# algorithms.statistics.rft¶

## Module: algorithms.statistics.rft¶

Inheritance diagram for nipy.algorithms.statistics.rft:

Random field theory routines

The theoretical results for the EC densities appearing in this module were partially supported by NSF grant DMS-0405970.

Taylor, J.E. & Worsley, K.J. (2012). “Detecting sparse cone alternatives

for Gaussian random fields, with an application to fMRI”. arXiv:1207.3840 [math.ST] and Statistica Sinica 23 (2013): 1629-1656.

Taylor, J.E. & Worsley, K.J. (2008). “Random fields of multivariate

test statistics, with applications to shape analysis.” arXiv:0803.1708 [math.ST] and Annals of Statistics 36( 2008): 1-27

## Classes¶

### ChiBarSquared¶

class nipy.algorithms.statistics.rft.ChiBarSquared(dfn=1, search=[1])

Bases: ChiSquared

__init__(dfn=1, search=[1])
density(x, dim)

The EC density in dimension dim.

integ(m=None, k=None)
pvalue(x, search=None)
quasi(dim)

(Quasi-)polynomial parts of EC density in dimension dim

• ignoring a factor of (2pi)^{-(dim+1)/2} in front.

### ChiSquared¶

class nipy.algorithms.statistics.rft.ChiSquared(dfn, dfd=inf, search=[1])

Bases: ECcone

EC densities for a Chi-Squared(n) random field.

__init__(dfn, dfd=inf, search=[1])
density(x, dim)

The EC density in dimension dim.

integ(m=None, k=None)
pvalue(x, search=None)
quasi(dim)

(Quasi-)polynomial parts of EC density in dimension dim

• ignoring a factor of (2pi)^{-(dim+1)/2} in front.

### ECcone¶

class nipy.algorithms.statistics.rft.ECcone(mu=[1], dfd=inf, search=[1], product=[1])

EC approximation to supremum distribution of var==1 Gaussian process

A class that takes the intrinsic volumes of a set and gives the EC approximation to the supremum distribution of a unit variance Gaussian process with these intrinsic volumes. This is the basic building block of all of the EC densities.

If product is not None, then this product (an instance of IntrinsicVolumes) will effectively be prepended to the search region in any call, but it will also affect the (quasi-)polynomial part of the EC density. For instance, Hotelling’s T^2 random field has a sphere as product, as does Roy’s maximum root.

__init__(mu=[1], dfd=inf, search=[1], product=[1])
density(x, dim)

The EC density in dimension dim.

integ(m=None, k=None)
pvalue(x, search=None)
quasi(dim)

(Quasi-)polynomial parts of EC density in dimension dim

• ignoring a factor of (2pi)^{-(dim+1)/2} in front.

### ECquasi¶

class nipy.algorithms.statistics.rft.ECquasi(c_or_r, r=0, exponent=None, m=None)

Bases: poly1d

Polynomials with premultiplier

A subclass of poly1d consisting of polynomials with a premultiplier of the form:

(1 + x^2/m)^-exponent

where m is a non-negative float (possibly infinity, in which case the function is a polynomial) and exponent is a non-negative multiple of 1/2.

These arise often in the EC densities.

Examples

>>> import numpy
>>> from nipy.algorithms.statistics.rft import ECquasi
>>> x = numpy.linspace(0,1,101)

>>> a = ECquasi([3,4,5])
>>> a
ECquasi(array([3, 4, 5]), m=inf, exponent=0.000000)
>>> a(3) == 3*3**2 + 4*3 + 5
True

>>> b = ECquasi(a.coeffs, m=30, exponent=4)
>>> numpy.allclose(b(x), a(x) * numpy.power(1+x**2/30, -4))
True

__init__(c_or_r, r=0, exponent=None, m=None)
property c

The polynomial coefficients

change_exponent(_pow)

Change exponent

Multiply top and bottom by an integer multiple of the self.denom_poly.

Examples

>>> import numpy
>>> b = ECquasi([3,4,20], m=30, exponent=4)
>>> x = numpy.linspace(0,1,101)
>>> c = b.change_exponent(3)
>>> c
ECquasi(array([  1.11111111e-04,   1.48148148e-04,   1.07407407e-02,
1.33333333e-02,   3.66666667e-01,   4.00000000e-01,
5.00000000e+00,   4.00000000e+00,   2.00000000e+01]), m=30.000000, exponent=7.000000)
>>> numpy.allclose(c(x), b(x))
True

property coef

The polynomial coefficients

property coefficients

The polynomial coefficients

property coeffs

The polynomial coefficients

compatible(other)

Check compatibility of degrees of freedom

Check whether the degrees of freedom of two instances are equal so that they can be multiplied together.

Examples

>>> import numpy
>>> b = ECquasi([3,4,20], m=30, exponent=4)
>>> x = numpy.linspace(0,1,101)
>>> c = b.change_exponent(3)
>>> b.compatible(c)
True
>>> d = ECquasi([3,4,20])
>>> b.compatible(d)
False
>>>

denom_poly()

Base of the premultiplier: (1+x^2/m).

Examples

>>> import numpy
>>> b = ECquasi([3,4,20], m=30, exponent=4)
>>> d = b.denom_poly()
>>> d
poly1d([ 0.03333333,  0.        ,  1.        ])
>>> numpy.allclose(d.c, [1./b.m,0,1])
True

deriv(m=1)

Evaluate derivative of ECquasi

Parameters:
mint, optional

Examples

>>> a = ECquasi([3,4,5])
>>> a.deriv(m=2)
ECquasi(array([6]), m=inf, exponent=0.000000)

>>> b = ECquasi([3,4,5], m=10, exponent=3)
>>> b.deriv()
ECquasi(array([-1.2, -2. ,  3. ,  4. ]), m=10.000000, exponent=4.000000)

integ(m=1, k=0)

Return an antiderivative (indefinite integral) of this polynomial.

Refer to polyint for full documentation.

polyint

equivalent function

property o

The order or degree of the polynomial

property order

The order or degree of the polynomial

property r

The roots of the polynomial, where self(x) == 0

property roots

The roots of the polynomial, where self(x) == 0

property variable

The name of the polynomial variable

### FStat¶

class nipy.algorithms.statistics.rft.FStat(dfn, dfd=inf, search=[1])

Bases: ECcone

EC densities for a F random field.

__init__(dfn, dfd=inf, search=[1])
density(x, dim)

The EC density in dimension dim.

integ(m=None, k=None)
pvalue(x, search=None)
quasi(dim)

(Quasi-)polynomial parts of EC density in dimension dim

• ignoring a factor of (2pi)^{-(dim+1)/2} in front.

### Hotelling¶

class nipy.algorithms.statistics.rft.Hotelling(dfd=inf, k=1, search=[1])

Bases: ECcone

Hotelling’s T^2

Maximize an F_{1,dfd}=T_dfd^2 statistic over a sphere of dimension k.

__init__(dfd=inf, k=1, search=[1])
density(x, dim)

The EC density in dimension dim.

integ(m=None, k=None)
pvalue(x, search=None)
quasi(dim)

(Quasi-)polynomial parts of EC density in dimension dim

• ignoring a factor of (2pi)^{-(dim+1)/2} in front.

### IntrinsicVolumes¶

class nipy.algorithms.statistics.rft.IntrinsicVolumes(mu=[1])

Bases: object

Compute intrinsic volumes of products of sets

A simple class that exists only to compute the intrinsic volumes of products of sets (that themselves have intrinsic volumes, of course).

__init__(mu=[1])

### MultilinearForm¶

class nipy.algorithms.statistics.rft.MultilinearForm(*dims, **keywords)

Bases: ECcone

Maximize a multivariate Gaussian form

Maximized over spheres of dimension dims. See:

Kuri, S. & Takemura, A. (2001). ‘Tail probabilities of the maxima of multilinear forms and their applications.’ Ann. Statist. 29(2): 328-371.

__init__(*dims, **keywords)
density(x, dim)

The EC density in dimension dim.

integ(m=None, k=None)
pvalue(x, search=None)
quasi(dim)

(Quasi-)polynomial parts of EC density in dimension dim

• ignoring a factor of (2pi)^{-(dim+1)/2} in front.

### OneSidedF¶

class nipy.algorithms.statistics.rft.OneSidedF(dfn, dfd=inf, search=[1])

Bases: ECcone

EC densities for one-sided F statistic

See:

Worsley, K.J. & Taylor, J.E. (2005). ‘Detecting fMRI activation allowing for unknown latency of the hemodynamic response.’ Neuroimage, 29,649-654.

__init__(dfn, dfd=inf, search=[1])
density(x, dim)

The EC density in dimension dim.

integ(m=None, k=None)
pvalue(x, search=None)
quasi(dim)

(Quasi-)polynomial parts of EC density in dimension dim

• ignoring a factor of (2pi)^{-(dim+1)/2} in front.

### Roy¶

class nipy.algorithms.statistics.rft.Roy(dfn=1, dfd=inf, k=1, search=[1])

Bases: ECcone

Roy’s maximum root

Maximize an F_{dfd,dfn} statistic over a sphere of dimension k.

__init__(dfn=1, dfd=inf, k=1, search=[1])
density(x, dim)

The EC density in dimension dim.

integ(m=None, k=None)
pvalue(x, search=None)
quasi(dim)

(Quasi-)polynomial parts of EC density in dimension dim

• ignoring a factor of (2pi)^{-(dim+1)/2} in front.

### TStat¶

class nipy.algorithms.statistics.rft.TStat(dfd=inf, search=[1])

Bases: ECcone

EC densities for a t random field.

__init__(dfd=inf, search=[1])
density(x, dim)

The EC density in dimension dim.

integ(m=None, k=None)
pvalue(x, search=None)
quasi(dim)

(Quasi-)polynomial parts of EC density in dimension dim

• ignoring a factor of (2pi)^{-(dim+1)/2} in front.

### fnsum¶

class nipy.algorithms.statistics.rft.fnsum(*items)

Bases: object

__init__(*items)

## Functions¶

nipy.algorithms.statistics.rft.Q(dim, dfd=inf)

Q polynomial

If dfd == inf (the default), then Q(dim) is the (dim-1)-st Hermite polynomial:

$H_j(x) = (-1)^j * e^{x^2/2} * (d^j/dx^j e^{-x^2/2})$

If dfd != inf, then it is the polynomial Q defined in [Worsley1994]

Parameters:
dimint

dimension of polynomial

dfdscalar
Returns:
q_polynp.poly1d instance

References

Worsley, K.J. (1994). ‘Local maxima and the expected Euler characteristic of excursion sets of chi^2, F and t fields.’ Advances in Applied Probability, 26:13-42.

A ball-shaped search region of radius r.

nipy.algorithms.statistics.rft.binomial(n, k)

Binomial coefficient

n!

c = ———

(n-k)! k!

Parameters:
nfloat

n of (n, k)

kfloat

k of (n, k)

Returns:
cfloat

Examples

First 3 values of 4 th row of Pascal triangle

>>> [binomial(4, k) for k in range(3)]
[1.0, 4.0, 6.0]

nipy.algorithms.statistics.rft.mu_ball(n, j, r=1)

jth curvature of n-dimensional ball radius r

Return mu_j(B_n(r)), the j-th Lipschitz Killing curvature of the ball of radius r in R^n.

nipy.algorithms.statistics.rft.mu_sphere(n, j, r=1)

jth curvature for n dimensional sphere radius r

Return mu_j(S_r(R^n)), the j-th Lipschitz Killing curvature of the sphere of radius r in R^n.

From Chapter 6 of

Adler & Taylor, ‘Random Fields and Geometry’. 2006.

nipy.algorithms.statistics.rft.scale_space(region, interval, kappa=1.0)

scale space intrinsic volumes of region x interval

See:

Siegmund, D.O and Worsley, K.J. (1995). ‘Testing for a signal with unknown location and scale in a stationary Gaussian random field.’ Annals of Statistics, 23:608-639.

and

Taylor, J.E. & Worsley, K.J. (2005). ‘Random fields of multivariate test statistics, with applications to shape analysis and fMRI.’

(available on http://www.math.mcgill.ca/keith

A spherical search region of radius r.

nipy.algorithms.statistics.rft.volume2ball(vol, d=3)

Approximate volume with ball

Approximate intrinsic volumes of a set with a given volume by those of a ball with a given dimension and equal volume.