Signal Statistics Guide¶
This guide describes the key time-domain statistics computed by the SignalStatistics class, explaining each metric’s meaning and practical applications in signal processing.
Overview¶
Time-domain statistics summarize raw signal samples using simple formulas. They provide insights into signal characteristics such as:
- Central tendency (mean, median)
- Dispersion (variance, standard deviation, MAD)
- Shape (skewness, kurtosis)
- Extremes (peak-to-peak, crest factor)
- Frequency proxies (zero-crossing rate)
These metrics are widely used for anomaly detection, quality assessment, feature extraction for machine learning, and diagnostic monitoring.
Metrics and Definitions¶
Given a sequence of \(N\) samples \(x_1, x_2, \dots, x_N\) with mean \(\mu\):
| Statistic | Definition | Interpretation / Use |
|---|---|---|
| Mean | \(\displaystyle \mu = \frac{1}{N}\sum_{i=1}^N x_i\) | DC offset or bias; used for baseline correction |
| Median | Middle value of sorted \(\{x_i\}\) | Robust central tendency; insensitive to outliers |
| Variance | \(\displaystyle \sigma^2 = \frac{1}{N}\sum_{i=1}^N (x_i - \mu)^2\) | Dispersion measure; basis for power and energy analyses |
| Standard Deviation (\(\sigma\)) | \(\displaystyle \sigma = \sqrt{\sigma^2}\) | Spread around mean; indicates noise level |
| Mean Absolute Deviation | \(\displaystyle \mathrm{MAD} = \frac{1}{N}\sum_{i=1}^N \lvert x_i - \mu\rvert\) | Robust dispersion; less sensitive to outliers |
| Power | \(\displaystyle P = \frac{1}{N}\sum_{i=1}^N x_i^2\) | Energy per sample; key for signal strength and SNR |
| Peak-to-Peak | \(\displaystyle \max_i x_i - \min_i x_i\) | Dynamic range; checks for clipping or saturation |
| Crest Factor | \(\displaystyle \frac{\max_i \lvert x_i\rvert}{\sqrt{P}}\) | Peak prominence relative to average power; important in audio |
| Skewness | \(\displaystyle \frac{1}{N\sigma^3}\sum_{i=1}^N (x_i - \mu)^3\) | Distribution asymmetry; indicates DC shifts or bursts |
| Kurtosis | \(\displaystyle \frac{1}{N\sigma^4}\sum_{i=1}^N (x_i - \mu)^4\) | Tail heaviness relative to Gaussian; spots impulses/spikes |
| Zero-Crossing Rate (ZCR) | \(\displaystyle \frac{1}{N-1}\sum_{i=1}^{N-1} \mathbf{1}[x_i x_{i+1}<0]\) | Proxy for frequency content; higher ZCR → higher frequencies |
| Zero Crossings | Total count of sign changes: \(\sum_{i=1}^{N-1} \mathbf{1}[x_i x_{i+1}<0]\) | Basic oscillation count; complements ZCR |
Practical Applications¶
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Anomaly Detection Sudden spikes in kurtosis or a high crest factor can indicate faults or transient events.
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Quality Assessment Noise level is quantified by \(\sigma\) and MAD; peak-to-peak highlights clipping or saturation.
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Feature Extraction These metrics serve as features in machine learning models for classification, regression, or clustering.
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Monitoring & Diagnostics Tracking mean and power over time helps detect drift, component aging, or environmental changes.
Tip: Always normalize or detrend your signal (remove its mean \(\mu\)) before computing higher-order moments (skewness, kurtosis) to avoid bias from DC offsets.