Least squares (least_squares) in SciPy - Cheat Sheet & Quick Revision

Recall & Review

beginner

What is the purpose of the least_squares function in SciPy?

The least_squares function finds the best fit parameters for a model by minimizing the sum of the squares of the residuals (differences between observed and predicted values).

beginner

What kind of problems is least_squares used to solve?

It is used to solve nonlinear or linear least squares problems, where you want to find parameters that minimize the difference between data and a model.

intermediate

What inputs does least_squares require?

It requires a function that computes residuals (differences between observed and predicted), an initial guess for parameters, and optionally bounds or method settings.

intermediate

What does the output of least_squares contain?

The output is an object with the optimized parameters, the cost (half the sum of squares), success status, and other information about the optimization process.

beginner

Why is least squares fitting important in data science?

Because it helps find the best model parameters that explain data, making predictions more accurate and understanding relationships clearer.

What does the least_squares function minimize?

A Sum of squared residuals

B Sum of absolute residuals

C Maximum residual

D Sum of residuals

The least_squares function minimizes the sum of squared residuals, which are the squared differences between observed and predicted values.

Which argument is required to start least_squares optimization?

A Final solution

B Initial guess of parameters

C Residual sum

D Data labels

You must provide an initial guess of parameters so the algorithm knows where to start searching for the best fit.

What type of problems can least_squares solve?

A Both linear and nonlinear problems

B Only nonlinear problems

C Only classification problems

D Only linear problems

least_squares can solve both linear and nonlinear least squares problems.

What does the 'cost' in the least_squares output represent?

A Number of iterations

B Sum of residuals

C Sum of squared residuals divided by 2

D Initial guess value

The 'cost' is half the sum of squared residuals, representing how well the model fits the data.

Which SciPy module contains the least_squares function?

A scipy.linalg

B scipy.stats

C scipy.interpolate

D scipy.optimize

least_squares is part of the scipy.optimize module used for optimization tasks.

Explain how the least_squares function works to fit a model to data.

Think about how differences between data and model are reduced.

Describe the key outputs you get from running least_squares and what they mean.

Consider what information helps you understand the fitting result.

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