Metrics catalogue

Every metric reduces a time series \(x = (x_1, \dots, x_N)\) of \(N\) observations to a single value. This page gives the formulation of each one. The live list is always available via metrics().

Basic family

Name	Description	Definition
max	Maximum	\(\max_i x_i\)
min	Minimum	\(\min_i x_i\)
mean	Arithmetic mean	\(\bar{x} = \tfrac{1}{N}\sum_i x_i\)
median	Median	middle value (Hazen \(Q_2\))
sum	Sum	\(\sum_i x_i\)
std	Standard deviation	\(\sqrt{\tfrac{1}{N-1}\sum_i (x_i-\bar{x})^2}\)
skew	Skewness	\(\dfrac{\sqrt{N(N-1)}}{N-2}\cdot\dfrac{m_3}{m_2^{3/2}}\)
kurt	Kurtosis (Pearson)	\(\dfrac{N\sum_i (x_i-\bar{x})^4}{\left(\sum_i (x_i-\bar{x})^2\right)^2}\)
amplitude	Range	\(\max_i x_i - \min_i x_i\)
fslope	Max. abs. first difference	\(\max_i \lvert x_{i+1}-x_i \rvert\)
abs_sum	Absolute sum	\(\sum_i \lvert x_i \rvert\)
amd	Mean abs. first difference	\(\tfrac{1}{N-1}\sum_i \lvert x_{i+1}-x_i \rvert\)
mse	Mean power spectrum	\(\tfrac{1}{N}\sum_k \lvert \hat{x}_k \rvert^2 = \sum_i x_i^2\)
fqr	First quartile	\(Q_1\) (Hazen)
tqr	Third quartile	\(Q_3\) (Hazen)
iqr	Interquartile range	\(Q_3 - Q_1\)

In the moment-based definitions above, \(\bar{x}\) is the mean and \(m_k = \tfrac{1}{N}\sum_i (x_i-\bar{x})^k\) is the \(k\)-th central moment.

A few conventions are worth stating, since implementations differ on them. The standard deviation is the sample deviation, ddof = 1. Skewness is the adjusted Fisher–Pearson coefficient, matching scipy.stats.skew(..., bias=False). Kurtosis follows Pearson's non-excess definition, so a normal distribution gives \(3.0\). Quantiles use the Hazen plotting position, matching numpy.quantile(..., method="hazen").

mse

By Parseval's theorem, the mean of \(\lvert \text{FFT} \rvert^2\) over all frequency bins equals the sum of squared samples, so mse is computed directly as \(\sum_i x_i^2\), with no FFT.

Polar family

Name	Description	Definition
area_ts	Polygon area	\(\tfrac{1}{2}\sin\!\left(\tfrac{2\pi}{N}\right)\sum_i r_i\, r_{i+1}\)
angle	Phase of the maximum	\(\theta_{\arg\max_i r_i}\), with \(\theta\) from \(\operatorname{linspace}(0, 2\pi, N)\)
gyration_radius	Spread of the shape	mean distance of \(P\)'s vertices to its area-centroid
csi	Cell shape index	\(\dfrac{\text{perimeter}(P)^2}{4\pi\,\text{area}(P)}\)
area_q1…area_q4	Seasonal quadrant areas	area of \(P\) within each Cartesian quadrant
polar_balance	Seasonal balance	standard deviation of the four quadrant areas

The polygon \(P\) these definitions refer to is the Körting polar plot representation used by stmetrics, against whose Shapely-based implementation these metrics are verified.

It is built by placing each observation on a circle: vertex \(i\) sits at angle \(\theta_i = \tfrac{2\pi i}{N}\) with radius \(r_i = \lvert x_i \rvert\), that is, at the Cartesian point \((r_i\cos\theta_i,\; r_i\sin\theta_i)\). Connecting consecutive vertices closes the polygon. Because radii are non-negative and the vertices are ordered by angle, \(P\) is star-shaped about the origin and therefore simple, never self-intersecting.

The quadrants partition the plane at the origin (area_q1 = upper-right, area_q2 = upper-left, area_q3 = lower-left, area_q4 = lower-right) and act as the four "seasons" of the cycle.

Not yet implemented

ecc_metric (eccentricity, via the minimum rotated rectangle) from stmetrics is not yet available.