Square root of a density
Penguin learning project
The project uses 342 complete Palmer penguin records: 240 training-pool and 102 held-out rows. The selected training rows determine feature scaling and model parameters. Source CSV
This project trains weights with softmax, unlike the homepage lab, which uses nearest species averages. Same penguin measurements; different classification methods. Predictions can differ.
Approximation and depth · Study approximation error and network size under stated conditions.
The project supplies finite observations and fitted parameters. A held-out sample does not establish population bounds or prove a theorem.
Consider this concept alongside the project; its source assumptions may differ. These observations do not prove a theorem.
Input: 12 × 2; Held-out evaluation: 102 · #58 · 40.6 mm, 193 mm. Known answer: Adélie; Prediction: Adélie · [≈0.7383, ≈0.2444, ≈0.01737].
Used in these source sections: 3.1 A rule based on distance
φ is the square root of the density of the probability measure μ. Thus φ² gives the density.
The square root converts a weighted L² expression into an unweighted one.
Source formula example
illustration · The finite example uses counting measure; the source uses a continuous Fourier-based construction.
Symbols
- A probability measure.
- The square root of its density.
- The density.
Source formula example
This source example uses its own inputs and assumptions. Editing it does not change the project.
Edit a value or the code, then run the example. This is a finite demonstration, not a proof of the paper’s theorem.
This example’s data flow
Code changed. Reset to restore this example’s data flow.
Initial values in code
density=[0.25, 0.75]
Calculated here
Result expression: {
density,
phiSquared: phi.map((v) => v * v),
total: density.reduce((a, b) => a + b, 0),
}
phi=density.map(Math.sqrt)Uses:density
Returns
density- array (2 items): [0.25, 0.75]
phiSquared- array (2 items): [0.25, 0.75]
total- 1
Read numbers from input and return the result. JavaScript basics.
Result
Technical statement
The numerical values are invented. The finite example uses counting measure; the source uses a continuous Fourier-based construction.
Source: Inline notation