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\centerline{\large The  Impact Toughness of C--Mn Steel Arc--Welds}
\centerline{\large -- A  Bayesian Neural Network Analysis}
\bigskip \singlespace
\centerline{ H. K. D. H. Bhadeshia$^*$, D. J. C. MacKay$^*$ and
L.--E. Svensson$^{\dag}$}
\medskip
\centerline{$^*$Mathematics and Physical Sciences Group}
\centerline{Darwin College, Silver Street, Cambridge CB3 9EU, U. K}
\medskip
\centerline{$^{\dag}$Central Research Laboratories, ESAB AB}
\centerline{Gothenburg, Sweden}
\bigskip \doublespace

\sec{ABSTRACT}

\x Charpy impact toughness data for  manual metal arc and
submerged arc weld metal samples have been analysed using a
neural network technique within a Bayesian framework. In this, the
toughness can be represented as a general empirical function of
variables that are commonly acknowledged to be important in
influencing the properties of steel welds. The method 
has limitations due to its empirical character, but it is
demonstrated here that it can be used in such a way that
the predicted trends make metallurgical sense. The method has been
used to examine the relative importance of the numerous variables
thought to control the toughness of welds.

\sec{INTRODUCTION}

\x Fusion welding is of the greatest importance in the 
fabrication of engineering structures. One of the most 
important requirements for such structures, including all the 
welded joints, is that they should be able to resist brittle 
fracture. The weld deposits therefore have to be \lqq tough" 
with a great deal of energy being absorbed by the metal 
during the process of fracture.  

A test used to characterise 
toughness is the Charpy test, in which a square sectioned, 
notched bar is fractured under specified conditions [1]. The 
energy absorbed during fracture is taken as a measure of 
toughness. The Charpy test is empirical in that the data cannot 
be used directly in engineering design. It is nevertheless a 
useful quality control test which is  
specified widely in international standards, and in the ranking of
samples in research and development experiments.

The toughness of a steel depends on many variables, and that 
of a weld on many more because of the complexity of the 
welding process. It is not yet possible to predict the Charpy 
toughness of a weld with any reliability. The usual approach 
is to correlate the results against chosen variables using 
linear regression analysis [\eg\ 2]. These methods are known to be 
severely limited in their application.

Therefore,  the most important 
mechanical property for welds has not been rationalised 
quantitatively as a function of the complex array of 
variables associated with welding. However, it is known from 
experience, and from the concepts of fracture mechanics, that
certain variables are more important than  others in their effect on
toughness. The purpose of the work  presented here was to see
whether an artificial neural  network [3] can be trained to
predict weld toughness as a nonlinear function of these
variables, and to see whether the patterns that emerge from the
work emulate metallurgical experience.

 In normal regression methods the analysis 
begins with the prior choice of a relationship (usually 
linear) between the output and input variables. A neural network is
capable of realising a greater variety of nonlinear relationships of
considerable complexity. Data are presented to the network in the
form of input and output parameters, and the optimum non--linear
relationship is found by minimising a penalized likelihood. The
network in effect tries out many kinds of  relationships in its search
for an optimum fit. As in  regression analysis, the results then
consist of a specification of the function, which in combination with
a series of  coefficients (called weights), relates the  inputs to
the outputs. The search for the optimum  representation
can be computer intensive, but once the  process is completed (\ie
the network trained) the estimation  of the outputs is very
rapid. In spite of its apparent  sophistication, the method is as blind
as regression  analysis, and neural nets can be susceptible to
overfitting. 

However, much of this danger can in principle be minimised or 
eliminated by combining the neural network approach with 
sound statistical and metallurgical theory.  MacKay  [4--7] has
developed a Bayesian framework for neural networks.  This
framework allows one to assess quantitatively the relative
probabilities of models of different complexity, and to put
quantitative error bars on the predictions of the models. We have
applied this work to the complex problem of predicting weld metal
toughness.

\sec{VARIABLES}

\x It is possible to choose a set of variables which should, 
using experience of welding metallurgy, have an influence of 
the Charpy toughness. These variables are listed in Table~1, 
and described below.


\midinsert \thicksize=1pt \thinsize=0.8pt 
\tablewidth=5.2truein \begintable 
Variable | Range & Mean & Standard \nr
               |                 &          &  Deviation \cr
Process | Submerged Arc & & \nr
             | Manual Metal Arc & & \nr
Yield Strength  MPa |  347--645 & 471 & 12.7 \cr
Carbon  wt.\% | 0.029--0.13  & 0.08 & 0.004 \nr
Silicon  wt.\% | 0.28--1.14 &  0.49 & 0.05 \nr
Manganese  wt.\%  | 0.77--2.50 & 1.32 & 0.07 \cr
Phosphorus  wt.\% | 0.008--0.028 & 0.015 & 0.001 \nr
Sulphur  wt.\% | 0.002--0.017 & 0.010 & 0.0005 \nr
Aluminium  wt.\%  | 0.001--0.04 & 0.014 & 0.002 \nr
Nitrogen p.p.m.w. | 26--119  & 67 & 4 \nr
Oxygen  p.p.m.w. | 234--821   & 412 & 30 \cr
Primary Microstructure \% | 0--91 & 34 & 4 \nr
Secondary Microstructure \% | 9--100  & 66 & 2 \nr
Allotriomorphic Ferrite \% | 16--62  & 31 & 2 \nr
Acicular Ferrite \% | 11--81 & 55 & 2 \nr
Widmanst\"atten Ferrite \% | 0--35 & 14 & 2 \cr
Temperature  K | 213--293  & 259 & 25 \nr
Charpy Toughness  J| 4--215   &         &
\endtable \tabtit {Table 1: }{  The variables. The
abbreviation p.p.m.w. stands for  parts per million by weight. }
\endinsert 

In general, the toughness decreases as the strength increases. 
This is because plastic deformation, which is the major 
energy absorbtion mechanism during fracture, becomes more 
difficult as the strength increases. Hence, the yield 
strength is included as a variable. The nature of the welding 
process itself may have a significant effect on toughness.  For
example, the submerged arc welding process is quite  different from
the manual metal arc welding process, leading  to the development
of different microstructures and  variations in impurity content.
However, heat input {\it per se} is not included since its effect is via
the microstructure [8--12], which is included in detail in the analysis. 

The major solute additions to steels, \ie C, Mn and Si, have 
large effects on the transformation behaviour and strength. 
Impurity elements (P, S, Al, N, O) are included because of 
their known tendency to embrittle or because of their 
importance in the formation of nonmetallic inclusions in 
welds. 

All fusion welding processes involve the deposition of a 
small amount of molten steel within a gap between the 
components to be joined. When the steel solidifies, it welds 
the components together.  The {\it fusion zone} represents 
both the deposited metal and the parts of the steel component 
melted during the process, and is a solidification 
microstructure, often called the {\it primary 
microstructure}. In practice, the gap between the components 
to be joined has to be filled by a sequence of several weld 
deposits. These multirun welds have a complicated 
microstructure.  The deposition of each successive layer
heat--treats the underlying microstructure. Some of the regions of 
original primary microstructure are reheated to temperatures  high
enough to cause the reformation of austenite, which  during the
cooling part of the thermal cycle transforms into  a different
microstructure. Other regions may simply be  tempered by the
deposition of subsequent runs. The  microstructure of the reheated
regions is called the {\it  reheated} or {\it secondary}
microstructure. The fractions of  the primary and secondary
microstructures are included as  input variables (Table~1).

In addition, the details of the primary microstructure are 
also included in the list of input variables, since the 
phases involved (allotriomorphic, Widmanst\"atten and 
acicular ferrite) are known to have a major influence on the 
weld properties. 

Iron undergoes a ductile--brittle transition as a function of 
temperature [1]. The flow stress of iron is sensitive to 
temperature, the strength increasing as the temperature 
decreases. At some critical temperature, it becomes easier to 
cleave iron without expending much energy. Below this 
critical temperature, the iron behaves in a very brittle 
manner. Hence, the test temperature is included as an 
important variable.

All of these input variables should to varying degrees 
influence the Charpy toughness, which is the output variable.

\sec{EXPERIMENTAL DATA}

\x All of the data used in the analysis are from experiments
conducted at the ESAB Central Research Laboratories [13--15].
These data represent a total of 181 different combinations of the
variables listed in Table~1, from all--weld metal tests in which the
joints were deposited to ISO2560 specification as described
elsewhere [11].  In this specification, the joint geometry
is such that there is a minimal dilution of the weld metal,
so that experiments can be conducted on all--weld samples. Hence,
the development of welding electrodes is usually carried out using
this joint design. References 13 and 15 contain data on manual
metal arc welds where as Ref.~14 deals with submerged arc welds.
All of these welds are typically classified as carbon--manganese
welds, because they do not contain deliberate additions of elements
such as nickel, boron \etc

The manual metal arc welds were deposited in the flat position
using the stringer bead technique, the parent plate thickness being
20~mm. The welding current and voltage used were 190~A and
23~V respectively, the weld consisting of some 27 runs with 3
runs per layer deposited at a speed of 0.004~${\rm m\,s^{-1}}$.
The interpass temperature was typically 250 \degg. The electrodes
used were 4~mm in diameter.

The submerged arc welds were fabricated to the same geometry,
but with a variety of proprietary wires and flux combinations. The
wires were of 4~mm diameter ({\it OK Autrod} 12.10, 12.22 \&\
12.32) and the fluxes included {\it OK Flux} 10.61, 10.71 and
10.81. The terms in italics are trade marks of ESAB AB -- there is
nothing special about these consumables but is  worth noting
that they are all commonly used in welding mild steels. The
significant welding parameters are listed in Table~2; the interpass
temperature was about 250 \deg\ in each case.



\midinsert \thicksize=1pt \thinsize=0.8pt
\tablewidth=5.2truein \begintable
Current  Amps | Voltage | No. of Beads | Welding Speed ${\rm
m\,s^{-1}}$ \cr
450 | 30 | 14--16 | 0.007 \nr
550 | 30 | 12 | 0.007 \nr
650 | 30 | 10 | 0.007
\endtable \tabtit{Table 2: } {Welding parameters for the submerged
arc welds.}\endinsert

The tests have been carried out over a number of
years with the systematic measurement of all of the important
parameters discussed earlier. However, the fraction of primary and
secondary microstructure was not measured for the data in [14].
These fractions were therefore calculated as described in [9,16]. 

The routine  method for quantitative metallography has been
described previously [11], and the mechanical property
measurements were carried out to ISO2560 specification. The
Charpy value used in the analysis represented the mean of between
three to five tests.

\sec{ANALYSIS}

\x The analysis was conducted using variables normalised between
+0.5 and -0.5; this normalisation is not essential to the neural
net approach but allows a convenient comparison of the influence
of individual input variables on an output. The normalisation
procedure is expressed quantitatively as follows:
$$ x_N = {{x - x_{min}}\over{x_{max} - x_{min}}} - 0.5 \numeqn
$$ where $x_N$ is the normalised value of $x$;  $x_{min}$ and
$x_{max}$ are respectively the minimum and maximum values of
$x$ in the entire dataset (Table~1).

 The normalisation is straightforward for all
the quantitative variables; however,  the welding process   was
represented by assigning a value of +0.5 to submerged arc welding,
and -0.5 to manual metal arc welding. 

The network consisted of fourteen input nodes, a number of hidden
nodes and an output node representing the
toughness (\fagg). The network was trained using a randomly
chosen 100 of the examples from a total of 181 available, the
remaining 81 examples being used as \lq new' experiments to test
the trained network. 

\picture{network} {114.0} {61.7} {700} {\tabtit{\figg : }{  A typical
network used in the analysis. Only the connections originating from
one input unit are illustrated, and the two bias units are not
illustrated.}}

Linear functions of the inputs $x_j$ are operated on by a hyperbolic
tangent transfer function:
$$h_i = \tanh \bl(\sum_j w^{(1)}_{ij}x_j + \theta^{(1)}_i\br)
\numeqn$$ so that each input contributes to every hidden unit. The
bias  is designated $\theta_i$ and is analogous to the constant
that appears in linear regression. The strength of the transfer
function is in each case determined by the weight $w_{ij}$. 
The transfer to the output $y$ is linear:
$$ y = \sum_i w_i^{(2)} h_i + \theta^{(2)} \numeqn $$
This  specification of the network structure, together with
the set of weights is a complete description of the formula relating
to the inputs to the output. The weights are determined by training
the network; the details are described elsewhere [4--7]. The
training involves a  minimisation of the regularised sum of squared
errors.  The term $\sigma_\nu$ used below is the framework
estimate of the noise level of the data. 

The complexity of the model is controlled by the
number of hidden units (\fagg), and the values of the
16 regularisation constants ($\sigma_w$), one associated with
each input, one for biases and one for all weights connected to the
output. 

\picture{sigmanu} {190.0} {98.7} {700} {\tabtit{\figg : }{ 
Variation in $\sigma_\nu$ as a function of the number of
hidden units. Several values are presented for each set of hidden
units because the training for each network was started with a
variety of random seeds.}}
\figg\  shows that the inferred noise level decreases
monotonically as the number of hidden units increases. However,
the complexity of the model also increases with the number of
hidden units. A high degree of complexity may not be justified, and
in an extreme case, the model may in a meaningless way attempt to
fit the noise in the experimental data. MacKay [4--7] has made a
detailed study of this problem and has defined a quantity (the
{\it evidence}) which comments on the probability of a model. In
circumstances where two models give similar results over the
known dataset, the more probable model would be predicted to be
that which is simpler; this simple model would have a higher value
of \lq evidence'.  The evidence framework was used to
control the regularisation constants and $\sigma_\nu$. The
number of hidden units was set by examining performance on test
data. A combination of Bayesian and pragmatic statistical techniques
were therefore used to control the model complexity. Four hidden
units were found to give a reasonable level of complexity to
represent the variations in toughness as a function of the input
variables. Larger numbers of hidden units did not give significantly
lower values of
$\sigma_\nu$; indeed,  the
test set error goes through a minimum at four hidden units (\fagg).

\picture{testenergy} {190.0} {87.0} {700} {\tabtit{\figg : }{ The
 test error as a function of the number of hidden units.}}

The optimum parameters for one trained network are presented in
Table~3; this listing would be required in order to reproduce the
predictions described, though not the error bars. The levels of
agreement for the training and test datasets are illustrated in
\fagg, which show good prediction in both instances. It should be
emphasized that the test data were not included in deriving the
weights given in Table~3 (except to choose the solution displayed),
so that the good fit is established to work well over the range of data
included in the analysis.


\picture{traintest} {117.0} {203.0} {700} {\tabtit{\figg : }{  Plot
of the estimated versus measured toughness; (a) training dataset;
(b) test dataset.}}

\vfill\eject
\sec{USE OF THE MODEL}

\x We now examine the
metallurgical significance of the results. We attempt  predictions out
of the range of the experimental data used during training, and
examine some aspects which cannot be studied experimentally.

\fagg\ illustrates the significance ($\sigma_w$) of each of the
input variables, as perceived by the neural network, in influencing
the toughness of the weld. The process clearly has a large
intrinsic effect, which complies with experience in that submerged
arc welds are in general of a lower quality than manual metal arc
welds. Note that this is a process effect which is independent of all
the other variables listed. The yield strength has a large effect and
that is well established [1]. It is also widely believed, as seen in
\figg, that acicular ferrite has a large effect on the toughness.
Nitrogen has a large effect, as is well established experimentally
[10,17--12]. Oxygen influences welds in both beneficial and
harmful ways, \eg by helping the nucleation of acicular ferrite
or contributing to fracture by nucleating oxides. 

It is surprising at first sight that carbon has such a small effect, but
what the results really demonstrate is that the influence of carbon
comes in via the strength and microstructure. 
Phosphorus and sulphur have only a small effect; the toughness
measured was in the as--welded condition whereas many of the
classical embrittlement effects manifest themselves in the
stress--relieved condition. It is also possible that the effects of P and
S are higher at strength levels larger than encountered here. All of
the welding consumables are commercially used so that they are not
expected to be embrittlement prone. Elements such as Mn and Si do
not feature greatly presumably because their effect comes in via
microstructure. Fig.~5 also shows a relatively small effect of
temperature on toughness, but it should be noted that the
temperature range considered is only 80 \deg (Table~1), and that
a part of the effect of temperature is to alter the yield strength,
which is identified by the model to be one of the important variables.

\picture{importance} {190.0} {98.7} {700} {\tabtit{\figg : }{
Bar chart showing a measure of the model--perceived significance
of each of the input variables in influencing toughness.}}

The model can be used to estimate the toughness if all of the
inputs listed in Table~1 are available. The amount of work required
to accumulate these inputs is not trivial, but the situation can be
ameliorated. A physical model [11,12,16] based on phase
transformation theory can be used to predict the values of all the
inputs from a knowledge of just the chemical composition and a
choice of welding conditions. This was done particularly to examine
the effects of carbon and manganese on weld toughness, given
that a lot of work on these lines has already been reported in the
literature.

\fagg a shows data generated using the neural network but with
all the inputs other than manganese calculated   using our
weld model [11,12,16]. In all cases, the calculated inputs are for
manual metal arc welds with 180~A, 34~V, a welding speed of
0.004~${\rm m\,s^{-1}}$, interpass temperature 200 \degg,
ISO2569 weld geometry. The manganese variations are for a basic
composition 
$${\rm Fe-0.07C-0.5Si-Mn-0.005S-0.0025O-0.0005N-0.025Al}$$.
It is  interesting that the toughness at relatively high temperatures
decreases as the manganese concentration is increased. This upper
shelf region involves ductile failure, and an increase in strength
leads to a reduction in the ductile fracture energy. The
calculated yield strength  [10,12,16] increases from 403--539 MPa
as the Mn concentration is changed from 0.5-2.0 wt.\%. However,
the cleavage toughness at low temperatures clearly increases with
Mn up to a concentration of 1.5 wt.\%. This is because the
calculated [10,12,16] acicular ferrite content increases from
35-67\%\ when Mn is changed over the range illustrated. The
low temperature toughness for 2 wt.\%\ Mn is nevertheless lower
than that for the 1.5 wt.\%\ alloy presumably because the
increased acicular ferrite content is not sufficient to compensate
for the increased strength. Indeed, an optimum manganese
concentration of about 1.5 wt.\%\ has been reported to achieve
the best toughness in manual metal arc welds of the type
discussed here [22,23].

\figg b shows similar data for carbon (the only difference being
that the Mn concentration is fixed at 1 wt.\%.). The explanation is
identical to that for the Mn data.

\picture{MnC} {174.0} {184.0} {700} {\tabtit{\figg : }{
(a) Variation in the normalised toughness as a function of the
manganese concentration. (b) Variation in the normalised
toughness as a function of the carbon concentration.}}

For welds similar to the carbon series, but with the carbon
concentration fixed at 0.07 wt.\%, the oxygen concentration alone
was varied to a range well outside of the training dataset. These
results are presented in \fagg\ along with the $\pm 1 $ standard
deviation predicted error bars. It is clear that any attempt to
extrapolate beyond the dataset on which the model is based gives
predictions which are not terribly useful. The fact that the
toughness increases with oxygen at low concentrations  is
strange since the acicular ferrite content (and indeed all the other
inputs) are kept constant. An increase in the oxygen content alone
should lead to a deterioration in toughness because of the
tendency for non--metallic oxide particles to initiate fracture. 

\picture{oxygen} {105.0} {86.4} {700} {\tabtit{\figg : }{
Variation in the normalised toughness as a function of the
oxygen concentration.  Oxygen is varied here without
changing any of the other inputs. The   oxygen
concentration in the training data was in the range 234--821
p.p.m.}}

Finally, it is possible using the model to examine effects which
cannot easily be produced experimentally. It has frequently been
argued that acicular ferrite is a better microstructure than \wid,
because the former with its less organised arrangement of ferrite
plates has a greater capacity to deflect cracks.  This was tested for
a manual metal arc weld containing 0.07 wt.\%\ carbon but of
otherwise identical composition to the carbon series of welds
(Fig.~6). The allotriomorphic ferrite fraction was set to zero and all
inputs except acicular ferrite and \wid\ were varied in a
complementary fashion. The results (\fagg) are exciting -- they
demonstrate that increased acicular ferrite leads to an improvement
of cleavage toughness but not of the upper shelf energy -- the latter
is not expected to change since the strengths of acicular and \wid\
are virtually identical [24].

\picture{acicwid} {190.0} {98.7} {700} {\tabtit{\figg : }{
Variation in the normalised toughness as a function of the
acicular ferrite/\wid\ content, everything else being kept
constant.}}

\sec{CONCLUSIONS}

\x An artifical neural network has been used to rationalise Charpy
impact toughness data on manual metal arc and submerged arc
steel weld deposits. The analysis is empirical but after appropriate
training, is found to reliably reproduce known metallurgical
experience. 

The method is useful in that the optimised network  summarises
knowledge in a quantitative manner and can be retrained as new
data become available.

\sec{ACKNOWLEDGEMENTS}

\x We are grateful to the Education and Research Committee of
Darwin College for fostering an environment in which
interdisciplinary learning flourishes.


\vfill\eject\sec{REFERENCES}

\parindent=0pt
\def\ref#1#2#3#4#5#6{\item{#1. } #2: {\it #3} {\bf #4} {(#5)} 
{#6}\vskip 0.5truemm} 

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\vfil\eject \nopagenumbers \vsize=26.5truecm

\sec{APPENDIX}

\x Table 3 contains the values for the weights obtained after
completing the training of the network.  These data can be used in
combination with Table 1 and equations 1--3 in order to use the
network to make predictions of weld metal toughness.


 \midinsert \thicksize=1pt \thinsize=0.8pt
\tablewidth=5.2truein \begintable
Description | Abbreviation | weight \cr
Bias to hidden unit 1| $\theta^{(1)}_1$ | -0.139995 \nr
Bias to hidden unit 2| $\theta^{(1)}_2$ | -0.248802 \nr
Bias to hidden unit 3| $\theta^{(1)}_3$ |  0.589258 \nr
Bias to hidden unit 4| $\theta^{(1)}_4$ | -0.080140 \nr
Bias to output unit   | $\theta^{(2)}   $ |  -1.04271 \cr

Process to hidden unit 1 | $w^{(1)}_{1,1}$|-0.953784 \nr
Process to hidden unit 2 | $w^{(1)}_{2,1}$| 0.455700 \nr
Process to hidden unit 3 | $w^{(1)}_{3,1}$| 0.010017 \nr
Process to hidden unit 4 | $w^{(1)}_{4,1}$| 0.355382 \cr

Yield strength to hidden unit 1 | $w^{(1)}_{1,2}$| 0.599265 \nr
Yield strength to hidden unit 2 | $w^{(1)}_{2,2}$| -0.396398 \nr
Yield strength to hidden unit 3 | $w^{(1)}_{3,2}$| -0.195261 \nr
Yield strength to hidden unit 4 | $w^{(1)}_{4,2}$| -0.051569 \cr

Carbon to hidden unit 1 | $w^{(1)}_{1,3}$| 0.003035\nr
Carbon to hidden unit 2 | $w^{(1)}_{2,3}$|  0.000652\nr
Carbon to hidden unit 3 | $w^{(1)}_{3,3}$| 0.002913 \nr
Carbon to hidden unit 4 | $w^{(1)}_{4,3}$|  0.004131\cr

Silicon to hidden unit 1 | $w^{(1)}_{1,4}$| -0.052339\nr
Silicon to hidden unit 2 | $w^{(1)}_{2,4}$|  0.069965\nr
Silicon to hidden unit 3 | $w^{(1)}_{3,4}$|  -0.088386\nr
Silicon to hidden unit 4 | $w^{(1)}_{4,4}$|  0.039075 \cr

Manganese to hidden unit 1 | $w^{(1)}_{1,5}$|  -0.009951\nr
Manganese to hidden unit 2 | $w^{(1)}_{2,5}$|  -0.006226\nr
Manganese to hidden unit 3 | $w^{(1)}_{3,5}$|  -0.010586\nr
Manganese to hidden unit 4 | $w^{(1)}_{4,5}$|  -0.008486\cr

Phosphorus to hidden unit 1 | $w^{(1)}_{1,6}$|  -0.108883\nr
Phosphorus to hidden unit 2 | $w^{(1)}_{2,6}$|   0.070553\nr
Phosphorus to hidden unit 3 | $w^{(1)}_{3,6}$|  0.029519\nr
Phosphorus to hidden unit 4 | $w^{(1)}_{4,6}$|  0.047509\cr

Sulphur to hidden unit 1 | $w^{(1)}_{1,7}$| -0.004894\nr
Sulphur to hidden unit 2 | $w^{(1)}_{2,7}$|  -0.008146\nr
Sulphur to hidden unit 3 | $w^{(1)}_{3,7}$|  -0.006305\nr
Sulphur to hidden unit 4 | $w^{(1)}_{4,7}$|  -0.009485
\endtable \tabtit {Table 3a: } {The weights describing the trained
network }\endinsert

 \midinsert \thicksize=1pt \thinsize=0.8pt
\tablewidth=5.2truein \begintable
Description | Abbreviation | weight \cr
Aluminium to hidden unit 1 | $w^{(1)}_{1,8}$| 0.003158\nr
Aluminium to hidden unit 2 | $w^{(1)}_{2,8}$|  0.004146\nr
Aluminium to hidden unit 3 | $w^{(1)}_{3,8}$|  0.001416\nr
Aluminium to hidden unit 4 | $w^{(1)}_{4,8}$|  0.005775\cr

Nitrogen to hidden unit 1 | $w^{(1)}_{1,9}$|  0.305801\nr
Nitrogen to hidden unit 2 | $w^{(1)}_{2,9}$|  0.123011\nr
Nitrogen to hidden unit 3 | $w^{(1)}_{3,9}$|  -0.533206\nr
Nitrogen to hidden unit 4 | $w^{(1)}_{4,9}$|  0.028576\cr

Oxygen to hidden unit 1 | $w^{(1)}_{1,10}$| -0.008650\nr
Oxygen to hidden unit 2 | $w^{(1)}_{2,10}$|  0.051781\nr
Oxygen to hidden unit 3 | $w^{(1)}_{3,10}$|  0.333115\nr
Oxygen to hidden unit 4 | $w^{(1)}_{4,10}$|  -0.204586\cr

Primary Mic. to hidden unit 1 | $w^{(1)}_{1,11}$|0.012150\nr 
Primary Mic. to hidden unit 2 |$w^{(1)}_{2,11}$| -0.020293\nr  
Primary Mic. to hidden unit 3 |$w^{(1)}_{3,11}$| -0.044580 \nr 
Primary Mic. to hidden unit 4 |$w^{(1)}_{4,11}$|  -0.008752\cr

Allotriomorphic to hidden unit 1 | $w^{(1)}_{1,12}$| 0.025770\nr
Allotriomorphic to hidden unit 2 | $w^{(1)}_{2,12}$|  -0.018466\nr
Allotriomorphic to hidden unit 3 | $w^{(1)}_{3,12}$|  0.016131\nr
Allotriomorphic to hidden unit 4 | $w^{(1)}_{4,12}$| 0.028425 \cr
Acicular to hidden unit 1 | $w^{(1)}_{1,13}$| -0.074019\nr
Acicular to hidden unit 2 | $w^{(1)}_{2,13}$|  0.623779\nr
Acicular to hidden unit 3 | $w^{(1)}_{3,13}$|  0.655171\nr
Acicular to hidden unit 4 | $w^{(1)}_{4,13}$|  -0.748358\cr
Temperature to hidden unit 1 | $w^{(1)}_{1,14}$| -0.077889\nr
Temperature to hidden unit 2 | $w^{(1)}_{2,14}$|  -0.077859\nr
Temperature to hidden unit 3 | $w^{(1)}_{3,14}$|  0.138927\nr
Temperature to hidden unit 4 | $w^{(1)}_{4,14}$|  0.084853\cr
Hidden unit 1 to output unit | $w^{(2)}_1$ | 7.474170 \nr
Hidden unit 2 to output unit | $w^{(2)}_2$ | 6.714920 \nr
Hidden unit 3 to output unit | $w^{(2)}_3$ | 7.727480 \nr
Hidden unit 4 to output unit | $w^{(2)}_4$ | 8.835680 
\endtable \tabtit {Table 3b: } {The weights describing the
trained network }\endinsert

\vfill\eject\bye

