Isolation Forest, LOF, one-class SVM

"You spent a week teaching me what a fault looks like. I found it faster by learning only what normal looks like, and flinching at everything else."

An Unsupervised AI Agent

These three methods share one framing: none of them knows what an anomaly looks like, only what normal looks like, and each turns the distance from normal into a continuous anomaly score rather than a hard label. You still need a threshold, and choosing it well is its own discipline covered in Section 12.7. Throughout, \(x \in \mathbb{R}^d\) is a feature vector, usually the windowed statistical and spectral features of Chapter 8, drawn from routine operation for training. Because all three learn "normal" from data, they inherit every leakage hazard of Chapter 5: a fault that sneaks into the training window gets learned as normal.

Isolation Forest: anomalies are cheap to isolate

Isolation Forest (Liu, Ting, and Zhou, 2008) inverts the usual density-estimation instinct: instead of modeling where the data is dense, it asks how hard each point is to isolate by random partitioning. Build a tree by repeatedly picking a random feature and a random split value between that feature's current min and max, recursing until every point sits alone in a leaf. A point in a dense cluster needs many splits to be cornered; an outlier far out on some axis gets carved off after only a few. The path length \(h(x)\), the number of edges from root to \(x\)'s leaf, is therefore short for anomalies and long for inliers; average over an ensemble of random trees for a stable estimate \(E[h(x)]\).

The score normalizes path length against \(c(n)\), the average path length of an unsuccessful search in a binary tree of \(n\) points, so that scores are comparable across sample sizes:

where \(H(i)\) is the \(i\)-th harmonic number. A score near 1 means "isolated almost immediately," a strong anomaly; near 0.5 means "as hard to isolate as a typical point." Isolation Forest's appeal for sensor fleets is practical: it is near-linear in \(n\), trivially parallel across trees, needs no distance metric or feature scaling (splits are axis-aligned and scale-invariant), and handles high-dimensional vibration and current-signature feature vectors without choking. Its blind spot is the flip side of axis-aligned splits: it struggles with anomalies defined only by a correlation between features rather than an extreme value on any single axis, a gap the Extended Isolation Forest (Hariri et al., 2019) closes with random-slope hyperplane cuts.

Local Outlier Factor: weird relative to your neighbors

Isolation Forest and most statistical detectors judge a point against the global distribution. That fails when normal operation has regions of genuinely different density: an HVAC chiller idling, ramping, and running at full load produce three legitimate clusters of very different tightness. A point on the sparse edge of the idle cluster might be perfectly healthy, while a point of identical global rarity inside the dense full-load cluster is a real fault. Local Outlier Factor (Breunig et al., 2000) makes rarity local. For each point it finds the \(k\) nearest neighbors and defines a smoothed local reachability density \(\mathrm{lrd}_k(x)\), the inverse of the average reachability distance to those neighbors. Reachability distance floors the raw distance \(d(x,o)\) at the neighbor's own \(k\)-distance, so a near-duplicate cannot inherit an artificially inflated density; that floor matters on sensor data, where back-to-back windows from a steady idle state sit almost on top of each other, and unfloored near-zero distances would otherwise destabilize the ratio below. LOF then compares your density to theirs:

An \(\mathrm{LOF}\) near 1 means you are about as dense as your neighbors: an inlier. Substantially above 1 means your neighborhood is much denser than you are, so you sit in a relative void: an outlier. Because the comparison is only ever local, LOF adapts to varying density automatically, which is its whole reason to exist. The costs are equally characteristic: it demands careful feature scaling (millivolts will drown kilohertz) and degrades in high dimensions where distances concentrate, pushing you toward the dimensionality reduction of Chapter 8 first. Naive LOF is also \(O(n^2)\) to score, and \(k\) matters: too small chases noise, too large washes out local structure.

One-class SVM: draw a boundary around normal

The third principle is geometric. A one-class SVM (Schölkopf et al., 2001) maps the data into a high-dimensional feature space through a kernel, usually the radial basis function \(k(x, x') = \exp(-\gamma \lVert x - x' \rVert^2)\), and finds the hyperplane that separates the data from the origin with maximum margin; points on the origin side score as anomalies. An equivalent, often more intuitive formulation, Support Vector Data Description (Tax and Duin, 2004), fits the smallest hypersphere enclosing the normal data, with anything outside anomalous. The key knob is \(\nu \in (0, 1]\), which simultaneously upper-bounds the fraction of training points allowed outside the boundary and lower-bounds the fraction of support vectors: set \(\nu = 0.05\) and you are asserting at most about 5 percent of training data are contaminants, a direct handle on your assumed anomaly rate.

One-class SVM shines when normal operation occupies a compact, curved region a good kernel can wrap tightly, producing a genuine decision boundary you can reason about. Its weaknesses are real: the RBF bandwidth \(\gamma\) is finicky and interacts with \(\nu\), training scales poorly past tens of thousands of points, and it is sensitive to feature scaling and training-set contamination. It is often the least robust of the three on messy, high-volume fleet telemetry, better kept as a boundary-aware complement than the primary detector.

The comparison code below fits all three on the same feature matrix and lines up their scores, which is exactly how you should start any new deployment: run all three, look at where they agree and disagree, and let that structure the investigation.

import numpy as np
from sklearn.ensemble import IsolationForest
from sklearn.neighbors import LocalOutlierFactor
from sklearn.svm import OneClassSVM
from sklearn.preprocessing import StandardScaler

# X_train: healthy windows (n, d).  X_test: recent windows to score.
scaler = StandardScaler().fit(X_train)          # matters for LOF and OCSVM
Xtr, Xte = scaler.transform(X_train), scaler.transform(X_test)

iforest = IsolationForest(n_estimators=200, contamination=0.02,
                          random_state=0).fit(Xtr)
ocsvm   = OneClassSVM(kernel="rbf", gamma="scale", nu=0.02).fit(Xtr)
lof     = LocalOutlierFactor(n_neighbors=20, novelty=True).fit(Xtr)

# Higher = more anomalous for all three (sign-flipped so they align).
scores = {
    "iforest": -iforest.score_samples(Xte),
    "ocsvm":   -ocsvm.score_samples(Xte),
    "lof":     -lof.score_samples(Xte),
}
consensus = np.mean([(s > np.quantile(s, 0.98)) for s in scores.values()], axis=0)
flagged = np.where(consensus >= 2/3)[0]          # >=2 of 3 detectors agree

What's Next

In Section 12.6, we confront the uncomfortable truth lurking under every score in this section: with few or no labels, how do you even know your detector is any good? We will build honest evaluation protocols for partially labeled anomaly data and dismantle the point-adjust trap, a scoring convention that has quietly inflated a decade of time-series anomaly-detection results.