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Ka Lok Chu & George P. H. Styan
Some illustrated comments on Anderson graphs and Greek mythology tables for classic magic squares

We present several new classifications for the 880 classic order-4 magic squares first enumerated by the French mathematician Bernard Fr\'enicle de Bessy (c.~1605--1675). We identify 382 distinct Anderson graphs (Bragdon's ``magic lines'' diagrams, Moran's ``sequence designs'') and study their symmetry and path lengths, building on results by Brigadier-General Sir Francis James Anderson (1860-1920), architect Claude Fayette Bragdon (1866--1946), mathematical puzzle aficionado Henry Ernest Dudeney (1857--1930), and publicist James Sterling ``Jim'' Moran (1908--1999), who in his 1980 book {\it The Wonders of Magic Squares,} identified a Greek deity for each Dudeney Type. Our ``Anderson graph'' is the graph produced by the lines joining the consecutive numbers in sequence. We also consider matrix factorizations due to the mathematicians Leonhard Euler (1707--1783) and Friedrich Fitting (1862--1945). We illustrate our findings as much as possible and whenever feasible with images of postage stamps or other philatelic items. This is joint research wth Reijo Sund (Helsinki) and Walter Trump (N\"urnberg). With this talk we are pleased to celebrate the International Year of Statistics 2013 and the special year for Mathematics of Planet Earth 2013.
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Amali Dassanayake
Local Orthogonal Polynomial Expansions For Density Estimations.

We propose a new method to estimate the density function of a univariate continuous random variable. This new method, local orthogonal polynomial expansions (LOrPE), draws similarities with kernel density estimation (KDE), orthogonal series density estimation (OSDE) and local likelihood density estimation (LLDE). It is most similar to LLDE in that it is a local method where the approximation is obtained at each point of the support. It is connected to the OSDE in that it is constructed by using an orthogonal polynomial series expansion at each point of the support. The order of the series (M) used is one of the method’s tuning parameters, a localized version of OSDE. Finally, LOrPE utilizes a bandwidth (h), the second tuning parameter, in order to construct the orthogonal polynomials over a localized window, and in this respect it is similar to KDE. Also, we show that under certain conditions, LOrPE is equivalent to KDE with a high order kernel. Comparisons of LOrPE with KDE are performed under a variety of conditions. We find that in terms of MISE, LOrPEperforms better than KDE when estimating densities with sharp boundaries and both LOrPE and KDE results remain same when estimating densities which decay slowly to zero at infinity.
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Sami Helle, Tampere University of Technology and University of Tampere
Error Bound for Metamodel Extreme Value Approximation of Black-Box Computer Experiments

One important field of statistical design of experiments is design of computationally expensive computer experiments (see, for example Chen et al. 2003, Fang et al. 1994, 2003, 2006, Cioppa and Lucas 2007, Koehler and Owen 1989, Lin 2003, Niederreiter 1992, Sacks et al. 1989). Theoretically computer models are deterministic and model is completely known. With stochastic simulations, like reliability or risk simulations, this is true in some sense but the simulation does include quasi-random components and it's difficult to treat it as deterministic model. Also computer model could be completely unknown "black-box" or it is too complicated. In many cases computationally inexpensive metamodel is fitted to simulation output data for approximating unknown model and metamodel is used for analyzing properties of a computer simulation (see, for example Chen et al. 2003, Fang et al. 2000, 2002, 2003, 2006, Lin 2003, Muller et al. 2010, 2012, 2013).
In this paper we analyze error bound of metamodel extreme value approximation of black-box computer experiments in deterministic and stochastic event simulation cases. Results show that error bound will be minimized by minimizing the uniformity measure of the point set if the model is unknown in deterministic case. In the stochastic event simulation case error is unbounded if random variables are unbounded. However, functions of simulation output parameter estimates could be used for estimating extreme values. The result will show that the expected value of the error bound of function of simulation output parameters estimates will be minimized by minimizing the uniformity measure of the point set in stochastic event simulation case under some assumptions.
References:
-Chen, V. C. P., Tsui, K.-L., Barton, R. R. and Allen, J. K. (2003), A Review of Design and Modeling in Computer Experiments, Handbook of Statistics Vol 22, Ed. Khattree, R. and Rao, C. R., Elsevier, Amsterdam, 231-262.
-Cioppa, T. M. and Lucas, T. W. (2007), Efficient Nearly Orthogonal and Space-Filling Latin Hypercubes, Technometrics, 49 (1), 45-55.
-Fang, K.-T. and Wang Y. (1994), Number-theoretic Methods in Statistics, Chapman & Hall, London.
-Fang, K.-T. Lin, D.K.J., Winker, P. and Zhang, Y. (2000), Uniform Design: Theory and Applications, Technometrics, 42, 237-248.
-Fang, K.-T. (2002), Theory, Method and Applications of the Uniform Design, International Journal of Reliability, Quality and Safety Engineering, 9 (4), 305-315.
-Fang K.-T. and Lin, D. K. J. (2003), Uniform Experimental Design and Their Applications in Industry, Handbook of Statistics Vol 22, Ed. Khattree, R. and Rao, C. R., Elsevier, Amsterdam, 131-170.
-Fang, K.-T. , Li, R. and Sudjianto, A. (2006), Design and Modeling for Computer Experiments, Chapman & Hall, New York.
-Koehler, J. R. and Owen, A. B., (1996), Computer Experiments, Handbook of Statistics Vol 13, Ed. -Ghosh, S. and Rao, C. R., Elsevier, Amsterdam, 261-308.
-Lin, D. K. J. (2003), Industrial Experimentation for Screening, Handbook of Statistics Vol 22, Ed. -Khattree, R. and Rao, C. R., Elsevier, Amsterdam, 33-74.
- Muller, J. and Piche, R. (2011), Mixture surrogate models based on Dempster-Shafer theory for global optimization problems, Journal of Global Optimization, 51, 79–104.
-Muller, J. (2012), Surrogate Model Algorithms for Computationally Expensive Black-Box Optimization Problems, Tampere University of Technology publication 1092, Juvenes Print TTY, Tampere.
-Muller, J, Shoemaker, C. A. and Piche, R. (2013), SO-MI: A surrogate model algorithm for computationally expensive nonlinear mixed-integer black-box global optimization problems, Computers & Operations Research, 40(5), 1383–1400.
- Niederreiter, H., (1992), Random Number Generation and Quasi-Monte Carlo Methods, Society for Industrial and Applied Mathematics (SIAM), Philadelphia.
- Sacks J., Welch, W. J. , Mitchell, T. J. and Wynn, H. P. (1989), Design and Analysis of Computer Experiments, Statistical Science, 4, 409-423.

Joonas Kauppinen, University of Tampere
Decomposing self-similarity matrices to infer structure in music
Coauthors: Anssi Klapuri and Tuomas Virtanen (Department of Signal Processing, Tampere University of Technology)

In the emerging field of music information retrieval, the goal of structure analysis is to discover the sectional form of musical works. This corresponds to locating segment boundaries and clustering the obtained segments to consistent parts, such as verse and chorus in popular music. The information on sectional form is useful for several applications, including cover song identification, music summarization, and music playback interfaces that allow jumping between the sections. An efficient approach to visualizing and analyzing the sectional form is to construct a symmetric self-similarity matrix (SSM) from a time series of acoustic feature vectors representing a song. Provided that we have used appropriate features, we can often discern two prominent structures in SSMs: rectangular blocks of high similarity and diagonal stripes off and parallel to the main diagonal. We explore some classic methods for inferring musical structure from SSMs. Furthermore, we illustrate novel methods to model the block and stripe-like structures in SSMs simultaneously. The common theme to the methods is that they are all based on the use of matrix decompositions, namely the singular value decomposition or nonnegative matrix factorization.

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Danielle Richer, McMaster University
Learning from Ecology: The Presence-Absence Matrix applied to Network Meta-Analysis
Coauthors: Joseph Beyene

Meta-analysis is a statistical practice that has been well-developed to combine results from trials comparing the same interventions. In an effort to address broader questions of comparative effectiveness, concepts used in traditional meta-analysis have been extended to create the field of network meta-analysis. A connected network of studies between multiple treatments can be represented geometrically with treatments as nodes and edges as studies. Several methods, both Bayesian and frequentist, have been proposed for network meta-analysis. Researchers are beginning to investigate the role that a network’s geometry might play in statistical inference. Recent simulations have determined that the shape of a network and number of studies per edge may affect the appropriateness of different models. Salanti (2008) proposes that the networks of treatments can be described using terms from ecology – diversity and co-occurrence. In particular, the presence-absence matrix used to measure co-occurrence of species in ecology can be used to identify possible biases in treatment networks. This poster highlights the concept of co-occurrence in ecology, the related c-score measurement with which it is quantified, its application to network meta-analysis, and its limitations.
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Suresh Kumar Sharma, Panjab University
Efficiency of Various Smoothers in Generalized Additive Models
Department of Statistics, Panjab University, Chandigarh, India
Coauthors: Kanchan Jain and Rashmi Aggarwal

A trend in the past few years has been to move away from linear functions and model the dependence of Y on X1,…,Xk in a nonparametric fashion. Generalized additive models try to expose the functional dependence without imposing rigid parametric assumptions about that dependence. Here, the data shows us the appropriate functional form and that is idea behind a scatter plot smoother. In Generalized additive models, smoother is a tool for summarizing the trend of a response measurement Y as a function of one or more predictor measurements X1,…,Xk. In literature, various smoothers, viz running mean, running line, bin, Kernel etc. are available. In this paper, our basic aim is to compare the efficiency of various smoothers in terms of Average Mean Squared Error (AMSE) for distributions including Binomial, Poisson, Normal and Exponential. For theoretical considerations, Bias-variance trades off for scatter plot smoothers and moment generating functions are also discussed.
Key Words: running mean, running lines, span, average mean squared error, bias-variance trade-off
References:
[1] Friedman, J.H. and Stuetzle, W.(1981): “Projection Pursuit Regression,” Journal of the American Statistical Association, 76, 817–823.
[2] Hastie, T. J. and Tibshirani, R. J. (1990): Generalized Additive Models, Chapman & Hall/CRC