Minkowski distance

A generalized metric that includes Euclidean and Manhattan distance as special cases.

Usage

minkowski(
  counts,
  margin = 1L,
  power = 1.5,
  norm = "none",
  pseudocount = NULL,
  pairs = NULL,
  cpus = n_cpus()
)

Arguments

counts

A numeric matrix of count data (samples $\times$ features). Typically contains absolute abundances (integer counts), though proportions are also accepted.

margin

The margin containing samples. 1 if samples are rows, 2 if samples are columns. Ignored when counts is a special object class (e.g. phyloseq). Default: 1

power

Scaling factor for the magnitude of differences between communities ($p$). Default: 1.5

norm

Normalize the incoming counts. Options are:

'none': No transformation.
'percent': Relative abundance (sample abundances sum to 1).
'binary': Unweighted presence/absence (each count is either 0 or 1).
'clr': Centered log ratio.

Default: 'none'.

pseudocount

Value added to counts to handle zeros when norm = 'clr'. Ignored for other normalization methods. See Pseudocount section.

pairs

Which combinations of samples should distances be calculated for? The default value (NULL) calculates all-vs-all. Provide a numeric or logical vector specifying positions in the distance matrix to calculate. See examples.

cpus

How many parallel processing threads should be used. The default, n_cpus(), will use all logical CPU cores.

Details

The Minkowski distance is defined as: $$\sqrt[p]{\sum_{i=1}^{n} (X_i - Y_i)^p}$$

Where:

$X_i$, $Y_i$ : Absolute abundances of the $i$-th feature.
$n$ : The number of features.
$p$ : The geometry of the space (power parameter).

Parameter: power

The power parameter (default 1.5) determines the value of $p$ in the equation.

Special Cases

Manhattan distance: When $p = 1$, the formula reduces to the sum of absolute differences.
Euclidean distance: When $p = 2$, the formula reduces to the standard straight-line distance.
Chebyshev distance: When $p \to \infty$, the formula reduces to the maximum absolute difference.

Base R Equivalent:

p <- 1.5
x <- ex_counts[1,]
y <- ex_counts[2,]
sum(abs(x - y)^p) ^ (1/p)

Input Types

The counts parameter is designed to accept a simple numeric matrix, but seamlessly supports objects from the following biological data packages:

phyloseq
rbiom
SummarizedExperiment
TreeSummarizedExperiment

For large datasets, standard matrix operations may be slow. See vignette('performance') for details on using optimized formats (e.g. sparse matrices) and parallel processing.

Pseudocount

The pseudocount parameter is only relevant when norm = 'clr'.

Zeros are undefined in the centered log-ratio (CLR) transformation. If norm = 'clr', pseudocount is NULL (the default), and zeros are detected, the function uses half the minimum non-zero value (min(x[x>0]) / 2) and issues a warning.

To suppress the warning, provide an explicit value (e.g., 1).

Why this matters: The choice of pseudocount is not neutral; it acts as a weighting factor that can significantly distort downstream results, especially for sparse datasets. See Gloor et al. (2017) and Kaul et al. (2017) for open-access discussions on the mathematical implications, or Costea et al. (2014) for the impact on community clustering.

See aitchison for references.

References

Deza, M. M., & Deza, E. (2009). Encyclopedia of distances. Springer.

Minkowski, H. (1896). Geometrie der Zahlen. Teubner.

Examples

    minkowski(ex_counts, power = 2) # Equivalent to Euclidean
#>          Saliva      Gums      Nose
#> Gums   637.8205                    
#> Nose   646.1300  983.5644          
#> Stool  654.8633 1001.5543  858.2296