squares(residuals, initial_guess, args=(x_data, y_data))

# Print optimized parameters
print(f"Optimized parameters: a={result.x[0]:.2f}, b={result.x[1]:.2f}, c={result.x[2]:.2f}")

Important Notes

The function you minimize should return residuals, not the sum of squares.

Good initial guesses help the optimizer find the best solution faster.

Least squares works well when errors are normally distributed and small.

Summary

Least squares optimization finds parameters that minimize the difference between model and data.

Use scipy.optimize.least_squares by defining a residuals function and giving an initial guess.

This method helps fit lines, curves, or complex models to data.

Practice

1. What is the main goal of using scipy.optimize.least_squares in data fitting?

easy

A. To sort the data points in ascending order

B. To maximize the difference between the model and data

C. To find parameters that minimize the difference between the model and data

D. To randomly select parameters for the model

Solution

1. Step 1: Understand the purpose of least squares

   Least squares optimization aims to find parameters that reduce the error between predicted and actual data.

2. Step 2: Connect to scipy.optimize.least_squares

   This function specifically minimizes the sum of squared residuals, which are differences between model and data.

3. Final Answer:

   To find parameters that minimize the difference between the model and data -> Option C

4. Quick Check:

   Least squares = minimize difference [OK]

Hint: Least squares means minimizing errors, not maximizing [OK]

Common Mistakes:

• Thinking it maximizes difference
• Confusing with sorting or random selection
• Assuming it changes data order

2. Which of the following is the correct way to call scipy.optimize.least_squares with a residual function fun and initial guess x0?

easy

A. least_squares(fun)

B. least_squares(x0, fun)

C. least_squares(fun=x0, x0=fun)

D. least_squares(fun, x0)

Solution

1. Step 1: Check the function signature

   The correct call is least_squares(fun, x0) where fun is the residual function and x0 is the initial guess.

2. Step 2: Verify argument order

   Arguments must be in order: first the function, then the initial guess.

3. Final Answer:

   least_squares(fun, x0) -> Option D

4. Quick Check:

   Function first, initial guess second [OK]

Hint: Function first, initial guess second in call [OK]

Common Mistakes:

• Swapping argument order
• Using keyword arguments incorrectly
• Omitting the initial guess

3. What will be the output of this code snippet?

import numpy as np
from scipy.optimize import least_squares

def residuals(x):
    return np.array([2*x[0] - 4, x[1] + 3])

result = least_squares(residuals, [0, 0])
print(result.x)

medium

A. [4.0, -3.0]

B. [2.0, -3.0]

C. [0.0, 0.0]

D. [-2.0, 3.0]

Solution

1. Step 1: Solve residual equations for zero residuals

   Set residuals to zero: 2*x0 - 4 = 0 => x0 = 2; x1 + 3 = 0 => x1 = -3.

2. Step 2: Confirm least_squares finds these values

   The optimizer finds x = [2, -3] minimizing residuals to zero.

3. Final Answer:

   [2.0, -3.0] -> Option B

4. Quick Check:

   2*2-4=0 and -3+3=0 [OK]

Hint: Set residuals to zero and solve for variables [OK]

Common Mistakes:

• Not solving equations correctly
• Confusing signs in residuals
• Assuming initial guess is output

4. Identify the error in this code snippet using least_squares:

from scipy.optimize import least_squares

def fun(x):
    return x**2 - 4

result = least_squares(fun)
print(result.x)

medium

A. Missing initial guess argument in least_squares call

B. Residual function returns scalar instead of array

C. Function fun should return x**2 + 4

D. Print statement syntax is incorrect

Solution

1. Step 1: Check least_squares function call

   The call lacks the required initial guess argument x0.

2. Step 2: Confirm residual function and print are correct

   The residual function returns an array-like (scalar is acceptable as 1D array), and print syntax is valid.

3. Final Answer:

   Missing initial guess argument in least_squares call -> Option A

4. Quick Check:

   least_squares needs initial guess [OK]

Hint: Always provide initial guess to least_squares [OK]

Common Mistakes:

• Forgetting initial guess
• Thinking scalar residuals cause error
• Misreading print syntax

5. You want to fit a line y = mx + c to data points x = [1, 2, 3] and y = [2, 3, 5] using least_squares. Which residual function correctly represents the difference between observed and predicted values?

hard

A. def residuals(p):\n m, c = p\n return [(m*x[i] + c) - y[i] for i in range(len(x))]

B. def residuals(p):\n m, c = p\n return [y[i] - (m*x[i] + c) for i in range(len(x))]

C. def residuals(p):\n m, c = p\n return [y[i] + (m*x[i] + c) for i in range(len(x))]

D. def residuals(p):\n m, c = p\n return [(m*x[i] - c) - y[i] for i in range(len(x))]

Solution

1. Step 1: Understand residual definition

   Residuals are predicted minus observed values: (model - data).

2. Step 2: Check each function

   def residuals(p):\n m, c = p\n return [(m*x[i] + c) - y[i] for i in range(len(x))] returns (m*x + c) - y, matching predicted minus observed.

3. Final Answer:

   def residuals(p):\n m, c = p\n return [(m*x[i] + c) - y[i] for i in range(len(x))] -> Option A

4. Quick Check:

   Residual = predicted - observed [OK]

Hint: Residual = predicted minus observed values [OK]

Common Mistakes:

• Swapping predicted and observed in residuals
• Adding instead of subtracting values
• Incorrect sign on intercept