Non-linear curve fitting in SciPy - Time & Space Complexity

Time Complexity: Non-linear curve fitting

O(n × k)

Understanding Time Complexity

When fitting a curve that is not a straight line, the computer tries many guesses to find the best fit.

We want to know how the time needed grows as we give more data points.

Scenario Under Consideration

Analyze the time complexity of the following code snippet.

import numpy as np
from scipy.optimize import curve_fit

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

xdata = np.linspace(0, 4, 50)
ydata = model_func(xdata, 2.5, 1.3) + 0.2 * np.random.normal(size=xdata.size)

popt, pcov = curve_fit(model_func, xdata, ydata)

This code fits an exponential curve to data points by trying to find the best parameters.

Identify Repeating Operations

Identify the loops, recursion, array traversals that repeat.

• Primary operation: Repeated evaluation of the model function and adjustment of parameters.
• How many times: Many iterations until the best fit is found, depending on data size and convergence.

How Execution Grows With Input

As the number of data points increases, the fitting process takes longer because it must consider more points each time it tests parameters.

Input Size (n)	Approx. Operations
10	Low number of function evaluations
100	About 10 times more evaluations
1000	About 100 times more evaluations

Pattern observation: The time grows roughly linearly with the number of data points multiplied by the number of iterations, which may depend on convergence.

Final Time Complexity

Time Complexity: O(n × k), where n is the number of data points and k is the number of iterations until convergence.

This means if you double the data points, the time to fit the curve roughly doubles, assuming the number of iterations stays constant.

Common Mistake

[X] Wrong: "The fitting time grows linearly with the number of data points because it just looks at each point once."

[OK] Correct: The fitting process repeats many times, each time using all data points, so the total work grows with both the number of points and the number of iterations.

Interview Connect

Understanding how curve fitting time grows helps you explain performance in real data analysis tasks and shows you can think about algorithm costs clearly.

Self-Check

"What if we used a simpler linear model instead of a non-linear one? How would the time complexity change?"

Practice

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

easy

A. To sort data points in ascending order

B. To find the best-fitting curve for data when the relationship is non-linear

C. To calculate the mean of a dataset

D. To generate random numbers for simulations

Solution

1. Step 1: Understand the function's purpose

   scipy.optimize.curve_fit is designed to fit a curve to data points, especially when the relationship is not a straight line.

2. Step 2: Compare options with the function's goal

   Options B, C, and D describe unrelated tasks like sorting, averaging, or random number generation, which are not the purpose of curve fitting.

3. Final Answer:

   To find the best-fitting curve for data when the relationship is non-linear -> Option B

4. Quick Check:

   Curve fitting = best-fitting curve [OK]

Hint: Curve fitting finds best curve, not sorting or averaging [OK]

Common Mistakes:

• Confusing curve fitting with data sorting
• Thinking curve_fit calculates averages
• Assuming curve_fit generates random data

2. Which of the following is the correct way to import the curve_fit function from SciPy?

easy

A. import curve_fit from scipy.optimize

B. import scipy.curve_fit

C. from scipy import curve_fit

D. from scipy.optimize import curve_fit

Solution

1. Step 1: Recall correct import syntax in Python

   To import a specific function from a module, use from module import function syntax.

2. Step 2: Match syntax with options

   from scipy.optimize import curve_fit matches the correct syntax: from scipy.optimize import curve_fit. Options B, C, and D use incorrect syntax or wrong module paths.

3. Final Answer:

   from scipy.optimize import curve_fit -> Option D

4. Quick Check:

   Correct import = from module import function [OK]

Hint: Use 'from module import function' to import specific functions [OK]

Common Mistakes:

• Using 'import scipy.curve_fit' which is invalid
• Trying 'from scipy import curve_fit' when it's in optimize submodule
• Incorrect order like 'import curve_fit from ...'

3. What will be the output of the following code snippet?

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])

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

medium

A. [1.00 1.00]

B. [1.02 1.00]

C. [1.00 0.99]

D. [0.99 1.00]

Solution

1. Step 1: Understand the model and data

   The model is an exponential function: a * exp(b * x). The ydata roughly follows this pattern with a near 1 for a and about 1 for b.

2. Step 2: Run curve_fit and round parameters

   Using curve_fit on given data returns parameters close to [1.00, 0.99]. Rounding to two decimals gives [1.00 0.99].

3. Final Answer:

   [1.00 0.99] -> Option C

4. Quick Check:

   Fitted params ≈ [1.00, 0.99] [OK]

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

Common Mistakes:

• Assuming parameters are exactly 1.00 and 1.00
• Confusing parameter order or values
• Ignoring rounding effects

4. Identify the error in the following code snippet for non-linear curve fitting:

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])

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

medium

A. Missing unpacking of the tuple returned by curve_fit

B. Model function has wrong parameters

C. xdata and ydata have different lengths

D. curve_fit is not imported correctly

Solution

1. Step 1: Check the return value of curve_fit

   curve_fit returns a tuple: (parameters, covariance). The code assigns this tuple to a single variable without unpacking.

2. Step 2: Identify the correct usage

   Correct usage unpacks the tuple: params, _ = curve_fit(...). Without unpacking, printing params shows the tuple, not just parameters.

3. Final Answer:

   Missing unpacking of the tuple returned by curve_fit -> Option A

4. Quick Check:

   curve_fit returns tuple, unpack it [OK]

Hint: Always unpack curve_fit output: params, _ = curve_fit(...) [OK]

Common Mistakes:

• Assigning curve_fit output to one variable without unpacking
• Assuming curve_fit returns only parameters
• Ignoring the covariance matrix returned

5. You want to fit a non-linear model y = a * x / (b + x) to data using curve_fit. Which of the following code snippets correctly defines the model and fits the data?

import numpy as np
from scipy.optimize import curve_fit

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

xdata = np.array([1, 2, 3, 4, 5])
ydata = np.array([0.5, 1.2, 1.8, 2.4, 2.9])

params, covariance = curve_fit(model, xdata, ydata)
print(np.round(params, 2))

hard

A. Correctly defines model and fits data using curve_fit

B. Model function should use addition instead of division

C. curve_fit requires initial guess parameters to work

D. xdata and ydata lengths must be different for curve_fit

Solution

1. Step 1: Check model function correctness

   The model y = a * x / (b + x) is correctly implemented as return a * x / (b + x).

2. Step 2: Verify curve_fit usage

   The code calls curve_fit(model, xdata, ydata) and unpacks parameters and covariance correctly. Initial guesses are optional here.

3. Final Answer:

   Correctly defines model and fits data using curve_fit -> Option A

4. Quick Check:

   Model and curve_fit usage correct [OK]

Hint: Define model exactly, call curve_fit with data and unpack results [OK]

Common Mistakes:

• Changing division to addition in model
• Thinking initial guesses are always required
• Using different lengths for xdata and ydata