Least squares optimization 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'

Aleast_squares

Bcurve_fit

Cminimize

Dlinprog

Attempts:

3 left

💡 Hint

Common Mistakes

Importing minimize instead of least_squares.

Confusing curve_fit with least squares function.

The least_squares function is used for least squares optimization in scipy.optimize.

2fill in blank

medium

Complete the code to define the residuals function for least squares optimization.

SciPy

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

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

Aparams / x

Bparams + x

Cparams * x

Dparams - x

Attempts:

3 left

💡 Hint

Common Mistakes

Using addition or division instead of multiplication for prediction.

Subtracting x from params instead of multiplying.

The residuals are the difference between observed y and the model prediction params * x.

3fill in blank

hard

Fix the error in the least squares call to correctly pass the residuals function and data.

SciPy

result = least_squares([1], 1.0, args=(x, y))

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

Aresiduals(params, x, y)

Bresiduals

Cresiduals(params)

Dresiduals(x, y)

Attempts:

3 left

💡 Hint

Common Mistakes

Calling the function instead of passing it.

Passing wrong number of arguments.

The function residuals should be passed without calling it, so it can be called internally by least_squares.

4fill in blank

hard

Fill both blanks to create a dictionary comprehension that maps words to their lengths only if length is greater than 3.

SciPy

lengths = {word: [1] for word in words if [2]

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Alen(word)

Blen(word) > 3

Cword > 3

Dlen(words) > 3

Attempts:

3 left

💡 Hint

Common Mistakes

Using word > 3 which compares string to number.

Using len(words) > 3 which checks length of the whole list.

The dictionary comprehension maps each word to its length using len(word), and filters words with length greater than 3 using len(word) > 3.

5fill in blank

hard

Fill all three blanks to create a dictionary comprehension that maps uppercase words to their values only if the value is positive.

SciPy

result = [1]: [2] for [3] in data.items() if [2] > 0

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

Aword.upper()

Bvalue

Cword, value

Dkey

Attempts:

3 left

💡 Hint

Common Mistakes

Using key instead of word for iteration variable.

Not filtering values greater than zero.

The comprehension maps uppercase words (word.upper()) to their values (value) for each word and value in data.items(), filtering values greater than zero.

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