DATA MINING Desktop Survival Guide by Graham Williams
Desktop Survival Project Home Introduction Getting Started The Business Problem Data Loading Data Exploring Data Interactive Graphics Statistical Tests Models Network Analysis Text Mining Decision Trees Random Forests Boosting Bagging Support Vector Machine Linear Regression Neural Network Naive Bayes Survival Analysis Evaluation and Deployment Transforming Data Deployment Troubleshooting Issues Moving into R R Getting Help Data Graphics in R Understanding Data Preparing Data Issues Evaluating Models Reporting Fraud Analysis Archetype Analysis Algorithms Bayes Classifier K-Nearest Neighbours Linear Models Open Products AlphaMiner Borgelt Data Mining Suite KNime R Rattle Weka Closed Products Clementine Equbits Foresight GhostMiner InductionEngine ODM Enterprise Miner Statistica Data Miner TreeNet Virtual Predict Installing Rattle Projects Bibliography Index Preface Goals Organisation Features Audience Typographical Conventions A Note on Languages Currency Acknowledgements Introduction Data Mining The Business Problem Types of Analysis Data Mining Applications A Framework for Modelling Agile Data Mining R Rattle Why R and Rattle? Data Preparation Number of Algorithms Repeatability Performance Open Source Data Mining Business Case Sample Business Case Pros and Cons Books on R Getting Started Initial Interaction with R Quitting Rattle and R First Contact Loading a Dataset Building a Model Understanding Our Data Evaluating the Model Evaluating the Model Interacting with R Interacting with Rattle Projects Toolbar Menus Interacting with Plots Keyboard Navigation Summary Command Summary The Business Problem Solar Panel Efficiency Water Collection Others Other Business Problems Fraud Detection Loan Approval Documenting the Business Problem Summary Resources Exercises Data Data Nomenclature Loading Data into Rattle CSV Data Datasets Reading Direct from URL Play Golf Weather Data Other Data Sources ARFF Data ODBC Sourced Data Setting Up a Data Source Name Netezza Setup Teradata Setup R Data R Dataset Data Entry Data Tab Options in Rattle Sampling Data Variable Roles Automatic Role Identification Weights Calculator Manipulating Data Loading Data CSV Data Locating and Loading Data Loading the File CSV Options Basic Data Summary ARFF Data ODBC Sourced Data R Dataset R Data Library Data Options Sampling Data Variable Roles Automatic Role Identification Weights Calculator Command Summary Exploring Data Summarising Data Summary Describe Basics Kurtosis Skewness Missing Exploring Distributions Box Plot Histogram Cumulative Distribution Plot Benford's Law Other Digits Stratified Benford Plots Bar Plot Dot Plot Mosaic Plot GGobi Scatterplot Data Viewer Brushing Identify Multivariate Outliers Other Options Quality Plots Using R Further GGobi Documentation Correlation Analysis Hierarchical Correlation Principal Components Single Variable Overviews Interactive Graphics Interactive Visualisations Latticist GGobi Scatterplot Multiple Plots Brushing Other Plots Data Viewer Brushing Identify Multivariate Outliers Other Options Quality Plots Using R Further GGobi Documentation Documenting Interactive Explorations Code Review Chapter Exercises Command Summary Statistical Tests Documenting Interactive Explorations Code Review Further Resources Chapter Exercises Command Summary Models A Framework for Modelling Descriptive Analytics Predictive Analytics Documenting Models Summary Code Review Exercises Further Resources Command Summary Cluster Analysis Summary Clusters Basic Clustering Hot Spots Alternative Clustering Other Cluster Examples KMeans Export KMeans Clusters Discriminant Coordinates Plot Number of Clusters Hierarchical Clusters Other Cluster Algorithms Association Analysis Summary Overview Algorithm Usage Read Transactions file format sep cols rm.duplicates Summary Apriori data parameter appearance control Inspect Examples Video Marketing Survey Data Other Examples Resources and Further Reading Basket Analysis General Rules Network Analysis Documenting Interactive Explorations Code Review Chapter Exercises Command Summary Text Mining Application to Text Text Mining with R Decision Trees Knowledge Representation Search Heuristic Measures Tutorial Example Rattle R Tuning Parameters Min Split (Rarg[]minsplit) Min Bucket (minbucket) Priors (prior) Loss Matrix Complexity (cp) Other Options Simple Example Convert Tree to Rules Predicting Salary Group Issues Summary Code Review Iris Wine Exercises Resources Command Summary Random Forests Formalities Tutorial Example Tuning Parameters Number of Trees Sample Size Number of Variables Summary Overview Algorithm Usage Random Forest importance classwt Examples Resources and Further Reading Summary Overview Example Algorithm Resources and Further Reading Boosting Formalities Tutorial Example Tuning Parameters Summary Overview AdaBoost Algorithm Examples Step by Step Using gbm Extensions and Variations Alternating Decision Tree Resources and Further Reading Documenting Code Review Further Resources Chapter Exercises Command Summary Bootstrapping Summary Usage Further Information Summary Overview Example Algorithm Resources and Further Reading Bagging Support Vector Machine Formalities Tutorial Example Tuning parameters Examples Resources and Further Reading Overview Examples Resources and Further Reading Linear Regression Linear Regression Formalities Tutorial Example Tuning parameters Generalized Regression Formalities Tutorial Example Tuning parameters Logistic Regression Formalities Tutorial Example Tuning parameters Discussion Probit Regression Formalities Tutorial Example Tuning Parameters Multinomial Regression Formalities Tutorial Example Tuning Parameters Neural Network Formalities Tutorial Example Tuning parameters Documenting Code Review Further Resources Chapter Exercises Command Summary Naive Bayes Summary Code Review Resources Exercises Command Summary Survival Analysis Sample Data Simple Lung Descriptive Analysis Regression survreg Simple Lung coxph Simple Lung Apply to New Data More Input Variables Decision Tree Example from Singer and Willett Other Approaches Design Package Random Survival Forests Prediction on Test Data Evaluation The Evaluate Tab Confusion Matrix Measures Graphical Measures Risk Charts Cost Curves Lift ROC Curves Area Under Curve Precision versus Recall Sensitivity versus Specificity Predicted versus Observed Scoring Documenting Interactive Explorations Code Review Chapter Exercises Command Summary Transforming Data Rescale Data Recenter Scale [0,1] Rank Median/MAD Peer Relativity Profiling Index Impute Zero/Missing Mean/Median/Mode Constant Remap Binning Indicator Variables Join Categorics Math Transforms Outliers Cleanup Delete Ignored Delete Selected Delete Missing Delete Obs with Missing Other Transformations Removing Duplicates Command Summary Deployment Documenting Deployment Code Review Chapter Exercises Command Summary Troubleshooting Cairo A factor has new levels Issues Model Selection Overfitting Imbalanced Classification Sampling Cost Based Learning Model Deployment and Interoperability SQL PMML XML for Data Bibliographic Notes Documenting Code Review Chapter Exercises Command Summary Moving into R Interacting with R Basic Command Line Windows, Icons, Mouse, Pointer--WIMP The Current Rattle State Samples Projects The Rattle Log Further Tuning Models Emacs and ESS Documenting Code Review Chapter Exercises Command Summary R Evaluation Exercises Assignment Libraries and Packages Searching for Objects Package Management Information About a Package Testing Package Availability Packages and Namespaces Basic Programming in R Principles Folders and Files Flow Control If Statement For Loop Functions Apply Methods Objects System Running System Commands System Parameters Misc Internet Memory Management Memory Usage Garbage Collection Errors Frivolous Sudoku Further Resources Using R Specific Purposes Survey Analysis Getting Help R Documentation Data Data Types Numbers Strings Building Strings Splitting Strings Substitution Trim Whitespace Evaluating Strings Logical Dates and Times Space Data Structures Vectors Arrays Lists Sets Matricies Exercises Data Frames Accessing Columns Removing Columns Exercises General Manipulation Factors Elements Rows and Columns Finding Index of Elements Partitions Head and Tail Reverse a List Sorting Unique Values Loading Data Interactive Responses Interactive Data Entry Available Datasets The Iris Dataset CSV Data Used In The Book The Wine Dataset The Cardiac Arrhythmia Dataset The Adult Survey Dataset Foreign Formats Stata Data Conversions Reading Variable Width Data Saving Data Formatted Output Automatically Generate Filenames Reading a Large File Manipulating Data Manipulating Data As SQL Using SQLite ODBC Data Database Connection Excel Access Clipboard Data Spatial Data Simple Map A Density Map Overlays and Point in Polygon Other Data Formats Fixed Width Data Global Positioning System Documenting a Dataset Common Data Problems Graphics in R Basic Plot Controlling Axes Arrow Axes Legends and Points Tables Within Plots Colour Labels in Plots Axis Labels Legend Labels Within Plots Maths in Labels Multiple Plots MatPlot Multiple Plots Using ggplot2 Using GGPlot Networks Symbols Other Graphic Elements Making an Animation Animated Mandelbrot Adding a Logo to a Graphic Graphics Devices Setup Screen Devices Multiple Devices File Devices Multiple Plots Copy and Print Devices Graphics Parameters Plotting Region Locating Points on a Plot Scientific Notation and Plots Understanding Data Single Variable Overviews Textual Summaries Multiple Line Plots Separate Line Plots Pie Chart Fan Plot Stem and Leaf Plots Histogram Barplot Trellis Histogram Histogram Uneven Distribution Bump Chart Density Plot Basic Histogram Basic Histogram with Density Curve Practical Histogram Multiple Variable Overviews Scatterplot Scatterplot with Marginal Histograms Multi-Dimension Scatterplot Correlation Plot Colourful Correlations Fluctuation Plot Heat Map Projection Pursuit RADVIZ Parallel Coordinates Categoric and Numeric Measuring Data Distributions Textual Summaries Boxplot Multiple Boxplots Boxplot by Class Tuning a Boxplot Boxplot Using Lattice Boxplot Using ggplot Violin Plot What Distribution Miscellaneous Plots Line and Point Plots Matrix Data Multiple Plots Aligned Plots Probability Scale Network Plot Sunflower Plot Stairs Plot Graphing Means and Error Bars Bar Charts With Segments Bar Plot With Means 3d Bar Plot Stacks Versus Lines Multi-Line Title Mathematics Plots for Normality Basic Bar Chart Bar Chart Displays Multiple Dot Plots Alternative Multiple Dot Plots 3D Plot Clustered Box Plot Perspective Plots Star Plot Residuals Plot Waterfall Plots Dates and Times Simple Time Series Multiple Time Series Plot Time Series Plot Time Series with Axis Labels Grouping Time Series for Box Plot Time Series Heatmap Textual Summaries Stem and Leaf Plots Histogram Barplot Density Plot Basic Histogram Basic Histogram with Density Curve Practical Histogram Correlation Plot Colourful Correlations Measuring Data Distributions Textual Summaries Boxplot Multiple Boxplots Boxplot by Class Box and Whisker Plot Box and Whisker Plot Clustered Box Plot Further Resources Map Displays Further Resources Preparing Data Data Selection and Extraction Training and Test Datasets Data Cleaning Review Data Selectively Changing Vector Values Replace Indices By Names Missing Values Remove Levels from a Factor Variable Manipulations Remove Columns Reorder Columns Remove Non-Numeric Columns Remove Variables with no Variance Cleaning the Wine Dataset Cleaning the Cardiac Dataset Cleaning the Survey Dataset Imputation Nearest Neighbours Multiple Imputation Data Linking Simple Linking Record Linkage Data Transformation Aggregation Sum of Columns Pivot Tables Normalising Data Binning Interpolation Variable Selection Classification Classification Classification Issues Incremental or Online Modelling Model Tuning Tuning rpart Unbalanced Classification Building Models Temporal Analysis Evaluation Basics Basic Measures Cross Validation Graphical Performance Measures Lift The ROC Curve Other Examples 10 Fold Cross Validation Area Under Curve Calibration Curves Reporting Generating Open Document Format Getting Started with odfWeave OpenOffice.org Macro Support Generating HTML Generating PDF with LATEX Configuration Figure Sizes Fraud Analysis Archetype Analysis

AdaBoost Algorithm

Boosting employs a weak learning algorithm (which we identify as the learner). Suppose the dataset (data) consists of $N$ entites described using $M$ variables (lines 1 and 2 of the meta-code below). The $M$th variable (i.e., the last variable of each observation) is assumed to be the classification of the observation. In the algorithm presented here we denote the training data (an $N$ by $M-1$ matrix) as $x$ (line 3) and the class associated with each observation in the training data (a vector of length $M$) as $y$ (line 4). Without loss of generality we can restrict the class to be either $1$ (perhaps representing $yes$) or $-1$ (representing $no$). This will simplify the mathematics. Each observation in the training data is initially assigned the same weight: $w_i = \frac{1}{N}$ (line 5).

The weak learner will need to use the weights associated with each observation. This may be handled directly by the learner (e.g., rpart takes an option to specify the Roption[]weights) or else by generating a modified dataset by sampling the original dataset based on the weights.

The first model, $\mathcal{M}_1$, is built by applying the weak $\id{learner}$ to the $\id{data}$ with weights $w$ (line 7). $\mathcal{M}_1$, predicting either $1$ or $-1$, is then used to identify the set of indicies of misclassified entities (i.e., where $\mathcal{M}_1(x_p)\not=y_p)$, denoted as $\id{ms}$ (line 8). For a completely accurate model we would have $\mathcal{M}_1(x_i) = y_i$. Of course the model is expected to be only slightly better than random so ms is unlikely to be empty.

A relative error $\epsilon_1$ for $\mathcal{M}_1$ is calculated as the relative sum of the weights of the misclassified entities (line 9). This is used to calculate $\alpha_1$ (line 10), used, in turn, to adjust the weights (line 11). All weights could be either decreased or increased depending on whether the model correctly classifies the corresponding observation, as proposed by (). However, this can be simplified to only increasing the weights of the misclassified entities, as proposed by (). These entities thus become more important.

The learning algorithm is then applied to the new weighted $\id{data}$ with the $\id{learner}$ expected to give more focus on the difficult entities whilst building this next model, $\mathcal{M}_2$. The weights are then modified again using the errors from $\mathcal{M}_2$. The model building and weight modification is then repeated until the new model performs no better than random (i.e., the error is 50% or more: $\epsilon_i \ge 0.5$), or is perfect (i.e., the error rate is 0% and $\id{ms}$ is empty), or perhaps after a fixed number of iterations.

$\parbox{0.985\textwidth}{\begin{codebox} \Procname{$\proc{AdaBoost}(... ... = \Sign\bigl(\sum_{j=1}^{T} \alpha_j \mathcal{M}_j(x)\bigr)$] \end{codebox}}$

The final model $\mathcal{M}$ (line 14) combines the other models using a weighted sum of the outputs of these other models. The weights, $\alpha_j$, reflect the accuracy of each of the constituent models.

A simple example can illustrate the process. Suppose the number of training entities, $N$, is 10. Each weight, $w_j$, is thus initially $0.1$ (line 5). Imagine the first model, $\mathcal{M}_1$, correctly classifies the first 6 of the 10 entities (e.g., $ms=\{7,8,9,10\}$), so that $\epsilon_1=0.1+0.1+0.1+0.1/1=0.4$. Then $\alpha_1=\log(0.6/0.4)=0.405$, and is the weight that will be used to multiply the results from this model to be added into the overall model score. The weights $w_7, \dots, w_{10}$ then become $0.1\times e^{0.405} = 0.1\times 1.5 = 0.15$. That is, they now have more importance for the next model build. Suppose now that $\mathcal{M}_2$ correctly classifies 8 of the entities (with $ms=\{1,8\}$), so that $\epsilon_2=(0.1+0.15)/1.2=0.208$ and $\alpha_2=\log(0.792/0.208)=1.337$. Thus $w_1 = 0.1\times e^{1.337} \approx 0.381$ and $w_8 = 0.15\times e^{1.337} \approx 0.571$. Note how record 8 is proving particularly troublesome and so its weight is now the highest.

We can understand the behaviour of the function used to weight the models ( $\log((1-\epsilon_i)/\epsilon_i)$) and to adjust the observation weights ($e^{\alpha_i}$) by plotting both of these for the range of errors we expect (from 0 to 0.5 or 50%).

plot(function(x) exp(log((1-x)/x)), xlim=c(0.01, 0.5), ylim=c(0.01, 200),
     xlab=expression(Error: epsilon), ylab="", log="y", lty=1, col=2, yaxt="n")
plot(function(x) log((1-x)/x), add=TRUE, lty=2, col=3)
axis(side=2, at=c(0.01, 0.1, 1, 10, 100), c(0.01, 0.1, 1, 10, 100))
grid()
exp.leg <- expression(alpha == log(over(1-epsilon, epsilon)), exp(alpha))
legend("topright", exp.leg, lty=2:1, col=3:2, inset=c(0.04, 0.1))

http://rattle.togaware.com/code/rplot-adaboost.R

First, looking at the value of the $\alpha$'s ( $\log((1-\epsilon_i)/\epsilon_i)$), we see that for errors close to zero, that is, for very accurate models, the $\alpha_i$ (i.e., the weight that is used to multiply the results from the model) is very high. For an error of about 5% the multiplier is almost 3, and for a 1% error the multiplier is about 4.6. With an error of approximately 27% (i.e., $\epsilon=0.26894$) the multiplier is close to 1. For errors greater than this the model gets weights less than 1, heading down to a weight of 0 at $\epsilon=0.5$.

In terms of building the models, the observation weights are multiplied by $e^{\alpha_i}$. If the model we have just built ($\mathcal{M}_i$) is quite accurate ($\epsilon$ close to 0) then fewer entities are being misclassified, and their weights are increased significantly (e.g., for a 5% error the weight is multiplied by 19). For inaccurate models (as $\epsilon$ approaches 0.5) the multiplier for the weights approaches 1. Note that if $\epsilon_i=0.5$ the multiplier is 1, and thus no change is made to the weights of the entities. Of course, building a model on the same dataset with the same weights will build the same model, thus the criteria for continuing to build a model tests that $\epsilon < 0.5$.

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