LIME：用簡單模型解釋附近的一次預測

pi_x(z) = exp(−D(x,z)^2 / sigma^2)

f(z1,z2) = 0.10 + 0.45z1 + 0.25z2 + 0.10z1z2

sqrt(exp(−d²/kernel_width²))
  = exp(−d²/(2×kernel_width²))

選 g，使下式較小：
  sum_i pi_x(z_i) × [f_j(z_i) − g(z'_i)]² + Omega(g)

前半：在選定鄰域與權重下，代理模型偏離黑箱的程度
後半：代理模型的複雜度

g(z′) = b0 + b1×z1 + b2×z2

g(z′) = 0.006537 + 0.545876×bright + 0.345876×repair

python3 lime_local_surrogate.py

#!/usr/bin/env python3
"""Deterministic LIME-style neighborhood and weighted-surrogate toy."""

from __future__ import annotations

import math


QUERY = (1.0, 1.0)  # two interpretable word-presence indicators
WORDS = ("bright", "repair")
SIGMA = 0.5


def black_box(z: tuple[float, float]) -> float:
    """Hand-specified nonlinear score, not a fitted classifier."""
    z1, z2 = z
    return 0.10 + 0.45 * z1 + 0.25 * z2 + 0.10 * z1 * z2


def cosine_distance(query: tuple[float, float], z: tuple[float, float]) -> float:
    """Cosine distance; define an all-zero neighbor's distance as 1."""
    q_norm = math.sqrt(sum(value * value for value in query))
    z_norm = math.sqrt(sum(value * value for value in z))
    if z_norm == 0.0:
        return 1.0
    cosine = sum(a * b for a, b in zip(query, z)) / (q_norm * z_norm)
    return 1.0 - cosine


def solve_linear_system(matrix: list[list[float]], vector: list[float]) -> list[float]:
    """Solve a small square system with Gaussian elimination and pivoting."""
    size = len(vector)
    augmented = [row[:] + [vector[i]] for i, row in enumerate(matrix)]
    for column in range(size):
        pivot = max(range(column, size), key=lambda row: abs(augmented[row][column]))
        if abs(augmented[pivot][column]) < 1e-12:
            raise ValueError("weighted design matrix is rank deficient")
        augmented[column], augmented[pivot] = augmented[pivot], augmented[column]
        scale = augmented[column][column]
        augmented[column] = [value / scale for value in augmented[column]]
        for row in range(size):
            if row == column:
                continue
            factor = augmented[row][column]
            augmented[row] = [
                a - factor * b for a, b in zip(augmented[row], augmented[column])
            ]
    return [augmented[row][-1] for row in range(size)]


def main() -> None:
    # Enumerate every subset of the two present words: no random sampling.
    neighbors = [(1.0, 1.0), (1.0, 0.0), (0.0, 1.0), (0.0, 0.0)]
    rows = []
    for z in neighbors:
        distance = cosine_distance(QUERY, z)
        weight = math.exp(-(distance * distance) / (SIGMA * SIGMA))
        rows.append((z, distance, weight, black_box(z)))

    # g(z') = beta0 + beta1*z1 + beta2*z2, fit by locality-weighted OLS.
    # This deliberately omits LIME's K-Lasso feature selection/sparsity step.
    design = [[1.0, z[0], z[1]] for z, _, _, _ in rows]
    targets = [prediction for _, _, _, prediction in rows]
    weights = [weight for _, _, weight, _ in rows]
    normal = [
        [sum(w * row[i] * row[j] for w, row in zip(weights, design)) for j in range(3)]
        for i in range(3)
    ]
    rhs = [
        sum(w * row[i] * y for w, row, y in zip(weights, design, targets))
        for i in range(3)
    ]
    beta = solve_linear_system(normal, rhs)

    residual = max(
        abs(sum(normal[i][j] * beta[j] for j in range(3)) - rhs[i])
        for i in range(3)
    )
    assert residual < 1e-10
    local_prediction = sum(beta[j] * [1.0, QUERY[0], QUERY[1]][j] for j in range(3))
    weighted_sse = sum(
        w * (y - sum(b * feature for b, feature in zip(beta, row))) ** 2
        for w, y, row in zip(weights, targets, design)
    )

    print("scope=hand-specified black box; deterministic exhaustive 2-word neighborhood")
    print(f"query_interpretable={QUERY}; words={WORDS}; sigma={SIGMA}")
    print("neighbor z1 z2 cosine_distance locality_weight black_box_score")
    for z, distance, weight, prediction in rows:
        print(f"{int(z[0])} {int(z[1])} {distance:.6f} {weight:.6f} {prediction:.6f}")
    print(f"surrogate=intercept + b1*{WORDS[0]} + b2*{WORDS[1]}")
    print(f"intercept={beta[0]:.6f}; b_{WORDS[0]}={beta[1]:.6f}; b_{WORDS[1]}={beta[2]:.6f}")
    print(f"black_box_at_query={black_box(QUERY):.6f}")
    print(f"surrogate_at_query={local_prediction:.6f}")
    print(f"weighted_squared_error={weighted_sse:.6f}")
    print(f"normal_equation_max_residual={residual:.3e}")
    print("limit=no trained model, corpus, randomized LIME sampling, feature selection, or stability study")


if __name__ == "__main__":
    main()

Copyright (c) 2016, Marco Tulio Correia Ribeiro
All rights reserved.

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# MIT License

Copyright (c) 2025 lime authors

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of this software and associated documentation files (the "Software"), to deal
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# DS56 image source and license notices

This record describes the selected source illustrations. Each source keeps its own grant; this notice does not combine or extend the licenses. All edits are limited to the changes described below. No source data or model outputs were regenerated for these images.

## ISMIR 2017 paper figures — CC BY 4.0

**Source:** Saumitra Mishra, Bob L. Sturm, and Simon Dixon, “Local Interpretable Model-Agnostic Explanations for Music Content Analysis,” ISMIR 2017, Figs. 2, 3, and 6. [Original paper](https://archives.ismir.net/ismir2017/paper/000216.pdf). The paper’s first page grants CC BY 4.0; the selected figure pages show no separate third-party credit. [CC BY 4.0 terms](https://creativecommons.org/licenses/by/4.0/).

- `fig-01-lime-overview.png`, `fig-01-lime-overview-mobile.png`, `cover-ds56-lime-overview.png`: Fig. 2. The desktop/cover preserve the complete source figure and caption after page-margin cropping; the mobile image places a complete original source-panel crop after the whole figure.
- `fig-02-slime-workflow.png`, `fig-02-slime-workflow-mobile.png`: Fig. 3. The desktop preserves the complete source figure/caption after page-margin cropping; the mobile image places six original node crops after the whole figure. Connections should be read in the complete figure; no connections were added.
- `fig-03-sample-budget.png`, `fig-03-sample-budget-mobile.png`: Fig. 6. The desktop preserves the full two-panel figure/caption after page-margin cropping; the mobile image places complete original panels after the whole figure.

Changes: page whitespace was cropped, desktop images were proportionally resized with white margins, and mobile variants stack source-only crops beneath the whole source image. The cover proportionally fits the full Fig. 2 on a white 1600×900 canvas. No labels, arrows, data, or visual marks were added or redrawn. Source page/crop coordinates and output hashes are recorded in `source-crop-audit.json`.

## `marcotcr/lime` multiclass output — BSD 2-Clause

**Creator:** Marco Tulio Correia Ribeiro. **Source:** pinned repository commit [`fd7eb2e6f760619c29fca0187c07b82157601b32`](https://github.com/marcotcr/lime/tree/fd7eb2e6f760619c29fca0187c07b82157601b32), [`doc/images/multiclass.png`](https://github.com/marcotcr/lime/blob/fd7eb2e6f760619c29fca0187c07b82157601b32/doc/images/multiclass.png). The repository-authored HTML output is covered by the pinned repository’s BSD 2-Clause license. The full BSD license, copyright notice and disclaimer are reproduced above in this article.

- `fig-04-multiclass-text.png` is byte-identical to the source PNG.
- `fig-04-multiclass-text-mobile.png` places the original whole image before three source-pixel crops (probabilities and two complete class-pair panels); it adds no labels or data.

## `tidymodels/lime` Iris README output — MIT

**Source:** pinned [`tidymodels/lime` commit `1ac166867a7b2b4af6e3f4717159cd59e30cf7b0`](https://github.com/tidymodels/lime/tree/1ac166867a7b2b4af6e3f4717159cd59e30cf7b0), README.Rmd-generated `man/figures/README-unnamed-chunk-2-1.png`. The pinned MIT license applies to the associated documentation and this author-generated plot; the complete MIT notice and terms are reproduced above in this article.

- `fig-05-iris-cases.png` is byte-identical to the source PNG.
- `fig-05-iris-cases-mobile.png` places the original whole image before five source-pixel case crops. The shared feature/weight axes and legend remain in the whole image; no values were reconstructed.

The source README uses R’s built-in `iris`. The UCI Iris resource has a separate CC BY 4.0 grant, but byte/row identity between those releases has not been established. No dataset file is included, and these figures must not be described as a verified exact reproduction of the UCI data.

## Exclusions and limitations

The original 2016 LIME paper figures are link-only because no positive commercial republication grant was established. The `marcotcr/lime` tabular screenshot with unresolved data provenance, photographic/email screenshots, and the Jamendo-dependent ISMIR figure are not included.