Time-domain features

"My gait classifier scored 98 percent in the lab and collapsed in the field. It had learned the mean of the accelerometer, which was just how each volunteer happened to clip the sensor to their belt."

An Overfit AI Agent

Windowing: the unit of computation

What a time-domain feature summarizes is not the whole stream but a window: a contiguous block of \(N\) samples, typically 0.5 to 5 seconds long for human-scale motion, drawn with a fixed stride so consecutive windows overlap. Why windows exist is that most sensing tasks are locally stationary but globally not: a person is "walking" for a few seconds at a time, a bearing is "healthy" for hours, so the label lives at the window level, and the features must too. How you pick the window length is a genuine tradeoff. Too short and a feature is dominated by noise and cannot span a full gait cycle or breath; too long and a single window straddles two different activities, blurring the boundary the model must find. A common default for wearable motion is a 2 second window with 50 percent overlap, giving one feature vector every second. Overlap matters for a subtle reason we return to in Chapter 5: overlapping windows from the same recording are highly correlated, so if some land in training and their neighbors in test, your accuracy is inflated by leakage.

Amplitude and energy features

The simplest descriptors ask "how big is the signal in this window." The mean \(\mu = \frac{1}{N}\sum_n x[n]\) captures the DC level, which for an accelerometer axis encodes the projection of gravity, and therefore sensor orientation. The root-mean-square,

measures overall magnitude including any offset, and its square is proportional to signal energy or power. For a zero-mean vibration or EMG burst, RMS is the workhorse intensity feature: it rises with muscle force, with machine load, with impact severity. Closely related are the peak \(\max_n |x[n]|\), the peak-to-peak range \(\max_n x[n] - \min_n x[n]\), and the crest factor (peak divided by RMS), which is small for a smooth sinusoid and large for a spiky, impulsive signal. Crest factor is a classic early warning in rotating machinery: a healthy bearing hums smoothly (low crest factor) while a spalled one rings with sharp impacts on every revolution (rising crest factor) long before RMS energy itself climbs.

Dispersion and distribution shape

Beyond size, the spread of the samples is diagnostic. The variance \(\sigma^2 = \frac{1}{N}\sum_n (x[n]-\mu)^2\) and its root, the standard deviation, measure how much the signal swings around its own level, and unlike RMS they are immune to a constant offset, which is often what you want. Two higher moments describe the shape of the amplitude distribution. Skewness measures asymmetry: a PPG pulse or an impact accelerometer has a fast rise and slow decay, so its sample distribution is skewed. Kurtosis measures tailedness, how peaky-and-heavy-tailed the distribution is relative to a Gaussian:

Kurtosis near 3 signals near-Gaussian, well-behaved noise; a value far above 3 signals rare, large excursions, the statistical fingerprint of impulsive faults. This is why kurtosis is a staple of vibration-based bearing diagnostics: it responds to the presence of sharp transients even when their total energy is small enough to leave RMS unmoved. Robust cousins, the median, the interquartile range, and quantiles, capture the same spread and shape while shrugging off the occasional saturated sample or dropout that would wreck a variance.

Temporal structure: the features raw statistics miss

Every feature so far ignores the order of the samples: shuffle the window and mean, RMS, variance, and kurtosis are unchanged. Yet order is where the signal dynamics live, and a second family of features reads it directly from the time axis. The zero-crossing rate, the number of times the signal changes sign per window, is a cheap proxy for dominant frequency and a mainstay of activity recognition and voice-activity detection. Mean absolute difference and the RMS of the first difference \(x[n]-x[n-1]\) measure how fast the signal moves, distinguishing a slow postural sway from a brisk shake at equal amplitude. Autocorrelation at lag \(\tau\),

reveals periodicity: the lag of its first strong peak is the period, so on an accelerometer it recovers step cadence and on a PPG or ECG it recovers heart rate, entirely in the time domain and cheaper than an FFT for a single dominant period. Signal-magnitude area, the summed absolute value over the window, and simple peak counts round out a vocabulary that has driven human-activity classifiers for two decades. These order-aware features are the bridge to Chapter 23, where inertial motion signals are the whole subject.

Computing a feature vector in one pass

The code below turns a window into a labeled feature dictionary, so the earlier discussion becomes something you can run. It computes amplitude, dispersion, shape, and temporal-structure features together, and it deliberately keeps the raw mean available so you can see how large a per-window offset can be, and why you may want to drop it.

import numpy as np

def time_domain_features(x, fs):
    x = np.asarray(x, dtype=float)
    dx = np.diff(x)                       # first difference: how fast it moves
    mu, sd = x.mean(), x.std()
    z = (x - mu) / (sd + 1e-12)           # standardized, for stable moments
    feats = {
        "mean":        mu,                # DC level (often orientation/placement)
        "rms":         np.sqrt((x**2).mean()),
        "std":         sd,
        "ptp":         x.ptp(),           # peak-to-peak range
        "crest":       np.abs(x).max() / (np.sqrt((x**2).mean()) + 1e-12),
        "skew":        (z**3).mean(),
        "kurtosis":    (z**4).mean(),     # ~3 for Gaussian, high for impulsive
        "iqr":         np.subtract(*np.percentile(x, [75, 25])),
        "zcr":         np.mean(np.abs(np.diff(np.sign(x - mu)))) / 2 * fs,
        "mean_absdiff": np.abs(dx).mean(),
    }
    return feats

fs = 50
t  = np.arange(2 * fs) / fs
walk = 1.0 + 0.6*np.sin(2*np.pi*1.8*t)    # ~1.8 Hz cadence on a 1 g offset
print({k: round(v, 3) for k, v in time_domain_features(walk, fs).items()})

The printout makes the leakage argument concrete: the mean dwarfs the dynamic std, so a model handed the raw mean is largely reading the gravity projection, not the gait. The remaining features (crest factor, kurtosis, zero-crossing rate) describe the motion itself and travel far better across subjects.

import numpy as np
from numpy.lib.stride_tricks import sliding_window_view

W, hop = 100, 50                          # 2 s window, 50% overlap at 50 Hz
win = sliding_window_view(signal, W)[::hop]     # (n_windows, W)
feat_matrix = np.stack([np.column_stack([
    w.mean(1), w.std(1), np.sqrt((w**2).mean(1))]) for w in [win]])[0]

What's Next

In Section 8.2, we cross from the time axis to the frequency axis, turning each window into spectral and time-frequency features: band powers, spectral centroid and bandwidth, dominant-frequency and its harmonics, and the spectrogram-derived descriptors that capture periodic structure a single autocorrelation lag cannot. Together with this section's amplitude, shape, and temporal-structure features, they form the raw material that the statistical and entropy features of Section 8.3 and the feature-selection methods of Section 8.5 then refine into a compact, leakage-safe representation.