WOLFRAM

Wolfram Language & System Documentation Center
Wolfram Language Home Page »

HoeffdingD
✖
HoeffdingD

✖
HoeffdingD[v1,v2]

gives Hoeffding's dependence measure for the vectors v1 and v2.

✖

gives Hoeffding's dependence measure for the matrix m.

✖
HoeffdingD[m1,m2]

gives Hoeffding's dependence measure for the matrices m1 and m2.

✖
HoeffdingD[dist]

gives Hoeffding's matrix for the multivariate symbolic distribution dist.

✖
HoeffdingD[dist,i,j]

gives the ^(th) element of for the multivariate symbolic distribution dist.

Details

  • HoeffdingD[v1,v2] gives Hoeffding's dependence measure between v1 and v2.
  • Hoeffding's is a measure of dependence based on the relative order of elements in the two lists.
  • Hoeffding's between v1 and v2 is given by 30 (-2 (n-2) R+(n-3) (n-2) S+Q)/TemplateBox[{{n, -, 4}, 5}, Pochhammer], where is the number of observations in v1, , , , for , is the rank of v1i, is the rank of v2i, and is equal to Boole[a<b].
  • The arguments v1 and v2 can be any real‐valued vectors of equal length greater than 5.
  • For a matrix m with columns, HoeffdingD[m] is a × matrix of the dependence measures between columns of m.
  • For an × matrix m1 and an × matrix m2, HoeffdingD[m1,m2] is a × matrix of the dependence measures between columns of m1 and columns of m2.
  • HoeffdingD[dist,i,j] is given by 30 Expectation[(F[x,y]-G[x]H[y])^2,{x,y}disti,j], where F[x,y], G[x], and H[y] are the CDFs of the ^(th), ^(th), and ^(th) marginals of dist respectively.
  • HoeffdingD[dist] gives a matrix where the ^(th) entry is given by HoeffdingD[dist,i,j].

Examples

open allclose all

Basic Examples  (4)Summary of the most common use cases

Hoeffding's for two vectors:

In[1]:=1
✖
https://wolfram.com/xid/0tqnwmszzmb9sv-gugb6d
In[3]:=3
✖
https://wolfram.com/xid/0tqnwmszzmb9sv-fulot0
Out[3]=3

Hoeffding's for a matrix:

In[1]:=1
✖
https://wolfram.com/xid/0tqnwmszzmb9sv-9un6t
In[2]:=2
✖
https://wolfram.com/xid/0tqnwmszzmb9sv-hz9gv1

Hoeffding's for two matrices:

In[1]:=1
✖
https://wolfram.com/xid/0tqnwmszzmb9sv-bv14aj
In[2]:=2
✖
https://wolfram.com/xid/0tqnwmszzmb9sv-ce2wv
In[3]:=3
✖
https://wolfram.com/xid/0tqnwmszzmb9sv-i51zwj

Compute Hoeffding's for a bivariate distribution:

In[1]:=1
✖
https://wolfram.com/xid/0tqnwmszzmb9sv-f7dh9d
In[2]:=2
✖
https://wolfram.com/xid/0tqnwmszzmb9sv-mzz2lg

Compare to a simulated value:

In[3]:=3
✖
https://wolfram.com/xid/0tqnwmszzmb9sv-nwpxl1

Scope  (7)Survey of the scope of standard use cases

Data  (4)

Exact input yields exact output:

In[1]:=1
✖
https://wolfram.com/xid/0tqnwmszzmb9sv-eta06h
Out[1]=1

Approximate input yields approximate output:

In[1]:=1
✖
https://wolfram.com/xid/0tqnwmszzmb9sv-usqqq
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0tqnwmszzmb9sv-httrs6
Out[2]=2

Works with large arrays:

In[1]:=1
✖
https://wolfram.com/xid/0tqnwmszzmb9sv-kt50m4
Out[1]=1

SparseArray data can be used:

In[1]:=1
✖
https://wolfram.com/xid/0tqnwmszzmb9sv-cz9dkm

Distributions and Processes  (3)

Hoeffding's matrix for a continuous multivariate distribution:

In[1]:=1
✖
https://wolfram.com/xid/0tqnwmszzmb9sv-fnv00k
In[2]:=2
✖
https://wolfram.com/xid/0tqnwmszzmb9sv-tekwa
Out[2]=2

Hoeffding's matrix for derived distributions:

In[1]:=1
✖
https://wolfram.com/xid/0tqnwmszzmb9sv-c3a9y
In[2]:=2
✖
https://wolfram.com/xid/0tqnwmszzmb9sv-cy97of
In[3]:=3
✖
https://wolfram.com/xid/0tqnwmszzmb9sv-etkwz

Data distribution:

In[4]:=4
✖
https://wolfram.com/xid/0tqnwmszzmb9sv-hxjash
In[5]:=5
✖
https://wolfram.com/xid/0tqnwmszzmb9sv-gm1a47
In[6]:=6
✖
https://wolfram.com/xid/0tqnwmszzmb9sv-edj4ah

Hoeffding's matrix for a random process at times and :

In[1]:=1
✖
https://wolfram.com/xid/0tqnwmszzmb9sv-hv1ygl

Applications  (3)Sample problems that can be solved with this function

Hoeffding's is typically used to detect non-monotonic dependency structures:

In[1]:=1
✖
https://wolfram.com/xid/0tqnwmszzmb9sv-dz3nkj
In[4]:=4
✖
https://wolfram.com/xid/0tqnwmszzmb9sv-dp75n
Out[4]=4

Hoeffding's tends to be larger for dependent vectors:

In[5]:=5
✖
https://wolfram.com/xid/0tqnwmszzmb9sv-bruog5
Out[5]=5

The value tends to 0 for independent vectors:

In[6]:=6
✖
https://wolfram.com/xid/0tqnwmszzmb9sv-lp2ywe
Out[6]=6

Hoeffding's can detect linear dependence:

In[1]:=1
✖
https://wolfram.com/xid/0tqnwmszzmb9sv-eyfz2q
In[2]:=2
✖
https://wolfram.com/xid/0tqnwmszzmb9sv-c1b9b4
Out[2]=2

SpearmanRho and KendallTau are more sensitive to linear dependence:

In[3]:=3
✖
https://wolfram.com/xid/0tqnwmszzmb9sv-deujkj
Out[3]=3

Hoeffding's can also detect many types of nonlinear dependence:

In[1]:=1
✖
https://wolfram.com/xid/0tqnwmszzmb9sv-iwph7r
In[2]:=2
✖
https://wolfram.com/xid/0tqnwmszzmb9sv-fgog5o
In[3]:=3
✖
https://wolfram.com/xid/0tqnwmszzmb9sv-e7iyc7
In[4]:=4
✖
https://wolfram.com/xid/0tqnwmszzmb9sv-dd08fp
Out[4]=4

Use HoeffdingDTest to determine if the value is statistically significant:

In[5]:=5
✖
https://wolfram.com/xid/0tqnwmszzmb9sv-cty8ts
Out[5]=5

Properties & Relations  (4)Properties of the function, and connections to other functions

Larger values of Hoeffding's indicate increasing dependence:

In[1]:=1
✖
https://wolfram.com/xid/0tqnwmszzmb9sv-zet3g
In[2]:=2
✖
https://wolfram.com/xid/0tqnwmszzmb9sv-o4ucu0
In[3]:=3
✖
https://wolfram.com/xid/0tqnwmszzmb9sv-l3nnwr
Out[3]=3

Hoeffding's matrix is symmetric:

In[1]:=1
✖
https://wolfram.com/xid/0tqnwmszzmb9sv-31bc2
In[2]:=2
✖
https://wolfram.com/xid/0tqnwmszzmb9sv-mmzf9d
Out[2]=2

The diagonal elements of Hoeffding's matrix are 1:

In[1]:=1
✖
https://wolfram.com/xid/0tqnwmszzmb9sv-o1vsb7
In[2]:=2
✖
https://wolfram.com/xid/0tqnwmszzmb9sv-h7qlge
Out[2]=2

Hoeffding's for a continuous bivariate distribution:

In[1]:=1
✖
https://wolfram.com/xid/0tqnwmszzmb9sv-bfewud
In[2]:=2
✖
https://wolfram.com/xid/0tqnwmszzmb9sv-ewdode
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0tqnwmszzmb9sv-grblhu
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0tqnwmszzmb9sv-hqcqde
Out[4]=4
Wolfram Research (2012), HoeffdingD, Wolfram Language function, https://reference.wolfram.com/language/ref/HoeffdingD.html.
✖
Wolfram Research (2012), HoeffdingD, Wolfram Language function, https://reference.wolfram.com/language/ref/HoeffdingD.html.

Text

Wolfram Research (2012), HoeffdingD, Wolfram Language function, https://reference.wolfram.com/language/ref/HoeffdingD.html.

✖
Wolfram Research (2012), HoeffdingD, Wolfram Language function, https://reference.wolfram.com/language/ref/HoeffdingD.html.

CMS

Wolfram Language. 2012. "HoeffdingD." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/HoeffdingD.html.

✖
Wolfram Language. 2012. "HoeffdingD." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/HoeffdingD.html.

APA

Wolfram Language. (2012). HoeffdingD. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/HoeffdingD.html

✖
Wolfram Language. (2012). HoeffdingD. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/HoeffdingD.html

BibTeX

@misc{reference.wolfram_2025_hoeffdingd, author="Wolfram Research", title="{HoeffdingD}", year="2012", howpublished="\url{https://reference.wolfram.com/language/ref/HoeffdingD.html}", note=[Accessed: 30-March-2025 ]}

✖
@misc{reference.wolfram_2025_hoeffdingd, author="Wolfram Research", title="{HoeffdingD}", year="2012", howpublished="\url{https://reference.wolfram.com/language/ref/HoeffdingD.html}", note=[Accessed: 30-March-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_hoeffdingd, organization={Wolfram Research}, title={HoeffdingD}, year={2012}, url={https://reference.wolfram.com/language/ref/HoeffdingD.html}, note=[Accessed: 30-March-2025 ]}

✖
@online{reference.wolfram_2025_hoeffdingd, organization={Wolfram Research}, title={HoeffdingD}, year={2012}, url={https://reference.wolfram.com/language/ref/HoeffdingD.html}, note=[Accessed: 30-March-2025 ]}