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3.2.3 Nelder-Mead Simplex Algorithm The Nelder-Mead algorithm (Nelder and Mead, 1965) is the most widely employed direct search method for unconstrained nonlinear optimisation (Lagarias et al., 1998; Kolda et al., 2003). The algorithm is commonly referred to as the Nelder-Mead simplex; however, it greatly varies from the popular simplex algorithm by Dantzig (Dantzig, 1987), which is solely used for linear programming. The Nelder-Mead method constitutes an extension of the simplex-based algorithm initially proposed by Spendley, Hext and Himsworth (Spendley et al., 1962). The Nelder-Mead algorithm approximates the optimal point by assessing the values of a given nonlinear cost function , without using any implicit or explicit derivative information (Kolda et al., 2003). In comparison to the fixed shape simplex of Spendley et al. (1962), the Nelder-Mead algorithm is based on a variable shape simplex, which allows adaptions both in size and shape. “A simplex in is a geometric figure (polytope) in dimensions of nonzero volume that is the convex hull of vertices” where is the total number of variables (Lagarias et al., 1998); for example, a simplex in has the form of a triangle (Nelder and Singer, 2009). In order to identify the optimal point (the optimal solution), the Nelder-Mead algorithm constructs a series of new simplices in an iterative manner. In each iteration, the vertices of the simplex are sorted according to their cost function values. The algorithm attempts to replace the worst vertex with a new point, which depends on the worst point and the centre of the best vertices using four possible steps: reflection, expansion, (outside and inside) contraction and shrink (Lagarias et al., 1998) as illustrated in Figure 3-3. A thorough review and step-by-step explanation of the Nelder-Mead simplex methodology can be found in Lagarias et al. (1998) and Nelder et al. (2009). The greatest appeal of the Nelder-Mead direct search algorithm according to Nelder and Singer (2009) is that it easily “adapts itself to the local landscape” such as the three-dimensional surface plots of Figure 2-9; the simplex is elongating itself down long slopes, it alters direction when encountering a valley at an angle, and it contacts as it approximates the minimum (Nelder and Singer, 2009). 63
Figure 3-3 The steps of the Nelder-Mead algorithm The Nelder-Mead steps include: reflection, expansion, (inside and outside) contraction and shrink. The figure was extracted from http://www.math.uiuc.edu/documenta/vol-ismp/42_wright-margaret.pdf 3.2.4 Box Constrained Simplex Algorithm The Box complex algorithm (Box, 1965) is a derivative-free method used for nonlinear constrained optimisation. Unlike the other direct search methods of Section 3.2.3, which are solely applied to unconstrained optimisation problems, Box extended the functionality of the algorithm by Spendley et al. (1962) by explicitly incorporating bounds and/or nonlinear inequality constraints into the search space via a constrained simplex, namely the “complex”. Similar to the Nelder-Mead simplex, a complex in is a flexible mathematical figure in dimensions made up of at least vertices, where is the total number of variables; the use of vertices is recommended by Box. Each point’s coordinates in the complex correspond to individual variables of the objective function. The complex moves around the solution space by expanding or contracting in any direction as long as it is feasible. The Box complex algorithm solves the following constrained minimisation problem Equation 19 Box complex algorithm 64
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CRANFIELD UNIVERSITY Eleni Anthippi
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ABSTRACT Muscle foods such as meat,
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TABLE OF CONTENTS ABSTRACT ........
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6.2.4 The iWebPlots package .......
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Figure 3-12 Speedup produced by the
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Figure 6-9 Sweave example for the d
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TABLE OF EQUATIONS Equation 1 Mean-
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RMSE RMSECV SSE SVD SVMs SYMBIOSIS-
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Figure 1-1 Evolution from molecular
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1.1.2.1 Fourier Transform Infrared
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1.1.3 Microbial Spoilage in Meat Sy
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The multivariate techniques applied
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1.4 Multivariate Analysis: Unsuperv
Page 27 and 28: 1.4.2 Cluster Analysis Cluster anal
Page 29 and 30: 1.5 Multivariate Analysis: Supervis
Page 31 and 32: In any linearly separable binary da
Page 33 and 34: Where ( ) are the Lagrange multipli
Page 35 and 36: Every kernel is characterised by a
Page 37 and 38: The “one-against-all” approach
Page 39 and 40: Furthermore, metrics such as the bi
Page 41 and 42: 1.6.3 Leave-One-Out Cross-Validatio
Page 43 and 44: In such cases, the model becomes qu
Page 45 and 46: 1.8 Aims and objectives The overall
Page 47 and 48: 2 Development of the multivariate a
Page 49 and 50: Sensory panel scores were of the hi
Page 51 and 52: DF (Hz) 2.2.1.5 Electronic Nose (e-
Page 53 and 54: Figure 2-3 Data intersection The fi
Page 55 and 56: 3. Validation Techniques In the cas
Page 57 and 58: Figure 2-4 The process of construct
Page 59 and 60: 2.3 Results and Discussion 2.3.1 Pr
Page 61 and 62: (b) HPLC data (c) e-nose data Figur
Page 63 and 64: In the case of FTIR, linear classif
Page 65 and 66: 2.3.2.2 Class Prediction Accuracies
Page 67 and 68: Figure 2-7 Class prediction rates o
Page 69 and 70: The topology of the three-dimension
Page 71 and 72: Figure 2-9 Three-dimensional error
Page 73 and 74: Furthermore, a thorough comparison
Page 75 and 76: In the master/slave architecture, a
Page 77: precision of the classification acc
Page 81 and 82: 3.3 Results and Discussion 3.3.1 Li
Page 83 and 84: 3.3.2 Nonlinear Models In order to
Page 85 and 86: 9) 10) 11) 12) 13) 14) 15) 16) Figu
Page 87 and 88: As an additional visual aid, these
Page 89 and 90: a) Number of iterations b) Number o
Page 91 and 92: Figure 3-10 Comparison of the execu
Page 93 and 94: Figure 3-12 Speedup produced by the
Page 95 and 96: 4 Integration of Heterogeneous Data
Page 97 and 98: 4.2.2 Procrustes Analysis Procruste
Page 99 and 100: symmetric method since the ordering
Page 101 and 102: Figure 4-3 Steps of CPCA for the da
Page 103 and 104: 4.2.5 Data Integration and Analysis
Page 105 and 106: Figure 4-4 Data integration workflo
Page 107 and 108: a) GPA b) CPCA Figure 4-6 The conse
Page 109 and 110: accuracies as the linear models; po
Page 111 and 112: Figure 4-8 Classification Results f
Page 113 and 114: Similar to FTIR, the PLS-DA ensembl
Page 115 and 116: Figure 4-9 Class prediction rates o
Page 117 and 118: 4.3.3 Permutation Tests Even though
Page 119 and 120: Even though the interpretation of t
Page 121 and 122: Figure 4-12 Distribution plots of t
Page 123 and 124: RBF SVMs Datasets Original %CC Mean
Page 125 and 126: Figure 4-14 Boxplots representing t
Page 127 and 128: 5 Application of the multivariate a
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Figure 5-1 Mean FTIR spectra for ca
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5.2.2.2 Fourier Transform Infrared
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5.2.2.4 Data Overview For each expe
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5.2.3.3 High Throughput Liquid Chro
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Figure 5-5 displays the PCA scores
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(a) GPA (b) CPCA Figure 5-6 The con
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observation confirms that indeed CP
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Figure 5-8 Class prediction rates o
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(a) RBF SVMs (b) PLS-DA Figure 5-9
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RBF SVMs Datasets Original %CC Mean
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Figure 5-12 Execution times of the
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(a) FTIR data (b) Raman data Figure
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The classification results for the
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Figure 5-16 Class prediction rates
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(a) RBF SVMs (b) PLS-DA Figure 5-17
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Figure 5-20 Execution times of the
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(a) FTIR data (b) HPLC data 148
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(a) GPA (b) CPCA Figure 5-22 The co
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The classification results of the f
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Figure 5-24 Classification Results
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Figure 5-25 Class prediction rates
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The permutation results for the dat
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Figure 5-28 Distribution plots of t
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Figure 5-30 Boxplots representing t
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5.4 Comparison of the individual ca
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Figure 5-32 Investigating the commo
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6 Development of improved visualisa
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6.2.2 Generating static graphs As d
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Figure 6-2 Construction process of
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6.2.2.2 Generating dynamic reports
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activity, while simultaneously the
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or disappeared (Wang et al., 2008).
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ectangles, circles and/or irregular
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In addition to the static figures g
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c) graphics package d) ggplot2 pack
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In addition to aesthetics, faceting
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Datasets FTIR HPLC e-nose PLS-DA (L
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Figure 6-10 Interactive dendrogram
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6.4 Conclusion The work reported in
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large number of individual models i
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The generated suite of tools, as pr
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In addition, the difficulty to corr
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REFERENCES Almeida, J. A. S., Barbo
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Processing., Proceedings of the 12t
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Clarke, B., Fokoué, E. and Zhang,
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spectroscopy and machine learning",
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Gower, J. C. (1975), "Generalized p
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Jardon, M. (2006), Systems Biology:
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Nicolaou, N., Xu, Y. and Goodacre,
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Shah, A. A., Barthel, D., Lukasiak,
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Vera, G., Jansen, R. and Suppi, R.
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Xu, Y. and Goodacre, R. (2012), "Mu
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