The entropic prior for distributions with positive and negative values

Hobson, M. P.; Lasenby, A. N.

doi:10.1046/j.1365-8711.1998.01707.x

Cited by 57 publications

(35 citation statements)

References 6 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Skilling (1988) defines the entropy S of a positive function H , over a domain Ω, as where d is a measure which can be interpreted as the default model obtained in the absence of any constraints (the so‐called ‘flat map’). From it is possible to define the entropy for functions H consisting of both positive and negative values (Gull & Skilling 1990; Hobson & Lasenby 1998): where (see also of Paper I). When looking at the limit of S [ H , d ] as d becomes very large compared to the typical amplitude of H , a second order Taylor series expansion of gives (e.g.…”

Section: Methods and Notationsmentioning

confidence: 99%

“…Further background details are presented in the companion paper by Jackson et al (2007), hereinafter referred as Paper I. First developed for positive images only, the maximum entropy method was later extended to non‐positive fields (Gull & Skilling 1990; Hobson & Lasenby 1998). Recently, Jackson (2003) applied it to reconstruct snapshots of the radial magnetic field at the CMB for the epochs 1980 and 2000, using the Magsat and Ørsted satellite data.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Maximum entropy regularization of time-dependent geomagnetic field models

Gillet¹,

Jackson²,

Finlay³

2007

Geophysical Journal International

View full text Add to dashboard Cite

S U M M A R YWe incorporate a maximum entropy image reconstruction technique into the process of modelling the time-dependent geomagnetic field at the core-mantle boundary (CMB). In order to deal with unconstrained small lengthscales in the process of inverting the data, some core field models are regularized using a priori quadratic norms in both space and time. This artificial damping leads to the underestimation of power at large wavenumbers, and to a loss of contrast in the reconstructed picture of the field at the CMB. The entropy norm, recently introduced to regularize magnetic field maps, provides models with better contrast, and involves a minimum of a priori information about the field structure. However, this technique was developed to build only snapshots of the magnetic field. Previously described in the spatial domain, we show here how to implement this technique in the spherical harmonic domain, and we extend it to the time-dependent problem where both spatial and temporal regularizations are required. We apply our method to model the field over the interval 1840-1990 from a compilation of historical observations. Applying the maximum entropy method in space-for a fit to the data similar to that obtained with a quadratic regularization-effectively reorganizes the magnetic field lines in order to have a map with better contrast. This is associated with a less rapidly decaying spectrum at large wavenumbers. Applying the maximum entropy method in time permits us to model sharper temporal changes, associated with larger spatial gradients in the secular variation, without producing spurious fluctuations on short timescales. This method avoids the smearing back in time of field features that are not constrained by the data. Perspectives concerning future applications of the method are also discussed.

show abstract

Section: Methods and Notationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Maximum entropy regularization of time-dependent geomagnetic field models

Gillet¹,

Jackson²,

Finlay³

2007

Geophysical Journal International

View full text Add to dashboard Cite

show abstract

“…We perform this optimization using the M em S ys package (Gull & Skilling 1999). This algorithm considers the parameters a to have prior probabilities proportional to e α S ( a ) , where S ( a ) is the positive–negative entropy functional (Hobson & Lasenby 1998). α is treated as a hyperparameter of the prior, and sets the scale over which variations in a are expected.…”

Section: Neural Networkmentioning

confidence: 99%

Constraints on fNL from Wilkinson Microwave Anisotropy Probe 7-year data using a neural network classifier

Casaponsa¹,

Bridges²,

Curto³

et al. 2011

Monthly Notices of the Royal Astronomical Society

Self Cite

View full text Add to dashboard Cite

We present a multiclass neural network (NN) classifier as a method to measure non‐Gaussianity, characterized by the local non‐linear coupling parameter fNL, in maps of the cosmic microwave background (CMB) radiation. The classifier is trained on simulated non‐Gaussian CMB maps with a range of known fNL values by providing it with wavelet coefficients of the maps; we consider both the HEALPix wavelet (HW) and the spherical Mexican hat wavelet (SMHW). When applied to simulated test maps, the NN classifier produces results in very good agreement with those obtained using standard χ2 minimization. The standard deviations of the fNL estimates for Wilkinson Microwave Anisotropy Probe1 like simulations were σ= 22 and 33 for the SMHW and the HW, respectively, which are extremely close to those obtained using classical statistical methods in Curto et al. and Casaponsa et al. Moreover, the NN classifier does not require the inversion of a large covariance matrix, thus avoiding any need to regularize the matrix when it is not directly invertible, and is considerably faster.

show abstract

“…Cyron et al 32 introduced a gap function to allow errors in the second‐order constraint while maintaining the convex condition and weak Kronecker‐delta property at the boundaries. Sukumar and Wright 29 introduced a higher order approximation using the entropy expression for the positive/negative distribution 33, but the approximation loses convexity and Kronecker‐delta property at the boundaries.…”

Section: Introductionmentioning

confidence: 99%

A generalized approximation for the meshfree analysis of solids

Park

Chen

2010

Numerical Meth Engineering

View full text Add to dashboard Cite

SUMMARYThis paper presents a new approach in the construction of meshfree approximations as well as the weak Kronecker-delta property at the boundary, referred to as a generalized meshfree (GMF) approximation. The GMF approximation introduces an enriched basis function in the original Shepard's method. This enriched basis function is introduced to meet the linear or higher order reproducing conditions and at the same time to offer great flexibility on the control of the smoothness and convexity of the approximation. The construction of the GMF approximation can be viewed as a special root-finding scheme of constraint equations that enforces that the basis functions are corrected and the reproducing conditions with certain orders are satisfied within a set of nodes. By choosing different basis functions, various convex and nonconvex approximations including moving least-squares (MLS), reproducing kernel (RK), and maximum entropy (ME) approximations can be obtained. Furthermore, the basis function can also be translated or blended with other functions to generate a particular approximation for a special purpose. One application in this paper is to incorporate a blending function at the boundary based on the concept of local convexity for the non-convex approximation, such as MLS, to acquire the weak Kronecker-delta property. To achieve the higher order GMF approximation, two possible methods are also introduced. Several examples are presented to examine the effectiveness of various GMF approximations.

show abstract

The entropic prior for distributions with positive and negative values

Cited by 57 publications

References 6 publications

Maximum entropy regularization of time-dependent geomagnetic field models

Maximum entropy regularization of time-dependent geomagnetic field models

Constraints on fNL from Wilkinson Microwave Anisotropy Probe 7-year data using a neural network classifier

A generalized approximation for the meshfree analysis of solids

Contact Info

Product

Resources

About