Superconductor regression explorer

A material that, below a certain temperature, carries an electric current with no measurable resistance and pushes magnetic field out of its interior. Each of the 21,263 rows in this dataset is a material for which that transition has been observed and reported.

The temperature at which that transition happens, in K. It is the one quantity this page predicts. Across this file it runs from 0.00021 to 185.0 K, with a median of 20.0 and a mean of 34.4: most materials superconduct only at low temperatures, and a few at much higher ones. A higher critical temperature means less extreme cooling is needed, which is why the hottest materials are the interesting ones.

The expulsion of magnetic field from inside a superconductor below its critical temperature. Together with zero resistance, it defines the superconducting state. The dataset records only the temperature of the transition.

The temperature unit used throughout. It starts at absolute zero, so no value in this column can be negative, and a value near zero means the material has only ever been seen to superconduct within a whisker of absolute zero.

The recipe of a material: which elements it contains and in what proportions. Compositions have different numbers of elements, so they cannot be handed to a regressor directly — which is the whole reason the engineered features below exist.

The 81 numeric columns that stand in for the composition. Each summarises a property of the elements in the material — atomic mass, first ionization energy, atomic radius, density, electron affinity, heat of fusion, thermal conductivity, valence — through a fixed recipe such as a mean, a weighted mean, a geometric mean, an entropy, a range or a standard deviation. Every material therefore gets the same 81 numbers whatever its formula.

An average in which each element counts in proportion to how much of it the material contains, rather than each element counting once. Seven of the eight columns included on this page are weighted means, which is why their names begin that way.

This demo, like the notebook it rebuilds, reads only the features file, so each material is identified by its row number in that file, counted from 0. The Dataset page shows the chemical formulas from the formulas file and rebuilds the features from them.

The 21,263 × 81 table of engineered features. Rows are samples (materials), columns are features. The measured temperature is not one of its columns.

The separate column of measured critical temperatures, one per row: the value the models learn to predict. It is not used in standardization or in the reduction.

X[i,:] is one material; X[:,j] is one feature across every material; x_ij is one number. In the Design matrix section each has its own colour, always shown next to its notation, so the meaning does not depend on seeing the colour.

Only 8 of the 81 feature columns are included in this page, chosen to cover the element properties the features summarise. Every branch was fitted on all 81; the 8 are what the Design matrix section draws.

Rows whose 81 features are identical to another row's; 6,093 of the 21,263 rows are like this. Identical rows sit on the same spot in the unsupervised layouts.

Where a value sits inside the distribution of the same column across the 14,884 training rows only. “Across one row” and the feature table in the Evaluation section use it so that eight columns in eight different units can share one axis.

The range the “Down one column” histogram is drawn over: the middle 98 % of a column's values, computed over all 21,263 rows. Without it, a few extreme values would flatten the whole distribution into one bar. A selected value outside the range is drawn at the nearest edge with a dashed marker.

There is none: the file has 0 rows with a missing value, and 0 rows are dropped. Nothing is filled in.

Learning a rule that maps inputs to a continuous number, from examples where the number is known. Here the inputs are the engineered features and the number is the measured critical temperature.

One random division of the rows, made before anything is fitted: 14,884 training rows and 6,379 held-out rows, a measured held-out fraction of 30.00 %. Every branch on this page reuses exactly that division, which is what lets their predictions be compared material by material.

A row kept aside for testing: no fitting step uses it. Every dot in the charts is one of these 6,379 materials, and every score on this page is computed on them alone.

Letting information from the held-out rows reach a fitting step — for example by standardizing or projecting the whole table before splitting it. The source notebook did that; this demo fits the scaler and the projection on the training rows only (the first item under “Our changes” in the Source evidence section).

Rescaling each column to a common centre and spread, so a column measured in thousands cannot dominate one measured in units. Fitted on the training rows only. Trees do not need it, which is why the all-feature forest branch skips it too.

A rotation of the standardized feature axes onto new axes ordered by how much of the variation each one carries. Keeping the first 17 of them is the reduced representation on this page; together they carry 95.54 % of the training variance.

One of those new axes: a weighted mixture of every original feature. It has no physical unit and its sign is arbitrary, so component scores can be compared with each other but not with physical quantities.

The share of the training data's total variation that a set of components reproduces. It does not measure how useful the components are for prediction, because the target is not used to compute them.

The experiment this page is built around: hold the rows, the split and the models fixed, change only how X is represented, and read the difference. The source notebook runs its models on the 17 components only; the two branches that keep all 81 features were added in this demo.

One model on one representation. The page compares five: linear regression and random forest on both representations, and polynomial regression on the 17 components. The Key Findings page calls them the five prediction models.

A branch differing from another in exactly one respect, so a difference between them can be attributed to that respect. Here: the same model on all 81 features instead of on 17 components.

Fits one weight per input plus an offset and predicts their sum. It can only express a straight-line relationship between each input and the target.

Adds every square and every pairwise product of the inputs, so a linear fit over them can bend. On 17 components that is a manageable number of terms; on all 81 features it is far larger, which is why the polynomial branch exists on the reduced representation only.

Many decision trees, each fitted on a different random view of the training rows, whose predictions are averaged. It can express interactions and non-linear effects without being told to look for them, and it needs no scaling.

Learning the training rows rather than the pattern. Its signature is a training score much better than the held-out score; the Conclusions section gives that gap for every branch.

The fixed number that makes a random choice repeatable. The split, the forests, each unsupervised layout and the k-means grouping all use one; the layout seeds are listed in that panel's notes, and the k-means seed is 1337.

Every branch was fitted once in advance by the analysis script tools/precompute.py, and its predictions for the held-out materials are stored in this page; the controls switch between them.

The share of the held-out variation a branch reproduces. 1 is perfect, 0 is no better than always predicting the mean, and it can go negative. It is a ratio, so it says nothing about how many K a typical miss is.

Mean absolute error, in K: the average size of a miss with its direction ignored. It answers the concrete question R² does not.

Root mean squared error, also in K. It weights large misses more heavily than MAE, so the gap between RMSE and MAE is a rough measure of how uneven the errors are.

The error (prediction minus measurement) with its sign kept, and its average over a group. Negative means the branch predicts below the measurement; the Evaluation section uses it to show every branch running low in the hottest band, which an absolute error alone would hide.

A slice of the held-out rows by their measured temperature — < 10 K holds 2,306, 10–30 K holds 1,408, 30–60 K holds 972, 60–90 K holds 1,044, ≥ 90 K holds 649 materials. Bands are defined on the measurement and never on the prediction, so the same materials are compared across branches.

The colour scale of the component scatter in the Transformation and modelling section. The measured temperature it shows is an ordered quantity, so the colours run along one ordered scale rather than being a set of unrelated hues.

The marker shape, not the colour, shows the representation: filled for the 17 components, hollow for all 81 features. Colour always identifies the model, so the same hue is the same model in every panel.

Every colour-coded chart comes with a table of the same numbers, beside it or in a collapsible section under it.

Hovering over a chart shows the material or bar under the pointer, with its numbers. In most scatter charts, clicking a dot opens that material in the Evaluation section.

The four-column diagram under the page title shows the whole analysis: inputs, representation, models and evaluation, with the steps in each.