Finite measure and L2
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.
Probability and expectation · Describe random variables and their averages under a law.
Project vectors have two features; the training matrix has n rows and two columns. Mathematical tools may operate on these values when their assumptions apply.
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: 1.2 Learning from examples
A measure μ assigns sizes to measurable sets. The L²(μ) norm squares function values, integrates with respect to μ, and takes the square root.
The measure determines how values contribute to the norm.
Source formula example
illustration · A finite measure is σ-finite; general measures can require integration over infinitely many values.
Symbols
- A σ-finite reference measure in the density example.
- The domain is a countable union of finite-measure sets.
- Square-integrable functions under μ.
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
values=[1, 3]weights=[0.5, 0.5]
Calculated here
Result expression: {
values,
weights,
norm: Math.sqrt(values.reduce((s, v, i) => s + weights[i] * v * v, 0)),
}
Returns
values- array (2 items): [1, 3]
weights- array (2 items): [0.5, 0.5]
norm- ≈2.236
Read numbers from input and return the result. JavaScript basics.
Result
Technical statement
The numerical values are invented. A finite measure is σ-finite; general measures can require integration over infinitely many values.
Source: Inline notation