Why fitting models to data reveals relationships in SciPy - Challenge Your Understanding

Challenge - 5 Problems

❓ Predict Output

intermediate

Output of linear fit coefficients

What is the output of the following code that fits a linear model to data points?

SciPy

import numpy as np
from scipy.optimize import curve_fit

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

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

params, _ = curve_fit(linear_model, x_data, y_data)
print(np.round(params, 2))

A [2.02 0.04]

B [2.00 0.10]

C [1.95 0.20]

D [2.10 -0.05]

💡 Hint

Think about how close the y_data values are to a line with slope 2 and intercept near zero.

The data roughly follows y = 2x with a small intercept. The curve_fit finds parameters close to slope 2 and intercept 0.04.

❓ data_output

intermediate

Number of points within 0.2 of fitted quadratic

After fitting a quadratic model to data, how many points lie within 0.2 units of the fitted curve?

SciPy

import numpy as np
from scipy.optimize import curve_fit

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

x = np.linspace(-2, 2, 10)
y = 3 * x**2 + 2 * x + 1 + np.random.normal(0, 0.1, x.size)
params, _ = curve_fit(quad_model, x, y)
y_fit = quad_model(x, *params)
close_points = np.sum(np.abs(y - y_fit) < 0.2)
print(close_points)

A 10

B 8

C 6

D 4

💡 Hint

The noise is small, so most points should be close to the fitted curve.

Because noise is low, all 10 points are within 0.2 of the fitted quadratic curve.

❓ visualization

advanced

Identify the plot showing best fit line

Which plot correctly shows the data points and the best fit line from a linear regression?

SciPy

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

def linear(x, m, c):
    return m * x + c

x = np.array([0, 1, 2, 3, 4])
y = np.array([1, 3, 5, 7, 9])
params, _ = curve_fit(linear, x, y)
y_fit = linear(x, *params)

plt.figure(figsize=(8, 5))
plt.scatter(x, y, label='Data points')
plt.plot(x, y_fit, label='Fitted line', color='red')
plt.legend()
plt.title('Linear Fit Example')
plt.xlabel('x')
plt.ylabel('y')
plt.show()

A Only a red line without scatter points

B Scatter points with a red line close but not exactly through all points

C Scatter points with a blue curve bending through points

D Scatter points with a red line passing exactly through all points

💡 Hint

Best fit lines minimize error but rarely pass exactly through all points.

The data lies perfectly on the line y=2x+1, so the fitted line passes exactly through all points.

🧠 Conceptual

advanced

Why does fitting a model reveal relationships?

Why does fitting a mathematical model to data help reveal relationships between variables?

A Because it removes all noise from data to show perfect relationships

B Because it forces data to match a predefined pattern regardless of actual trends

C Because it finds parameters that best explain how variables change together

D Because it randomly assigns values to variables to create correlations

💡 Hint

Think about what fitting tries to optimize in the model.

Fitting finds parameters that minimize differences between model predictions and data, showing how variables relate.

🔧 Debug

expert

Identify the error in model fitting code

What error will this code raise when trying to fit a model to data?

SciPy

import numpy as np
from scipy.optimize import curve_fit

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

x = np.linspace(0, 2 * np.pi, 50)
y = np.sin(x) + np.random.normal(0, 0.1, x.size)

params, _ = curve_fit(model, x, y, p0=[1, 1])

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

B ValueError: initial guess p0 must have length 2

C RuntimeWarning: overflow encountered in sin

D No error, code runs successfully

💡 Hint

Check the number of parameters in the model and the initial guess length.

The model has two parameters (a, b), but p0 provides only one initial guess, causing ValueError.