vowel.train: Vowel Recognition (Deterding data)
In ElemStatLearn: Data Sets, Functions and Examples from the Book: "The Elements of Statistical Learning, Data Mining, Inference, and Prediction" by Trevor Hastie, Robert Tibshirani and Jerome Friedman
Description
Usage
Format
Details
Source
References
Examples
Description
Speaker independent recognition of the eleven steady state vowels
of British English using a specified training set of lpc derived log area
ratios.
Usage
1
Format
A data frame with 528 observations on the following 11 variables.
- y
a numeric vector
- x.1
a numeric vector
- x.2
a numeric vector
- x.3
a numeric vector
- x.4
a numeric vector
- x.5
a numeric vector
- x.6
a numeric vector
- x.7
a numeric vector
- x.8
a numeric vector
- x.9
a numeric vector
- x.10
a numeric vector
Details
For a more detailed explanation of the problem, see the excerpt from Tony
Robinson's Ph.D. thesis in the COMMENTS section. In Robinson's opinion,
connectionist problems fall into two classes, the possible and the
impossible. He is interested in the latter, by which he means problems
that have no exact solution. Thus the problem here is not to see how fast
a network can be trained (although this is important), but to maximise a
less than perfect performance.
METHODOLOGY:
Report the number of test vowels classified correctly, (i.e. the number of
occurences when distance of the correct output to the actual output was the
smallest of the set of distances from the actual output to all possible
target outputs).
Though this is not the focus of Robinson's study, it would also be useful
to report how long the training took (measured in pattern presentations or
with a rough count of floating-point operations required) and what level of
success was achieved on the training and testing data after various amounts
of training. Of course, the network topology and algorithm used should be
precisely described as well.
VARIATIONS:
This benchmark is proposed to encourage the exploration of different node
types. Please theorise/experiment/hack. The author (Robinson) will try to
correspond by email if requested. In particular there has been some
discussion recently on the use of a cross-entropy distance measure, and it
would be interesting to see results for that.
RESULTS:
Here is a summary of results obtained by Tony Robinson. A more complete
explanation of this data is given in the exceprt from his thesis in the
COMMENTS section below.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
+-------------------------+--------+---------+---------+
| | no. of | no. | percent |
| Classifier | hidden | correct | correct |
| | units | | |
+-------------------------+--------+---------+---------+
| Single-layer perceptron | - | 154 | 33 |
| Multi-layer perceptron | 88 | 234 | 51 |
| Multi-layer perceptron | 22 | 206 | 45 |
| Multi-layer perceptron | 11 | 203 | 44 |
| Modified Kanerva Model | 528 | 231 | 50 |
| Modified Kanerva Model | 88 | 197 | 43 |
| Radial Basis Function | 528 | 247 | 53 |
| Radial Basis Function | 88 | 220 | 48 |
| Gaussian node network | 528 | 252 | 55 |
| Gaussian node network | 88 | 247 | 53 |
| Gaussian node network | 22 | 250 | 54 |
| Gaussian node network | 11 | 211 | 47 |
| Square node network | 88 | 253 | 55 |
| Square node network | 22 | 236 | 51 |
| Square node network | 11 | 217 | 50 |
| Nearest neighbour | - | 260 | 56 |
+-------------------------+--------+---------+---------+
Notes:
1. Each of these numbers is based on a single trial with random starting
weights. More trials would of course be preferable, but the computational
facilities available to Robinson were limited.
2. Graphs are given in Robinson's thesis showing test-set performance vs.
epoch count for some of the training runs. In most cases, performance
peaks at around 250 correct, after which performance decays to different
degrees. The numbers given above are final performance figures after about
3000 trials, not the peak performance obtained during the run.
REFERENCES:
[Deterding89] D. H. Deterding, 1989, University of Cambridge, "Speaker
Normalisation for Automatic Speech Recognition", submitted for PhD.
[NiranjanFallside88] M. Niranjan and F. Fallside, 1988, Cambridge University
Engineering Department, "Neural Networks and Radial Basis Functions in
Classifying Static Speech Patterns", CUED/F-INFENG/TR.22.
[RenalsRohwer89-ijcnn] Steve Renals and Richard Rohwer, "Phoneme
Classification Experiments Using Radial Basis Functions", Submitted to
the International Joint Conference on Neural Networks, Washington,
1989.
[RabinerSchafer78] L. R. Rabiner and R. W. Schafer, Englewood Cliffs, New
Jersey, 1978, Prentice Hall, "Digital Processing of Speech Signals".
[PragerFallside88] R. W. Prager and F. Fallside, 1988, Cambridge University
Engineering Department, "The Modified Kanerva Model for Automatic
Speech Recognition", CUED/F-INFENG/TR.6.
[BroomheadLowe88] D. Broomhead and D. Lowe, 1988, Royal Signals and Radar
Establishment, Malvern, "Multi-variable Interpolation and Adaptive
Networks", RSRE memo, \#4148.
[RobinsonNiranjanFallside88-tr] A. J. Robinson and M. Niranjan and F.
Fallside, 1988, Cambridge University Engineering Department,
"Generalising the Nodes of the Error Propagation Network",
CUED/F-INFENG/TR.25.
[Robinson89] A. J. Robinson, 1989, Cambridge University Engineering
Department, "Dynamic Error Propagation Networks".
[McCullochAinsworth88] N. McCulloch and W. A. Ainsworth, Proceedings of
Speech'88, Edinburgh, 1988, "Speaker Independent Vowel Recognition
using a Multi-Layer Perceptron".
[RobinsonFallside88-neuro] A. J. Robinson and F. Fallside, 1988, Proceedings
of nEuro'88, Paris, June, "A Dynamic Connectionist Model for Phoneme
Recognition.
COMMENTS:
(By Tony Robinson)
The program supplied is slow. I ran it on several MicroVaxII's for many
nights. I suspect that if I had spent more time on it, it would have been
possible to get better results. Indeed, my later code has a slightly
better adaptive step size algotithm, but the old version is given here for
comatability with the stated performance values. It is interesting that,
for this problem, the nearest neighbour clasification outperforms any of
the connectionist models. This can be seen as a challange to improve the
connectionist performance.
The following problem description results and discussion is taken from my
PhD thesis. The aim was to demonstrate that many types of node can be
trained using gradient descent. The full thesis will be available from me
when it has been examined, say maybe July 1989.
Application to Vowel Recognition
——————————–
This chapter describes the application of a variety of feed-forward networks
to the task of recognition of vowel sounds from multiple speakers. Single
speaker vowel recognition studies by Renals and Rohwer [RenalsRohwer89-ijcnn]
show that feed-forward networks compare favourably with vector-quantised
hidden Markov models. The vowel data used in this chapter was collected by
Deterding [Deterding89], who recorded examples of the eleven steady state
vowels of English spoken by fifteen speakers for a speaker normalisation
study. A range of node types are used, as described in the previous section,
and some of the problems of the error propagation algorithm are discussed.
The Speech Data
(An ascii approximation to) the International Phonetic Association (I.P.A.)
symbol and the word in which the eleven vowel sounds were recorded is given in
table 4.1. The word was uttered once by each of the fifteen speakers. Four
male and four female speakers were used to train the networks, and the other
four male and three female speakers were used for testing the performance.
1
2
3
4
5
6
7
8
9
10
Table 4.1: Words used in Recording the Vowels
Front End Analysis
The speech signals were low pass filtered at 4.7kHz and then digitised to 12
bits with a 10kHz sampling rate. Twelfth order linear predictive analysis was
carried out on six 512 sample Hamming windowed segments from the steady part
of the vowel. The reflection coefficients were used to calculate 10 log area
parameters, giving a 10 dimensional input space. For a general introduction
to speech processing and an explanation of this technique see Rabiner and
Schafer [RabinerSchafer78].
Each speaker thus yielded six frames of speech from eleven vowels. This gave
528 frames from the eight speakers used to train the networks and 462 frames
from the seven speakers used to test the networks.
Details of the Models
All the models had common structure of one layer of hidden units and two
layers of weights. Some of the models used fixed weights in the first layer
to perform a dimensionality expansion [Robinson89:sect3.1], and the remainder
modified the first layer of weights using the error propagation algorithm for
general nodes described in [Robinson89:chap2]. In the second layer the hidden
units were mapped onto the outputs using the conventional weighted-sum type
nodes with a linear activation function. When Gaussian nodes were used the
range of influence of the nodes, w_{ij1}, was set to the standard deviation of
the training data for the appropriate input dimension. If the locations of
these nodes, w_{ij0}, are placed randomly, then the model behaves like a
continuous version of the modified Kanerva model [PragerFallside88]. If the
locations are placed at the points defined by the input examples then the
model implements a radial basis function [BroomheadLowe88]. The first layer
of weights remains constant in both of these models, but can be also trained
using the equations of [Robinson89:sect2.4]. Replacing the Gaussian nodes
with the conventional type gives a multilayer perceptron and replacing them
with conventional nodes with the activation function f(x) = x^2 gives a
network of square nodes. Finally, dispensing with the first layer altogether
yields a single layer perceptron.
The scaling factor between gradient of the energy and the change made to the
weights (the ‘learning rate’, ‘eta’) was dynamically varied during training,
as described in [Robinson89:sect2.5]. If the energy decreased this factor was
increased by 5%, if it increased the factor was halved. The networks changed
the weights in the direction of steepest descent which is susceptible to
finding a local minimum. A ‘momentum’ term [RumelhartHintonWilliams86] is
often used with error propagation networks to smooth the weight changes and
‘ride over’ small local minima. However, the optimal value of this term is
likely to be problem dependent, and in order to provide a uniform framework,
this additional term was not used.
Recognition Results
This experiment was originally carried out with only two frames of data from
each word [RobinsonNiranjanFallside88-tr]. In the earlier experiment some
problems were encountered with a phenomena termed ‘overtraining’ whereby the
recognition rate on the test data peaks part way through training then decays
significantly. The recognition rates for the six frames per word case are
given in table 4.2 and are generally higher and show less variability than the
previously presented results. However, the recognition rate on the test set
still displays large fluctuations during training, as shown by the plots in
[Robinson89:fig3.2] Some fluctuations will arise from the fact that the
minimum in weight space for the training set will not be coincident with the
minima for the test set. Thus, half the possible trajectories during learning
will approach the test set minimum and then move away from it again on the way
to the training set minima [Mark Plumbley, personal communication]. In
addition, continued training sharpens the class boundaries which makes the
energy insensitive to the class boundary position [Mahesan Niranjan, personal
communiation]. For example, there are a large number planes defined with
threshold units which will separate two points in the input space, but only
one least squares solution for the case of linear units.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
+-------------------------+--------+---------+---------+
| | no. of | no. | percent |
| Classifier | hidden | correct | correct |
| | units | | |
+-------------------------+--------+---------+---------+
| Single-layer perceptron | - | 154 | 33 |
| Multi-layer perceptron | 88 | 234 | 51 |
| Multi-layer perceptron | 22 | 206 | 45 |
| Multi-layer perceptron | 11 | 203 | 44 |
| Modified Kanerva Model | 528 | 231 | 50 |
| Modified Kanerva Model | 88 | 197 | 43 |
| Radial Basis Function | 528 | 247 | 53 |
| Radial Basis Function | 88 | 220 | 48 |
| Gaussian node network | 528 | 252 | 55 |
| Gaussian node network | 88 | 247 | 53 |
| Gaussian node network | 22 | 250 | 54 |
| Gaussian node network | 11 | 211 | 47 |
| Square node network | 88 | 253 | 55 |
| Square node network | 22 | 236 | 51 |
| Square node network | 11 | 217 | 50 |
| Nearest neighbour | - | 260 | 56 |
+-------------------------+--------+---------+---------+
Table 4.2: Vowel classification with different non-linear classifiers
Discussion
From these vowel classification results it can be seen that minimising the
least mean square error over a training set does not guarantee good
generalisation to the test set. The best results were achieved with nearest
neighbour analysis which classifies an item as the class of the closest
example in the training set measured using the Euclidean distance. It is
expected that the problem of overtraining could be overcome by using a larger
training set taking data from more speakers. The performance of the Gaussian
and square node network was generally better than that of the multilayer
perceptron. In other speech recognition problems which attempt to classify
single frames of speech, such as those described by McCulloch and Ainsworth
[McCullochAinsworth88] and that of [Robinson89:chap7 and
RobinsonFallside88-neuro], the nearest neighbour algorithm does not perform as
well as a multilayer perceptron. It would be interesting to investigate this
difference and apply a network of Gaussian or square nodes to these problems.
The initial weights to the hidden units in the Gaussian network can be given a
physical interpretation in terms of matching to a template for a set of
features. This gives an advantage both in shortening the training time and
also because the network starts at a point in weight space near a likely
solution, which avoids some possible local minima which represent poor
solutions.
The results of the experiments with Gaussian and square nodes are promising.
However, it has not been the aim of this chapter to show that a particular
type of node is necessarily ‘better’ for error propagation networks than the
weighted sum node, but that the error propagation algorithm can be applied
successfully to many different types of node.
Source
David Deterding (data and non-connectionist analysis)
Mahesan Niranjan (first connectionist analysis)
Tony Robinson (description, program, data, and results)
References
neural-bench@cs.cmu.edu
Examples
1
Example output
'data.frame': 528 obs. of 11 variables:
$ y : int 1 2 3 4 5 6 7 8 9 10 ...
$ x.1 : num -3.64 -3.33 -2.12 -2.29 -2.6 ...
$ x.2 : num 0.418 0.496 0.894 1.809 1.938 ...
$ x.3 : num -0.67 -0.694 -1.576 -1.498 -0.846 ...
$ x.4 : num 1.779 1.365 0.147 1.012 1.062 ...
$ x.5 : num -0.168 -0.265 -0.707 -1.053 -1.633 ...
$ x.6 : num 1.627 1.933 1.559 1.06 0.764 ...
$ x.7 : num -0.388 -0.363 -0.579 -0.567 0.394 0.217 0.322 -0.435 -0.512 -0.466 ...
$ x.8 : num 0.529 0.51 0.676 0.235 -0.15 -0.246 0.45 0.992 0.928 0.702 ...
$ x.9 : num -0.874 -0.621 -0.809 -0.091 0.277 0.238 0.377 0.575 -0.167 0.06 ...
$ x.10: num -0.814 -0.488 -0.049 -0.795 -0.396 -0.365 -0.366 -0.301 -0.434 -0.836 ...
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Description Usage Format Details Source References Examples
Description
Speaker independent recognition of the eleven steady state vowels of British English using a specified training set of lpc derived log area ratios.
Usage
1 |
Format
A data frame with 528 observations on the following 11 variables.
- y
a numeric vector
- x.1
a numeric vector
- x.2
a numeric vector
- x.3
a numeric vector
- x.4
a numeric vector
- x.5
a numeric vector
- x.6
a numeric vector
- x.7
a numeric vector
- x.8
a numeric vector
- x.9
a numeric vector
- x.10
a numeric vector
Details
For a more detailed explanation of the problem, see the excerpt from Tony Robinson's Ph.D. thesis in the COMMENTS section. In Robinson's opinion, connectionist problems fall into two classes, the possible and the impossible. He is interested in the latter, by which he means problems that have no exact solution. Thus the problem here is not to see how fast a network can be trained (although this is important), but to maximise a less than perfect performance.
METHODOLOGY:
Report the number of test vowels classified correctly, (i.e. the number of occurences when distance of the correct output to the actual output was the smallest of the set of distances from the actual output to all possible target outputs).
Though this is not the focus of Robinson's study, it would also be useful to report how long the training took (measured in pattern presentations or with a rough count of floating-point operations required) and what level of success was achieved on the training and testing data after various amounts of training. Of course, the network topology and algorithm used should be precisely described as well.
VARIATIONS:
This benchmark is proposed to encourage the exploration of different node types. Please theorise/experiment/hack. The author (Robinson) will try to correspond by email if requested. In particular there has been some discussion recently on the use of a cross-entropy distance measure, and it would be interesting to see results for that.
RESULTS:
Here is a summary of results obtained by Tony Robinson. A more complete explanation of this data is given in the exceprt from his thesis in the COMMENTS section below.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 | +-------------------------+--------+---------+---------+
| | no. of | no. | percent |
| Classifier | hidden | correct | correct |
| | units | | |
+-------------------------+--------+---------+---------+
| Single-layer perceptron | - | 154 | 33 |
| Multi-layer perceptron | 88 | 234 | 51 |
| Multi-layer perceptron | 22 | 206 | 45 |
| Multi-layer perceptron | 11 | 203 | 44 |
| Modified Kanerva Model | 528 | 231 | 50 |
| Modified Kanerva Model | 88 | 197 | 43 |
| Radial Basis Function | 528 | 247 | 53 |
| Radial Basis Function | 88 | 220 | 48 |
| Gaussian node network | 528 | 252 | 55 |
| Gaussian node network | 88 | 247 | 53 |
| Gaussian node network | 22 | 250 | 54 |
| Gaussian node network | 11 | 211 | 47 |
| Square node network | 88 | 253 | 55 |
| Square node network | 22 | 236 | 51 |
| Square node network | 11 | 217 | 50 |
| Nearest neighbour | - | 260 | 56 |
+-------------------------+--------+---------+---------+
|
Notes:
1. Each of these numbers is based on a single trial with random starting weights. More trials would of course be preferable, but the computational facilities available to Robinson were limited.
2. Graphs are given in Robinson's thesis showing test-set performance vs. epoch count for some of the training runs. In most cases, performance peaks at around 250 correct, after which performance decays to different degrees. The numbers given above are final performance figures after about 3000 trials, not the peak performance obtained during the run.
REFERENCES:
[Deterding89] D. H. Deterding, 1989, University of Cambridge, "Speaker Normalisation for Automatic Speech Recognition", submitted for PhD.
[NiranjanFallside88] M. Niranjan and F. Fallside, 1988, Cambridge University Engineering Department, "Neural Networks and Radial Basis Functions in Classifying Static Speech Patterns", CUED/F-INFENG/TR.22.
[RenalsRohwer89-ijcnn] Steve Renals and Richard Rohwer, "Phoneme Classification Experiments Using Radial Basis Functions", Submitted to the International Joint Conference on Neural Networks, Washington, 1989.
[RabinerSchafer78] L. R. Rabiner and R. W. Schafer, Englewood Cliffs, New Jersey, 1978, Prentice Hall, "Digital Processing of Speech Signals".
[PragerFallside88] R. W. Prager and F. Fallside, 1988, Cambridge University Engineering Department, "The Modified Kanerva Model for Automatic Speech Recognition", CUED/F-INFENG/TR.6.
[BroomheadLowe88] D. Broomhead and D. Lowe, 1988, Royal Signals and Radar Establishment, Malvern, "Multi-variable Interpolation and Adaptive Networks", RSRE memo, \#4148.
[RobinsonNiranjanFallside88-tr] A. J. Robinson and M. Niranjan and F. Fallside, 1988, Cambridge University Engineering Department, "Generalising the Nodes of the Error Propagation Network", CUED/F-INFENG/TR.25.
[Robinson89] A. J. Robinson, 1989, Cambridge University Engineering Department, "Dynamic Error Propagation Networks".
[McCullochAinsworth88] N. McCulloch and W. A. Ainsworth, Proceedings of Speech'88, Edinburgh, 1988, "Speaker Independent Vowel Recognition using a Multi-Layer Perceptron".
[RobinsonFallside88-neuro] A. J. Robinson and F. Fallside, 1988, Proceedings of nEuro'88, Paris, June, "A Dynamic Connectionist Model for Phoneme Recognition.
COMMENTS:
(By Tony Robinson)
The program supplied is slow. I ran it on several MicroVaxII's for many nights. I suspect that if I had spent more time on it, it would have been possible to get better results. Indeed, my later code has a slightly better adaptive step size algotithm, but the old version is given here for comatability with the stated performance values. It is interesting that, for this problem, the nearest neighbour clasification outperforms any of the connectionist models. This can be seen as a challange to improve the connectionist performance.
The following problem description results and discussion is taken from my PhD thesis. The aim was to demonstrate that many types of node can be trained using gradient descent. The full thesis will be available from me when it has been examined, say maybe July 1989.
Application to Vowel Recognition ——————————–
This chapter describes the application of a variety of feed-forward networks to the task of recognition of vowel sounds from multiple speakers. Single speaker vowel recognition studies by Renals and Rohwer [RenalsRohwer89-ijcnn] show that feed-forward networks compare favourably with vector-quantised hidden Markov models. The vowel data used in this chapter was collected by Deterding [Deterding89], who recorded examples of the eleven steady state vowels of English spoken by fifteen speakers for a speaker normalisation study. A range of node types are used, as described in the previous section, and some of the problems of the error propagation algorithm are discussed.
The Speech Data
(An ascii approximation to) the International Phonetic Association (I.P.A.) symbol and the word in which the eleven vowel sounds were recorded is given in table 4.1. The word was uttered once by each of the fifteen speakers. Four male and four female speakers were used to train the networks, and the other four male and three female speakers were used for testing the performance.
1 2 3 4 5 6 7 8 9 10 |
Table 4.1: Words used in Recording the Vowels
Front End Analysis
The speech signals were low pass filtered at 4.7kHz and then digitised to 12 bits with a 10kHz sampling rate. Twelfth order linear predictive analysis was carried out on six 512 sample Hamming windowed segments from the steady part of the vowel. The reflection coefficients were used to calculate 10 log area parameters, giving a 10 dimensional input space. For a general introduction to speech processing and an explanation of this technique see Rabiner and Schafer [RabinerSchafer78].
Each speaker thus yielded six frames of speech from eleven vowels. This gave 528 frames from the eight speakers used to train the networks and 462 frames from the seven speakers used to test the networks.
Details of the Models
All the models had common structure of one layer of hidden units and two layers of weights. Some of the models used fixed weights in the first layer to perform a dimensionality expansion [Robinson89:sect3.1], and the remainder modified the first layer of weights using the error propagation algorithm for general nodes described in [Robinson89:chap2]. In the second layer the hidden units were mapped onto the outputs using the conventional weighted-sum type nodes with a linear activation function. When Gaussian nodes were used the range of influence of the nodes, w_{ij1}, was set to the standard deviation of the training data for the appropriate input dimension. If the locations of these nodes, w_{ij0}, are placed randomly, then the model behaves like a continuous version of the modified Kanerva model [PragerFallside88]. If the locations are placed at the points defined by the input examples then the model implements a radial basis function [BroomheadLowe88]. The first layer of weights remains constant in both of these models, but can be also trained using the equations of [Robinson89:sect2.4]. Replacing the Gaussian nodes with the conventional type gives a multilayer perceptron and replacing them with conventional nodes with the activation function f(x) = x^2 gives a network of square nodes. Finally, dispensing with the first layer altogether yields a single layer perceptron.
The scaling factor between gradient of the energy and the change made to the weights (the ‘learning rate’, ‘eta’) was dynamically varied during training, as described in [Robinson89:sect2.5]. If the energy decreased this factor was increased by 5%, if it increased the factor was halved. The networks changed the weights in the direction of steepest descent which is susceptible to finding a local minimum. A ‘momentum’ term [RumelhartHintonWilliams86] is often used with error propagation networks to smooth the weight changes and ‘ride over’ small local minima. However, the optimal value of this term is likely to be problem dependent, and in order to provide a uniform framework, this additional term was not used.
Recognition Results
This experiment was originally carried out with only two frames of data from each word [RobinsonNiranjanFallside88-tr]. In the earlier experiment some problems were encountered with a phenomena termed ‘overtraining’ whereby the recognition rate on the test data peaks part way through training then decays significantly. The recognition rates for the six frames per word case are given in table 4.2 and are generally higher and show less variability than the previously presented results. However, the recognition rate on the test set still displays large fluctuations during training, as shown by the plots in [Robinson89:fig3.2] Some fluctuations will arise from the fact that the minimum in weight space for the training set will not be coincident with the minima for the test set. Thus, half the possible trajectories during learning will approach the test set minimum and then move away from it again on the way to the training set minima [Mark Plumbley, personal communication]. In addition, continued training sharpens the class boundaries which makes the energy insensitive to the class boundary position [Mahesan Niranjan, personal communiation]. For example, there are a large number planes defined with threshold units which will separate two points in the input space, but only one least squares solution for the case of linear units.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 | +-------------------------+--------+---------+---------+
| | no. of | no. | percent |
| Classifier | hidden | correct | correct |
| | units | | |
+-------------------------+--------+---------+---------+
| Single-layer perceptron | - | 154 | 33 |
| Multi-layer perceptron | 88 | 234 | 51 |
| Multi-layer perceptron | 22 | 206 | 45 |
| Multi-layer perceptron | 11 | 203 | 44 |
| Modified Kanerva Model | 528 | 231 | 50 |
| Modified Kanerva Model | 88 | 197 | 43 |
| Radial Basis Function | 528 | 247 | 53 |
| Radial Basis Function | 88 | 220 | 48 |
| Gaussian node network | 528 | 252 | 55 |
| Gaussian node network | 88 | 247 | 53 |
| Gaussian node network | 22 | 250 | 54 |
| Gaussian node network | 11 | 211 | 47 |
| Square node network | 88 | 253 | 55 |
| Square node network | 22 | 236 | 51 |
| Square node network | 11 | 217 | 50 |
| Nearest neighbour | - | 260 | 56 |
+-------------------------+--------+---------+---------+
|
Table 4.2: Vowel classification with different non-linear classifiers
Discussion
From these vowel classification results it can be seen that minimising the least mean square error over a training set does not guarantee good generalisation to the test set. The best results were achieved with nearest neighbour analysis which classifies an item as the class of the closest example in the training set measured using the Euclidean distance. It is expected that the problem of overtraining could be overcome by using a larger training set taking data from more speakers. The performance of the Gaussian and square node network was generally better than that of the multilayer perceptron. In other speech recognition problems which attempt to classify single frames of speech, such as those described by McCulloch and Ainsworth [McCullochAinsworth88] and that of [Robinson89:chap7 and RobinsonFallside88-neuro], the nearest neighbour algorithm does not perform as well as a multilayer perceptron. It would be interesting to investigate this difference and apply a network of Gaussian or square nodes to these problems.
The initial weights to the hidden units in the Gaussian network can be given a physical interpretation in terms of matching to a template for a set of features. This gives an advantage both in shortening the training time and also because the network starts at a point in weight space near a likely solution, which avoids some possible local minima which represent poor solutions.
The results of the experiments with Gaussian and square nodes are promising. However, it has not been the aim of this chapter to show that a particular type of node is necessarily ‘better’ for error propagation networks than the weighted sum node, but that the error propagation algorithm can be applied successfully to many different types of node.
Source
David Deterding (data and non-connectionist analysis) Mahesan Niranjan (first connectionist analysis) Tony Robinson (description, program, data, and results)
References
neural-bench@cs.cmu.edu
Examples
1 |
Example output
'data.frame': 528 obs. of 11 variables:
$ y : int 1 2 3 4 5 6 7 8 9 10 ...
$ x.1 : num -3.64 -3.33 -2.12 -2.29 -2.6 ...
$ x.2 : num 0.418 0.496 0.894 1.809 1.938 ...
$ x.3 : num -0.67 -0.694 -1.576 -1.498 -0.846 ...
$ x.4 : num 1.779 1.365 0.147 1.012 1.062 ...
$ x.5 : num -0.168 -0.265 -0.707 -1.053 -1.633 ...
$ x.6 : num 1.627 1.933 1.559 1.06 0.764 ...
$ x.7 : num -0.388 -0.363 -0.579 -0.567 0.394 0.217 0.322 -0.435 -0.512 -0.466 ...
$ x.8 : num 0.529 0.51 0.676 0.235 -0.15 -0.246 0.45 0.992 0.928 0.702 ...
$ x.9 : num -0.874 -0.621 -0.809 -0.091 0.277 0.238 0.377 0.575 -0.167 0.06 ...
$ x.10: num -0.814 -0.488 -0.049 -0.795 -0.396 -0.365 -0.366 -0.301 -0.434 -0.836 ...
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