Table 1 Interpretation of persistence statistics. An advantage of using persistent homology with a SEDT filtration is its interpretability. Here we list the persistent statistics that we use and their morphological interpretation. We use ‘Q.’ for quadrant, ‘p.’ for percentile, ‘st. dev.’ for the standard deviation, IQR for the interquartile range, and micro-hole radius or size refers to the radius of the largest inscribed (open) circle. The inter-micro-hole distance is half the distance from one micro-hole to the next nearest micro-hole.
\(H_*\) | Q | Persistent statistic | Interpretation |
|---|---|---|---|
\(H_0\) | 2 | Number of points | Number of micro-holes |
\(H_0\) | 2 | Mean birth | Mean micro-hole radius (of largest inscribed circle) |
\(H_0\) | 2 | Median birth | Median micro-hole radius |
\(H_0\) | 2 | 25p. birth | Minimum radius of largest 25% of micro-holes |
\(H_0\) | 2 | IQR birth | Range in radii of middle 50% of micro-holes |
\(H_0\) | 2 | St. dev. birth | Spread of micro-hole radii |
\(H_0\) | 2 | Mean death | Average micro-hole crowding (mean of inter-micro-hole distance) |
\(H_0\) | 2 | Median death | Median inter-micro-hole distance |
\(H_0\) | 2 | 25p. death | Maximum of smallest 25% of the inter-micro-hole distances |
\(H_0\) | 2 | IQR death | Range in the middle 50% of the inter-micro-hole distances |
\(H_0\) | 2 | St. dev. death | Spread of inter-micro-hole distances |
\(H_0\) | 2 | Persistent entropy | Diversity of micro-hole lifetimes |
\(H_0\) | 2 | Number micro-holes \(r \ge 2\) | Number of micro-holes with radius greater than or equal to 2 |
\(H_0\) | 2 | Number micro-holes \(r < 2\) | Number of micro-holes with radius less than 2 |
\(H_1\) | 1 | Number of points | Number of bone regions surrounded by micro-holes |
\(H_1\) | 1 | Mean birth | Mean distance from micro-holes at which loop is formed |
\(H_1\) | 1 | Median birth | Middle of the distances from micro-holes at which loop is formed |
\(H_1\) | 1 | 25p. birth | Distance 25% through distances from micro-holes when loop is formed |
\(H_1\) | 1 | IQR birth | Range of middle 50% of distances from micro-holes when loop is formed |
\(H_1\) | 1 | St. dev. birth | Spread of distances from micro-holes at which loop is formed |
\(H_1\) | 1 | Mean death | Mean of the radii of regions surrounded by micro-holes |
\(H_1\) | 1 | Median death | Radius of middle-sized bone region surrounded by micro-holes |
\(H_1\) | 1 | 25p. death | Max radius of smallest 25% of bone regions surrounded by micro-holes |
\(H_1\) | 1 | IQR death | Range in radii of middle 50% of bone regions surrounded by micro-holes |
\(H_1\) | 1 | St. dev. death | Spread of radii of bone regions surrounded by micro-holes |
\(H_1\) | 1 | Persistent entropy | Diversity of lifetimes for bone regions surrounded by micro-holes |
