Polynomial Regression of Infrared Distance Sensor Calibration Curves on a Mobile Robot

Project info

• Field of studyRobotics and Mechatronics
• LevelUndergraduate thesis (S1)
• Method typeStatistics and Regression
• Estimated build time± 3 working days
• Toolspolyfit, polyval, cvpartition (Statistics and Machine Learning Toolbox)
• PriceUS$19 - US$125

Research goal

Model the true distance as a function of the sensor output voltage and the sensor temperature using polynomial regression and assess the model fit.

Variables

Example result

• 3Selected degree
• 0.621CV RMSE
• 0.609Training RMSE
• 0.863R-squared
• -0.584Leading coefficient
• 220Samples

Five-fold cross-validation selects a degree-3 polynomial for the True distance versus Sensor output voltage relation with a validation RMSE of 0.621 and R-squared of 0.863; degree 9 fits the training data better (RMSE 0.596) but risks overfitting.

Computed from illustrative synthetic data, not from a real study.

About Polynomial Regression and Curves

Polynomial regression models curved relationships by adding powers of the predictor. It suits experimental data with simple curved patterns and when the theoretical functional form is unknown.

Steps

1. 1Plot the data to judge the shape of the relation
2. 2Fit polynomials of degree 1 to a maximum with polyfit
3. 3Choose the degree by k-fold cross-validation
4. 4Compute R2 and RMSE at the chosen degree
5. 5Compute an uncertainty band with polyval and the S structure
6. 6Plot the fitted curve and data

Assumptions to check

• High degrees overfit easily so they must be limited
• The predictor is centred and scaled for numerical stability
• Extrapolation beyond the data range is unreliable

Reading and reporting the results

Report the chosen degree with its cross-validation rationale, R2, RMSE, and the curve equation. Warn that polynomials are not mechanistic models and should not be used for extrapolation.

Example MATLAB code

A short starting script with this project's variable names. The delivered version is adapted to your data and fully commented.

rng(2);
n = 80;
sensor_output_voltage = sort(10*rand(n, 1));
true_distance = 2 + 1.5*sensor_output_voltage - 0.35*sensor_output_voltage.^2 + 0.02*sensor_output_voltage.^3 + 0.8*randn(n, 1);

maxdeg = 6;  K = 5;
cv = cvpartition(n, 'KFold', K);
cvrmse = zeros(maxdeg, 1);
for d = 1:maxdeg
    e = 0;
    for k = 1:K
        tr = training(cv, k);  te = test(cv, k);
        p = polyfit(sensor_output_voltage(tr), true_distance(tr), d);
        e = e + sum((true_distance(te) - polyval(p, sensor_output_voltage(te))).^2);
    end
    cvrmse(d) = sqrt(e / n);
end
[~, dbest] = min(cvrmse);
fprintf('CV RMSE per degree: %s\n', mat2str(cvrmse', 3));
fprintf('Chosen degree: %d\n', dbest);

[p, S, mu] = polyfit(sensor_output_voltage, true_distance, dbest);
[yfit, delta] = polyval(p, sensor_output_voltage, S, mu);
r2 = 1 - sum((true_distance - yfit).^2) / sum((true_distance - mean(true_distance)).^2);
fprintf('R2 = %.3f, RMSE = %.3f\n', r2, sqrt(mean((true_distance - yfit).^2)));

figure;
plot(sensor_output_voltage, true_distance, 'o'); hold on;
plot(sensor_output_voltage, yfit, 'r', 'LineWidth', 1.5);
plot(sensor_output_voltage, yfit + 2*delta, 'r--', sensor_output_voltage, yfit - 2*delta, 'r--');
xlabel('x'); ylabel('y'); legend('Data', 'Fit', '95% band');

Packages and prices

Choose how much help you need. Each higher package includes everything in the lower ones. Prices exclude 11% VAT.

• Payment: 50% down payment before work starts, balance after the deliverables are handed over
• Tax: Prices exclude 11% VAT
• Data and academic integrity: Results are computed from your own research data; you write and stand behind your manuscript, we help with the analysis and your understanding of it
• Revisions: Up to 2 revisions are included in the price (Package 3: 3 rounds)

Frequently asked questions