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Symmetric Extremal Dependence Index

Source: R/class-sedi.R
sedi.Rd

Symmetric Extremal Dependence Index (SEDI) is a skill metric for classification that remains reliable at extreme prevalence levels where traditional metrics (TSS, MCC, Kappa) degrade. It is defined using the hit rate (sensitivity) and false alarm rate (1 - specificity):

$$\text{SEDI} = \frac{\ln F - \ln H - \ln(1-F) + \ln(1-H)} {\ln F + \ln H + \ln(1-F) + \ln(1-H)}$$

where \(H\) is sensitivity (hit rate) and \(F\) is the false alarm rate (1 - specificity).

Usage

sedi(data, ...)

# S3 method for class 'data.frame'
sedi(
  data,
  truth,
  estimate,
  estimator = NULL,
  na_rm = TRUE,
  case_weights = NULL,
  event_level = yardstick_event_level(),
  ...
)

sedi_vec(
  truth,
  estimate,
  estimator = NULL,
  na_rm = TRUE,
  case_weights = NULL,
  event_level = yardstick_event_level(),
  ...
)

Arguments

data

Either a data.frame containing the columns specified by the truth and estimate arguments, or a table/matrix where the true class results should be in the columns of the table.

...

Not currently used.

truth

The column identifier for the true class results (that is a factor). This should be an unquoted column name although this argument is passed by expression and supports quasiquotation (you can unquote column names). For _vec() functions, a factor vector.

estimate

The column identifier for the predicted class results (that is also factor). As with truth this can be specified different ways but the primary method is to use an unquoted variable name. For _vec() functions, a factor vector.

estimator

One of: "binary", "macro", "macro_weighted", or "micro" to specify the type of averaging to be done. "binary" is only relevant for the two class case. The other three are general methods for calculating multiclass metrics. The default will automatically choose "binary" or "macro" based on estimate.

na_rm

A logical value indicating whether NA values should be stripped before the computation proceeds.

case_weights

The optional column identifier for case weights. This should be an unquoted column name that evaluates to a numeric column in data. For _vec() functions, a numeric vector, hardhat::importance_weights(), or hardhat::frequency_weights().

event_level

A single string. Either "first" or "second" to specify which level of truth to consider as the "event". This argument is only applicable when estimator = "binary". The default uses an internal helper that defaults to "first".

Value

A tibble with columns .metric, .estimator, and .estimate and 1 row of values.

For grouped data frames, the number of rows returned will be the same as the number of groups.

For sedi_vec(), a single numeric value (or NA).

Details

Suppose a 2x2 table with notation:

Reference
PredictedPositiveNegative
PositiveAB
NegativeCD

The formulas used here are:

$$H = \text{Sensitivity} = \frac{A}{A + C}$$

$$F = 1 - \text{Specificity} = \frac{B}{B + D}$$

SEDI is a metric that should be maximized. The output ranges from -1 to 1, with 1 indicating perfect discrimination.

SEDI is base-rate independent: its value depends only on sensitivity and specificity (class-conditional rates), not on prevalence. The logarithmic transformation ensures the metric remains discriminating even when events are extremely rare (prevalence < 2.5%), where j_index() (TSS) converges to the hit rate alone and mcc() exhibits denominator suppression.

When sensitivity or specificity is exactly 0 or 1, the logarithm is undefined. A small constant (1e-9) is used to clamp values away from these boundaries.

Prevalence guidance

  • Prevalence >= 10%: MCC, TSS, and SEDI all perform well.

  • Prevalence 2.5-10%: SEDI preferred; MCC and TSS still usable.

  • Prevalence < 2.5%: SEDI strongly recommended; MCC and TSS unreliable.

Multiclass

Macro, micro, and macro-weighted averaging is available for this metric. The default is to select macro averaging if a truth factor with more than 2 levels is provided. Otherwise, a standard binary calculation is done. See vignette("multiclass", "yardstick") for more information.

For multiclass problems, SEDI is computed via one-vs-all decomposition: each class is treated as a binary problem against all other classes, and a per-class SEDI is calculated. Macro averaging (the default) weights all classes equally, which is recommended since SEDI's log transform already handles class imbalance internally. Macro-weighted averaging weights by class prevalence. Micro averaging pools counts across classes before computing a single SEDI value.

Relevant Level

There is no common convention on which factor level should automatically be considered the "event" or "positive" result when computing binary classification metrics. In yardstick, the default is to use the first level. To alter this, change the argument event_level to "second" to consider the last level of the factor the level of interest. For multiclass extensions involving one-vs-all comparisons (such as macro averaging), this option is ignored and the "one" level is always the relevant result.

References

Ferro, C.A.T. and Stephenson, D.B. (2011). "Extremal Dependence Indices: Improved Verification Measures for Deterministic Forecasts of Rare Binary Events". Weather and Forecasting. 26 (5): 699-713.

Wunderlich, R.F., Lin, Y.-P., Anthony, J. and Petway, J.R. (2019). "Two alternative evaluation metrics to replace the true skill statistic in the assessment of species distribution models". Nature Conservation. 35: 97-116.

See also

All class metrics

Other class metrics: accuracy(), bal_accuracy(), detection_prevalence(), f_meas(), fall_out(), j_index(), kap(), markedness(), mcc(), miss_rate(), npv(), ppv(), precision(), recall(), roc_dist(), sens(), spec()

Author

Simon Dedman

Examples

# Two class
data("two_class_example")
sedi(two_class_example, truth, predicted)
#> # A tibble: 1 × 3
#>   .metric .estimator .estimate
#>   <chr>   <chr>          <dbl>
#> 1 sedi    binary         0.823

# Multiclass
library(dplyr)
data(hpc_cv)

hpc_cv |>
  filter(Resample == "Fold01") |>
  sedi(obs, pred)
#> # A tibble: 1 × 3
#>   .metric .estimator .estimate
#>   <chr>   <chr>          <dbl>
#> 1 sedi    macro          0.633

# Groups are respected
hpc_cv |>
  group_by(Resample) |>
  sedi(obs, pred)
#> # A tibble: 10 × 4
#>    Resample .metric .estimator .estimate
#>    <chr>    <chr>   <chr>          <dbl>
#>  1 Fold01   sedi    macro          0.633
#>  2 Fold02   sedi    macro          0.620
#>  3 Fold03   sedi    macro          0.735
#>  4 Fold04   sedi    macro          0.647
#>  5 Fold05   sedi    macro          0.645
#>  6 Fold06   sedi    macro          0.621
#>  7 Fold07   sedi    macro          0.580
#>  8 Fold08   sedi    macro          0.667
#>  9 Fold09   sedi    macro          0.615
#> 10 Fold10   sedi    macro          0.621

# Weighted macro averaging
hpc_cv |>
  group_by(Resample) |>
  sedi(obs, pred, estimator = "macro_weighted")
#> # A tibble: 10 × 4
#>    Resample .metric .estimator     .estimate
#>    <chr>    <chr>   <chr>              <dbl>
#>  1 Fold01   sedi    macro_weighted     0.713
#>  2 Fold02   sedi    macro_weighted     0.699
#>  3 Fold03   sedi    macro_weighted     0.773
#>  4 Fold04   sedi    macro_weighted     0.688
#>  5 Fold05   sedi    macro_weighted     0.702
#>  6 Fold06   sedi    macro_weighted     0.671
#>  7 Fold07   sedi    macro_weighted     0.633
#>  8 Fold08   sedi    macro_weighted     0.710
#>  9 Fold09   sedi    macro_weighted     0.630
#> 10 Fold10   sedi    macro_weighted     0.676

# Vector version
sedi_vec(
  two_class_example$truth,
  two_class_example$predicted
)
#> [1] 0.8227266

# Making Class2 the "relevant" level
sedi_vec(
  two_class_example$truth,
  two_class_example$predicted,
  event_level = "second"
)
#> [1] 0.8227266