Least squares optimization in SciPy - Mini Project: Build & Apply

Least squares optimization

📖 Scenario: You are working as a data scientist helping a small business understand the relationship between advertising spending and sales. You have collected data on advertising budgets and sales figures. Your goal is to find the best straight line that fits this data using least squares optimization.

🎯 Goal: Build a simple least squares optimization model using scipy.optimize.least_squares to find the best line parameters (slope and intercept) that fit the sales data.

📋 What You'll Learn

Create arrays for advertising budgets and sales data

Define a residual function for the least squares method

Use scipy.optimize.least_squares to find the best slope and intercept

Print the optimized slope and intercept values

💡 Why This Matters

🌍 Real World

Least squares optimization is used in many fields like economics, engineering, and science to find the best fit model for data.

💼 Career

Data scientists and analysts often use least squares methods to build predictive models and understand relationships between variables.

1

Create the data arrays

Create two numpy arrays called advertising and sales with these exact values: advertising = [5, 10, 15, 20, 25] and sales = [7, 12, 14, 22, 27].

SciPy

import numpy as np
# Create the advertising and sales arrays
# Your code here

Hint

Use np.array([...]) to create numpy arrays with the given values.

2

Define the residual function

Define a function called residuals that takes params as input. Inside, unpack params into slope and intercept. Return the difference between the predicted sales (slope * advertising + intercept) and the actual sales.

SciPy

import numpy as np
advertising = np.array([5, 10, 15, 20, 25])
sales = np.array([7, 12, 14, 22, 27])

# Define the residuals function
# Your code here

Hint

The residuals function calculates how far off the predicted sales are from the actual sales for given slope and intercept.

3

Run least squares optimization

Import least_squares from scipy.optimize. Use least_squares with the residuals function and an initial guess [1, 0] for slope and intercept. Store the result in a variable called result.

SciPy

import numpy as np
advertising = np.array([5, 10, 15, 20, 25])
sales = np.array([7, 12, 14, 22, 27])

def residuals(params):
    slope, intercept = params
    return slope * advertising + intercept - sales

# Import least_squares and run optimization
# Your code here

Hint

Use least_squares(residuals, [1, 0]) to start with slope=1 and intercept=0.

4

Print the optimized parameters

Print the optimized slope and intercept from result.x using print(f"Slope: {result.x[0]:.2f}") and print(f"Intercept: {result.x[1]:.2f}").

SciPy

import numpy as np
advertising = np.array([5, 10, 15, 20, 25])
sales = np.array([7, 12, 14, 22, 27])

def residuals(params):
    slope, intercept = params
    return slope * advertising + intercept - sales

from scipy.optimize import least_squares
result = least_squares(residuals, [1, 0])

# Print the optimized slope and intercept
# Your code here

Hint

Use print(f"Slope: {result.x[0]:.2f}") and print(f"Intercept: {result.x[1]:.2f}") to show results rounded to two decimals.

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