Curve fitting (polyfit, fit) in MATLAB - Deep Dive

Overview - Curve fitting (polyfit, fit)

What is it?

Curve fitting is a way to find a smooth line or curve that best matches a set of points on a graph. In MATLAB, functions like polyfit and fit help you find these curves by calculating the best mathematical formula. This helps you understand trends or predict new values based on your data. It is like drawing a line that goes through or near your points to show the pattern.

Why it matters

Without curve fitting, it would be hard to see patterns or make predictions from scattered data points. For example, if you measure temperature over days, curve fitting helps you find the trend and guess future temperatures. It solves the problem of noisy or incomplete data by summarizing it with a simple formula. This makes data easier to understand and use in real life.

Where it fits

Before learning curve fitting, you should know basic plotting and how to work with arrays in MATLAB. After this, you can learn about more advanced modeling like machine learning or nonlinear regression. Curve fitting is a bridge between raw data and complex analysis, helping you move from numbers to insights.

Mental Model

Core Idea

Curve fitting finds the best mathematical curve that closely follows your data points to reveal patterns or make predictions.

Think of it like...

Imagine you have a set of scattered pebbles on a table and you want to lay a flexible wire so it touches or stays close to most pebbles, showing their general shape. Curve fitting is like bending that wire to best match the pebbles' arrangement.

Data points: *   *     *  *  *
Curve fit:  _________

  X-axis →

The curve tries to pass near the stars (*) representing data points.

Build-Up - 7 Steps

1

Foundation

Understanding data points and plotting

Concept: Learn what data points are and how to plot them in MATLAB.

Data points are pairs of numbers, like (x, y), showing measurements or observations. In MATLAB, you can plot them using plot(x, y, 'o') to see their positions on a graph.

Result

A scatter plot showing your data points on a graph.

Seeing data visually helps you understand its spread and whether a curve might fit well.

2

Foundation

Basics of polynomial curves

Concept: Introduce polynomial curves as smooth lines made by adding powers of x with coefficients.

A polynomial curve looks like y = a0 + a1*x + a2*x^2 + ... where a0, a1, a2 are numbers to find. These curves can bend to fit data better than straight lines.

Result

Understanding that curves can be simple (line) or more complex (parabola, cubic).

Knowing polynomial shapes helps you choose the right curve complexity for your data.

3

Intermediate

Using polyfit to find polynomial fits

🤔Before reading on: do you think polyfit finds a curve that passes exactly through all points or one that best fits overall? Commit to your answer.

Concept: polyfit calculates coefficients of a polynomial that best fits data in a least squares sense.

In MATLAB, polyfit(x, y, n) finds coefficients for a polynomial of degree n that minimizes the total squared distance from the data points to the curve. For example, polyfit(x, y, 2) fits a parabola.

Result

An array of coefficients representing the best-fit polynomial.

Understanding least squares fitting explains why the curve balances errors instead of hitting every point exactly.

4

Intermediate

Evaluating fitted curves with polyval

🤔Before reading on: do you think polyval uses the coefficients to predict y for new x values or just plots the original data? Commit to your answer.

Concept: polyval uses polynomial coefficients to calculate y values for any x, letting you draw or predict the curve.

After polyfit, use y_fit = polyval(coeffs, x_new) to get y values on the curve for new or original x points. This helps visualize or predict data.

Result

A smooth curve line that you can plot or analyze.

Knowing how to generate curve points from coefficients lets you apply the fit beyond original data.

5

Intermediate

Using fit for flexible curve fitting

🤔Before reading on: do you think fit can only do polynomials or other curve types too? Commit to your answer.

Concept: The fit function in MATLAB can fit many curve types, not just polynomials, using built-in or custom models.

fit(x, y, 'poly2') fits a quadratic polynomial, but you can also fit exponential, Gaussian, or custom models by changing the type string. This gives more flexibility for different data shapes.

Result

A fit object representing the chosen curve model and parameters.

Understanding fit's flexibility helps you choose the best model for your data shape.

6

Advanced

Assessing fit quality with residuals and R-square

🤔Before reading on: do you think a higher R-square always means a better model? Commit to your answer.

Concept: Residuals show errors between data and fit; R-square measures how well the fit explains data variance.

Calculate residuals as differences between actual y and fitted y. R-square near 1 means a good fit. But very high R-square can mean overfitting, where the curve fits noise, not just trend.

Result

Metrics to judge if your curve fit is good or too complex.

Knowing how to evaluate fit quality prevents trusting misleading curves.

7

Expert

Handling overfitting and model selection

🤔Before reading on: do you think increasing polynomial degree always improves predictions? Commit to your answer.

Concept: Higher-degree polynomials fit data points more closely but can overfit, capturing noise instead of true trends.

Choosing the right polynomial degree balances fit accuracy and simplicity. Techniques like cross-validation or visual inspection help avoid overfitting. Sometimes simpler models predict better on new data.

Result

A well-chosen curve that generalizes beyond the sample data.

Understanding overfitting is key to building reliable models that work in real situations.

Under the Hood

polyfit uses linear algebra to solve a system of equations that minimizes the sum of squared vertical distances (errors) between data points and the polynomial curve. It builds a matrix of powers of x and solves for coefficients using least squares. The fit function extends this by using optimization algorithms to adjust parameters for various model types, sometimes iteratively.

Why designed this way?

Least squares fitting was chosen because it provides a mathematically stable and efficient way to find the best fit, balancing errors across all points. Alternatives like minimizing absolute errors are harder to compute. The design allows fast computation and works well for noisy data, which is common in real measurements.

Data points (x,y) → [Construct matrix of x powers] → Solve linear system → Polynomial coefficients → Use coefficients to predict y

fit function: Data points → Choose model type → Optimization algorithm → Model parameters → Predict y

Myth Busters - 4 Common Misconceptions

Quick: Does a polynomial fit always pass through all data points exactly? Commit yes or no.

Common Belief: A polynomial fit always goes through every data point perfectly.

Reality: Polynomial fits usually do not pass through all points; they minimize overall error but balance distances to all points.

Why it matters: Expecting exact fits can lead to confusion when the curve misses points, causing mistrust in the method.

Quick: Is a higher-degree polynomial always better for fitting data? Commit yes or no.

Common Belief: Using a higher-degree polynomial always improves the fit and predictions.

Reality: Higher-degree polynomials can overfit, capturing noise and reducing prediction accuracy on new data.

Why it matters: Blindly increasing degree can produce misleading models that fail in real applications.

Quick: Does a high R-square guarantee a good model? Commit yes or no.

Common Belief: A high R-square means the model is perfect and reliable.

Reality: High R-square can result from overfitting and does not guarantee good predictions on new data.

Why it matters: Relying only on R-square can cause overconfidence and poor decision-making.

Quick: Can fit only handle polynomial models? Commit yes or no.

Common Belief: The fit function only fits polynomial curves.

Reality: fit can handle many model types like exponential, Gaussian, and custom models.

Why it matters: Limiting fit to polynomials restricts modeling power and misses better fits.

Expert Zone

1

The numerical stability of polyfit depends on scaling and centering data; large x values can cause inaccurate coefficients.

2

fit objects store metadata and allow easy extraction of goodness-of-fit statistics and confidence intervals, useful for advanced analysis.

3

Choosing model types in fit requires understanding data nature; for example, exponential fits suit growth data better than polynomials.

When NOT to use

Avoid polynomial fitting when data shows periodic or non-polynomial behavior; use Fourier or spline fitting instead. For very noisy data, robust fitting methods or smoothing may be better than least squares.

Production Patterns

In real projects, curve fitting is used for sensor calibration, trend analysis in finance, and preprocessing for machine learning. Professionals combine fit with residual analysis and cross-validation to ensure model reliability.

Connections

Linear Regression

Curve fitting with polyfit is a form of linear regression on polynomial features.

Understanding curve fitting helps grasp how linear regression extends to nonlinear relationships by transforming inputs.

Signal Processing

Curve fitting relates to smoothing noisy signals to reveal underlying trends.

Knowing curve fitting aids in designing filters and smoothing techniques in signal analysis.

Physics - Experimental Data Analysis

Curve fitting is used to model physical phenomena from experimental measurements.

Recognizing curve fitting as a tool in physics shows its role in validating theories and extracting constants from data.

Common Pitfalls

#1 Choosing too high polynomial degree causing overfitting.

Wrong approach:

coeffs = polyfit(x, y, 10); % Using degree 10 for small noisy data

Correct approach:

coeffs = polyfit(x, y, 2); % Using degree 2 to avoid overfitting

Root cause:

Misunderstanding that higher degree always means better fit, ignoring noise and model complexity.

#2 Using polyfit without checking data scale causing numerical errors.

Wrong approach:

coeffs = polyfit(x_large, y, 3); % x_large has very big values

Correct approach:

x_scaled = (x_large - mean(x_large)) / std(x_large);
coeffs = polyfit(x_scaled, y, 3); % Scale data first

Root cause:

Ignoring that large input values can cause unstable calculations in polynomial fitting.

#3 Assuming fit function only fits polynomials and missing better models.

Wrong approach:

f = fit(x, y, 'poly3'); % Using polynomial when data is exponential

Correct approach:

f = fit(x, y, 'exp1'); % Using exponential model for growth data

Root cause:

Lack of knowledge about fit's flexibility and model selection.

Key Takeaways

Curve fitting finds a smooth curve that best represents scattered data points to reveal patterns or make predictions.

polyfit fits polynomial curves by minimizing squared errors, but the curve usually does not pass through all points exactly.

The fit function in MATLAB offers flexible models beyond polynomials, allowing better fits for different data shapes.

Evaluating fit quality with residuals and R-square helps avoid overfitting and ensures reliable models.

Choosing the right model complexity is crucial; higher-degree polynomials can overfit and reduce prediction accuracy.

Practice

1. What does the MATLAB function polyfit(x, y, n) do?

easy

A. It returns a fit object that can be used to predict new values.

B. It plots the data points x and y on a graph.

C. It finds the coefficients of a polynomial of degree n that fits the data points x and y.

D. It calculates the correlation coefficient between x and y.

Solution

1. Step 1: Understand the purpose of polyfit

   polyfit calculates polynomial coefficients that best fit the data points given by x and y.

2. Step 2: Differentiate from other functions

   Unlike fit, polyfit returns just coefficients, not a fit object or plot.

3. Final Answer:

   It finds the coefficients of a polynomial of degree n that fits the data points x and y. -> Option C

4. Quick Check:

   polyfit = polynomial coefficients [OK]

Hint: Remember: polyfit returns coefficients, fit returns a fit object [OK]

Common Mistakes:

• Confusing polyfit with fit function
• Thinking polyfit plots data
• Assuming polyfit returns a fit object

2. Which of the following is the correct syntax to fit a quadratic polynomial to data vectors x and y using polyfit?

easy

A. p = polyfit(x, y, 2);

B. p = polyfit(y, x, 2);

C. p = polyfit(x, y, 3);

D. p = polyfit(x, 2, y);

Solution

1. Step 1: Identify correct argument order

   The syntax is polyfit(x, y, n) where n is the polynomial degree.

2. Step 2: Confirm degree for quadratic

   Quadratic means degree 2, so the third argument must be 2.

3. Final Answer:

   p = polyfit(x, y, 2); -> Option A

4. Quick Check:

   polyfit(x, y, 2) fits quadratic [OK]

Hint: Remember order: x, y, degree in polyfit [OK]

Common Mistakes:

• Swapping x and y arguments
• Using wrong polynomial degree
• Incorrect argument order

3. What is the output of the following MATLAB code?

x = [1 2 3 4 5];
y = [2.2 4.1 6.0 8.1 10.2];
p = polyfit(x, y, 1);
y_fit = polyval(p, 6);

medium

A. y_fit = 12.1

B. y_fit = 12.3

C. y_fit = 13.0

D. y_fit = 10.2

Solution

1. Step 1: Fit a line to data

   Using polyfit with degree 1 fits a line y = mx + b to points.

2. Step 2: Calculate predicted value at x=6

   Using polyval, we evaluate the fitted line at x=6. The slope and intercept from polyfit give approximately y_fit = 12.1.

3. Final Answer:

   y_fit = 12.1 -> Option A

4. Quick Check:

   polyval(polyfit(x,y,1),6) ≈ 12.1 [OK]

Hint: polyval evaluates polynomial at given x [OK]

Common Mistakes:

• Confusing polyfit output with predicted value
• Using wrong degree in polyfit
• Calculating y_fit without polyval

4. Identify the error in this MATLAB code snippet for fitting data with fit:

x = (1:5)';
y = [2 4 6 8 10]';
f = fit(x, y, 'poly2');

medium

A. The fit type string should be 'poly1' for linear fit, not 'poly2'.

B. The fit function requires the Curve Fitting Toolbox to be installed.

C. The input vectors x and y must be row vectors, not column vectors.

D. The syntax is correct; there is no error.

Solution

1. Step 1: Check syntax and input shapes

   Vectors x and y as column vectors are valid inputs for fit.

2. Step 2: Verify toolbox requirement

   The fit function requires the Curve Fitting Toolbox; without it, the code will error.

3. Final Answer:

   The fit function requires the Curve Fitting Toolbox to be installed. -> Option B

4. Quick Check:

   fit needs toolbox installed [OK]

Hint: fit needs Curve Fitting Toolbox installed [OK]

Common Mistakes:

• Assuming fit works without toolbox
• Confusing poly1 and poly2 as errors
• Thinking vector orientation causes error

5. You have noisy data vectors x and y. You want to fit a cubic polynomial and then predict the value at x = 7. Which sequence of MATLAB commands correctly does this?

hard

A. p = polyfit(x, y, 3); y_pred = p(7);

B. f = fit(x, y, 'poly2'); y_pred = feval(f, 7);

C. p = polyfit(y, x, 3); y_pred = polyval(p, 7);

D. f = fit(x, y, 'poly3'); y_pred = feval(f, 7);

Solution

1. Step 1: Choose correct fit function and degree

   Using fit with 'poly3' fits a cubic polynomial to x and y.

2. Step 2: Predict value at x=7 using fit object

   Use feval(f, 7) or f(7) to get predicted y value from the fit object.

3. Final Answer:

   f = fit(x, y, 'poly3'); y_pred = feval(f, 7); -> Option D

4. Quick Check:

   fit with 'poly3' and feval predicts y at 7 [OK]

Hint: Use fit with 'poly3' and feval to predict [OK]

Common Mistakes:

• Swapping x and y in polyfit
• Using wrong polynomial degree
• Using poly2 instead of poly3 for cubic fit