Data Modeling with Regression and Interpolation in MATLAB

Understanding Regression vs. Interpolation in MATLAB

Data modeling is a cornerstone of both computer science and data science. In MATLAB, two fundamental techniques—regression and interpolation—are used to create mathematical models that describe or predict data. This course explains when to use each method, how to implement them, and how to visualize the results effectively.

When to Choose Interpolation Over Regression

Interpolation is the preferred approach when the goal is to reproduce the exact values of the original data points within the observed range. Unlike regression, which seeks a best‑fit curve that may deviate from individual points, interpolation guarantees that the generated curve passes through every data point.

• Use interpolation when you need a smooth curve that honors the measured values.
• It is ideal for generating intermediate values (e.g., estimating a temperature at a time between two recorded measurements).
• Interpolation should not be used for extrapolation—predicting values outside the range of the data—because the curve’s behavior beyond the known points is undefined.

Regression: Capturing Trends in Noisy Data

Regression, on the other hand, is designed to capture the overall trend of a dataset, especially when the data contains noise or measurement error. A regression model provides a simplified representation (often a polynomial) that may not pass through every point but minimizes the overall error.

• Choose regression when you want to understand the underlying relationship between variables.
• It is useful for forecasting, hypothesis testing, and reducing dimensionality.
• Higher‑order polynomials can fit more complex patterns, but they also increase the risk of over‑fitting.

Implementing Regression in MATLAB

MATLAB’s polyfit function computes the coefficients of a polynomial regression model, while polyval evaluates the polynomial at desired points.

Third‑Order Regression Example

To fit a third‑order (cubic) regression curve, use the following command pair:

coeffs = polyfit(x, y, 3);
ynew   = polyval(coeffs, xnew);

This pair correctly calculates the coefficients for a cubic polynomial (n = 3) and evaluates the model at the new x‑coordinates (xnew).

Note that the number of regression coefficients is always n+1. For a third‑order fit, you obtain four coefficients (a₃, a₂, a₁, a₀).

Visualizing Regression Curves

When plotting regression results, the LineWidth property controls the thickness of the line. For example:

plot(xnew, ynew3, 'k-', 'LineWidth', 2);

This command draws a black solid line with a thickness of 2 points.

Implementing Interpolation in MATLAB

MATLAB provides several interpolation methods via the interp1 function. The choice of method determines the smoothness and shape of the resulting curve.

Spline Interpolation for a Smooth Third‑Degree Curve

If you have data such as:

x = [0,1,2,3,4,5];
y = [15,10,9,6,2,0];

and you need a smooth curve that passes through every point using a third‑degree polynomial, the spline method is appropriate:

y_interp = interp1(x, y, xnew, 'spline');

Spline interpolation constructs piecewise cubic polynomials between each pair of points, ensuring continuity of the first and second derivatives, which yields a visually smooth curve.

Other Interpolation Options

• 'linear' – straight‑line segments between points; less smooth.
• 'nearest' – assigns the value of the nearest data point; creates a step‑like appearance.
• 'pchip' – shape‑preserving piecewise cubic Hermite interpolation; avoids overshoot.

Combining Regression and Interpolation in a Single Plot

It is often instructive to display both a regression model and an interpolation curve on the same graph to compare their behavior.

% Compute regression curves
coeffs1 = polyfit(x, y, 1);   % 1st‑order (linear)
coeffs3 = polyfit(x, y, 3);   % 3rd‑order (cubic)

y1 = polyval(coeffs1, xnew);
y3 = polyval(coeffs3, xnew);

% Interpolation curve (spline)
y_spline = interp1(x, y, xnew, 'spline');

% Plot everything
figure; hold on;
plot(x, y, 'ko', 'MarkerFaceColor', 'b'); % original data
plot(xnew, y1, 'r--', 'LineWidth', 2);    % linear regression
plot(xnew, y3, 'k-',  'LineWidth', 2);    % cubic regression
plot(xnew, y_spline, 'g:', 'LineWidth', 2); % spline interpolation
legend('Data','Linear Reg','Cubic Reg','Spline Interp');
hold off;

This script demonstrates how to add multiple lines with distinct styles using the plot function. The line styles 'r--' (red dashed) and 'k-' (black solid) clearly differentiate the regression models.

Customizing Markers and Lines

MATLAB offers extensive control over marker appearance. To set a blue marker face, a red edge, and a marker size of 5, use:

plot(x, y, 'o', 'MarkerFaceColor','b', 'MarkerEdgeColor','r', 'MarkerSize',5);

Notice that LineWidth is not required for marker styling; it affects only the line thickness.

Key Takeaways

• Interpolation reproduces every original data point within the range; ideal for generating smooth curves that honor measured values.
• Regression captures the overall trend, often ignoring individual point deviations; useful for noisy data and predictive modeling.
• Use polyfit and polyval for regression; the number of coefficients equals order + 1.
• Use interp1 with methods such as 'spline' for smooth, third‑degree interpolation.
• Customize plots with LineWidth for line thickness and MarkerFaceColor, MarkerEdgeColor, MarkerSize for marker aesthetics.
• When displaying multiple models, assign distinct line styles (e.g., 'r--' vs. 'k-') to improve readability.

Frequently Asked Questions (FAQ)

Can I use a regression model for extrapolation?

Yes, regression models can be evaluated outside the original data range, but the reliability depends on the model’s assumptions and the underlying physics of the problem.

What happens if I increase the polynomial order too much?

Increasing the order adds more coefficients (n+1) and can lead to over‑fitting, where the model captures noise rather than the true signal. This often results in oscillatory behavior between points.

Is spline interpolation the same as cubic‑spline interpolation?

Yes, the 'spline' option in interp1 implements a cubic‑spline interpolation, constructing piecewise cubic polynomials that ensure smooth first and second derivatives.

How do I choose between 'spline' and 'pchip'?

'spline' provides a very smooth curve but may introduce overshoot near steep gradients. 'pchip' preserves the shape of the data and avoids overshoot, making it preferable when monotonicity is important.