Random Polynomial Regression Benchmark Generator

Description

Generate known ascending-coefficient polynomial targets on a bounded evenly spaced grid, add seeded normal noise, and delegate separate requested degree-1–6 normal-equation fits and predictions; reverse coefficients only at the highest-first evaluator boundary, retain singular/nonfinite fit outcomes, and make no exact-solver or stability guarantee.

Synthetic statistical data separates a specified generating model from the values observed in a finite draw. A model parameter, a sample summary, and the result of a fitted analysis answer different questions; retaining them separately makes numerical and statistical checks easier to interpret.

When to use this tool

Use this generator to create reproducible polynomial regression benchmark cases from the supplied model or bank. Inspect the returned evidence before using cases in a test or exercise.

How to use the tool and how it works

Supply coefficients, count, minimumX, maximumX, noiseStandardDeviation, fittingDegrees in the field shapes below. Set seed when you need an explicit replay.
Run the tool and correct any input error. For generated controls, distinguish a successful fixture containing an invalid case from failure to generate the fixture.
Read model, observations, fits. Use the conventions below and keep answer-bearing fields separate when presenting an exercise.

Evaluate the supplied polynomial on a regular predictor grid, add Gaussian noise, and fit every distinct requested degree to the same observations. Return each fit's predictions and distinguish finite, singular, and numeric-overflow statuses.^[1]

Delegated operations used for this result: evaluate polynomial; polynomial. Their calculation or parsing conventions apply to the corresponding result fields.^{[2, 3]}

Input field	Meaning and units	Accepted input
coefficients	Generating coefficients in ascending powers c0..cd.	Required. Array of Finite number.; minimum items 1; maximum items 7.
count	Requested count; the mechanics explain whether one draw expands into several returned records.	Required. Safe integer; minimum 2; maximum 1024.
minimumX	Minimum X	Required. Finite number.
maximumX	Maximum X	Required. Finite number.
noiseStandardDeviation	Noise Standard Deviation	Required. Finite number; minimum 0.
fittingDegrees	Fitting Degrees	Required. Array of Safe integer; minimum 1; maximum 6.; minimum items 1; maximum items 6.
seed	Optional numeric seed; takes precedence over context randomness. Without either, use the fixed tool.io default seed.	Optional. Safe integer.

Output field	Interpretation	Returned shape
model	Model	Required. Object; no unknown keys.
model.coefficientsAscending	Generating coefficients in ascending powers c0..cd.	Required. Array of Finite number.; minimum items 1; maximum items 7.
model.noiseStandardDeviation	Noise Standard Deviation	Required. Finite number; minimum 0.
observations	Observations	Required. Array of Object; no unknown keys.; minimum items 2; maximum items 1024.
observations[].x	X	Required. Finite number.
observations[].target	Target	Required. Finite number.
observations[].observed	Observed	Required. Finite number.
fits	Fits	Required. Array of Object; no unknown keys.; minimum items 1; maximum items 6.
fits[].degree	Degree	Required. Safe integer; minimum 1; maximum 6.
fits[].status	Named outcome; inspect null and error fields before interpreting associated values.	Required. Text; one of "finite", "singular", "numeric-overflow".
fits[].result	Result	Required. Object; no unknown keys. or Null.
fits[].predictions	Predictions	Required. Array of Finite number.; maximum items 1024.
randomness	Replay metadata: source is seed or context; seed is the normalized 32-bit seed or null for a context source.	Required. Object; no unknown keys.
randomness.seed	Optional numeric replay seed; normalization and precedence are explained below.	Required. Safe integer. or Null.
randomness.source	Source	Required. Text; one of "context", "seed".

An explicit safe-integer seed takes precedence over injected context randomness and is normalized to an unsigned 32-bit integer. With no seed and no context source, replay uses the fixed tool.io default seed. A context source must return finite values from zero inclusive to one exclusive; its advancing state must be retained separately. randomness records the source and normalized seed, or null for context. These generators are not cryptographic random sources.

json

{
  "input": {
    "coefficients": [
      0,
      0,
      1
    ],
    "count": 3,
    "minimumX": -1,
    "maximumX": 1,
    "noiseStandardDeviation": 0,
    "fittingDegrees": [
      1,
      2
    ],
    "seed": 42
  },
  "output": {
    "model": {
      "coefficientsAscending": [
        0,
        0,
        1
      ],
      "noiseStandardDeviation": 0
    },
    "observations": [
      {
        "x": -1,
        "target": 1,
        "observed": 1
      },
      {
        "x": 0,
        "target": 0,
        "observed": 0
      },
      {
        "x": 1,
        "target": 1,
        "observed": 1
      }
    ],
    "fits": [
      {
        "degree": 1,
        "status": "finite",
        "result": {
          "coefficients": [
            0.6666666666666666,
            0
          ],
          "rSquared": 0
        },
        "predictions": [
          0.6666666666666666,
          0.6666666666666666,
          0.6666666666666666
        ]
      },
      {
        "degree": 2,
        "status": "finite",
        "result": {
          "coefficients": [
            0,
            0,
            1
          ],
          "rSquared": 1
        },
        "predictions": [
          1,
          0,
          1
        ]
      }
    ],
    "randomness": {
      "seed": 42,
      "source": "seed"
    }
  }
}

Limitations and assumptions

Input coefficients are ascending in power, while delegated evaluation reverses them internally. Each fit degree must be below count. In-sample fit quality is not cross-validation or evidence for a real generating model.
A replay requires the same complete input and implementation. Randomized construction does not guarantee uniform coverage of every feasible output; finite samples can repeat unless a distinct-selection rule is explicitly stated.
The field tables state structural bounds; cross-field constraints and bounded search or output budgets also apply. Unknown keys are rejected where the object contract disallows them. Input errors are distinct from null fields and deliberate invalid controls in a successful output.

Alternative or Complementary methods

For a supplied example, inspect the returned construction or delegated check directly. To evaluate a consumer, also test cases chosen for its own boundaries; a generated finite corpus does not establish exhaustive coverage.

References

Tool behavior and example: Random Polynomial Regression Benchmark Generator — tool.io
Calculation or validation: algebra/evaluate-polynomial — tool.io
Calculation or validation: regression/polynomial — tool.io

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