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Neuron Digest Volume 05 Number 26
Neuron Digest Saturday, 10 Jun 1989 Volume 5 : Issue 26
Today's Topics:
Neural Net Applications in Chemistry
Re: Neural Net Applications in Chemistry
Re: Neural Net Applications in Chemistry
Re: Neural Net Applications in Chemistry
Re: Neural Net Applications in Chemistry
Re: Neural Net Applications in Chemistry
Re: Neural Net Applications in Chemistry
Re: Neural Net Applications in Chemistry
Re: Neural Net Applications in Chemistry
Re: Neural Net Applications in Chemistry
computer composition
Info request on parallel implementation & invarient object recognition
RE: Neurological Topology (leading to poll)
Re: Neuron Digest V5 #20
RE: Neuron Digest V5 #24
Re: Neuron Digest V5 #24
Send submissions, questions, address maintenance and requests for old issues to
"neuron-request@hplabs.hp.com" or "{any backbone,uunet}!hplabs!neuron-request"
ARPANET users can get old issues via ftp from hplpm.hpl.hp.com (15.255.16.205).
------------------------------------------------------------
Subject: Neural Net Applications in Chemistry
From: dtinker@gpu.utcs.utoronto.ca (Prof. David Tinker)
Organization: University of Toronto Computing Services
Date: Wed, 10 May 89 15:09:44 +0000
An interesting (non-tech) account of Neural Network applications in
Chemistry appeared in "Chemical and Engineering News", April 24, 1989. The
article describes papers given at a symposium on NN's held by the Division
of Computers in Chemistry of the American Chemical Society (the article
dosn't give detail on this symposium). Besides a thumbnail sketch of NN
theory, several applications are described.
D.W. Elrod, G.M. Maggiora and R.G. Trenary used the "ANSim" commercial
NN simulator(*) to predict reaction products involving nitrations of
monosubstututed benzenes; a 25-element connection table incorporated data on
identity, connectivity and charges of non-H atoms. The net, a back-prop
algorithm was trained and tested with 13 test compounds - it predicted 10 of
the 13 product distributions correctly, about as well as "three organic
chemists at Upjohn". An "artificial intelligence" program (not specified)
did not do as well. No reference was cited. (* Source: "Science
Applications International Corp", running on a 386 PC-AT clone with math
co-processor).
D.F. Stubbs described use of a NN program (not described in detail) to
predict adverse gastrointestinal applications of non-steroidal anti
inflammatory drugs, using data on pK, blood half-life, molecular weight and
dosage. The net was able to predict adverse drug reaction frequency to an
accuracy of 1%.
J.D. Bryngelson and J.J. Hopfield discussed use of a NN to predict
protein secondary structure from data on amino-acid properties. Here, a
couple of recent references are given: N. Qian & T.J. Sejnowski, J. Mol.
Biol. 202(4), 865, 1988; L.H. Holley & M. Karplus, Proc. Natl. Acad. Sci.
86(1), 152, 1989. Another approach was described by M.N. Liebman.
If anyone has more details on the meeting or the work described,
or further references to chemistry/biochemistry applications, please
post!
Disclaimer: I'm just learning about all this stuff!
- ---------------------------------------------------------------------------
! David O. Tinker ! ^ ^ ! UUCP: dtinker@gpu.utcs.utoronto.ca !
! Department of Biochemistry !< O O >! BITNET: dtinker@vm.utcs.utoronto.ca !
! University of Toronto ! ^ ! BIX: dtinker !
! TORONTO, Ontario, Canada ! ##### ! Voice: (416) 978-3636 !
! M5S 1A8 ! ! And so on. !
! ! Hi ho ! !
------------------------------
Subject: Re: Neural Net Applications in Chemistry
From: ted@nmsu.edu (Ted Dunning)
Organization: NMSU Computer Science
Date: Wed, 10 May 89 19:07:44 +0000
alan lapedes and co at los alamos were able to predict whether short
sequences of dna coded for specific proteins with good accuracy using
modified neural nets.
much of the work at lanl using neural net methods has been supplanted by
doyne farmers local approximation method which (for many problems) is
several orders of magnitude more computationally efficient. the use of
radial basis functions improves the value of this method considerably (in
addition to making the link to nn techniques even stronger).
------------------------------
Subject: Re: Neural Net Applications in Chemistry
From: andrew@berlioz (Lord Snooty @ The Giant Poisoned Electric Head)
Organization: National Semiconductor, Santa Clara
Date: Thu, 11 May 89 01:44:05 +0000
"Science News", April 29th, p271 "Neural Network Predicts Reactions". The
key quote for me was "When tested with 13 [..] not in the training set [of
32] the network predicted [..] proportions within 20% of actual values in 10
cases. That equals the performance by a small set of human chemists and
beats out by three an existing conventional computer expert system for
predicting reaction outcomes."
Go Nets!
Andrew Palfreyman USENET: ...{this biomass}!nsc!logic!andrew
National Semiconductor M/S D3969, 2900 Semiconductor Dr., PO Box 58090,
Santa Clara, CA 95052-8090 ; 408-721-4788 there's many a slip
'twixt cup and lip
------------------------------
Subject: Re: Neural Net Applications in Chemistry
From: aboulang@bbn.com (Albert Boulanger)
Date: Thu, 11 May 89 15:26:47 +0000
I should mention that besides Doyne Farmer's method, James Crutchfield and
Bruce S. McNamara have a method for recovering the equations of motion from
a time series. The reference is:
"Equations of Motion from a Data Series", James Crutchfield & Bruce
McNamara, Complex Systems, 1(1987) 417-452.
Unfortunately, I never did get a reference to Doyne's method.
Chaotically yours,
Albert Boulanger
BBN Systems & Technologies Corp.
aboulanger@bbn.com
------------------------------
Subject: Re: Neural Net Applications in Chemistry
From: ted@nmsu.edu (Ted Dunning)
Organization: NMSU Computer Science
Date: Thu, 11 May 89 20:15:34 +0000
the best reference is the los alamos tech report LA-UR-88-901. i asssume
that this is available from lanl, somehow (i got mine by hand).
(interestingly, on the last page i not a us gov printing office number:
1988-0-573-034/80049). (you might also try doyne whose net address is
jdf@lanl.gov)
an extract of the abstract follows:
Exploiting Chaos to Predict the Future and Reduce Noise
J. Doyne Farmer and John J. Sidorowich
We discuss new approaches to forecasting, noise reduction, and the analysis
of experimental data. The basic idea is to embed the data in a state space
and then use straightforward numerical techniques to build a nonlinear
dynamical model. We pick an ad hoc nonlinear representation, and fit it to
the data. For higher dimensional problems we find that breaking the domain
into neighborhoods using local approxiamtion is usually better than using an
arbitrary global representation. When random behavior is caused by low
dimensional chaos our short term forecasts can be several orders of
magnitude better than thos of standard linear methods. We derive error
estimates for the accuracy of approximatin in terms of attractor dimension
and Lyapunov exponents, the number of data points, and the extrapolation
time. We demonstrate that for a given extrapolation time T iterating a
short-term estimate is superior to computing an estimate for T directly.
...
We propose a nonlinear averaging scheme for separating noise from
deterministic dynamics. For chaotic time series the noise reduction
possible depends exponentially on the length of the time series, whereas for
non-chaotic behavior it is proportional to the square root. WHen the
equations of motion are known exactly, we can achieve noise reductions of
more than ten orders of magnitude. Wehn the equations are not known the
limitation comes from predication error, but for low dimensional systems
noise reductions of several orders of magnitude are still possible.
The basic principles underlying our methods are similar to those of neural
nets, but are more straightforward. For forecasting, we get equivalent or
better results with vastly less computer time. We suggest that these ideas
can be applied to a much larger class of problems.
hope that this helps.
------------------------------
Subject: Re: Neural Net Applications in Chemistry
From: mbkennel@phoenix.Princeton.EDU (Matthew B. Kennel)
Organization: Princeton University, NJ
Date: Thu, 11 May 89 20:34:22 +0000
The best reference for Farmers' local approximation method is in the LANL
preprint: "Exploiting Chaos to Predict the Future and Reduce Noise" by
Farmer and Sidrowich.
Note that this paper only deals with predicting chaotic dynamical systems.
Having just done a thesis on the same sort of prediction using neural
networks, I might be able to give an outline of various methods. There are
basically two major categories:
1) Global methods. In this scheme, one tries to fit some complex function
to all the data. There are various functional forms.
A) Linear methods. By "linear" I mean the output of the function
depends linearly on the _free parameters_; all of these functions
can represent nonlinear transformations from input to output, of
course. If the output is linear in the free parameters (weights),
there is a well-known guaranteed, optimal & deterministic
learning algorithm for the least squared error measure.
1) Global polynomials. Basically a taylor expansion of degree
d in m unknowns. This is what Crutchfield & MacNamara use.
Advantages: easy to compute. Disadvantages, for complicated
functions, you need large values of d and the number of
free weights increases very rapidly, approx = d^m.
High-degree polynomials tend to blow up away from the domains
on which they are fitted. (trained)
2) Rational quotients of polynomials. Similar to above, but if
the degrees of num. and denom. are same, they don't blow up.
3) Radial basis functions, with fixed centers. Here you choose
the centers of your radial functions, and then learn the
weights of the output layer using linear least squares, for
example. This can put put in terms of a network with a single
hidden layer of neurons, where the first layer of weights is
fixed, and the second layer of weights leads to a linear
output unit. See a recent paper by Broomhead & (somebody-or
other) in Complex Systems. There's also a pre-print
by Martin Casdagli, using this to predict chaotic systems.
ED note: this method looks promising and should probably
be investigated further, especially to see whether it works
in applications other than chaotic systems prediction.
B) Nonlinear methods. Now, you have to use an iterative
gradient-descent type of method.
1) Standard sigmoidal back-prop net. Well known learning algorithm.
2) Radial basis function, with adjustable centers. (This is what I
used). Lets you represent the mapping more accurately using
the same size network (=free parameters) compared to sigmoidal
net, I found. Learn with conjugate gradient.
II) Local methods.
Now, instead of trying to fit a global function to all examples,
you make a simple individual fit _every_ time you need an output.
Here's how Farmer's method works:
Given some input vector, find the N closest points in the
data base to this input vector. For each of these stored points,
you know what the output was. With these input-output pairs,
make a linear or quadratic approximation, and fit the free parameters
using linear least squares. This should be fast because there aren't
that many examples (10-30 say) and free parameters.
"Learning" simply consists of putting the input data base into a k-d
tree that permits you to retrieve nearest neighbors is O(log N) time, as
needed to make predictions.
This method has a nice intuitive interpretation: given some new
input, you look around to see what things like that input did, and you do
something like that.
Advantages: Much faster computationally than an iterative gradient-descent
optimization, especially for large data bases of examples. Doesn't require
any hard crunching to learn some function. Probably more accurate for large
data bases, too, because most people wouldn't have the patience or computer
power to learn a large network to high accuracy.
Disadvantages: The mapping isn't in a nice analytic functional form. You
can't realize it in silicon. You need to carry around the data base of
examples, and making predictions is slower (a search & small fit vs. a
simple functional evaluation).
=================================
Personal opinion:
If you have a large data base (say over 1K examples) of noise-free
continuous-valued examples, local linear and local quadratic methods will
probably be the best. I don't know what the effect input noise would have
on this method, compared to neural networks, but I don't think it would be
that bad. For binary values, it may not be as good, for the whole method is
rooted in the field of dynamical systems. But I don't think anybody's tried
yet, so I have no real evidence one way or the other.
==================
Get the preprint (it's very good) from Doyne Farmer at LANL. I believe his
e-mail address is "jdf%heretic@lanl.gov". An earlier version of this
appeared in Physical Review Letters, so it's definitely real.
Matt Kennel
mbkennel@phoenix.princeton.edu
------------------------------
Subject: Re: Neural Net Applications in Chemistry
From: hsg@romeo.cs.duke.edu (Henry Greenside)
Organization: Duke University CS Dept.; Durham, NC
Date: Fri, 12 May 89 03:15:49 +0000
In these discussions of Farmer et al's methods versus neural nets, has
anyone addressed the real issue, how to treat high-dimensional data?
In his paper, Farmer et al point out the crucial fact that one can learn
only low dimensional chaotic systems (where low is rather vague, say of
dimension less than about 5). High dimensional systems require huge amounts
of data for learning. Presumably many interesting data sets (weather, stock
markets, chemical patterns, etc.) are not low-dimensional and neither
method will be useful.
------------------------------
Subject: Re: Neural Net Applications in Chemistry
From: aboulang@bbn.com (Albert Boulanger)
Date: Fri, 12 May 89 19:17:56 +0000
[ Regarding article above by Henry Greenside: ]
I think you have a legit concern here, but in many cases where
high-dimensionality is suspected, it turns out to be low-dimensional.
Farmer, Crutchfield and others have developed a way of estimating the
dimensionality of the dynamics underlying time-series data by using a
sliding-time window technique where data-values within the window are
treated as independent dimensions. For example a 10000 point sample would be
broken-up as 1000 10-vectors for a 10-dimensional embedding. One then
varies the size of the window and compares the dimensionality of the
attractor with its embedding space. When they diverge is an estimate of the
dimensionality of the underlying dynamics. (I may have this wrong somewhat,
but this is the gist.)
What is truly amazing to an observer of nature, is the tremendous
dimensionality reduction that occurs in many-body systems. I don't really
understand why this is so, but it is this property that mean-field
approaches to NNs capitalize on.
People have looked the situation of low-dimensionality chaos coupled with
noise. For example see:
"Symbolic Dynamics of Noisy Chaos"
J.P. Crutchfield and N.H. Packard
Physica R-9D (1983) (I don't have the page numbers, since I have a
preprint copy, sorry.)
I have also heard of a growing interest to treat attractors in a
measure-theoretical way to deal with dynamical systems coupled with noise.
If anybody has pointers to this, please let me know.
Albert Boulanger
BBN Systems & Technologies Corporation
aboulanger@bbn.com
------------------------------
Subject: Re: Neural Net Applications in Chemistry
From: "Matthew B. Kennel" <phoenix!mbkennel@PRINCETON.EDU>
Organization: Princeton University, NJ
Date: 12 May 89 19:55:05 +0000
[ Regarding above article by hsg@romeo.UUCP (Henry Greenside) ]
Quite true. This is definitely a fundamental problem. As the dimension
gets higher and higher, the data series looks more and more like true random
noise, and so predicion becomes impossible. Note, for example, that the
output of your favorite "random number generator" is most likely
deterministic, but probably has such a high dimension that you can't predict
with any accuracy without knowing the exact algorithm used.
The real problem comes down to finding a representation with both smooth
mappings and low (fractal) dimensional input spaces. This requires plain
old hard work and clever insight.
Matt Kennel
mbkennel@phoenix.princeton.edu
------------------------------
Subject: Re: Neural Net Applications in Chemistry
From: aboulang@bbn.com (Albert Boulanger)
Date: Fri, 12 May 89 23:44:48 +0000
In article <8393@phoenix.Princeton.EDU>, Matthew B. Kennel writes:
Quite true. This is definitely a fundamental problem. As the
dimension gets higher and higher, the data series looks more and more
like true random noise, and so predicion becomes impossible. Note, for
example, that the output of your favorite "random number generator" is
most likely deterministic, but probably has such a high dimension that
you can't predict with any accuracy without knowing the exact algorithm
used.
Actually they are low dimensional chaotic discrete maps. They have to be low
dimensional to be efficient. The reason that they are cyclic is that there
is no computer representation for irrationals. For a nice picture of this
see page 90 of Knuth Vol 2. "The Numbers fall mainly in the planes." I have
been playing with increasing the dimensionality of random number generators
by asynchronous iterative methods on MIMD parallel machines. If anybody is
interested in the dynamical apsects of discrete iterations, a good book to
start with is:
Discrete Iterations: A Metric Study
Francios Robert
Springer-Verlag 1986.
The following is a nice think-piece on "discrete" randomness and
"continuous" randomness:
2. Ford, Joseph. "How Random is a Coin Toss?". Physics Today (April
1983), 40-47. Very good introduction to the notions of chaos and
deterministic randomness.
Now, this is off the subject of NNs, and we could follow up on this in the
group comp.theory.dynamic-sys.
Albert Boulanger
BBN Systems & Technologies Corp.
aboulanger@bbn.com
------------------------------
Subject: computer composition
From: Eric Harnden <EHARNDEN%AUVM.BITNET@CORNELLC.cit.cornell.edu>
Date: Tue, 23 May 89 15:08:38 -0400
i know this is a little off the wall for the topic of this digest, but the
reference to JimiMatrix leads me to think that there may be a resource
here...
i am sending this simultaneously to several of the lists to which i am
subscribed. my apologies to those of my compatriots who have parallel
subscriptions, and thus will see this more than once.
there are many approaches to computer-generated music that i am aware of.
in general they seem to fall into two categories.
1) analyze known works for event patterns and meta-patterns. this allows
long probability sequences to be developed by direct derivation.
2) synthesize sequences with stochastic models with short memory.
i'm interested in the idea of constructing an event tree, whose depth may
well equal the number of events to be specified. i want to assign weight to
an event based, not just on the occurence of its predecessor, but on the
development of the string of which it is a member. i don't want to derive
the rules for weighting from analysis of other strings. so i'm not writing
a mozart program, and i'm not doing pink tunes, and i'm not implementing
markov chains. before i go crashing headlong into this, does anybody have
any thoughts, references, things i should know, people i should talk to...
etc.?
Eric Harnden (Ronin)
<EHARNDEN@AUVM>
The American University Physics Dept.
(202) 885-2758
------------------------------
Subject: Info request on parallel implementation & invarient object recognition
From: plonski@aerospace.aero.org
Date: Mon, 15 May 89 13:43:50 -0700
I am looking for information on simulating neural networks on a parallel
processing architecture such as a Sequent or a network of SUNS. My interest
is in how one breaks up the processing for various networks, but in
particular backpropogation, when you have small number of processors (~10)
available. Should processing be divided by layer, by node subdivision
within a layer, by task, etc? What division yields the minimal
communication overhead? Also, what are the effects of asynchronous
processing. For Associate Memory models, asynchronous processing can be
used to avoid limit cycles, but at what cost in speed of convergence.
My other interest is in using higher order interconnections (ala Giles, et
al.) to get a network that is invariant to geometric transformations in 2D
images. It seems to me that the net effect is equivalent to using an
initial layer to generate an invariant feature space and then applying an
ordinary net to select features in this space. If this is the case then do
higher order networks yield any advantages over preproccessing the data
first to get an invariant feature space and using that as your input.
I would be interested in any references or work that you have done, or know
about regarding these issues. You can send the replies directly to me and I
will summarize the replies and send them to all interested parties. Thank
You.
. . .__. The opinions expressed herin are soley
|\./| !__! Michael Plonski those of the author and do not represent
| | | "plonski@aero.org" those of The Aerospace Corporation.
------------------------------
Subject: RE: Neurological Topology (leading to poll)
From: worden@emx.utexas.edu (Sue J. Worden)
Date: Fri, 02 Jun 89 05:31:32 -0500
James Salsman says:
"... I have read that in humans, the grey matter lobes act as insulation
around a somewhat planar network called the "white matter" that is "crunched
up," to form a ball. The white mater is not perfectly planar -- it is
arranged in layers. The grey matter has fewer neurons than the white
matter, but quite a few more capilaries..."
Dorland's Illustrated Medical Dictionary says:
"substantia alba, white substance: the white nervous tissue, constituting
the conducting portion of the brain and spinal cord, and composed mostly of
myelinated nerve fibers."
"substantia grisea, gray substance: the gray nervous tissue, composed of
nerve cell bodies, unmyelinated nerve fibers, and supportive tissue."
Simplistically speaking, gray matter is a mass of neurons' cell bodies and
white matter is a mass of neurons' axons. The cerebral cortex, for example,
is several (six major) layers of cell bodies and, hence, is gray matter.
Axons from cell bodies in the cortical layers (and elsewhere) form an
underlying white matter region.
Unfortunately, simple distinctions between white and gray matter, simple
maps of anatomical structures, simple neuron models, simple axon-bundle
"wiring diagrams", et cetera, don't appear to contribute much to a solid
advancement of knowlege. It seems to me that, for the most part, we're all
mucking about, spinning our wheels, getting nowhere fast. When I ponder my
own work (with systems of randomly interconnected McCulloch-Pitts neurons),
and the work of so many others with everything from sodium channels to
psychopharmacology, from snails to psychotics, from perceptrons to
artificial cochleae, et cetera, what I sense is a modern form of alchemy.
It is as though we're all fluttering about on the periphery of our
equivalent of a periodic table.
I am, naturally, curious to know where my opinion falls on the bell curve of
your honest thoughts. Ignoring for a moment the practicalities of
publish-or-perish, winning grants/contracts, and so on:
(1) Do any of you REALLY believe that any present
line of research will directly lead to either:
(a) widely-accepted understanding of our
nervous systems, our intelligence,
our personalities, our selves?
(b) machine capabilities widely accepted
to be equivalent to human capabilities?
(2) If so, which line(s) of research?
(3) If not, do you believe:
(a) it is impossible or highly improbable?
(b) it is possible, but we don't yet have
the necessary "key(s)"? (my opinion)
(c) something else?
- Sue Worden
Electrical and Computer Engineering
University of Texas at Austin
------------------------------
Subject: Re: Neuron Digest V5 #20
From: Robert Morelos-Zaragoza <robert@wiliki.eng.hawaii.edu>
Organization: University of Hawaii, College of Engineering
Date: Wed, 24 May 89 16:20:38 -1000
Ladies/Gentlemen:
I am looking for papers that indicate the conection between Neural- Networks
and Coding Theory. In particular, how can Neural Network Theory help in the
construction of new Error-Correcting Codes?
Robert Morelos-Zaragoza
(robert@wiliki.eng.hawaii.edu)
Tel. (808) 938-6094.
------------------------------
Subject: RE: Neuron Digest V5 #24
From: edstrom%UNCAEDU.BITNET@CORNELLC.cit.cornell.edu
Date: Wed, 31 May 89 08:31:38 -0600
> From: Ian Parberry <omega.cs.psu.edu!ian@PSUVAX1.CS.PSU.EDU>
>
>At NIPS last year, one of the workshop attendees told me that, assuming one
>models neurons as performing a discrete or analog thresholding operation on
>weighted sums of its inputs, the summation appears to be done in the axons
>and the thresholding in the soma. ...
>
>
>It's now time to write up the fault-tolerance result. I'd like to include
>some references to "accepted" neurobiological sources which back up the
>attendee's observation. Trouble is, I am not a neurobiologist, and do not
>know where to look. Can somebody knowledgeable please advise me?
>
THe best reference that comes to mind is "The Synaptic Organization of the
Brain an Introduction" by Gordon M. Shepherd, Oxford University Press, 1974
Its a bit outof date but still excellent. Many details have been filled in
since 1974 but our basic understanding of the integrative properties of the
neuron has not changed all that much. Also, any attempt to jump into
contemporary literature on the subject will be much more difficult without
this "classic" core knowledge.
This book begins by discussing electrotonic phenomena, space constants,
current flow in dendritic trees, etc... and then applies the principles to
different neurons (cerebral pyramidal cells, cerebellar purkinje cells,
spinal motorneurons, etc...). He also discusses some of the local circuits
in the various parts of the vertbrate brain from which the sample neurons
come.
This is the best introduction to the subject that Iknow of and I recommend
it highly.
John Edstrom
+-- In the Real World ----------+--- Elsewhere ---------+
|Dr. John P. Edstrom |EDSTROM@UNCAEDU Bitnet |
|Div. Neuroscience |7641,21 CIS |
|3330 Hospital Drive NW |JPEDstrom BIX |
|Calgary, ALberta T2N 4N1 | |
|CANADA (403) 220 4493 | |
+-------------------------------+-----------------------+
------------------------------
Subject: re: Neuron Digest V5 #24
From: stiber@CS.UCLA.EDU (Michael D Stiber)
Date: Wed, 31 May 89 13:00:30 -0700
>Subject: Re: wanted: neurobiology references
>From: Mark Robert Thorson <portal!cup.portal.com!mmm@uunet.uu.net>
>Organization: The Portal System (TM)
>Date: 22 Apr 89 00:13:57 +0000
>
>I was taught, 10 years ago, that action potentials are believed to originate
>at the axon hillock, which might be considered the transition between the
>axon and the soma (cell body). See FROM NEURON TO BRAIN by Kuffler and
>Nichols (Sinauer 1976), page 349.
>
>I would expect synaptic weights to be proportional to the axon circumference
>where it joins the cell body, but I have no evidence to support that belief.
Actually, this is a simplified model. Purkinje cells, for example, are
hypothesized to have trigger zones in their dendritic arborization. See
Segundo, J.P., "What Can Neurons do to Serve as Integrating Devices?", J.
Theoret. Neurobiol. 5, 1-59 (1986).
------------------------------
End of Neurons Digest
*********************
