Least squares (least_squares) in SciPy - Interactive Code Practice

Practice - 5 Tasks

1fill in blank

easy

Complete the code to import the least_squares function from scipy.optimize.

SciPy

from scipy.optimize import [1]

Drag options to blanks, or click blank then click option'

Aminimize

Blinprog

Cleast_squares

Dcurve_fit

Attempts:

3 left

💡 Hint

Common Mistakes

Importing the wrong function like minimize or curve_fit.

Forgetting to import from scipy.optimize.

The least_squares function is imported from scipy.optimize to solve least squares problems.

2fill in blank

medium

Complete the code to define the residual function for least squares fitting.

SciPy

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

Drag options to blanks, or click blank then click option'

Aparams + x

Bx * params[0] + params[1]

Cparams * x

Dparams[0] * y + params[1]

Attempts:

3 left

💡 Hint

Common Mistakes

Using params as a scalar instead of indexing.

Adding params and x directly without multiplication.

The residual function returns the difference between observed y and the model prediction x * params[0] + params[1], which is a linear model.

3fill in blank

hard

Fix the error in the least squares call to fit the model.

SciPy

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

Drag options to blanks, or click blank then click option'

A[0, 0]

Bx_data

Cy_data

Dresiduals

Attempts:

3 left

💡 Hint

Common Mistakes

Passing data arrays instead of initial guess.

Passing the residual function again instead of initial guess.

The initial guess for parameters must be a list or array, here [0, 0] for slope and intercept.

4fill in blank

hard

Fill both blanks to extract the optimized parameters and print the slope.

SciPy

optimized_params = result.[1]
print('Slope:', optimized_params[2])

Drag options to blanks, or click blank then click option'

Ax

By

Cparams

D[0]

Attempts:

3 left

💡 Hint

Common Mistakes

Using wrong attribute names like params or jac.

Not indexing the parameter array to get the slope.

The optimized parameters are stored in result.x. The slope is the first parameter, accessed by [0].

5fill in blank

hard

Fill all three blanks to create a dictionary of parameter names and values for slope and intercept.

SciPy

param_dict = { [1]: result.[2][0], [3]: result.x[1]}

Drag options to blanks, or click blank then click option'

A'slope'

Bx

C'intercept'

Attempts:

3 left

💡 Hint

Common Mistakes

Using wrong keys or missing quotes.

Not accessing the correct indices in x.

The dictionary keys are strings: 'slope' and 'intercept'. The values come from result.x at indices 0 and 1.

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