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Big Data Science in Finance
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Big Data Science in Finance
by
Big Data Science in Finance
Cover
Title Page
Copyright
Preface
REFERENCE
Chapter 1: Why Big Data?
Introduction
Appendix 1.A Coding Big Data in Python
Reference
Notes
Chapter 2: Neural Networks in Finance
Introduction
Neural Network Construction Methodology
The Architecture of Neural Networks
Choosing the Activation Function
Construction and Training of Neural Networks
Model Selection via Dropout
Overfitting
Adding Complexity
Big Data in Machine Learning
Coding a Simple Neural Network for One Instrument from Daily Data
Defining Target Outputs
Testing Performance
Adding Activation Levels
Convergence
Choosing Input Variables
Conclusion
Appendix 2.A Building a Neural Network in Python
References
Chapter 3: Supervised Learning
Introduction
Supervised Learning
Conclusion
Appendix 3.A Python for Supervised Models
References
Chapter 4: Modeling Human Behavior with Semi-Supervised Learning
Introduction
Performance Evaluation via Cross-Validation
Generative Models
Other SSL Models and Enhancements
Conclusion
Appendix 4.A Python for Semi-Supervised Models
References
Chapter 5: Letting the Data Speak with Unsupervised Learning
Introduction
Dimensionality Reduction in Finance
Conclusion
Appendix 5.A PCA and SVD in Python
References
Chapter 6: Big Data Factor Models
Why PCA and SVD Deliver Optimal Factorization
Eigenportfolios
Using Factors to Predict Returns
Factor Discovery
Instrumented PCA
The Three-Pass Model
Risk-Premium PCA
Nonlinear Factorization
Correlation-Based Factors
Hierarchical PCA (HPCA)
Disadvantages of PCA and SVD
Conclusion
Appendix 6.A Python for Big Data Factor Models
References
Note
Chapter 7: Data as a Signal versus Noise
Introduction
Random Data Shows in Eigenvalue Distribution
Application: What's in the Data Bag?
The Marčenko-Pastur Theorem
Spike Model: Which Value to Pick on the “Elbow”?
Dealing with Highly Correlated Data
Deconstructing the Mona Lisa
What's in the Data Bag?
Applications
The Karhunen-Loève Transform
Data Imputation
Missing Eigenvalues
The Tracy-Widom Distribution
Identifying (and Replacing) Missing Values in Streaming Data (the Johnson-Lindenstrauss Lemma)
Conclusion
Appendix 7 Finding the Optimal Number of Eigenvectors in Python
References
Chapter 8: Applications: Unsupervised Learning in Option Pricing and Stochastic Modeling
Introduction
Application 1: Unsupervised Learning in Options Pricing
Application 2: Optimizing Markov Chains with the Perron-Frobenius Theorem
Conclusion
Appendix 8.A Determining the Percentage of Variation Explained by the Top Principal Components in Python
References
Note
Chapter 9: Data Clustering
Introduction
Clustering Methodology
Clustering Financial Data
Empirical Results
Conclusion
Appendix 9.A Clustering with Python
References
Conclusion
Index
End User License Agreement
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Prev
Previous Chapter
Cover
Next
Next Chapter
Title Page
Table of Contents
Cover
Title Page
Copyright
Preface
REFERENCE
Chapter 1: Why Big Data?
Introduction
Appendix 1.A Coding Big Data in Python
Reference
Notes
Chapter 2: Neural Networks in Finance
Introduction
Neural Network Construction Methodology
The Architecture of Neural Networks
Choosing the Activation Function
Construction and Training of Neural Networks
Model Selection via Dropout
Overfitting
Adding Complexity
Big Data in Machine Learning
Coding a Simple Neural Network for One Instrument from Daily Data
Defining Target Outputs
Testing Performance
Adding Activation Levels
Convergence
Choosing Input Variables
Conclusion
Appendix 2.A Building a Neural Network in Python
References
Chapter 3: Supervised Learning
Introduction
Supervised Learning
Conclusion
Appendix 3.A Python for Supervised Models
References
Chapter 4: Modeling Human Behavior with Semi-Supervised Learning
Introduction
Performance Evaluation via Cross-Validation
Generative Models
Other SSL Models and Enhancements
Conclusion
Appendix 4.A Python for Semi-Supervised Models
References
Chapter 5: Letting the Data Speak with Unsupervised Learning
Introduction
Dimensionality Reduction in Finance
Conclusion
Appendix 5.A PCA and SVD in Python
References
Chapter 6: Big Data Factor Models
Why PCA and SVD Deliver Optimal Factorization
Eigenportfolios
Using Factors to Predict Returns
Factor Discovery
Instrumented PCA
The Three-Pass Model
Risk-Premium PCA
Nonlinear Factorization
Correlation-Based Factors
Hierarchical PCA (HPCA)
Disadvantages of PCA and SVD
Conclusion
Appendix 6.A Python for Big Data Factor Models
References
Note
Chapter 7: Data as a Signal versus Noise
Introduction
Random Data Shows in Eigenvalue Distribution
Application: What's in the Data Bag?
The Marčenko-Pastur Theorem
Spike Model: Which Value to Pick on the “Elbow”?
Dealing with Highly Correlated Data
Deconstructing the Mona Lisa
What's in the Data Bag?
Applications
The Karhunen-Loève Transform
Data Imputation
Missing Eigenvalues
The Tracy-Widom Distribution
Identifying (and Replacing) Missing Values in Streaming Data (the Johnson-Lindenstrauss Lemma)
Conclusion
Appendix 7 Finding the Optimal Number of Eigenvectors in Python
References
Chapter 8: Applications: Unsupervised Learning in Option Pricing and Stochastic Modeling
Introduction
Application 1: Unsupervised Learning in Options Pricing
Application 2: Optimizing Markov Chains with the Perron-Frobenius Theorem
Conclusion
Appendix 8.A Determining the Percentage of Variation Explained by the Top Principal Components in Python
References
Note
Chapter 9: Data Clustering
Introduction
Clustering Methodology
Clustering Financial Data
Empirical Results
Conclusion
Appendix 9.A Clustering with Python
References
Conclusion
Index
End User License Agreement
List of Tables
Chapter 3
Table 3.1 Correlations of daily changes of selected U.S. Treasuries.
Table 3.2 Coefficients determined by linear regression, ridge regression, LAS...
Chapter 4
Table 4.1 Average and standard deviation of RMSE scores deployed in optimal m...
Chapter 5
Table 5.1 Traditional covariance matrix of security returns.
Table 5.2 Factorized covariance matrix of security returns.
Table 5.3 Proportions of individual stock returns in the A-AEF Group (100 sto...
Table 5.4 The most common top positive factors in the first singular value ou...
Table 5.5 Correlation of the top-ten eigenvectors of the Russell 3000 stocks ...
Table 5.6 Covid crisis performance of the ETFs most correlated with the top-t...
Chapter 6
Table 6.1 Results of the first stage of the HPCA analysis.
Chapter 7
Table 7.1 The high-precision mean (
), variance (
), skewness (
), and kurtos...
Chapter 8
Table 8.1 Top 15 underlying constituents and bottom 15 underlying constituent...
Table 8.2 Average values across all remaining assets (around 3,300) and those...
Table 8.3 Morningstar realized transition probabilities, 2010–2016.
Chapter 9
Table 9.1 In-sample highly liquid January 2017–May 2020 returns for various c...
Table 9.2 Annualized mean returns and Sharpe ratios of out-of-sample performa...
Table 9.3 Annualized mean returns and Sharpe ratios of out-of-sample performa...
List of Illustrations
Chapter 1
Figure 1.1 Natural data relationships: inputs
x
correspond to responses
y
.
Figure 1.2 Differences in data interpretation between traditional data model...
Figure 1.A.1 Selecting the user-friendly Python editor upon installation.
Figure 1.A.4 Error dialogue box.
Chapter 2
Figure 2.1 A sample neural network.
Figure 2.2 Sigmoid function (a) and its derivative (b).
Figure 2.3 ReLU function (a) and its derivative (b).
Figure 2.4 Tanh function (a) and its derivative (b).
Figure 2.5 Linear function (a) and its derivative (b).
Figure 2.6 Performance of the neural network with one hidden layer, tanh act...
Figure 2.7 Weights W1 and W2 from the one-day-ahead SPY return NN prediction...
Figure 2.8 Predictability of the SPY return based on the previous day's SPY ...
Figure 2.9 Predictability of the SPY return based on the previous day's SPY ...
Figure 2.10 A monthly neural network strategy on NYSE:PFE with input variabl...
Figure 2.11 A monthly neural network strategy with linear activation functio...
Figure 2.12 A monthly neural network strategy with tanh activation function ...
Figure 2.13 A monthly neural network strategy with tanh activation function ...
Figure 2.14 A monthly neural network strategy with linear activation functio...
Figure 2.15 A monthly neural network strategy with tanh activation function ...
Figure 2.16 (a–k) Loss function convergence with increased number of iterati...
Chapter 3
Figure 3.1 Raw equity data sample. The data record a limit buy order for 100...
Figure 3.2 Out-of-sample t + 1 prediction of 20-Year U.S. Treasury rates wit...
Figure 3.3 Out-of-sample t + 1 prediction of 20-Year U.S. Treasury rates wit...
Figure 3.4 Out-of-sample t + 1 prediction of 20-Year U.S. Treasury rates wit...
Figure 3.5 In-sample Root Mean Squared Error (RMSE) for 30-second NYSE:ABT p...
Figure 3.6 Out-of-sample performance of K-NN on intraday S&P 500 data foreca...
Figure 3.7 Out-of-sample performance of K-NN on intraday S&P 500 data foreca...
Figure 3.8 A simple decision tree illustrating predictability of AUD/EUR giv...
Figure 3.9 Single decision tree process.
Figure 3.10 Single decision tree (left) vs. random decision forest, an illus...
Figure 3.11 Single decision tree (left) vs. Extra Trees, an illustration.
Figure 3.12 Cumulative performance paths for the S&P 500 constituents follow...
Figure 3.13 Distribution of cumulative end-of-day returns for the S&P 500 re...
Figure 3.14 Prediction of 30-second returns of the S&P 500 stocks using SVM....
Figure 3.15 Cumulative error computed by Eq (3.1) for each constituent of th...
Chapter 4
Figure 4.1 An illustration of 3-fold cross-validation. During each of the
k
...
Figure 4.2 Semi-supervised learning. Source: Adapted from Chakrabortty (2016...
Figure 4.3 Raw number of news announcements recorded for each mean analyst r...
Figure 4.4 SSL with ridge regression, estimation performance with various sa...
Figure 4.5 Out-of-sample prediction of analysts' forecasts using previous mo...
Figure 4.6 Average predictions of analysts' forecasts based on stock-specifi...
Figure 4.7 SSL ratings forecast for seven out-of-sample stocks produced by v...
Figure 4.8 Out-of-sample prediction of analysts' forecasts using previous mo...
Figure 4.9 Average predictions of analysts' forecasts based on stock-specifi...
Figure 4.10 SSL with Ridge regression, out-of-sample predictions of ratings ...
Figure 4.11 Average predictions of analysts; forecasts based on stock-specif...
Figure 4.15 Discriminant analysis using ridge regression with output in {0,1...
Figure 4.16 Discriminant analysis using K-nearest neighbors with output in {...
Chapter 5
Figure 5.1 Original sample image.
Figure 5.2 Scree plot corresponding to the image in Figure 5.1.
Figure 5.3 Reconstruction of the image of Figure 5.1 with just the first sin...
Figure 5.4 Reconstruction of the image in Figure 5.1 with the first 2 singul...
Figure 5.5 Reconstruction of the image of Figure 5.1 with the first 5 singul...
Figure 5.6 Reconstruction of the image in Figure 5.1 with the first 10 singu...
Figure 5.7 Reconstruction of the image in Figure 5.1 with the first 20 singu...
Figure 5.8 Reconstruction of the image in Figure 5.1 with the first 50 singu...
Figure 5.9 Reconstruction of the image in Figure 5.1 with the first 200 sing...
Figure 5.10 Rolling 250-day correlations between intraday downward volatilit...
Figure 5.11 2018 returns' eigenvectors' correlation with contemporaneous Fam...
Figure 5.12 January–April 2020 Performance of the issues most positively cor...
Figure 5.13 January–April 2020 Performance of the issues most negatively cor...
Figure 5.14 January 2019-April 2020 Performance of the issues most positivel...
Figure 5.15 January 2019-April 2020 Performance of the issues most negativel...
Figure 5.16 Distribution of Excess Sharpe ratios (SR of Portfolio less SR of...
Figure 5.17 Average performance of 1,000 of positive-only components of 1–50...
Figure 5.18 Average performance of 1,000 of negative-only components of 1–10...
Figure 5.19 Average performance of 1,000 of positive and negative components...
Figure 5.20 An illustration of the relationship between distance of point to...
Chapter 6
Figure 6.1 In-sample explanatory power of the first “eigenfactor” or princip...
Figure 6.2 In-sample explanatory power of the market portfolio (here, the da...
Figure 6.3 In-sample regression coefficients
of the Equally Weighted S&P 5...
Figure 6.4 First principal component vs. daily average return on the S&P 500...
Figure 6.5 Comparison of aggregate industry-based portfolios constructed wit...
Figure 6.6 Comparison of aggregate industry-based portfolios constructed wit...
Figure 6.7 Comparison of aggregate industry-based portfolios constructed wit...
Figure 6.8 Comparison of aggregate industry-based portfolios constructed wit...
Figure 6.9 Comparison of aggregate industry-based portfolios constructed wit...
Figure 6.10 Comparison of aggregate industry-based portfolios constructed wi...
Figure 6.11 Auto and truck dealerships, industry eigenportfolios, first eige...
Figure 6.12 Rental and leasing, industry eigenportfolios, first three eigenv...
Figure 6.13 Biotechnology, industry eigenportfolios, first three eigenvector...
Figure 6.14 Diagnostics and research, industry eigenportfolios, first eigenv...
Figure 6.15 Distribution of year-to-year changes in top-10 eigenfactors, Bas...
Figure 6.16 Distribution of year-to-year changes in top-10 eigenfactors, Com...
Figure 6.17 Distribution of year-to-year changes in top-10 eigenfactors, Com...
Figure 6.18 Distribution of year-to-year changes in top-10 eigenfactors, Con...
Figure 6.19 Distribution of year-to-year changes in top-10 eigenfactors, Ene...
Figure 6.20 Distribution of year-to-year changes in top-10 eigenfactors, Fin...
Figure 6.21 Distribution of year-to-year changes in top-10 eigenfactors, Hea...
Figure 6.22 Distribution of year-to-year changes in top-10 eigenfactors, Ind...
Figure 6.23 Distribution of year-to-year changes in top-10 eigenfactors, Rea...
Figure 6.24 Distribution of year-to-year changes in top-10 eigenfactors, Tec...
Figure 6.25 Distribution of year-to-year changes in top-10 eigenfactors, Uti...
Figure 6.26 OOS cumulative performance, Basic Materials.
Figure 6.27 OOS cumulative performance, Communication Services.
Figure 6.28 OOS cumulative performance, Consumer Cyclical.
Figure 6.29 OOS cumulative performance, Consumer Defensive.
Figure 6.30 OOS cumulative performance, Energy.
Figure 6.31 OOS cumulative performance, Financial Services.
Figure 6.32 OOS cumulative performance, Healthcare.
Figure 6.33 OOS cumulative performance, Industrials.
Figure 6.34 OOS cumulative performance, Real Estate.
Figure 6.35 OOS cumulative performance, Technology.
Figure 6.36 OOS cumulative performance, Utilities.
Chapter 7
Figure 7.1 Wigner Semicircle Law: the distribution of eigenvalues of a Gauss...
Figure 7.2 Distribution of eigenvalues of covariance of a randomly generated...
Figure 7.3 Distribution of eigenvalues of covariance of a randomly generated...
Figure 7.4 Distribution of eigenvalues of covariance of a randomly generated...
Figure 7.5 Theoretical and empirical distribution of eigenvalues of the cova...
Figure 7.6a These eigenvalues “pop out” from the Marčenko-Pastur distributio...
Figure 7.6b Log scale of the histogram in Figure 7.6a. Here, the log scale s...
Figure 7.7 Marčenko-Pastur for a signal of Eq (7.10) with
for N = 1000 and...
Figure 7.8 Marčenko-Pastur for a signal of Eq (7.10) with
for N = 1000 and...
Figure 7.9 Mona Lisa, grayscale.
Figure 7.10 Histogram of correlations of the Mona Lisa's columns.
Figure 7.11 The Marčenko-Pastur “elbow” of the detrended and descaled covari...
Figure 7.12 Mona Lisa reconstruction with just six significant eigenvalues....
Figure 7.13 Marčenko-Pastur cut-off on the unnormalized (raw) Mona Lisa imag...
Figure 7.14 Subsequent SVD reconstruction of the image with 120 significant ...
Figure 7.15 Marčenko-Pastur and the empirical distribution of the eigenvalue...
Figure 7.16 The scree plot of the singular values corresponding to the data ...
Figure 7.17 Marčenko-Pastur and the empirical distribution of the eigenvalue...
Figure 7.18 The scree plot of the singular values corresponding to the data ...
Figure 7.19 Marčenko-Pastur and the empirical distribution of the eigenvalue...
Figure 7.20 The scree plot of the singular values corresponding to the data ...
Figure 7.21 Marčenko-Pastur and the empirical distribution of the eigenvalue...
Figure 7.22 The scree plot of the singular values corresponding to the data ...
Figure 7.23 Marčenko-Pastur and the empirical distribution of the eigenvalue...
Figure 7.24 The scree plot of the singular values corresponding to the data ...
Figure 7.25 Marčenko-Pastur and the empirical distribution of the eigenvalue...
Figure 7.26 The scree plot of the singular values corresponding to the data ...
Figure 7.27 Marčenko-Pastur and the empirical distribution of the eigenvalue...
Figure 7.28 The scree plot of the singular values corresponding to the data ...
Figure 7.29 Histogram of correlations of daily returns of the S&P 500, 1,250...
Figure 7.30 Histogram of correlations of daily returns of the S&P 500,
T =
...
Figure 7.31 Histogram of correlations of intraday 30-minute returns of the S...
Figure 7.32 Histogram of errors of intraday returns and predicted returns de...
Figure 7.33 Histogram of correlations of KLT prediction, all eigenvectors pr...
Figure 7.34 Histogram of correlations of residuals, all eigenvectors present...
Figure 7.35 Scree plot of errors between the intraday 30-second S&P 500 retu...
Figure 7.36 Marčenko-Pastur and the empirical distribution of the eigenvalue...
Figure 7.37 The scree plot of the singular values corresponding to the corre...
Figure 7.38 Marčenko-Pastur and the empirical distribution of the eigenvalue...
Figure 7.39 The scree plot of the singular values corresponding to the data ...
Figure 7.40 Theoretical Marčenko-Pastur distribution for the eigenvalues of ...
Figure 7.41 Log scale of the distribution of the eigenvalues of the covarian...
Figure 7.42 Theoretical Marčenko-Pastur distribution for the eigenvalues of ...
Figure 7.43 Face completion with various supervised methods (“Olivetti Faces...
Figure 7.44 Original Mona Lisa image.
Figure 7.45 Distribution of eigenvalues for Mona Lisa with 0, 50, 100, and 2...
Figure 7.46 Eigenvalue changes for various whiteout regimes. Eigenvalues har...
Figure 7.47 Distribution of eigenvalues for Mona Lisa with 0, 50, 100, and 2...
Figure 7.48 Eigenvalues for different whiteout regimes. The eigenvalues hard...
Figure 7.49 Absolute differences in eigenvalues between 50-point blackout an...
Figure 7.50 Relative changes in the eigenvalues for different blackout level...
Figure 7.51 Differences in sequential eigenvalues between whiteout Mona Lisa...
Figure 7.52 Relative changes in eigenvalues from random data replacement in ...
Figure 7.53 Relative changes in eigenvalues from random data replacement in ...
Figure 7.54 Top 5 eigenvalues of the S&P 500 correlation matrix (20-day non-...
Figure 7.55 Top 5 eigenvalues of the S&P 500 correlation matrix (20-day non-...
Figure 7.56 Relationship between Wigner and Tracy-Widom distributions. Sourc...
Chapter 8
Figure 8.1 Implied volatility surface for SPX using call options, December 1...
Figure 8.2 Implied volatility surfaces for call options on SPX computed on s...
Figure 8.3 The number of significant top eigenvalues as determined by the Ma...
Figure 8.4 Percent of variation explained by the first three components for ...
Figure 8.5 Percent of variation explained by the first component for the 20 ...
Figure 8.6a-e Distribution of ratings of different corporate credit rating a...
Figure 8.7 Convergence of a sample 2×2 transition probability matrix.
Figure 8.8 Steady-state ratings distribution in Morningstar ratings as predi...
Figure 8.9 Normalized empirical distribution of actual Morningstar ratings. ...
Figure 8.10 The raw eigenvalues of the Morningstar empirical credit rating t...
Figure 8.11a 2010 prediction of 2011 Morningstar ratings distribution and re...
Figure 8.11b 2011 prediction of 2012 Morningstar ratings distribution and re...
Figure 8.11c 2012 prediction of 2013 Morningstar ratings distribution and re...
Figure 8.11d 2013 prediction of 2014 Morningstar ratings distribution and re...
Figure 8.11e 2014 prediction of 2015 Morningstar ratings distribution and re...
Figure 8.11f 2015 prediction of 2016 Morningstar ratings distribution and re...
Chapter 9
Figure 9.1 An illustration of K-means algorithm with K-4 on four convex data...
Figure 9.2 Clustering of the lower-left triangular correlation matrix of the...
Figure 9.3 Cumulative January 2017–May 2020 returns per cryptocurrency tradi...
Figure 9.4 K-means clustering of cryptocurrency returns, January 2017–May 20...
Figure 9.5 Spectral clustering of cryptocurrency returns, January 2017–May 2...
Figure 9.6 K-means clustering of cryptocurrency returns, January 2017–May 20...
Figure 9.7 Spectral clustering of cryptocurrency returns, January 2017–May 2...
Figure 9.8 K-means clustering of cryptocurrency returns, January 2017–May 20...
Figure 9.9 Spectral clustering of cryptocurrency returns, January 2017–May 2...
Figure 9.10 Spectral clustering of cryptocurrency returns, January 2017-May ...
Figure 9.11 Spectral clustering of cryptocurrency returns, detail, January 2...
Figure 9.12 K-means clustering of cryptocurrencies, 2 clusters.
Figure 9.13 K-means clustering of cryptocurrencies, 3 clusters, March 2018....
Figure 9.14 K-means clustering of cryptocurrencies, 5 clusters, March 2018....
Figure 9.15 K-means clustering of cryptocurrencies, 7 clusters, March 2018....
Figure 9.16 K-means clustering of cryptocurrencies, 10 clusters, March 2018....
Figure 9.17 K-means clustering of cryptocurrencies, 15 clusters, March 2018....
Figure 9.18 K-means clustering of cryptocurrencies, 20 clusters.
Figure 9.19 Spectral clustering of cryptocurrencies, 2 clusters.
Figure 9.20 Spectral clustering of cryptocurrencies, 3 clusters.
Figure 9.21 Spectral clustering of cryptocurrencies, 5 clusters, March 2018....
Figure 9.22 Spectral clustering of cryptocurrencies, 7 clusters.
Figure 9.23 Spectral clustering of cryptocurrencies, 10 clusters, March 2018...
Figure 9.24 Spectral clustering of cryptocurrencies, 15 clusters.
Figure 9.25 Spectral clustering of cryptocurrencies, 20 clusters.
Figure 9.26 Monthly out-of-sample K-means and spectral clustering for crypto...
Figure 9.27 Monthly out-of-sample K-means and spectral clustering for crypto...
Figure 9.28 Monthly out-of-sample K-means and spectral clustering for crypto...
Figure 9.29 Monthly out-of-sample K-means and spectral clustering for crypto...
Figure 9.30 Monthly out-of-sample K-means and spectral clustering for crypto...
Figure 9.31 Monthly out-of-sample K-means and spectral clustering for crypto...
Figure 9.32 Monthly out-of-sample K-means and spectral clustering for crypto...
Figure 9.33 K-means clustering for commodities, 2 clusters.
Figure 9.34 K-means clustering for commodities, 3 clusters.
Figure 9.35 K-means clustering for commodities, 5 clusters.
Figure 9.36 K-means clustering for commodities, 7 clusters.
Figure 9.37 K-means clustering for commodities, 10 clusters.
Figure 9.38 K-means clustering for commodities, 15 clusters.
Figure 9.39 K-means clustering for commodities, 20 clusters.
Figure 9.40 K-means clustering for commodities, 25 clusters.
Figure 9.41 K-means clustering for commodities, 35 clusters.
Figure 9.42 K-means clustering for commodities, 50 clusters.
Figure 9.43 Spectral clustering for commodities, 2 clusters.
Figure 9.44 Spectral clustering for commodities, 3 clusters.
Figure 9.45 Spectral clustering for commodities, 5 clusters.
Figure 9.46 Spectral clustering for commodities, 7 clusters.
Figure 9.47 Spectral clustering for commodities, 10 clusters.
Figure 9.48 Spectral clustering for commodities, 15 clusters.
Figure 9.49 Spectral clustering for commodities, 20 clusters.
Figure 9.50 Spectral clustering for commodities, 25 clusters.
Figure 9.51 Spectral clustering for commodities, 35 clusters.
Figure 9.52 Spectral clustering for commodities, 50 clusters.
Figure 9.53 Monthly out-of-sample K-means and spectral clustering for commod...
Figure 9.54 Monthly out-of-sample K-means and spectral clustering for commod...
Figure 9.55 Monthly out-of-sample K-means and spectral clustering for commod...
Figure 9.56 Monthly out-of-sample K-means and spectral clustering for commod...
Figure 9.57 Monthly out-of-sample K-means and spectral clustering for commod...
Figure 9.58 Monthly out-of-sample K-means and spectral clustering for commod...
Figure 9.59 Monthly out-of-sample K-means and spectral clustering for commod...
Figure 9.60 Monthly out-of-sample K-means and spectral clustering for commod...
Figure 9.61 Monthly out-of-sample K-means and spectral clustering for commod...
Figure 9.62 Monthly out-of-sample K-means and spectral clustering for commod...
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