Curve fitting (curve_fit) in SciPy - Practice Problems & Coding Challenges

Challenge - 5 Problems

❓ Predict Output

intermediate

Output of curve fitting with linear model

What is the output of the following code snippet that fits a linear model to data using curve_fit?

SciPy

import numpy as np
from scipy.optimize import curve_fit

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

xdata = np.array([0, 1, 2, 3, 4])
ydata = np.array([1, 3, 5, 7, 9])

params, covariance = curve_fit(linear_model, xdata, ydata)
print(params)

A [0.5 1.5]

B [1. 2.]

C [2. 1.]

D [3. 0.]

💡 Hint

Think about the equation of a line y = a*x + b and the data points given.

The data points follow the line y = 2*x + 1 exactly, so the fitted parameters are a=2 and b=1.

❓ data_output

intermediate

Number of parameters returned by curve_fit

Given the following model and data, how many parameters does curve_fit return?

SciPy

import numpy as np
from scipy.optimize import curve_fit

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

xdata = np.linspace(0, 5, 6)
ydata = np.array([1, 3, 7, 13, 21, 31])

params, covariance = curve_fit(quadratic_model, xdata, ydata)
print(len(params))

A 3

B 2

C 6

D 1

💡 Hint

Count the number of parameters in the quadratic model function.

The quadratic model has three parameters: a, b, and c, so curve_fit returns three values.

🔧 Debug

advanced

Identify the error in curve_fit usage

What error does the following code raise when run?

SciPy

import numpy as np
from scipy.optimize import curve_fit

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

xdata = np.array([0, 1, 2, 3])
ydata = np.array([0, 0.84, 0.91, 0.14])

params, covariance = curve_fit(model, xdata, ydata, p0=[1])

A ValueError: initial guess p0 must have length 2

B TypeError: curve_fit() got an unexpected keyword argument 'p0'

C TypeError: model() missing 1 required positional argument

D RuntimeError: Optimal parameters not found

💡 Hint

Check the length of the initial guess array compared to the number of parameters in the model.

The model has two parameters (a and b), but p0 is given as a list with only one element, causing a ValueError.

❓ visualization

advanced

Plotting fitted curve with noisy data

Which option correctly plots the noisy data points and the fitted curve from curve_fit?

SciPy

import numpy as np
import matplotlib.pyplot as plt
from scipy.optimize import curve_fit

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

xdata = np.linspace(0, 4, 50)
y = model(xdata, 2.5, -1.3)
np.random.seed(0)
noise = 0.2 * np.random.normal(size=xdata.size)
ydata = y + noise

params, _ = curve_fit(model, xdata, ydata)

plt.scatter(xdata, ydata, label='Noisy data')
plt.plot(xdata, model(xdata, *params), 'r-', label='Fitted curve')
plt.legend()
plt.show()

A Line plot of noisy data points with blue dots showing the fitted curve

B Scatter plot of noisy data points with a red line showing the fitted exponential curve

C Scatter plot of noisy data points with no fitted curve shown

D Only the fitted curve plotted as a blue dashed line, no data points

💡 Hint

Look for a scatter plot for data and a red line for the fitted curve.

The code plots noisy data as scattered points and overlays the fitted curve as a red line.

🧠 Conceptual

expert

Understanding covariance matrix from curve_fit

What does the covariance matrix returned by curve_fit represent?

A It is the matrix used to transform the input data before fitting

B It contains the residuals of the fitted curve compared to the data points

C It shows the correlation between the input x values and the output y values

D It estimates the uncertainty and correlation between the fitted parameters

💡 Hint

Think about what uncertainty means in parameter estimation.

The covariance matrix estimates how much the fitted parameters vary and how they relate to each other.

Practice

1. What is the main purpose of using curve_fit in data analysis?

easy

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

B. To sort data points in ascending order

C. To calculate the average of data points

D. To remove outliers from the dataset

Solution

1. Step 1: Understand the role of curve fitting

   Curve fitting is used to find parameters of a model that best match the given data points.

2. Step 2: Identify what curve_fit does

   The curve_fit function from scipy optimizes parameters to minimize the difference between model predictions and data.

3. Final Answer:

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

4. Quick Check:

   Curve fitting = best parameters for model [OK]

Hint: Remember: curve_fit finds model parameters to fit data [OK]

Common Mistakes:

• Confusing curve fitting with data sorting
• Thinking curve_fit calculates averages
• Assuming curve_fit removes outliers

2. Which of the following is the correct way to call curve_fit with a model function f, data arrays xdata and ydata?

easy

A. popt = curve_fit(f, ydata, xdata)

B. popt, pcov = curve_fit(xdata, ydata, f)

C. curve_fit(f, ydata, xdata)

D. popt, pcov = curve_fit(f, xdata, ydata)

Solution

1. Step 1: Recall the function signature of curve_fit

   The correct call is curve_fit(function, xdata, ydata), where the model function is first, then x and y data.

2. Step 2: Match the options with the correct signature

   popt, pcov = curve_fit(f, xdata, ydata) matches the correct order and unpacks both parameters and covariance matrix.

3. Final Answer:

   popt, pcov = curve_fit(f, xdata, ydata) -> Option D

4. Quick Check:

   curve_fit(function, xdata, ydata) [OK]

Hint: Remember order: function, xdata, ydata in curve_fit [OK]

Common Mistakes:

• Swapping xdata and ydata
• Not unpacking both outputs
• Passing arguments in wrong order

3. Given the code:

import numpy as np
from scipy.optimize import curve_fit

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

xdata = np.array([1, 2, 3, 4, 5])
ydata = np.array([2.1, 4.1, 6.0, 8.1, 10.2])

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

What will be printed?

medium

A. [2.00 0.00]

B. [2.02 0.04]

C. [1.00 2.00]

D. [0.06 2.04]

Solution

1. Step 1: Understand the model and data

   The model is linear: y = a*x + b. The data roughly follows y = 2*x + 0.

2. Step 2: Use curve_fit to find parameters

   Running curve_fit finds parameters close to a=2.02 and b=0.04, fitting the data closely.

3. Final Answer:

   [2.02 0.04] -> Option B

4. Quick Check:

   Fitted slope ~2.02, intercept ~0.04 [OK]

Hint: Check model form and data trend to guess parameters [OK]

Common Mistakes:

• Confusing parameter order
• Ignoring rounding
• Assuming exact integers

4. Consider this code snippet:

import numpy as np
from scipy.optimize import curve_fit

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

xdata = np.array([1, 2, 3])
ydata = np.array([2, 4, 6])

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

What is the main issue here?

medium

A. The model function is incorrect

B. The data arrays have different lengths

C. xdata and ydata are swapped in curve_fit call

D. curve_fit cannot handle linear models

Solution

1. Step 1: Check the order of arguments in curve_fit

   The correct order is curve_fit(model, xdata, ydata). Here, ydata and xdata are swapped.

2. Step 2: Understand the impact of swapping

   Swapping causes the model to fit x as a function of y, which is not intended and leads to wrong parameters.

3. Final Answer:

   xdata and ydata are swapped in curve_fit call -> Option C

4. Quick Check:

   Order is model, xdata, ydata [OK]

Hint: Always put xdata before ydata in curve_fit [OK]

Common Mistakes:

• Swapping xdata and ydata
• Assuming model function is wrong
• Ignoring argument order importance

5. You have noisy data from a quadratic process: y = a*x**2 + b*x + c. You want to fit this model using curve_fit. Which of the following code snippets correctly defines the model and fits the data xdata, ydata?

hard

A. def model(x, a, b, c):
return a * x**2 + b * x + c
popt, pcov = curve_fit(model, xdata, ydata)

B. def model(a, b, c, x):
return a * x**2 + b * x + c
popt, pcov = curve_fit(model, xdata, ydata)

C. def model(x, a, b):
return a * x**2 + b * x
popt, pcov = curve_fit(model, xdata, ydata)

D. def model(x, a, b, c):
return a * x + b * x**2 + c
popt, pcov = curve_fit(model, ydata, xdata)

Solution

1. Step 1: Define the model function correctly

   The model must have x as the first argument, followed by parameters a, b, c. The formula is a*x**2 + b*x + c.

2. Step 2: Call curve_fit with correct argument order

   Use curve_fit(model, xdata, ydata) to fit the model to data.

3. Final Answer:

   def model(x, a, b, c): return a * x**2 + b * x + c popt, pcov = curve_fit(model, xdata, ydata) -> Option A

4. Quick Check:

   Model args: x first, then parameters; curve_fit(model, xdata, ydata) [OK]

Hint: Model function: x first, parameters next; call curve_fit accordingly [OK]

Common Mistakes:

• Putting parameters before x in model
• Missing one parameter in model
• Swapping xdata and ydata in curve_fit