In many data mining applications, it is a priori known
that the target function should satisfy certain constraints imposed
by, for example, economic theory or a human-decision maker. In this
paper we consider partially monotone prediction problems, where the
target variable depends monotonically on some of the input variables
but not on all. We propose a novel method to construct prediction
models, where monotone dependences with respect to some of
the input variables are preserved by virtue of construction. Our
method belongs to the class of mixture models. The basic idea is to
convolute monotone neural networks with weight (kernel) functions
to make predictions. By using simulation and real case studies,
we demonstrate the application of our method. To obtain sound
assessment for the performance of our approach, we use standard
neural networks with weight decay and partially monotone linear
models as benchmark methods for comparison. The results show that
our approach outperforms partially monotone linear models in terms
of accuracy. Furthermore, the incorporation of partial monotonicity
constraints not only leads to models that are in accordance with the
decision maker's expertise, but also reduces considerably the model
variance in comparison to standard neural networks with weight
decay.
[1] O. Harrison and D. Rubinfeld. Hedonic prices and the demand for clean
air. Journal of Environmental Economics and Management, 53:81–102,
1978.
[2] H. Mukarjee and S. Stern. Feasible nonparametric estimation of
multiargument monotone functions. Journal of the American Statistical
Association, 89(425):77–80, 1994.
[3] H.A.M. Daniels and B. Kamp. Application of MLP networks to bond
rating and house pricing. Neural Computing & Applications, 8(3):226–
234, 1999.
[4] H. Kay and L. H. Ungar. Estimating monotonic functions and their
bounds. American Institute of Chemical Engineers (AIChE) Journal,
46(12):2426–2434, 2000.
[5] J. Sill. Monotonic networks. Advances in Neural Information Processing
Systems, 10:661–667, 1998.
[6] S. Wang. A neural network method of density estimation for univariate
unimodal data. Neural Computing & Applications, 2(3):160–167, 1994.
[7] M. Sarfraz, M. Al-Mulhem, and F. Ashraf. Preserving monotonic shape
of the data by using piecewise rational cubic functions. Computers and
Graphics, 21:5–14, 1997.
[8] M. Velikova and H.A.M. Daniels. Decision trees for monotone price
models. Computational Management Science, 1(3–4):231–244, 2004.
[9] A. Ben-David. Monotonicity maintenance in information-theoretic
machine learning algorithms. Machine Learning, 19(1):29–43, 1995.
[10] R. Potharst and A. Feelders. Classication trees for problems with
monotonicity constraints. SIGKDD Explorations Newsletter, 4(1):1–10,
2002.
[11] P. J. Rousseeuw. Silhouettes: a graphical aid to the interpretation and
validation of cluster analysis. Journal of Computational and Applied
Mathematics, 20:53–65, 1987.
[12] E. A. Nadaraya. On estimating regression. Theory of Probability & Its
Applications, 9(1):141–142, 1964.
[13] G. S. Watson. Smooth regression analysis. Sankhya: The Indian Journal
of Statistics, Series A, 26(4):359–372, 1964.
[14] S. Geman, E. Bienenstock, and R. Doursat. Neural networks and the
bias/variance dilemma. Neural Computation, 4(1):1–58, 1992.
[15] C. F. J. Wu and M. Hamada. Experiments: Planning, Analysis, and
Parameter Design Optimization. John Wiley & Sons, New York, wiley
series in probability and statistics edition, 2000.
[16] M. Velikova, H.A.M. Daniels, and A. Feelders. Solving partially
monotone problems with neural networks. In Proceedings of the twelfth
International Conference on Computer Science, Vienna, Austria, pages
82–87. World Enformatika Society, Turkey, 2006.
[17] D. J. Newman, S. Hettich, C. L. Blake, and C. J. Merz. UCI Repository
of Machine Learning Databases, 1998. http://www.ics.uci.
edu/┬ÿmlearn/MLRepository.html.
[18] S. Waugh. Extending and benchmarking cascade-correlation. Phd
dissertation, University of Tasmania, Tasmania, Australia, 1995.
[19] D. Clark, Z. Schreter, and A. Adams. A quantitative comparison of
Dystal and backpropagation. In Proceedings of the seventh Australian
Conference Neural Networks, Canberra, Australia, pages 132–137.
1996.
[20] A. M. Abdelbar. Achieving superior generalisation with a high order
neural network. Neural Computing & Applications, 7(2):141–146, 1998.
[21] W. J. Nash, T. L. Sellers, S. R. Talbot, and W. B. Cawthorn, A.
J. anf Ford. The population biology of abalone (halotis species) in
tasmania. i blacklip abalone (h. rubra) from the north coast and islands of
bass strait. Technical Report 48, Sea Fisheries Division, Marine Research
Laboratories-Taroona, Department of Primary Industry and Fisheries,
Tasmania, Australia, 1994.
[1] O. Harrison and D. Rubinfeld. Hedonic prices and the demand for clean
air. Journal of Environmental Economics and Management, 53:81–102,
1978.
[2] H. Mukarjee and S. Stern. Feasible nonparametric estimation of
multiargument monotone functions. Journal of the American Statistical
Association, 89(425):77–80, 1994.
[3] H.A.M. Daniels and B. Kamp. Application of MLP networks to bond
rating and house pricing. Neural Computing & Applications, 8(3):226–
234, 1999.
[4] H. Kay and L. H. Ungar. Estimating monotonic functions and their
bounds. American Institute of Chemical Engineers (AIChE) Journal,
46(12):2426–2434, 2000.
[5] J. Sill. Monotonic networks. Advances in Neural Information Processing
Systems, 10:661–667, 1998.
[6] S. Wang. A neural network method of density estimation for univariate
unimodal data. Neural Computing & Applications, 2(3):160–167, 1994.
[7] M. Sarfraz, M. Al-Mulhem, and F. Ashraf. Preserving monotonic shape
of the data by using piecewise rational cubic functions. Computers and
Graphics, 21:5–14, 1997.
[8] M. Velikova and H.A.M. Daniels. Decision trees for monotone price
models. Computational Management Science, 1(3–4):231–244, 2004.
[9] A. Ben-David. Monotonicity maintenance in information-theoretic
machine learning algorithms. Machine Learning, 19(1):29–43, 1995.
[10] R. Potharst and A. Feelders. Classication trees for problems with
monotonicity constraints. SIGKDD Explorations Newsletter, 4(1):1–10,
2002.
[11] P. J. Rousseeuw. Silhouettes: a graphical aid to the interpretation and
validation of cluster analysis. Journal of Computational and Applied
Mathematics, 20:53–65, 1987.
[12] E. A. Nadaraya. On estimating regression. Theory of Probability & Its
Applications, 9(1):141–142, 1964.
[13] G. S. Watson. Smooth regression analysis. Sankhya: The Indian Journal
of Statistics, Series A, 26(4):359–372, 1964.
[14] S. Geman, E. Bienenstock, and R. Doursat. Neural networks and the
bias/variance dilemma. Neural Computation, 4(1):1–58, 1992.
[15] C. F. J. Wu and M. Hamada. Experiments: Planning, Analysis, and
Parameter Design Optimization. John Wiley & Sons, New York, wiley
series in probability and statistics edition, 2000.
[16] M. Velikova, H.A.M. Daniels, and A. Feelders. Solving partially
monotone problems with neural networks. In Proceedings of the twelfth
International Conference on Computer Science, Vienna, Austria, pages
82–87. World Enformatika Society, Turkey, 2006.
[17] D. J. Newman, S. Hettich, C. L. Blake, and C. J. Merz. UCI Repository
of Machine Learning Databases, 1998. http://www.ics.uci.
edu/┬ÿmlearn/MLRepository.html.
[18] S. Waugh. Extending and benchmarking cascade-correlation. Phd
dissertation, University of Tasmania, Tasmania, Australia, 1995.
[19] D. Clark, Z. Schreter, and A. Adams. A quantitative comparison of
Dystal and backpropagation. In Proceedings of the seventh Australian
Conference Neural Networks, Canberra, Australia, pages 132–137.
1996.
[20] A. M. Abdelbar. Achieving superior generalisation with a high order
neural network. Neural Computing & Applications, 7(2):141–146, 1998.
[21] W. J. Nash, T. L. Sellers, S. R. Talbot, and W. B. Cawthorn, A.
J. anf Ford. The population biology of abalone (halotis species) in
tasmania. i blacklip abalone (h. rubra) from the north coast and islands of
bass strait. Technical Report 48, Sea Fisheries Division, Marine Research
Laboratories-Taroona, Department of Primary Industry and Fisheries,
Tasmania, Australia, 1994.
@article{"International Journal of Information, Control and Computer Sciences:51379", author = "Marina Velikova and Hennie Daniels and Ad Feelders", title = "Mixtures of Monotone Networks for Prediction", abstract = "In many data mining applications, it is a priori known
that the target function should satisfy certain constraints imposed
by, for example, economic theory or a human-decision maker. In this
paper we consider partially monotone prediction problems, where the
target variable depends monotonically on some of the input variables
but not on all. We propose a novel method to construct prediction
models, where monotone dependences with respect to some of
the input variables are preserved by virtue of construction. Our
method belongs to the class of mixture models. The basic idea is to
convolute monotone neural networks with weight (kernel) functions
to make predictions. By using simulation and real case studies,
we demonstrate the application of our method. To obtain sound
assessment for the performance of our approach, we use standard
neural networks with weight decay and partially monotone linear
models as benchmark methods for comparison. The results show that
our approach outperforms partially monotone linear models in terms
of accuracy. Furthermore, the incorporation of partial monotonicity
constraints not only leads to models that are in accordance with the
decision maker's expertise, but also reduces considerably the model
variance in comparison to standard neural networks with weight
decay.", keywords = "mixture models, monotone neural networks, partially monotone models, partially monotone problems.", volume = "1", number = "10", pages = "2994-11", }