CRANFIELD UNIVERSITY Eleni Anthippi Chatzimichali ...

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The first PC places all points the farthest apart from each other, extracting thus the highest variance. Similarly, a perpendicular to the first PC axis is drawn for the second PC, which accounts for the second highest variance. The process is repeated to get multiple orthogonal principal components. Each successive orthogonal axis displays a decreasing amount of the total variance. Figure 1-4 Extracting the Principal Components PCA projects the input data into a subspace of reasonable and meaningful dimension by setting new directions in the pattern space. Thus, the projected cloud of points is as dispersed as possible. The figure has been extracted from Kavraki (2007). In matrix notation, suppose that variables have been observed on instances. The generated multivariate dataset forms an data matrix with rows (observations) and columns (variables). Thus, a cloud of points is created in an -dimensional space, where a new axis is used per variable. The PCA algorithm reduces the size of possibly correlated variables into new uncorrelated variables (PCs), where . Each PC can be expressed mathematically as an orthogonal linear combination of the original variables . In PCA, the original matrix can be decomposed into the scores matrix , loadings matrix and a residuals matrix . Several algorithms can be used for data decomposition, the most widely applied of which are Singular Value Decomposition (SVD) and the Nonlinear Iterative Partial Least Squares (NIPALS) (Wold, 1975) algorithm. In general, the mathematical equation for PCA can be described by Equation 3 PCA scores and loadings 11
1.4.2 Cluster Analysis Cluster analysis consists of a set of unsupervised methods that are used in numerous data mining tasks. The clustering algorithms attempt to partition a dataset into several subsets – the clusters – so that data belonging to the same cluster are mutually similar, providing a sense of homogeneity. 1.4.2.1 Hierarchical Cluster Analysis (HCA) Hierarchical clustering (HCA) is based on calculating the distances between elements found in a given matrix of size . The distances represent the degree of similarity/dissimilarity between these objects. The shorter the distance, the more similar the objects are with each other. HCA is based on two important categories of algorithms – distance and linkage algorithms. Distance algorithms determine how the similarity or “distance measure” between two given objects is calculated. The most widely used distance algorithms include Euclidean and Mahalanobis distance, among others. For instance, for two objects and in , the Euclidean distance in -dimensional space satisfies the equation √∑ Equation 4 Euclidean distance algorithm Hierarchical clustering is graphically represented in tree structures, also known as dendrograms. Linkage algorithms determine how the clustering is performed. A bottom-up linkage algorithm includes the following steps: 1. Each object forms and belongs to its own cluster 2. The two closest clusters are linked together 3. The two linked clusters are aggregated into a single new cluster 4. The algorithm keeps iterating from Step 2 until the number of clusters is one 12
Page 1 and 2: CRANFIELD UNIVERSITY Eleni Anthippi
Page 3 and 4: ABSTRACT Muscle foods such as meat,
Page 5 and 6: TABLE OF CONTENTS ABSTRACT ........
Page 7 and 8: 6.2.4 The iWebPlots package .......
Page 9 and 10: Figure 3-12 Speedup produced by the
Page 11 and 12: Figure 6-9 Sweave example for the d
Page 13 and 14: TABLE OF EQUATIONS Equation 1 Mean-
Page 15 and 16: RMSE RMSECV SSE SVD SVMs SYMBIOSIS-
Page 17 and 18: Figure 1-1 Evolution from molecular
Page 19 and 20: 1.1.2.1 Fourier Transform Infrared
Page 21 and 22: 1.1.3 Microbial Spoilage in Meat Sy
Page 23 and 24: The multivariate techniques applied
Page 25: 1.4 Multivariate Analysis: Unsuperv
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
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precision of the classification acc
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Figure 3-3 The steps of the Nelder-
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3.3 Results and Discussion 3.3.1 Li
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3.3.2 Nonlinear Models In order to
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9) 10) 11) 12) 13) 14) 15) 16) Figu
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As an additional visual aid, these
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a) Number of iterations b) Number o
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Figure 3-10 Comparison of the execu
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Figure 3-12 Speedup produced by the
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4 Integration of Heterogeneous Data
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4.2.2 Procrustes Analysis Procruste
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symmetric method since the ordering
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Figure 4-3 Steps of CPCA for the da
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4.2.5 Data Integration and Analysis
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Figure 4-4 Data integration workflo
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a) GPA b) CPCA Figure 4-6 The conse
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accuracies as the linear models; po
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Figure 4-8 Classification Results f
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Similar to FTIR, the PLS-DA ensembl
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Figure 4-9 Class prediction rates o
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4.3.3 Permutation Tests Even though
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Even though the interpretation of t
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Figure 4-12 Distribution plots of t
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RBF SVMs Datasets Original %CC Mean
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Figure 4-14 Boxplots representing t
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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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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.
Page 235:
Xu, Y. and Goodacre, R. (2012), "Mu
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