(Maximum) topological dimension or homological degree

The degree of the homology group to vectorize, or the degree at which to stop vectorizing.

get_blur_range(object, x, ...)

hom_degree(range = c(0L, unknown()), trans = NULL)

max_hom_degree(range = c(0L, unknown()), trans = NULL)

get_hom_range(object, x, max_dim = 2L, ...)

Arguments

object

A param object or a list of param objects.

x

The predictor data. In some cases (see below) this should only include numeric data.

...

Other arguments to pass to the underlying parameter finalizer functions. For example, for get_rbf_range(), the dots are passed along to kernlab::sigest().

range

A two-element vector holding the defaults for the smallest and largest possible values, respectively. If a transformation is specified, these values should be in the transformed units.

trans

A trans object from the scales package, such as scales::transform_log10() or scales::transform_reciprocal(). If not provided, the default is used which matches the units used in range. If no transformation, NULL.

max_dim

Bound on the maximum dimension determined from the data.

Value

A param object or list of param objects.

Details

Topological features have whole number dimensions that determine the degrees of homology that encode them. Any finite point cloud will have finite topological dimension, but most practical applications exploit features of degree at most 3.

Steps may vectorize features of a single degree (hom_degree()) or of degrees zero through some maximum (max_hom_degree()).

In case the (maximum) degree is not provided, get_hom_range() queries each list-column for the maximum dimension of its point cloud and returns the smaller of this maximum and max_dim (which defaults to 2L, the highest homological degree of interest in most practical applications).

Examples

# toy data set
klein_sampler <- function(n, prob = .5) {
  if (rbinom(1, 1, prob) == 0) {
    tdaunif::sample_klein_flat(n)
  } else {
    tdaunif::sample_klein_tube(n)
  }
}
sample_data <- data.frame(
  id = LETTERS[seq(4L)],
  sample = I(c(replicate(4L, klein_sampler(60), simplify = FALSE)))
)

# options to calibrate homological degree
hom_degree(range = c(2, 5))
#> Homological Degree (quantitative)
#> Range: [2, 5]
hom_degree() %>% get_hom_range(x = sample_data[, 2, drop = FALSE])
#> Homological Degree (quantitative)
#> Range: [0, 1]
hom_degree() %>% get_hom_range(x = sample_data[, 2, drop = FALSE], max_dim = 5)
#> Homological Degree (quantitative)
#> Range: [0, 3]

# heterogeneous data types
hetero_data <- tibble(dataset = list(mtcars, nhtemp, eurodist, HairEyeColor))
hetero_data %>% 
  mutate(class = vapply(dataset, function(x) class(x)[[1L]], ""))
#> # A tibble: 4 × 2
#>   dataset             class     
#>   <list>              <chr>     
#> 1 <df [32 × 11]>      data.frame
#> 2 <ts [60]>           ts        
#> 3 <dist [210]>        dist      
#> 4 <table [4 × 4 × 2]> table     
get_hom_range(
  hom_degree(),
  hetero_data,
  max_dim = 60
)
#> Homological Degree (quantitative)
#> Range: [0, 20]