qad | R Documentation |

Quantification of (asymmetric and directed) dependence structures between two random variables X and Y.

qad(x, ...) ## S3 method for class 'data.frame' qad( x, resolution = NULL, p.value = TRUE, nperm = 1000, p.value_asymmetry = FALSE, nboot = 1000, print = TRUE, remove.00 = FALSE, ... ) ## S3 method for class 'numeric' qad( x, y, resolution = NULL, p.value = TRUE, nperm = 1000, p.value_asymmetry = FALSE, nboot = 1000, print = TRUE, remove.00 = FALSE, ... )

`x` |
a data.frame containing two columns with the observations of the bi-variate sample or a (non-empty) numeric vector of data values |

`...` |
Further arguments passed to 'qad' will be ignored |

`resolution` |
an integer indicating the number of strips for the checkerboard aggregation (see ECBC). We recommend to use the default value (resolution = NULL) |

`p.value` |
a logical indicating whether to return a p-value of rejecting independence (based on permutation). |

`nperm` |
an integer indicating the number of permutation runs (if p.value = TRUE) |

`p.value_asymmetry` |
a logical indicating whether to return a (heuristic) p-value for the measure of asymmetry (based on bootstrap). |

`nboot` |
an integer indicating the number of runs for the bootstrap. |

`print` |
a logical indicating whether the result of qad is printed. |

`remove.00` |
a logical indicating whether double 0 entries should be excluded (default = FALSE) |

`y` |
a (non-empty) numeric vector of data values. |

qad is the implementation of a strongly consistent estimator of the copula based dependence measure zeta_1 introduced in Trutschnig 2011. We first compute the empirical copula of a two-dimensional sample, aggregate it to the so called empirical checkerboard copula (ECBC), and calculate zeta_1 of the ECBC and its transpose. In order to test for independence (in both directions), a built-in p-value is implemented (a permutation test with nperm permutation runs to estimate the p-value). Furthermore, a (heuristic) bootstrap test with nboot runs can be applied to estimate a p-value for the measure of asymmetry a.

qad returns an object of class qad containing the following components:

`data` |
a data.frame containing the input data. |

`q(X,Y)` |
influence of X on Y |

`q(Y,X)` |
influence of Y on X |

`max.dependence` |
maximal dependence |

`results` |
a data.frame containing the results of the dependence measures. |

`mass_matrix` |
a matrix containing the mass distribution of the empirical checkerboard copula. |

`resolution` |
an integer containing the used resolution of the checkerboard aggregation. |

`n` |
Sample size. |

Trutschnig, W. (2011). On a strong metric on the space of copulas and its induced dependence measure, Journal of Mathematical Analysis and Applications 384, 690-705.

Junker, R., Griessenberger, F. and Trutschnig, W. (2021). Estimating scale-invariant directed dependence of bivariate distributions. Computational Statistics and Data Analysis, 153.

A tutorial can be found at http://www.trutschnig.net/software.html.

#Example 1 (independence) n <- 100 x <- runif(n,0,1) y <- runif(n,0,1) sample <- data.frame(x,y) qad(sample) ### #Example 2 (mutual complete dependence) n <- 500 x <- runif(n,0,1) y <- x^2 sample <- data.frame(x,y) qad(sample) #Example 3 (complete dependence) n <- 1000 x <- runif(n,-10,10) y <- sin(x) sample <- data.frame(x,y) qad(sample) #Example 4 (Asymmetry) n <- 100 x <- runif(n,0,1) y <- (2*x) %% 1 qad(x, y)

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