Spline Interpolation of Titration Curves for Equivalence Point Determination from Limited Data

Project info

• Field of studyChemistry and Materials
• LevelUndergraduate thesis (S1)
• Method typeCommunications and Forecasting
• Estimated build time± 3 working days
• Toolsspline, pchip, interp1; csaps (Curve Fitting Toolbox)
• PriceUS$19 - US$125

Research goal

Interpolate solution pH with splines and assess the error under varying titrant volume and number of data points.

Variables

Example result

• 12Knots
• 0.373Max error linear
• 0.184Max error spline
• 0.214Max error PCHIP
• 0.052RMSE spline
• 1.032Max slope error spline

With 12 knots the cubic spline cuts the maximum error of Solution pH from 0.373 (linear) to 0.184 with an RMSE of 0.0516; PCHIP preserves shape at an error of 0.214 and the spline keeps a smooth first derivative.

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

About Spline Interpolation

Spline interpolation joins data points with smooth piecewise cubic polynomials, while a smoothing spline (csaps) filters noise. It is used to fill between measurement points, compute derivatives, or build smooth curves from experimental data.

Steps

1. 1Prepare sorted x points and y values
2. 2Interpolate with spline, pchip, and linear
3. 3Build a dense evaluation grid
4. 4Compare error against the true function if known
5. 5Use csaps for noisy data with a smoothing parameter
6. 6Plot the resulting curves and data

Assumptions to check

• The x points are unique and sorted
• An interpolating spline follows data exactly so it is unsuitable for noisy data
• The underlying function is reasonably smooth

Reading and reporting the results

Report the interpolation error (RMSE) for each method and the reason for the spline type chosen. For noisy data, explain how the smoothing parameter was chosen.

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(9);
truef = @(x) sin(x) + 0.3*x;
titrant_volume = linspace(0, 10, 12)';                      % coarse sample points
solution_ph = truef(titrant_volume);
xi = linspace(0, 10, 500)';

ys = spline(titrant_volume, solution_ph, xi);
yp = pchip(titrant_volume, solution_ph, xi);
yl = interp1(titrant_volume, solution_ph, xi, 'linear');
rm = @(a) sqrt(mean((a - truef(xi)).^2));
fprintf('RMSE spline = %.4f, pchip = %.4f, linear = %.4f\n', rm(ys), rm(yp), rm(yl));

xn = linspace(0, 10, 80)';
yn = truef(xn) + 0.2*randn(size(xn));               % noisy data
sp = csaps(xn, yn, 0.9);
ysm = fnval(sp, xi);
fprintf('RMSE smoothing spline = %.4f\n', rm(ysm));

figure;
subplot(1,2,1);
plot(xi, truef(xi), 'k', xi, ys, 'r', xi, yp, 'b--', xi, yl, 'g:', titrant_volume, solution_ph, 'ko');
legend('True', 'Spline', 'PCHIP', 'Linear', 'Data'); title('Interpolation');
subplot(1,2,2);
plot(xn, yn, '.', xi, ysm, 'r', 'LineWidth', 1.5); title('Smoothing spline');

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)

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