Least squares (least_squares) in SciPy

Introduction

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

You want to find a line that best fits scattered data points.

You need to estimate parameters in a model from noisy measurements.

You want to solve equations that don't have exact solutions.

You want to reduce the difference between predicted and actual values.

You want to analyze trends in data with some randomness.

Syntax

SciPy

scipy.optimize.least_squares(fun, x0, args=(), method='trf', jac='2-point', bounds=(-np.inf, np.inf), ...)

fun is the function that calculates residuals (differences).

x0 is the initial guess for the parameters to find.

Examples

Finds x where x² - 4 is close to zero, starting from 1.

SciPy

from scipy.optimize import least_squares

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

result = least_squares(fun, x0=[1])

Fits a line y = m*x + b to points (1,2), (2,4), (3,6).

SciPy

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

x_data = [1, 2, 3]
y_data = [2, 4, 6]

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

Sample Program

This code fits a straight line to points that roughly follow y = 2x + 1. It finds the best slope (m) and intercept (b) that minimize the difference between the line and points.

SciPy

from scipy.optimize import least_squares
import numpy as np

# Define residuals function for line y = m*x + b
# params = [m, b]
def residuals(params, x, y):
    m, b = params
    return m * x + b - y

# Sample data points
x_data = np.array([0, 1, 2, 3, 4])
y_data = np.array([1, 3, 5, 7, 9])  # roughly y = 2*x + 1

# Initial guess for m and b
initial_guess = [0, 0]

# Run least squares to fit line
result = least_squares(residuals, initial_guess, args=(x_data, y_data))

# Extract fitted parameters
m_fit, b_fit = result.x

print(f"Fitted line: y = {m_fit:.2f}*x + {b_fit:.2f}")

Important Notes

Least squares finds parameters that make residuals (differences) as small as possible.

Initial guess affects how fast and well the solution is found.

It works well when the model is close to linear or smooth.

Summary

Least squares finds the best fit by minimizing errors between model and data.

Use scipy.optimize.least_squares with a residual function and initial guess.

It helps estimate parameters for models from noisy or imperfect data.

Practice

1. What is the main purpose of using scipy.optimize.least_squares in data science?

easy

A. To find the best fit parameters by minimizing the difference between model predictions and data

B. To sort data points in ascending order

C. To calculate the mean of a dataset

D. To generate random numbers for simulations

Solution

1. Step 1: Understand the purpose of least squares

   Least squares is used to find parameters that minimize the error between a model and observed data.

2. Step 2: Match the purpose with the options

   Only To find the best fit parameters by minimizing the difference between model predictions and data describes minimizing differences to find best fit parameters.

3. Final Answer:

   To find the best fit parameters by minimizing the difference between model predictions and data -> Option A

4. Quick Check:

   Least squares = minimize error [OK]

Hint: Least squares minimizes errors to fit data best [OK]

Common Mistakes:

• Confusing least squares with sorting or averaging
• Thinking it generates random data
• Assuming it calculates statistics like mean

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=x0, x0=fun)

B. least_squares(x0, fun)

C. least_squares(fun, x0)

D. least_squares(x0)

Solution

1. Step 1: Recall 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: Check each option

   least_squares(fun, x0) matches the correct order and parameters. Others have wrong order or missing arguments.

3. Final Answer:

   least_squares(fun, x0) -> Option C

4. Quick Check:

   Function first, initial guess second [OK]

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

Common Mistakes:

• Swapping the order of arguments
• Passing only one argument
• Using keyword arguments incorrectly

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, x0=[0, 0])
print(result.x)

medium

A. [-2.0, 3.0]

B. [4.0, 3.0]

C. [0.0, 0.0]

D. [2.0, -3.0]

Solution

1. Step 1: Understand the residual function

   The residuals are [2*x0 - 4, x1 + 3]. We want to find x that makes residuals close to zero.

2. Step 2: Solve equations for zero residuals

   Set 2*x0 - 4 = 0 => x0 = 2; and x1 + 3 = 0 => x1 = -3.

3. Final Answer:

   [2.0, -3.0] -> Option D

4. Quick Check:

   Zero residuals at x=[2, -3] [OK]

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

Common Mistakes:

• Not solving residual equations correctly
• Confusing signs in equations
• Assuming initial guess is the answer

4. Identify the error in this code using least_squares:

import numpy as np
from scipy.optimize import least_squares

def residuals(x):
    return 2*x - 5

result = least_squares(residuals, x0=3)
print(result.x)

medium

A. Initial guess x0 should be a list or array, not a scalar

B. Residual function returns a scalar instead of an array

C. Missing import statement for numpy

D. least_squares requires a Jacobian function

Solution

1. Step 1: Check residual function output

   The residual function returns 2*x - 5, which is a scalar, but least_squares expects an array-like residual.

2. Step 2: Verify other parts

   x0 as scalar is allowed; numpy is imported; Jacobian is optional.

3. Final Answer:

   Residual function returns a scalar instead of an array -> Option B

4. Quick Check:

   Residuals must be array-like [OK]

Hint: Residuals must be array, not single number [OK]

Common Mistakes:

• Returning scalar residual instead of array
• Thinking initial guess must be array
• Assuming Jacobian is mandatory

5. You have noisy data points for a line: x = [0,1,2,3], y = [1.1, 2.0, 2.9, 4.2]. Using least_squares, which residual function best fits a line model y = m*x + c to estimate m and c?

hard

A. def residuals(p): return y - (p[0]*x + p[1])

B. def residuals(p): return p[0]*x + p[1]

C. def residuals(p): return (p[0]*x + p[1]) * y

D. def residuals(p): return y / (p[0]*x + p[1])

Solution

1. Step 1: Understand residuals for least squares

   Residuals are differences between observed y and model predictions m*x + c.

2. Step 2: Check residual function forms

   def residuals(p): return y - (p[0]*x + p[1]) returns y - model prediction (m*x + c), the standard residuals to minimize. def residuals(p): return p[0]*x + p[1] returns only the model predictions without subtracting y, so it minimizes the sum of squared model values instead of fitting errors.

3. Step 3: Eliminate incorrect options

   Options C and D multiply or divide, which is incorrect for residuals.

4. Final Answer:

   def residuals(p): return y - (p[0]*x + p[1]) -> Option A

5. Quick Check:

   Residual = observed - predicted [OK]

Hint: Residual = observed minus predicted values [OK]

Common Mistakes:

• Using multiplication or division instead of subtraction
• Forgetting to subtract the observed values
• Ignoring residuals should be array differences