Keep the calculation
open to inspection.

Three kinds of result

The applications serve different scientific purposes. Keep their roles explicit.

An interpretation can organize a research programme without changing the prediction of a benchmark experiment. A distinctive test requires an explicit law and a calibration that remains fixed across independent observations.

Prepare data for the comparison workbench

Use a UTF-8 text file with a comma-separated header. Numeric fields accept ordinary decimals or scientific notation.

x,y,sigma,split
0,1.0,0.1,train
1,1.6,0.1,train
2,2.1,0.1,test
3,2.9,0.1,train
4,3.5,0.1,train
5,4.2,0.1,test

Only x and y are required. If sigma is included, every row must contain a positive standard deviation in the units of y. If using the split column, every row must say train or test. Other automatic split options ignore the split column. No rows are silently discarded.

All x values must use one unit, and all y values another consistent unit. For harmonic models, x is in radians. Rescale large values before importing: |x|≤10⁶, |y|≤10¹² and 10⁻¹²≤sigma≤10¹². Missing values, mixed separators and multiline quoted fields must be cleaned before import.

The synthetic examples use a fixed illustrative oscillatory perturbation; they are deterministic demonstrations, not independent draws from a fitted noise distribution.

What the workbench computes

For a declared basis fⱼ(x), the prediction is Σⱼ βⱼfⱼ(x). The coefficients minimize Σᵢ[(yᵢ−ŷᵢ)/sigmaᵢ]² on the training rows only.

Without sigma, all weights are one and the residual variance estimate is RSS/(n−p). With supplied sigma, uncertainties are treated as known independent standard deviations: the covariance is (XᵀWX)⁻¹. Without sigma, it is multiplied by the estimated residual variance. Reported standard errors are square roots of the covariance diagonal; they are not confidence intervals or model-selection probabilities.

The implementation solves a column-scaled, reorthogonalized QR system. It rejects rank-deficient or nearly indistinguishable basis columns. Each model needs more training observations than fitted coefficients. The complete export contains the covariance matrix, degrees of freedom, predictions and residuals.

RMSE is √[Σ residual²/n] in y units. Reduced χ² is reported only when sigma is supplied, using training degrees of freedom n−p. Test rows contribute neither to coefficient fitting nor to the training covariance estimate. Automatic row allocations are conveniences; use an independently acquired test set when the scientific protocol requires one.

NIST weighted least-squares criterion and residual-analysis guidance provide the statistical background.

Retain enough to reproduce the result

Save the exported JSON alongside the original input file and a short description of the experiment or model.

• Page, model version and export date.
• Input values, units and approximation conventions.
• Training/test allocation and acquisition protocol.
• Fitted coefficients, uncertainty assumptions and residuals.
• Any supplied reference constants or prior calibration.
• The claim being tested and the outcome that would contradict it.

All calculators and CSV analysis execute in the browser. Exports are downloads you choose to save. The new comparison and custom-memory tools do not upload or automatically retain your input data.