Table 9 List of computed metrics of height and cover
Metric | Description/statistic | Symbol for all returns (HM/AR) | Symbol for first returns (HM/FR) |
|---|---|---|---|
Height Metrics (HM) | |||
Location “center” | |||
Mean | \( \overline{x}=\frac{1}{n}{\displaystyle \sum }{x}_i \) | HME | HMEf |
Quadratic mean | \( \sqrt{\frac{1}{n}{\displaystyle \sum }{x}_i^2} \) | HSQ | HSQf |
Cubic mean | \( \sqrt[3]{\frac{1}{n}{\displaystyle \sum }{x}_i^3} \) | HC | HCf |
Mode | Mo = Most frequent value in data set | HMO | HMOf |
Location “percentile” | |||
Percentile | Height separating lowest y% of data from remainder | HPy | HPyf |
Dispersion | |||
Standard deviation | \( \sqrt{\frac{1}{n-1}{\displaystyle \sum }{\left({x}_i-\overline{x}\right)}^2} \) | HSD | HSDf |
Coefficient of variation | \( \frac{\sqrt{\frac{1}{n-1}{\displaystyle \sum }{\left({x}_i-\overline{x}\right)}^2}}{\overline{x}} \) | HCV | HCVf |
Skewness | \( \frac{\frac{1}{n}{\displaystyle \sum }{\left({x}_i-\overline{x}\right)}^3}{{\left(\frac{1}{n}{\displaystyle \sum }{\left({x}_i-\overline{x}\right)}^2\right)}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}} \) | HS | HSf |
Kurtosis | \( \frac{\frac{1}{n}{\displaystyle \sum }{\left({x}_i-\overline{x}\right)}^4}{{\left(\frac{1}{n}{\displaystyle \sum }{\left({x}_i-\overline{x}\right)}^2\right)}^2}-3 \) | HK | HKf |
Cover metrics (CM) | |||
All returns differentiated by height (CM/AR) | |||
Percentage of all returns above a specified height | \( \frac{N\left(>{h}_j\right)}{N}\times 100 \) | PARA2 | |
Percentage of all returns above the mean height | \( \frac{N\left(>\overline{h}\right)}{N}\times 100 \) | PARAM | |
Percentage of all returns above the mode height | \( \frac{N\left(>Mo(h)\right)}{N}\times 100 \) | PARAMO | |
First returns differentiated by height (CM/FR) | |||
Percentage of first returns above a specified height | \( \frac{N_1\left(>{h}_j\right)}{N}\times 100 \) | PFRA2 | |
Percentage of first returns above the mean height | \( \frac{N_1\left(>\overline{h}\right)}{N}\times 100 \) | PFRAM | |
Percentage of first returns above the mode height | \( \frac{N_1\left(>Mo(h)\right)}{N}\times 100 \) | PFRAMO | |
Percentage of first returns above the specified height related to first returns | \( \frac{N_1\left(>{h}_j\right)}{N_1}\times 100 \) | FR2FR | |