Curve fitting (curve_fit) in SciPy - Step-by-Step Execution

Concept Flow - Curve fitting (curve_fit)

Start with data points (x, y)

↓

Define model function f(x, params)

↓

Call curve_fit with f, x, y

↓

curve_fit adjusts params to minimize error

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Output best-fit parameters and covariance

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Use params to predict or plot fitted curve

We start with data points and a model function. curve_fit finds parameters that make the model best match the data.

Execution Sample

SciPy

import numpy as np
from scipy.optimize import curve_fit

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

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

params, cov = curve_fit(model, x, y)

This code fits a straight line y = a*x + b to the given x and y data points.

Execution Table

Step	Action	Parameters Tried	Error (Sum of Squares)	Result
1	Initial guess (default)	a=1.0, b=0.0	12.06	Try next params
2	Try params	a=2.0, b=0.1	0.06	Better fit, try refining
3	Try params	a=2.02, b=0.08	0.04	Improved fit
4	Try params	a=2.04, b=0.06	0.03	Improved fit
5	Try params	a=2.03, b=0.07	0.025	Best fit found
6	Return	a=2.03, b=0.07	0.025	Output params and covariance

💡 curve_fit stops when error is minimized and parameters converge

Variable Tracker

Variable	Start	After Step 2	After Step 3	After Step 4	After Step 5	Final
a	1.0	2.0	2.02	2.04	2.03	2.03
b	0.0	0.1	0.08	0.06	0.07	0.07
Error	12.06	0.06	0.04	0.03	0.025	0.025

Key Moments - 3 Insights

Why does curve_fit try many different parameters instead of just one?

curve_fit searches for parameters that minimize error. It tries many values step-by-step (see execution_table rows 1-5) to find the best fit.

What do the output parameters represent?

The output parameters are the best values for 'a' and 'b' that make the model line fit the data closely (see final row in execution_table).

Why is the error value important?

The error shows how far the model predictions are from actual data. curve_fit tries to make this error as small as possible (see error column in execution_table).

Visual Quiz - 3 Questions

Look at the execution_table, what is the value of parameter 'a' after step 3?

A 2.04

B 2.02

C 2.0

D 1.0

💡 Hint

Check the 'Parameters Tried' column at step 3 in execution_table.

At which step does curve_fit find the best fit parameters?

A Step 2

B Step 1

C Step 5

D Step 4

💡 Hint

Look for the row where 'Result' says 'Best fit found' in execution_table.

If the initial guess was closer to the true parameters, how would the error at step 1 change?

A It would be smaller

B It would stay the same

C It would be larger

D It would be zero

💡 Hint

Refer to the 'Error' column in execution_table and think about how starting closer affects error.

Concept Snapshot

curve_fit(model, x, y) fits data points (x,y) to a model function.
It finds parameters that minimize difference between model and data.
Returns best-fit parameters and covariance matrix.
Useful for fitting lines, curves, or custom functions.
Requires defining a model function with parameters.
Works by iterative optimization to reduce error.

Full Transcript

Curve fitting with curve_fit starts by having data points and a model function. The function curve_fit tries different parameters to make the model match the data as closely as possible. It measures error as the difference between predicted and actual data. It changes parameters step-by-step to reduce this error. When the error is as small as possible, it stops and returns the best parameters. These parameters can then be used to predict or plot the fitted curve. This process helps us find the line or curve that best represents the data.

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