Regression Toolkit (linear/poly/exp/log) mechanism

Model scope

The listed methods are standard only after the model forms, logarithm base, residual space, R² convention, polynomial degree, domain restrictions, and singular-fit behavior are specified. The exact ambiguity requiring review is whether exponential and power fits report transformed-space or original-space least-squares and R²; the specification also does not state the actual power-model formula.

• The normal-equation statement is correct for ordinary least squares, but the polynomial degree and coefficient ordering are unspecified.
• Exponential log-linearization is standard for fitting y=a*exp(bx) only when y>0; it minimizes squared residuals in ln(y), not generally squared residuals in y.
• Logarithmic fitting must specify whether the model is y=a+b*ln(x) or uses another logarithm base; x must be strictly positive.
• Power fitting is not explicitly parameterized; the standard model y=a*x^b requires x>0 and y>0 and log-linearizes to ln(y)=ln(a)+b*ln(x).
• R² is ambiguous for transformed exponential and power fits: it may mean R² in transformed log space or R² computed from residuals in the original y-space. These values are generally different.
• The residual-plot definition is unspecified, including whether residuals are y-observed minus y-predicted in original space or transformed space.

Units and conventions

• Linear and polynomial predictors and responses are numeric quantities in consistent units; coefficients inherit the corresponding units.
• For exponential models, x is treated as a numeric predictor and b has reciprocal-x units; y and fitted a must use the same response units.
• For logarithmic and power models, the logarithm argument must be dimensionless or normalized by a stated reference unit; the implementation appears to assume raw positive numeric x.
• Natural logarithms are assumed for all log-linearizations; changing the log base changes intermediate coefficients but not an exactly transformed fit if handled consistently.
• R² is assumed to be 1-SSE/SST using the same residual space as the reported fit, but the specification does not state whether that space is original or transformed.

Inspectable vector

One of 4 source-bound review vectors:

{
  "inputs": {
    "model_type": "linear",
    "data": [
      [
        1,
        2
      ],
      [
        2,
        3
      ],
      [
        3,
        5
      ]
    ],
    "residual_definition": "observed_minus_predicted"
  },
  "expected_outputs": {
    "coefficients": {
      "intercept": "-8/3",
      "slope": "3"
    },
    "predicted_y": [
      "1/3",
      "10/3",
      "19/3"
    ],
    "residuals": [
      "5/3",
      "-1/3",
      "-4/3"
    ],
    "R2": "0",
    "prediction_at_x_4": "28/3"
  },
  "reference_data": {
    "model": "y=a+bx",
    "R2_definition": "1-SSE/SST in original y-space"
  },
  "working": "x̄=(1+2+3)/3=2. ȳ=(2+3+5)/3=10/3. Sxx=(1-2)^2+(2-2)^2+(3-2)^2=2. Sxy=(1-2)(2-10/3)+(2-2)(3-10/3)+(3-2)(5-10/3)=4/3+5/3=3. Therefore b=Sxy/Sxx=3 and a=ȳ-bx̄=10/3-6=-8/3. Predictions are 3(1)-8/3=1/3, 3(2)-8/3=10/3, and 3(3)-8/3=19/3. Residuals are 2-1/3=5/3, 3-10/3=-1/3, and 5-19/3=-4/3. SSE=(25+1+16)/9=42/9=14/3. SST=(16+1+25)/9=42/9=14/3. R²=1-(14/3)/(14/3)=0. At x=4, ŷ=12-8/3=28/3."
}

Failure boundary

• Normal equations become singular or ill-conditioned when predictors are collinear, polynomial degree is too high relative to the number or spread of observations, or there are insufficient data points.
• A linear fit needs at least two non-identical x values; a degree-d polynomial needs enough distinct, non-degenerate x values to make the design matrix full rank.
• Exponential fitting must reject or explicitly handle y<=0; logarithmic fitting must reject x<=0; power fitting must reject x<=0 and y<=0 under the stated log-linearization.
• Division by zero occurs when Sxx=0 for linear regression or when SST=0 for R²; the output policy for constant y is unspecified.
• Log transformation changes the objective and can create bias when predictions are back-transformed; a log-linearized fit is not generally the same as nonlinear least squares in original y-space.
• Large magnitudes, extreme polynomial degrees, and exponentials can overflow or suffer severe numerical conditioning problems; QR or SVD would generally be more stable than explicit normal equations.
• Residual plots need a stated residual convention and transformed-versus-original response space.
• The empty honest_label is a concern: the user-facing label should disclose domain restrictions, transformed-error fitting, R²-space ambiguity, and that power fitting is only listed in computes but not parameterized in formulas.

Other records

Sources and evidence

These pointers bind the page to evidence. They do not grant authority or collapse verification into deployment.

• VERIFIED_LEDGER.json
• evidence-tranche8/tranche8-results.json