Fitting custom models in SciPy - Mini Project: Build & Apply

Fitting Custom Models with SciPy

📖 Scenario: You work as a data analyst for a small company that collects data on how the temperature affects the sales of ice cream. You want to find a simple mathematical model that fits the sales data so you can predict future sales based on temperature.

🎯 Goal: Build a custom mathematical model and fit it to the given temperature and sales data using SciPy's curve fitting tools. Then, display the fitted model's predicted sales for the given temperatures.

📋 What You'll Learn

Create a dataset of temperatures and corresponding ice cream sales.

Define a custom model function that relates temperature to sales.

Use SciPy's curve_fit function to find the best parameters for the model.

Print the predicted sales using the fitted model.

💡 Why This Matters

🌍 Real World

Fitting custom models helps businesses understand relationships in their data, like how temperature affects sales, so they can make better decisions.

💼 Career

Data scientists and analysts often fit models to data to predict outcomes and find trends, which is essential in many industries.

1

Create the temperature and sales data

Create two lists called temperatures and sales with these exact values: temperatures = [15, 18, 21, 24, 27, 30] and sales = [100, 150, 200, 250, 300, 350].

SciPy

# Create the temperature and sales data lists
# Your code here

Hint

Use square brackets to create lists and separate values with commas.

2

Define a custom linear model function

Define a function called linear_model that takes three parameters: x, a, and b. The function should return the value of a * x + b.

SciPy

temperatures = [15, 18, 21, 24, 27, 30]
sales = [100, 150, 200, 250, 300, 350]

# Define the linear_model function
# Your code here

Hint

Define a function using def and return the expression a * x + b.

3

Fit the linear model to the data

Import curve_fit from scipy.optimize. Use curve_fit with the linear_model function, temperatures, and sales to find the best parameters. Store the parameters in a variable called params.

SciPy

temperatures = [15, 18, 21, 24, 27, 30]
sales = [100, 150, 200, 250, 300, 350]

def linear_model(x, a, b):
    return a * x + b

# Import curve_fit and fit the model
# Your code here

Hint

Use from scipy.optimize import curve_fit and call curve_fit(linear_model, temperatures, sales).

4

Print the predicted sales using the fitted model

Use the fitted parameters in params to calculate predicted sales for each temperature in temperatures by calling linear_model. Store the results in a list called predicted_sales. Then print predicted_sales.

SciPy

from scipy.optimize import curve_fit

temperatures = [15, 18, 21, 24, 27, 30]
sales = [100, 150, 200, 250, 300, 350]

def linear_model(x, a, b):
    return a * x + b

params, _ = curve_fit(linear_model, temperatures, sales)

# Calculate predicted sales and print them
# Your code here

Hint

Use a list comprehension to apply linear_model to each temperature with the fitted parameters, then print the list.

Practice

1. What is the main purpose of using scipy.optimize.curve_fit in fitting custom models?

easy

A. To find the best parameters that make the model fit the data

B. To plot the data points automatically

C. To generate random data for testing

D. To calculate the mean of the dataset

Solution

1. Step 1: Understand the role of curve_fit

   curve_fit is used to adjust parameters of a model function so that it best fits the given data points.

2. Step 2: Identify the correct purpose

   It does not plot data, generate random data, or calculate means. Its main job is parameter estimation for fitting.

3. Final Answer:

   To find the best parameters that make the model fit the data -> Option A

4. Quick Check:

   curve_fit finds best parameters [OK]

Hint: Remember: curve_fit adjusts parameters to fit data [OK]

Common Mistakes:

• Thinking curve_fit plots data automatically
• Confusing curve_fit with data generation functions
• Assuming curve_fit calculates statistics like mean

2. Which of the following is the correct way to define a custom model function for curve_fit that fits a line y = m*x + c?

easy

A. def model(m, c, x): return m + c * x

B. def model(x, m, c): return m * x + c

C. def model(x): return m * x + c

D. def model(x, m, c): return m + c / x

Solution

1. Step 1: Check parameter order for curve_fit

   The model function must have the independent variable as the first argument, followed by parameters to fit.

2. Step 2: Verify function matches y = m*x + c

   def model(x, m, c): return m * x + c correctly defines model(x, m, c) returning m * x + c. Others have wrong order or formula.

3. Final Answer:

   def model(x, m, c): return m * x + c -> Option B

4. Quick Check:

   Model args: x first, then parameters [OK]

Hint: Model function: x first, then parameters [OK]

Common Mistakes:

• Swapping parameter and variable order
• Missing parameters in function definition
• Using wrong formula inside the function

3. Given the code below, what will be the output of print(popt)?

import numpy as np
from scipy.optimize import curve_fit

def model(x, a, b):
    return a * np.exp(b * x)

xdata = np.array([0, 1, 2, 3])
ydata = np.array([1, 2.7, 7.4, 20.1])

popt, _ = curve_fit(model, xdata, ydata)
print(np.round(popt, 2))

medium

A. [1.0 0.99]

B. [1.0 1.0]

C. [1.0 1.0] but with a runtime error

D. [1.01 1.0]

Solution

1. Step 1: Understand the model and data

   The model is a * exp(b * x). Given ydata roughly follows exponential growth, parameters a and b will be close to 1.

2. Step 2: Check output of curve_fit

   Running the code fits parameters close to a=1.0 and b=0.99 (data approximates e^x but slightly less). Rounded to two decimals, popt is approximately [1.0 0.99].

3. Final Answer:

   [1.0 0.99] -> Option A

4. Quick Check:

   Exponential fit params ~ [1.0, 0.99] [OK]

Hint: Run curve_fit and round parameters to check values [OK]

Common Mistakes:

• Misestimating parameters as [1.0 1.0] due to data approximation
• Confusing parameter order
• Expecting runtime errors without cause

4. What is wrong with the following code snippet for fitting a quadratic model using curve_fit?

import numpy as np
from scipy.optimize import curve_fit

def quad(x, a, b, c):
    return a * x**2 + b * x + c

xdata = np.array([1, 2, 3, 4])
ydata = np.array([3, 7, 13, 21])

popt, pcov = curve_fit(quad, ydata, xdata)
print(popt)

medium

A. Missing initial guess for parameters

B. The model function has wrong formula for quadratic

C. The independent and dependent variables are swapped in curve_fit call

D. The print statement is incorrect

Solution

1. Step 1: Check curve_fit arguments

   curve_fit expects the model, xdata (independent), then ydata (dependent). Here, ydata and xdata are swapped.

2. Step 2: Identify the error impact

   Swapping causes wrong fitting or runtime errors because the model expects x values first.

3. Final Answer:

   The independent and dependent variables are swapped in curve_fit call -> Option C

4. Quick Check:

   curve_fit(xdata, ydata) order matters [OK]

Hint: Remember: curve_fit(model, xdata, ydata) [OK]

Common Mistakes:

• Swapping xdata and ydata in curve_fit
• Assuming model formula is incorrect
• Thinking initial guess is always required

5. You want to fit a custom model y = a * sin(b * x) + c to noisy data. Which approach correctly fits the model and plots the result?

hard

A. Plot data first, then call curve_fit without storing parameters

B. Use curve_fit without defining a model function, just pass np.sin

C. Fit the model by manually guessing parameters without curve_fit

D. Define model with def model(x, a, b, c): return a * np.sin(b * x) + c, use curve_fit with data, then plot original and fitted curves

Solution

1. Step 1: Define the correct model function

   Model must be defined as model(x, a, b, c) returning a * np.sin(b * x) + c.

2. Step 2: Use curve_fit and plot results

   Call curve_fit(model, xdata, ydata) to get parameters, then plot original data and fitted curve for comparison.

3. Final Answer:

   Define model with def model(x, a, b, c): return a * np.sin(b * x) + c, use curve_fit with data, then plot original and fitted curves -> Option D

4. Quick Check:

   Model function + curve_fit + plot = correct approach [OK]

Hint: Always define model function before curve_fit and plot results [OK]

Common Mistakes:

• Passing np.sin directly without parameters
• Skipping model function definition
• Not storing or using fitted parameters for plotting