Understanding results and analysis

Equarith reports a set of trade-offs rather than declaring one universally best formula. Validation, residual structure, interpretability, and domain knowledge all matter.

The Pareto front

The Search result table uses the columns Complexity, Train Error, optional Test Error, and Formula. A formula is Pareto-efficient when no available alternative is both simpler and better on the training objective.

The archive is bounded to 256 entries and keeps one best representative per complexity. You can:

• sort by complexity, training error, test error, or a balanced view;
• filter displayed formulas;
• choose an automatic best result based on training or test behavior; and
• select a row to open its detailed views.

A tiny improvement at much higher complexity is often not useful. Look for an elbow where error stops falling rapidly, then compare nearby formulas.

When test rows are available, the initial ordering and automatic selection favor test error; otherwise they favor training error. Selecting a row temporarily follows that formula. Click outside the rows to return to automatic best-result tracking as new solutions arrive.

Training and test interpretation

Training objective measures the rows that guided selection. Test objective measures held-out rows and is shown only when the split contains a test set.

• Similar train and test behavior supports, but does not prove, generalization.
• Much worse test behavior suggests overfitting, data shift, leakage, or a test set too small to be stable.
• Both poor suggests inadequate variables, unsuitable functions, insufficient search, noisy data, or no strong deterministic relationship.
• Unusually better test behavior can happen by chance or because the partitions have different difficulty.

The displayed direction depends on the metric. Error metrics are minimized; R², C², and Pearson are maximized. Compare only values calculated with the same metric, row set, and preprocessing.

Solution detail views

Selecting a solution opens five complementary views:

• Fit shows observed and predicted values across the chosen x-axis.
• Observed vs Predicted shows distance from an ideal one-to-one relationship.
• Residuals shows prediction error patterns.
• Pareto places the solution in the error-complexity trade-off.
• Search History shows how published quality evolved during the run.

Chart controls can choose source row or an input as the x-axis, combine training and test series, show signed or percentage residuals, switch supported axes between linear and logarithmic scale, and toggle the legend.

Point at a plotted value to read its details. Scroll or pinch to zoom around the pointer, drag to pan, and hold Shift while dragging a rectangle to zoom into one area. Double-click the plot or use Fit, Reset, Home, or 0 to restore all visible data. In version 1.0.3, Fit and Reset both fit the viewport to the visible data. With keyboard focus, + and - zoom while the arrow keys pan. Toolbar controls change the grid, lines, and point size; Copy, PNG, and CSV copy the image or export the image and plotted series.

Logarithmic axes omit non-positive values because they have no real logarithmic coordinate. Percentage residuals can be unstable near a zero target. Always read the chart status and axes.

The toolbar exports a PNG image and the plotted source series as CSV. For very large datasets the interactive chart uses a bounded representative point budget; this affects rendering density, not the full-precision solution evaluation used for metrics and normal prediction exports.

Diagnostic numbers

Solution details include RMSE, MAE, R², Pearson correlation, median absolute error, maximum error, and invalid-prediction counts where applicable.

Residual sign in the Search and Analysis charts follows observed - predicted. Percentage residual is 100 × (observed - predicted) / observed; it is undefined when the observed value is zero and can be unstable near zero. A cloud centered near zero is desirable, but inspect for:

• curvature, indicating missing structure;
• a widening fan, indicating non-constant error scale;
• bands or steps, indicating regimes or rounded inputs;
• clusters separated by time or category;
• large isolated residuals; and
• invalid values concentrated in one input region.

A high correlation can coexist with biased scale or offset. R² can be negative. A low average error can hide rare severe failures. No single number replaces the views.

Formula actions

From a solution you can copy its readable formula, use Copy as for a supported target syntax, export one or all solutions, load it as a seed, or open it in Prediction. Copy as copies the expression only; use file export when executable code needs a complete function or helper definitions.

Copy and export use the displayed published expression. If constant rounding is configured, its measured score corresponds to that restricted expression.

Analysis workspace

The Analysis workspace separates exploratory views from the Search selection.

Error vs Complexity visualizes the Pareto trade-off and lets you inspect formulas around the elbow.

Dataset Statistics groups numerical summaries into:

• data quality: finite, missing, invalid, and distinct counts;
• sign and zero counts;
• extremes and quantiles from P1 through P99;
• central tendency: arithmetic, trimmed, geometric, harmonic, and root-mean-square means plus the sum;
• dispersion: range, IQR, population and sample variance and standard deviation, standard error, coefficient of variation, mean and median absolute deviation, and robust standard deviation;
• shape: skewness and kurtosis;
• uncertainty: a 95 percent Student t confidence interval for the mean; and
• Tukey fences and outlier count.

For very large columns, some quantile or distinct estimates are deterministic approximations and are marked with ≈. Undefined statistics remain unavailable rather than being presented as valid zeros.

Correlations provides a matrix and selected-pair scatter preview. Correlation describes linear association and does not identify cause. Wide datasets should be narrowed to relevant columns for a readable matrix.

Solution Diagnostics has its own ordered selection of one to three Pareto formulas. It can temporarily edit valid formula text without changing Search, the project, or the Pareto archive. Its five charts are Fit, Observed vs Predicted with a one-to-one reference, Residual vs X, Residual vs Predicted, and Residual Histogram.

Choosing a formula

1. Reject formulas with implausible operations, units, or domains.
2. Compare train and held-out test behavior.
3. Inspect residuals, invalid predictions, and boundary regions.
4. Prefer the simpler formula when added complexity has no meaningful validation benefit.
5. Test sensitivity to split seed and search seed.
6. Validate on data from the real deployment regime.
7. Export and independently test the exact published expression.

For new inputs, continue with Predictions. For language targets, see Exporting data and formulas.