From e335db1e841a93a9c7a86288d5979d8b09149475 Mon Sep 17 00:00:00 2001 From: superuser mintkde Date: Thu, 16 Feb 2017 13:34:04 -0600 Subject: [PATCH 1/5] rename args(train.default) to args(train). add variable `type` to trainPC. Otherwise, it doesn't work. --- .../Practical Machine Learning Course Notes.Rmd | 14 +++++++++----- 1 file changed, 9 insertions(+), 5 deletions(-) diff --git a/8_PREDMACHLEARN/Practical Machine Learning Course Notes.Rmd b/8_PREDMACHLEARN/Practical Machine Learning Course Notes.Rmd index 96fae3d..65c47fd 100644 --- a/8_PREDMACHLEARN/Practical Machine Learning Course Notes.Rmd +++ b/8_PREDMACHLEARN/Practical Machine Learning Course Notes.Rmd @@ -2,13 +2,13 @@ title: "Practical Machine Learning Course Notes" author: "Xing Su" output: - pdf_document: - toc: yes - toc_depth: 3 html_document: highlight: pygments theme: spacelab toc: yes + pdf_document: + toc: yes + toc_depth: 3 header-includes: \usepackage{graphicx} \usepackage{mathtools} --- @@ -427,7 +427,8 @@ folds$test[[1]] ```{r} # returns the arguments of the default train function -args(train.default) +# args(train.default) <- this dosn't work anymore in caret +args(train) ``` * `train` function has a large set of parameters, below are the default options @@ -774,8 +775,11 @@ testing <- spam[-inTrain,] preProc <- preProcess(log10(training[,-58]+1),method="pca",pcaComp=2) # calculate PCs for training data trainPC <- predict(preProc,log10(training[,-58]+1)) +# add variable `type` to trainPC +type <- training$type +trainPC <- data.frame(trainPC, type) # join variable `type` # run model on outcome and principle components -modelFit <- train(training$type ~ .,method="glm",data=trainPC) +modelFit <- train(type ~ .,method="glm",data=trainPC) # calculate PCs for test data testPC <- predict(preProc,log10(testing[,-58]+1)) # compare results From 53ed16b190dae24d7b51c0576dd82be3cb98df4f Mon Sep 17 00:00:00 2001 From: superuser mintkde Date: Thu, 16 Feb 2017 13:37:09 -0600 Subject: [PATCH 2/5] change #training$type# to just "type" to make this analysis work. --- 8_PREDMACHLEARN/Practical Machine Learning Course Notes.Rmd | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/8_PREDMACHLEARN/Practical Machine Learning Course Notes.Rmd b/8_PREDMACHLEARN/Practical Machine Learning Course Notes.Rmd index 65c47fd..036b722 100644 --- a/8_PREDMACHLEARN/Practical Machine Learning Course Notes.Rmd +++ b/8_PREDMACHLEARN/Practical Machine Learning Course Notes.Rmd @@ -793,7 +793,7 @@ confusionMatrix(testing$type,predict(modelFit,testPC)) ```{r message = FALSE, warning = FALSE} # construct model -modelFit <- train(training$type ~ .,method="glm",preProcess="pca",data=training) +modelFit <- train(type ~ .,method="glm",preProcess="pca",data=training) # print results of model confusionMatrix(testing$type,predict(modelFit,testing)) ``` From 58ddba9d7562c63607a9c9cbf7f0c7e59c353d68 Mon Sep 17 00:00:00 2001 From: superuser mintkde Date: Thu, 16 Feb 2017 13:52:59 -0600 Subject: [PATCH 3/5] add note that package "klaR" is required. --- 8_PREDMACHLEARN/Practical Machine Learning Course Notes.Rmd | 1 + 1 file changed, 1 insertion(+) diff --git a/8_PREDMACHLEARN/Practical Machine Learning Course Notes.Rmd b/8_PREDMACHLEARN/Practical Machine Learning Course Notes.Rmd index 036b722..0b505fa 100644 --- a/8_PREDMACHLEARN/Practical Machine Learning Course Notes.Rmd +++ b/8_PREDMACHLEARN/Practical Machine Learning Course Notes.Rmd @@ -1325,6 +1325,7 @@ pred.lda - ***example: `caret` package*** ```{r message = F, warning = F} +# package needed: klaR # using the same data from iris, run naive Bayes on training data nb <- train(Species ~ ., data=training,method="nb") # predict test outcomes using naive Bayes model From 9e8ae3c2fe54b4ca4517e1438639b742a2e77343 Mon Sep 17 00:00:00 2001 From: superuser mintkde Date: Thu, 16 Feb 2017 14:37:57 -0600 Subject: [PATCH 4/5] add knitr header to prevent comments and stop on errors. --- 8_PREDMACHLEARN/.directory | 4 + ...ractical Machine Learning Course Notes.Rmd | 4 + ...actical_Machine_Learning_Course_Notes.html | 1467 +++++++++-------- ...ractical_Machine_Learning_Course_Notes.pdf | Bin 4442947 -> 4441858 bytes 4 files changed, 783 insertions(+), 692 deletions(-) create mode 100644 8_PREDMACHLEARN/.directory diff --git a/8_PREDMACHLEARN/.directory b/8_PREDMACHLEARN/.directory new file mode 100644 index 0000000..f88ed35 --- /dev/null +++ b/8_PREDMACHLEARN/.directory @@ -0,0 +1,4 @@ +[Dolphin] +Timestamp=2017,2,16,14,12,56 +Version=3 +ViewMode=1 diff --git a/8_PREDMACHLEARN/Practical Machine Learning Course Notes.Rmd b/8_PREDMACHLEARN/Practical Machine Learning Course Notes.Rmd index 0b505fa..deec457 100644 --- a/8_PREDMACHLEARN/Practical Machine Learning Course Notes.Rmd +++ b/8_PREDMACHLEARN/Practical Machine Learning Course Notes.Rmd @@ -13,6 +13,10 @@ header-includes: \usepackage{graphicx} \usepackage{mathtools} --- +```{r setup, include=FALSE, error=TRUE, message=FALSE} +knitr::opts_chunk$set(echo = TRUE, comment = NA, error = TRUE) +``` + $\pagebreak$ diff --git a/8_PREDMACHLEARN/Practical_Machine_Learning_Course_Notes.html b/8_PREDMACHLEARN/Practical_Machine_Learning_Course_Notes.html index 1c9b791..243f198 100644 --- a/8_PREDMACHLEARN/Practical_Machine_Learning_Course_Notes.html +++ b/8_PREDMACHLEARN/Practical_Machine_Learning_Course_Notes.html @@ -8,38 +8,58 @@ + Practical Machine Learning Course Notes - + - - - - + + + + + + + @@ -63,17 +110,43 @@ color: inherit; background-color: rgba(0, 0, 0, 0.04); } -img { - max-width:100%; - height: auto; +img { + max-width:100%; + height: auto; +} +.tabbed-pane { + padding-top: 12px; +} +button.code-folding-btn:focus { + outline: none; } + + +
+ + + + + + + -
@@ -329,7 +403,7 @@

Prediction Study Design

  • important to not tune model to quiz set specifically
  • -

    +

    Sample Division Guidelines for Prediction Study Design

      @@ -373,13 +447,13 @@

      Picking the Right Data

    • use like data to predict like
    • to predict a variable/process X, use the data that’s as closely related to X as possible
    • weighting the data/variables by understanding and intuition can help to improve accuracy of prediction
    • -
    • data properties matter \(\rightarrow\) knowing how the data connects to what you are trying to measure
    • +
    • data properties matter \(\rightarrow\) knowing how the data connects to what you are trying to measure
    • predicting on unrelated data is the most common mistake
      • if unrelated data must be used, be careful about interpreting the model as to why it works/doesn’t work
    -

    \(\pagebreak\)

    +

    \(\pagebreak\)

    @@ -404,9 +478,9 @@

    Types of Errors

    Notable Measurements for Error – Binary Variables

    • accuracy = weights false positives/negatives equally
    • -
    • concordance = for multi-class cases, \[\kappa = \frac{accuracy - P(e)}{1 - P(e)}\] where \[P(e) = \frac{TP+FP}{total} \times \frac{TP+FN}{total} + \frac{TN+FN}{total} \times \frac{FP+TN}{total}\]
    • +
    • concordance = for multi-class cases, \[\kappa = \frac{accuracy - P(e)}{1 - P(e)}\] where \[P(e) = \frac{TP+FP}{total} \times \frac{TP+FN}{total} + \frac{TN+FN}{total} \times \frac{FP+TN}{total}\]
    -

    +

    • example
        @@ -414,13 +488,13 @@

        Notable Measurements for Error – Binary Variables

      • what about 10% prevalence?
    -

    +

    Notable Measurements for Error – Continuous Variables

      -
    • Mean squared error (MSE) = \[\frac{1}{n} \sum_{i=1}^n (Prediction_i - Truth_i)^2\]
    • -
    • Root mean squared error (RMSE) - \[\sqrt{\frac{1}{n} \sum_{i=1}^n(Prediction_i - Truth_i)^2}\] +
    • Mean squared error (MSE) = \[\frac{1}{n} \sum_{i=1}^n (Prediction_i - Truth_i)^2\]
    • +
    • Root mean squared error (RMSE) - \[\sqrt{\frac{1}{n} \sum_{i=1}^n(Prediction_i - Truth_i)^2}\]
      • in the same units as variable
      • most commonly used error measure for continuous data
      • @@ -429,9 +503,9 @@

        Notable Measurements for Error – Continuous Variables

      • one large value may significantly raise the RMSE
    -
  • median absolute deviation = \[median(|Prediction_i - Truth_i|)\]
  • +
  • median absolute deviation = \[median(|Prediction_i - Truth_i|)\]
  • -

    \(\pagebreak\)

    +

    \(\pagebreak\)

    @@ -440,11 +514,11 @@

    Receiver Operating Characteristic Curves

  • are commonly used techniques to measure the quality of a prediction algorithm.
  • predictions for binary classification often are quantitative (i.e. probability, scale of 1 to 10)
      -
    • different cutoffs/threshold of classification (> 0.8 \(\rightarrow\) one outcome) yield different results/predictions
    • +
    • different cutoffs/threshold of classification (> 0.8 \(\rightarrow\) one outcome) yield different results/predictions
    • Receiver Operating Characteristic curves are generated to compare the different outcomes
  • -

    +

    • ROC Curves
        @@ -453,22 +527,22 @@

        Receiver Operating Characteristic Curves

      • points plotted = cutoff/combination
      • areas under curve = quantifies whether the prediction model is viable or not
          -
        • higher area \(\rightarrow\) better predictor
        • -
        • area = 0.5 \(\rightarrow\) effectively random guessing (diagonal line in the ROC curve)
        • -
        • area = 1 \(\rightarrow\) perfect classifier
        • -
        • area = 0.8 \(\rightarrow\) considered good for a prediction algorithm
        • +
        • higher area \(\rightarrow\) better predictor
        • +
        • area = 0.5 \(\rightarrow\) effectively random guessing (diagonal line in the ROC curve)
        • +
        • area = 1 \(\rightarrow\) perfect classifier
        • +
        • area = 0.8 \(\rightarrow\) considered good for a prediction algorithm
    -

    +

    • example
      • each point on the graph corresponds with a specificity and sensitivity
    -

    -

    \(\pagebreak\)

    +

    +

    \(\pagebreak\)

    Cross Validation

    @@ -504,7 +578,7 @@

    Cross Validation

    Random Subsampling

    -

    +

    • a randomly sampled test set is subsetted out from the original training set
    • the predictor is built on the remaining training data and applied to the test set
    • @@ -522,11 +596,11 @@

      Random Subsampling

    K-Fold

    -

    +

      -
    • break training set into \(K\) subsets (above is a 3-fold cross validation)
    • +
    • break training set into \(K\) subsets (above is a 3-fold cross validation)
    • build the model/predictor on the remaining training data in each subset and applied to the test subset
    • -
    • rebuild the data \(K\) times with the training and test subsets and average the findings
    • +
    • rebuild the data \(K\) times with the training and test subsets and average the findings
    • considerations
      • larger k = less bias, more variance
      • @@ -536,13 +610,13 @@

        K-Fold

    Leave One Out

    -

    +

    • leave out exactly one sample and build predictor on the rest of training data
    • predict value for the left out sample
    • repeat for each sample
    -

    \(\pagebreak\)

    +

    \(\pagebreak\)

    @@ -550,10 +624,10 @@

    caret Package (\(\rightarrow\) preProcess() -
  • cross validation/data splitting \(\rightarrow\) createDataPartition(), createResample(), createTimeSlices()
  • -
  • train algorithms on training data and apply to test sets \(\rightarrow\) train(), predict()
  • -
  • model comparison (evaluate the accuracy of model on new data) \(\rightarrow\) confusionMatrix()
  • +
  • preprocessing/cleaning data \(\rightarrow\) preProcess()
  • +
  • cross validation/data splitting \(\rightarrow\) createDataPartition(), createResample(), createTimeSlices()
  • +
  • train algorithms on training data and apply to test sets \(\rightarrow\) train(), predict()
  • +
  • model comparison (evaluate the accuracy of model on new data) \(\rightarrow\) confusionMatrix()
  • machine learning algorithms in caret package
  • +
    List of 10
    + $ Fold01: int [1:460] 15 16 18 40 45 62 68 81 82 102 ...
    + $ Fold02: int [1:459] 1 41 55 58 67 75 117 123 151 175 ...
    + $ Fold03: int [1:461] 3 14 66 69 70 80 90 112 115 135 ...
    + $ Fold04: int [1:460] 5 19 25 65 71 83 85 88 91 93 ...
    + $ Fold05: int [1:460] 6 10 17 21 26 56 57 104 107 116 ...
    + $ Fold06: int [1:459] 7 8 13 39 52 54 76 89 99 106 ...
    + $ Fold07: int [1:461] 4 23 27 29 32 33 34 38 49 51 ...
    + $ Fold08: int [1:460] 2 9 30 31 36 37 43 46 47 48 ...
    + $ Fold09: int [1:461] 12 20 24 44 53 59 60 64 84 98 ...
    + $ Fold10: int [1:460] 11 22 28 35 42 61 72 86 92 118 ...
    +
    # return first 10 elements of the first training set
    +folds[[1]][1:10]
    +
     [1]  1  2  3  4  6  7  8  9 10 12
    • createResample(y=data$var, times=10, list=TRUE) = create 10 resamplings from the given data with replacement
        @@ -674,21 +748,21 @@

        Data Slicing

      • times=10 = number of samples to create
    -
    # create 10 resamples
    +
    # create 10 resamples
     resamples <- createResample(y=spam$type,times=10,list=TRUE)
     # structure of the resamples (note some samples are repeated)
    -str(resamples)
    -
    ## List of 10
    -##  $ Resample01: int [1:4601] 1 4 4 4 7 8 12 13 13 14 ...
    -##  $ Resample02: int [1:4601] 3 3 5 7 10 12 12 13 13 14 ...
    -##  $ Resample03: int [1:4601] 1 2 2 3 4 5 8 10 11 12 ...
    -##  $ Resample04: int [1:4601] 1 3 3 4 7 8 8 9 10 14 ...
    -##  $ Resample05: int [1:4601] 2 4 5 6 7 7 8 8 9 12 ...
    -##  $ Resample06: int [1:4601] 3 6 6 7 8 9 12 13 13 14 ...
    -##  $ Resample07: int [1:4601] 1 2 2 5 5 6 7 8 9 10 ...
    -##  $ Resample08: int [1:4601] 2 2 3 4 4 7 7 8 8 9 ...
    -##  $ Resample09: int [1:4601] 1 4 7 8 8 9 12 13 15 15 ...
    -##  $ Resample10: int [1:4601] 1 3 4 4 7 7 9 9 10 11 ...
    +str(resamples)
    +
    List of 10
    + $ Resample01: int [1:4601] 1 4 4 4 7 8 12 13 13 14 ...
    + $ Resample02: int [1:4601] 3 3 5 7 10 12 12 13 13 14 ...
    + $ Resample03: int [1:4601] 1 2 2 3 4 5 8 10 11 12 ...
    + $ Resample04: int [1:4601] 1 3 3 4 7 8 8 9 10 14 ...
    + $ Resample05: int [1:4601] 2 4 5 6 7 7 8 8 9 12 ...
    + $ Resample06: int [1:4601] 3 6 6 7 8 9 12 13 13 14 ...
    + $ Resample07: int [1:4601] 1 2 2 5 5 6 7 8 9 10 ...
    + $ Resample08: int [1:4601] 2 2 3 4 4 7 7 8 8 9 ...
    + $ Resample09: int [1:4601] 1 4 7 8 8 9 12 13 15 15 ...
    + $ Resample10: int [1:4601] 1 3 4 4 7 7 9 9 10 11 ...
    • createTimeSlices(y=data, initialWindow=20, horizon=10) = creates training sets with specified window length and the corresponding test sets
        @@ -701,32 +775,30 @@

        Data Slicing

    -
    # create time series data
    +
    # create time series data
     tme <- 1:1000
     # create time slices
     folds <- createTimeSlices(y=tme,initialWindow=20,horizon=10)
     # name of lists
    -names(folds)
    -
    ## [1] "train" "test"
    -
    # first training set
    -folds$train[[1]]
    -
    ##  [1]  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 19 20
    -
    # first test set
    -folds$test[[1]]
    -
    ##  [1] 21 22 23 24 25 26 27 28 29 30
    +names(folds)
    +
    [1] "train" "test" 
    +
    # first training set
    +folds$train[[1]]
    +
     [1]  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 19 20
    +
    # first test set
    +folds$test[[1]]
    +
     [1] 21 22 23 24 25 26 27 28 29 30

    Training Options (tutorial)

    • train(y ~ x, data=df, method="glm") = function to apply the machine learning algorithm to construct model from training data
    -
    # returns the arguments of the default train function
    -args(train.default)
    -
    ## function (x, y, method = "rf", preProcess = NULL, ..., weights = NULL, 
    -##     metric = ifelse(is.factor(y), "Accuracy", "RMSE"), maximize = ifelse(metric == 
    -##         "RMSE", FALSE, TRUE), trControl = trainControl(), tuneGrid = NULL, 
    -##     tuneLength = 3) 
    -## NULL
    +
    # returns the arguments of the default train function
    +# args(train.default)   <- this dosn't work anymore in caret
    +args(train)
    +
    function (x, ...) 
    +NULL
    • train function has a large set of parameters, below are the default options
    -
    # returns the default arguments for the trainControl object
    -args(trainControl)
    -
    ## function (method = "boot", number = ifelse(grepl("cv", method), 
    -##     10, 25), repeats = ifelse(grepl("cv", method), 1, number), 
    -##     p = 0.75, initialWindow = NULL, horizon = 1, fixedWindow = TRUE, 
    -##     verboseIter = FALSE, returnData = TRUE, returnResamp = "final", 
    -##     savePredictions = FALSE, classProbs = FALSE, summaryFunction = defaultSummary, 
    -##     selectionFunction = "best", preProcOptions = list(thresh = 0.95, 
    -##         ICAcomp = 3, k = 5), index = NULL, indexOut = NULL, timingSamps = 0, 
    -##     predictionBounds = rep(FALSE, 2), seeds = NA, adaptive = list(min = 5, 
    -##         alpha = 0.05, method = "gls", complete = TRUE), trim = FALSE, 
    -##     allowParallel = TRUE) 
    -## NULL
    +
    # returns the default arguments for the trainControl object
    +args(trainControl)
    +
    function (method = "boot", number = ifelse(grepl("cv", method), 
    +    10, 25), repeats = ifelse(grepl("cv", method), 1, number), 
    +    p = 0.75, search = "grid", initialWindow = NULL, horizon = 1, 
    +    fixedWindow = TRUE, skip = 0, verboseIter = FALSE, returnData = TRUE, 
    +    returnResamp = "final", savePredictions = FALSE, classProbs = FALSE, 
    +    summaryFunction = defaultSummary, selectionFunction = "best", 
    +    preProcOptions = list(thresh = 0.95, ICAcomp = 3, k = 5, 
    +        freqCut = 95/5, uniqueCut = 10, cutoff = 0.9), sampling = NULL, 
    +    index = NULL, indexOut = NULL, indexFinal = NULL, timingSamps = 0, 
    +    predictionBounds = rep(FALSE, 2), seeds = NA, adaptive = list(min = 5, 
    +        alpha = 0.05, method = "gls", complete = TRUE), trim = FALSE, 
    +    allowParallel = TRUE) 
    +NULL

    Preprocessing (tutorial)

    • some predictors may have strange distributions (i.e. skewed) and may need to be transformed to be more useful for prediction algorithm
        -
      • particularly true for model based algorithms \(\rightarrow\) naive Bayes, linear discriminate analysis, linear regression
      • +
      • particularly true for model based algorithms \(\rightarrow\) naive Bayes, linear discriminate analysis, linear regression
    • centering = subtracting the observations of a particular variable by its mean
    • scaling = dividing the observations of a particular variable by its standard deviation
    • -
    • normalizing = centering and scaling the variable \(\rightarrow\) effectively converting each observation to the number of standard deviations away from the mean +
    • normalizing = centering and scaling the variable \(\rightarrow\) effectively converting each observation to the number of standard deviations away from the mean
    -
    # load spam data
    +
    +
                  mean      std
    +train 6.097035e-18 1.000000
    +test  7.548133e-02 1.633866
    -
    # set up BoxCox transforms
    +
    # set up BoxCox transforms
     preObj <- preProcess(training[,-58],method=c("BoxCox"))
     # perform preprocessing on training data
     trainCapAveS <- predict(preObj,training[,-58])$capitalAve
     # plot histogram and QQ Plot
     # Note: the transformation definitely helped to
     # normalize the data but it does not produce perfect result
    -par(mfrow=c(1,2)); hist(trainCapAveS); qqnorm(trainCapAveS)
    -

    +par(mfrow=c(1,2)); hist(trainCapAveS); qqnorm(trainCapAveS)
    +

    -
    # Make some values NA
    +
    +
               0%           25%           50%           75%          100% 
    +-1.656344e+00  2.377772e-05  1.286900e-03  1.880821e-03  3.174413e-01 
    +

    \(\pagebreak\)

    @@ -998,7 +1072,7 @@

    Covariate Creation/Feature Extraction

  • generally, more knowledge and understanding you have of the system/data, the easier it will be to extract the summarizing features
      -
    • when in doubt, more features is always safer \(\rightarrow\) lose less information and the features can be filtered during model construction
    • +
    • when in doubt, more features is always safer \(\rightarrow\) lose less information and the features can be filtered during model construction
  • this process can be automated (i.e. PCA) but generally have to be very careful, as one very useful feature in the training data set may not have as much effect on the test data set
  • Note: science is the key here, Google “feature extraction for [data type]” for more guidance @@ -1021,31 +1095,31 @@

    Covariate Creation/Feature Extraction

    Creating Dummy Variables

      -
    • convert factor variables to indicator/dummy variable \(\rightarrow\) qualitative become quantitative
    • +
    • convert factor variables to indicator/dummy variable \(\rightarrow\) qualitative become quantitative
    • dummyVars(outcome~var, data=training) = creates a dummy variable object that can be used through predict function to create dummy variables
      • predict(dummyObj, newdata=training) = creates appropriate columns to represent the factor variable with appropriate 0s and 1s
          -
        • 2 factor variable \(\rightarrow\) two columns which have 0 or 1 depending on the outcome
        • -
        • 3 factor variable \(\rightarrow\) three columns which have 0, 0, and 1 representing the outcome
        • +
        • 2 factor variable \(\rightarrow\) two columns which have 0 or 1 depending on the outcome
        • +
        • 3 factor variable \(\rightarrow\) three columns which have 0, 0, and 1 representing the outcome
        • Note: only one of the columns can have values of 1 for each observation
    -
    # setting up data
    +
    # setting up data
     inTrain <- createDataPartition(y=Wage$wage,p=0.7, list=FALSE)
     training <- Wage[inTrain,]; testing <- Wage[-inTrain,]
     # create a dummy variable object
     dummies <- dummyVars(wage ~ jobclass,data=training)
     # create the dummy variable columns
    -head(predict(dummies,newdata=training))
    -
    ##        jobclass.1. Industrial jobclass.2. Information
    -## 231655                      1                       0
    -## 86582                       0                       1
    -## 161300                      1                       0
    -## 155159                      0                       1
    -## 11443                       0                       1
    -## 376662                      0                       1
    +head(predict(dummies,newdata=training))
    +
           jobclass.1. Industrial jobclass.2. Information
    +231655                      1                       0
    +86582                       0                       1
    +161300                      1                       0
    +155159                      0                       1
    +11443                       0                       1
    +376662                      0                       1
  • Removing Zero Covariates

    @@ -1061,21 +1135,21 @@

    Removing Zero Covariates

  • Note: when nzv = TRUE, those variables should be thrown out
  • -
    # print nearZeroVar table
    -nearZeroVar(training,saveMetrics=TRUE)
    -
    ##            freqRatio percentUnique zeroVar   nzv
    -## year        1.017647    0.33301618   FALSE FALSE
    -## age         1.231884    2.85442436   FALSE FALSE
    -## sex         0.000000    0.04757374    TRUE  TRUE
    -## maritl      3.329571    0.23786870   FALSE FALSE
    -## race        8.480583    0.19029496   FALSE FALSE
    -## education   1.393750    0.23786870   FALSE FALSE
    -## region      0.000000    0.04757374    TRUE  TRUE
    -## jobclass    1.070936    0.09514748   FALSE FALSE
    -## health      2.526846    0.09514748   FALSE FALSE
    -## health_ins  2.209160    0.09514748   FALSE FALSE
    -## logwage     1.011765   18.83920076   FALSE FALSE
    -## wage        1.011765   18.83920076   FALSE FALSE
    +
    # print nearZeroVar table
    +nearZeroVar(training,saveMetrics=TRUE)
    +
               freqRatio percentUnique zeroVar   nzv
    +year        1.017647    0.33301618   FALSE FALSE
    +age         1.231884    2.85442436   FALSE FALSE
    +sex         0.000000    0.04757374    TRUE  TRUE
    +maritl      3.329571    0.23786870   FALSE FALSE
    +race        8.480583    0.19029496   FALSE FALSE
    +education   1.393750    0.23786870   FALSE FALSE
    +region      0.000000    0.04757374    TRUE  TRUE
    +jobclass    1.070936    0.09514748   FALSE FALSE
    +health      2.526846    0.09514748   FALSE FALSE
    +health_ins  2.209160    0.09514748   FALSE FALSE
    +logwage     1.011765   18.83920076   FALSE FALSE
    +wage        1.011765   18.83920076   FALSE FALSE

    Creating Splines (Polynomial Functions)

    @@ -1086,7 +1160,7 @@

    Creating Splines (Polynomial Functions)

  • gam() function can also be used and it allows for smoothing of multiple variables with different values for each variable
  • Note: the same polynomial operations must be performed on the test sets using the predict function
  • -
    # load splines package
    +
    # load splines package
     library(splines)
     # create polynomial function
     bsBasis <- bs(training$age,df=3)
    @@ -1095,17 +1169,17 @@ 

    Creating Splines (Polynomial Functions)

    # plot all age vs wage data plot(training$age,training$wage,pch=19,cex=0.5) # plot the fitted polynomial function -points(training$age,predict(lm1,newdata=training),col="red",pch=19,cex=0.5)
    -

    -
    # predict on test values
    -head(predict(bsBasis,age=testing$age))
    -
    ##              1          2           3
    -## [1,] 0.0000000 0.00000000 0.000000000
    -## [2,] 0.2368501 0.02537679 0.000906314
    -## [3,] 0.4163380 0.32117502 0.082587862
    -## [4,] 0.4308138 0.29109043 0.065560908
    -## [5,] 0.3625256 0.38669397 0.137491189
    -## [6,] 0.3063341 0.42415495 0.195763821
    +points(training$age,predict(lm1,newdata=training),col="red",pch=19,cex=0.5)
    +

    +
    # predict on test values
    +head(predict(bsBasis,age=testing$age))
    +
                 1          2           3
    +[1,] 0.0000000 0.00000000 0.000000000
    +[2,] 0.2368501 0.02537679 0.000906314
    +[3,] 0.4163380 0.32117502 0.082587862
    +[4,] 0.4308138 0.29109043 0.065560908
    +[5,] 0.3625256 0.38669397 0.137491189
    +[6,] 0.3063341 0.42415495 0.195763821

    Multicore Parallel Processing

    @@ -1122,7 +1196,7 @@

    Multicore Parallel Processing

  • Note: once registered, you should see in your task manager/activity monitor that 4 “R Session” appear when you run your code
  • -

    \(\pagebreak\)

    +

    \(\pagebreak\)

    @@ -1152,7 +1226,7 @@

    prcomp Function

    • pr<-prcomp(data) = performs PCA on all variables and returns a prcomp object that contains information about standard deviations and rotations
        -
      • pr$rotations = returns eigenvectors for the linear combinations of all variables (coefficients that variables are multiplied by to come up with the principal components) \(\rightarrow\) how the principal components are created
      • +
      • pr$rotations = returns eigenvectors for the linear combinations of all variables (coefficients that variables are multiplied by to come up with the principal components) \(\rightarrow\) how the principal components are created
      • often times, it is useful to take the log transformation of the variables and adding 1 before performing PCA
        • helps to reduce skewness or strange distribution in data
        • @@ -1165,23 +1239,23 @@

          prcomp Function

    -
    # load  spam data
    +
    # load  spam data
     data(spam)
     # perform PCA on dataset
     prComp <- prcomp(log10(spam[,-58]+1))
     # print out the eigenvector/rotations first 5 rows and PCs
    -head(prComp$rotation[, 1:5], 5)
    -
    ##                 PC1           PC2         PC3         PC4          PC5
    -## make    0.019370409  0.0427855959 -0.01631961  0.02798232 -0.014903314
    -## address 0.010827343  0.0408943785  0.07074906 -0.01407049  0.037237531
    -## all     0.040923168  0.0825569578 -0.03603222  0.04563653  0.001222215
    -## num3d   0.006486834 -0.0001333549  0.01234374 -0.01005991 -0.001282330
    -## our     0.036963221  0.0941456085 -0.01871090  0.05098463 -0.010582039
    -
    # create new variable that marks spam as 2 and nospam as 1
    +head(prComp$rotation[, 1:5], 5)
    +
                    PC1           PC2         PC3         PC4          PC5
    +make    0.019370409  0.0427855959 -0.01631961  0.02798232 -0.014903314
    +address 0.010827343  0.0408943785  0.07074906 -0.01407049  0.037237531
    +all     0.040923168  0.0825569578 -0.03603222  0.04563653  0.001222215
    +num3d   0.006486834 -0.0001333549  0.01234374 -0.01005991 -0.001282330
    +our     0.036963221  0.0941456085 -0.01871090  0.05098463 -0.010582039
    +
    # create new variable that marks spam as 2 and nospam as 1
     typeColor <- ((spam$type=="spam")*1 + 1)
     # plot the first two principal components
    -plot(prComp$x[,1],prComp$x[,2],col=typeColor,xlab="PC1",ylab="PC2")
    -

    +plot(prComp$x[,1],prComp$x[,2],col=typeColor,xlab="PC1",ylab="PC2")
    +

    caret Package

    @@ -1201,7 +1275,7 @@

    caret Package

  • Note: the same PCA must be performed on the test set
  • -
    # create train and test sets
    +
    # create train and test sets
     inTrain <- createDataPartition(y=spam$type,p=0.75, list=FALSE)
     training <- spam[inTrain,]
     testing <- spam[-inTrain,]
    @@ -1209,117 +1283,120 @@ 

    caret Package

    preProc <- preProcess(log10(training[,-58]+1),method="pca",pcaComp=2) # calculate PCs for training data trainPC <- predict(preProc,log10(training[,-58]+1)) +# add variable `type` to trainPC +type <- training$type +trainPC <- data.frame(trainPC, type) # join variable `type` # run model on outcome and principle components -modelFit <- train(training$type ~ .,method="glm",data=trainPC) +modelFit <- train(type ~ .,method="glm",data=trainPC) # calculate PCs for test data testPC <- predict(preProc,log10(testing[,-58]+1)) # compare results -confusionMatrix(testing$type,predict(modelFit,testPC))
    -
    ## Confusion Matrix and Statistics
    -## 
    -##           Reference
    -## Prediction nonspam spam
    -##    nonspam     656   41
    -##    spam         82  371
    -##                                           
    -##                Accuracy : 0.893           
    -##                  95% CI : (0.8737, 0.9103)
    -##     No Information Rate : 0.6417          
    -##     P-Value [Acc > NIR] : < 2.2e-16       
    -##                                           
    -##                   Kappa : 0.7724          
    -##  Mcnemar's Test P-Value : 0.0003101       
    -##                                           
    -##             Sensitivity : 0.8889          
    -##             Specificity : 0.9005          
    -##          Pos Pred Value : 0.9412          
    -##          Neg Pred Value : 0.8190          
    -##              Prevalence : 0.6417          
    -##          Detection Rate : 0.5704          
    -##    Detection Prevalence : 0.6061          
    -##       Balanced Accuracy : 0.8947          
    -##                                           
    -##        'Positive' Class : nonspam         
    -## 
    +confusionMatrix(testing$type,predict(modelFit,testPC))
    +
    Confusion Matrix and Statistics
    +
    +          Reference
    +Prediction nonspam spam
    +   nonspam     656   41
    +   spam         82  371
    +                                          
    +               Accuracy : 0.893           
    +                 95% CI : (0.8737, 0.9103)
    +    No Information Rate : 0.6417          
    +    P-Value [Acc > NIR] : < 2.2e-16       
    +                                          
    +                  Kappa : 0.7724          
    + Mcnemar's Test P-Value : 0.0003101       
    +                                          
    +            Sensitivity : 0.8889          
    +            Specificity : 0.9005          
    +         Pos Pred Value : 0.9412          
    +         Neg Pred Value : 0.8190          
    +             Prevalence : 0.6417          
    +         Detection Rate : 0.5704          
    +   Detection Prevalence : 0.6061          
    +      Balanced Accuracy : 0.8947          
    +                                          
    +       'Positive' Class : nonspam         
    +                                          
    • alternatively, PCA can be directly performed with the train method
      • train(outcome ~ ., method="glm", preProcess="pca", data=training) = performs PCA first on the training set and then runs the specified model
          -
        • effectively the same procedures as above (preProcess \(\rightarrow\) predict)
        • +
        • effectively the same procedures as above (preProcess \(\rightarrow\) predict)
    • Note: in both cases, the PCs were able to achieve 90+% accuracy
    -
    # construct model
    -modelFit <- train(training$type ~ .,method="glm",preProcess="pca",data=training)
    +
    # construct model
    +modelFit <- train(type ~ .,method="glm",preProcess="pca",data=training)
     # print results of model
    -confusionMatrix(testing$type,predict(modelFit,testing))
    -
    ## Confusion Matrix and Statistics
    -## 
    -##           Reference
    -## Prediction nonspam spam
    -##    nonspam     668   29
    -##    spam         59  394
    -##                                           
    -##                Accuracy : 0.9235          
    -##                  95% CI : (0.9066, 0.9382)
    -##     No Information Rate : 0.6322          
    -##     P-Value [Acc > NIR] : < 2.2e-16       
    -##                                           
    -##                   Kappa : 0.8379          
    -##  Mcnemar's Test P-Value : 0.001992        
    -##                                           
    -##             Sensitivity : 0.9188          
    -##             Specificity : 0.9314          
    -##          Pos Pred Value : 0.9584          
    -##          Neg Pred Value : 0.8698          
    -##              Prevalence : 0.6322          
    -##          Detection Rate : 0.5809          
    -##    Detection Prevalence : 0.6061          
    -##       Balanced Accuracy : 0.9251          
    -##                                           
    -##        'Positive' Class : nonspam         
    -## 
    -

    \(\pagebreak\)

    +confusionMatrix(testing$type,predict(modelFit,testing))
    +
    Confusion Matrix and Statistics
    +
    +          Reference
    +Prediction nonspam spam
    +   nonspam     668   29
    +   spam         59  394
    +                                          
    +               Accuracy : 0.9235          
    +                 95% CI : (0.9066, 0.9382)
    +    No Information Rate : 0.6322          
    +    P-Value [Acc > NIR] : < 2.2e-16       
    +                                          
    +                  Kappa : 0.8379          
    + Mcnemar's Test P-Value : 0.001992        
    +                                          
    +            Sensitivity : 0.9188          
    +            Specificity : 0.9314          
    +         Pos Pred Value : 0.9584          
    +         Neg Pred Value : 0.8698          
    +             Prevalence : 0.6322          
    +         Detection Rate : 0.5809          
    +   Detection Prevalence : 0.6061          
    +      Balanced Accuracy : 0.9251          
    +                                          
    +       'Positive' Class : nonspam         
    +                                          
    +

    \(\pagebreak\)

    Predicting with Regression

      -
    • prediction with regression = fitting regression model (line) to data \(\rightarrow\) multiplying each variable by coefficients to predict outcome
    • +
    • prediction with regression = fitting regression model (line) to data \(\rightarrow\) multiplying each variable by coefficients to predict outcome
    • useful when the relationship between the variables can be modeled as linear
    • the model is easy to implement and the coefficients are easy to interpret
    • if the relationships are non-linear, the regression model may produce poor results/accuracy
      • Note: linear regressions are generally used in combination with other models
    • -
    • model \[Y_i = \beta_0 + \beta_1 X_{1i} + \beta_2 X_{2i} + \ldots + \beta_p X_{pi} + e_i \] +
    • model \[Y_i = \beta_0 + \beta_1 X_{1i} + \beta_2 X_{2i} + \ldots + \beta_p X_{pi} + e_i \]
        -
      • where \(\beta_0\) is the intercept (when all variables are 0)
      • -
      • \(\beta_1, \beta_2, \ldots, \beta_p\) are the coefficients
      • -
      • \(X_{1i}, X_{2i}, \ldots, X_{pi}\) are the variables/covariates
      • -
      • \(e_i\) is the error
      • -
      • \(Y_i\) is the outcome
      • +
      • where \(\beta_0\) is the intercept (when all variables are 0)
      • +
      • \(\beta_1, \beta_2, \ldots, \beta_p\) are the coefficients
      • +
      • \(X_{1i}, X_{2i}, \ldots, X_{pi}\) are the variables/covariates
      • +
      • \(e_i\) is the error
      • +
      • \(Y_i\) is the outcome
    • -
    • prediction \[\hat Y_i = \hat \beta_0 + \hat \beta_1 X_{1i} + \hat \beta_2 X_{2i} + \ldots + \hat \beta_p X_{pi}\] +
    • prediction \[\hat Y_i = \hat \beta_0 + \hat \beta_1 X_{1i} + \hat \beta_2 X_{2i} + \ldots + \hat \beta_p X_{pi}\]
        -
      • where \(\hat \beta_0\) is the estimated intercept (when all variables are 0)
      • -
      • \(\hat \beta_1, \hat \beta_2, \ldots, \hat \beta_p\) are the estimated coefficients
      • -
      • \(X_{1i}, X_{2i}, \ldots, X_{pi}\) are the variables/covariates
      • -
      • \(\hat Y_i\) is the predicted outcome
      • +
      • where \(\hat \beta_0\) is the estimated intercept (when all variables are 0)
      • +
      • \(\hat \beta_1, \hat \beta_2, \ldots, \hat \beta_p\) are the estimated coefficients
      • +
      • \(X_{1i}, X_{2i}, \ldots, X_{pi}\) are the variables/covariates
      • +
      • \(\hat Y_i\) is the predicted outcome

    R Commands and Examples

      -
    • lm<-lm(y ~ x, data=train) = runs a linear model of outcome y on predictor x \(\rightarrow\) univariate regression +
    • lm<-lm(y ~ x, data=train) = runs a linear model of outcome y on predictor x \(\rightarrow\) univariate regression
        -
      • summary(lm) = returns summary of the linear regression model, which will include coefficients, standard errors, \(t\) statistics, and p values
      • +
      • summary(lm) = returns summary of the linear regression model, which will include coefficients, standard errors, \(t\) statistics, and p values
      • lm(y ~ x1+x2+x3, data=train) = run linear model of outcome y on predictors x1, x2, and x3
      • lm(y ~ ., data=train = run linear model of outcome y on all predictors
    • -
    • predict(lm, newdata=df) = use the constructed linear model to predict outcomes (\(\hat Y_i\)) for the new values +
    • predict(lm, newdata=df) = use the constructed linear model to predict outcomes (\(\hat Y_i\)) for the new values
      • newdata data frame must have the same variables (factors must have the same levels) as the training data
      • newdata=test = predict outcomes for the test set based on linear regression model from the training
      • @@ -1327,10 +1404,10 @@

        R Commands and Examples

    • RSME can be calculated to measure the accuracy of the linear model
        -
      • Note: \(RSME_{test}\), which estimates the out-of-sample error, is almost always GREATER than \(RSME_{train}\)
      • +
      • Note: \(RSME_{test}\), which estimates the out-of-sample error, is almost always GREATER than \(RSME_{train}\)
    -
    # load data
    +
    # load data
     data(faithful)
     # create train and test sets
     inTrain <- createDataPartition(y=faithful$waiting, p=0.5, list=FALSE)
    @@ -1338,31 +1415,31 @@ 

    R Commands and Examples

    # build linear model lm1 <- lm(eruptions ~ waiting,data=trainFaith) # print summary of linear model -summary(lm1)
    -
    ## 
    -## Call:
    -## lm(formula = eruptions ~ waiting, data = trainFaith)
    -## 
    -## Residuals:
    -##      Min       1Q   Median       3Q      Max 
    -## -1.24867 -0.36292  0.00002  0.35768  1.19858 
    -## 
    -## Coefficients:
    -##              Estimate Std. Error t value Pr(>|t|)    
    -## (Intercept) -2.165648   0.227486   -9.52   <2e-16 ***
    -## waiting      0.079396   0.003146   25.24   <2e-16 ***
    -## ---
    -## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
    -## 
    -## Residual standard error: 0.5013 on 135 degrees of freedom
    -## Multiple R-squared:  0.8251, Adjusted R-squared:  0.8238 
    -## F-statistic: 636.9 on 1 and 135 DF,  p-value: < 2.2e-16
    -
    # predict eruptions for new waiting time
    +summary(lm1)
    +
    
    +Call:
    +lm(formula = eruptions ~ waiting, data = trainFaith)
    +
    +Residuals:
    +     Min       1Q   Median       3Q      Max 
    +-1.30246 -0.40746  0.03955  0.40465  1.19221 
    +
    +Coefficients:
    +             Estimate Std. Error t value Pr(>|t|)    
    +(Intercept) -1.858966   0.237636  -7.823 1.33e-12 ***
    +waiting      0.075444   0.003287  22.952  < 2e-16 ***
    +---
    +Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
    +
    +Residual standard error: 0.5272 on 135 degrees of freedom
    +Multiple R-squared:  0.796, Adjusted R-squared:  0.7945 
    +F-statistic: 526.8 on 1 and 135 DF,  p-value: < 2.2e-16
    +
    # predict eruptions for new waiting time
     newdata <- data.frame(waiting=80)
    -predict(lm1,newdata)
    -
    ##     1 
    -## 4.186
    -
    # create 1 x 2 panel plot
    +predict(lm1,newdata)
    +
           1 
    +4.176566 
    +
    # create 1 x 2 panel plot
     par(mfrow=c(1,2))
     # plot train data with the regression line
     plot(trainFaith$waiting,trainFaith$eruptions,pch=19,col="blue",xlab="Waiting",
    @@ -1371,30 +1448,30 @@ 

    R Commands and Examples

    # plot test data with the regression line plot(testFaith$waiting,testFaith$eruptions,pch=19,col="blue",xlab="Waiting", ylab="Duration", main = "Test") -lines(testFaith$waiting,predict(lm1,newdata=testFaith),lwd=3)
    -

    -
    # Calculate RMSE on training and test sets
    +lines(testFaith$waiting,predict(lm1,newdata=testFaith),lwd=3)
    +

    +
    # Calculate RMSE on training and test sets
     c(trainRMSE = sqrt(sum((lm1$fitted-trainFaith$eruptions)^2)),
    -    testRMSE = sqrt(sum((predict(lm1,newdata=testFaith)-testFaith$eruptions)^2)))
    -
    ## trainRMSE  testRMSE 
    -##  5.824859  5.788547
    + testRMSE = sqrt(sum((predict(lm1,newdata=testFaith)-testFaith$eruptions)^2)))
    +
    trainRMSE  testRMSE 
    + 6.125867  5.388723 
    • pi<-predict(lm, newdata=test, interval="prediction") = returns 3 columns for fit (predicted value, same as before), lwr (lower bound of prediction interval), and upr (upper bound of prediction interval)
      • matlines(x, pi, type="l") = plots three lines, one for the linear fit and two for upper/lower prediction interval bounds
    -
    # calculate prediction interval
    +
    # calculate prediction interval
     pred1 <- predict(lm1,newdata=testFaith,interval="prediction")
     # plot data points (eruptions, waiting)
     plot(testFaith$waiting,testFaith$eruptions,pch=19,col="blue")
     # plot fit line and prediction interval
    -matlines(testFaith$waiting,pred1,type="l",,col=c(1,2,2),lty = c(1,1,1), lwd=3)
    -

    +matlines(testFaith$waiting,pred1,type="l",,col=c(1,2,2),lty = c(1,1,1), lwd=3)
    +

      -
    • lm <- train(y ~ x, method="lm", data=train) = run linear model on the training data \(\rightarrow\) identical to lm function +
    • lm <- train(y ~ x, method="lm", data=train) = run linear model on the training data \(\rightarrow\) identical to lm function
        -
      • summary(lm$finalModel) = returns summary of the linear regression model, which will include coefficients, standard errors, \(t\) statistics, and p values \(\rightarrow\) identical to summary(lm) for a lm object
      • +
      • summary(lm$finalModel) = returns summary of the linear regression model, which will include coefficients, standard errors, \(t\) statistics, and p values \(\rightarrow\) identical to summary(lm) for a lm object
      • train(y ~ ., method="lm", data=train) = run linear model on all predictors in training data
        • multiple predictors (dummy/indicator variables) are created for factor variables
        • @@ -1409,7 +1486,7 @@

          R Commands and Examples

    -
    # create train and test sets
    +
    # create train and test sets
     inTrain <- createDataPartition(y=Wage$wage,p=0.7, list=FALSE)
     training <- Wage[inTrain,]; testing <- Wage[-inTrain,]
     # fit linear model for age jobclass and education
    @@ -1419,14 +1496,14 @@ 

    R Commands and Examples

    # set up 2 x 2 panel plot par(mfrow = c(2, 2)) # construct diagnostic plots for model -plot(finMod,pch=19,cex=0.5,col="#00000010")
    -

    +plot(finMod,pch=19,cex=0.5,col="#00000010")
    +

    • plotting residuals by fitted values and coloring with a variable not used in the model helps spot a trend in that variable.
    -
    # plot fitted values by residuals 
    -qplot(finMod$fitted, finMod$residuals, color=race, data=training)
    -

    +
    # plot fitted values by residuals 
    +qplot(finMod$fitted, finMod$residuals, color=race, data=training)
    +

    • plotting residuals by index (ie; row numbers) can be helpful in showing missing variables
        @@ -1437,24 +1514,24 @@

        R Commands and Examples

    -
    # plot residual by index
    -plot(finMod$residuals,pch=19,cex=0.5)
    -

    +
    # plot residual by index
    +plot(finMod$residuals,pch=19,cex=0.5)
    +

    • here the residuals increase linearly with the index, and the highest residuals are concentrated in the higher indexes, so there must be a missing variable
    -

    \(\pagebreak\)

    +

    \(\pagebreak\)

    Prediction with Trees

      -
    • prediction with trees = iteratively split variables into groups (effectively constructing decision trees) \(\rightarrow\) produces nonlinear model +
    • prediction with trees = iteratively split variables into groups (effectively constructing decision trees) \(\rightarrow\) produces nonlinear model
        -
      • the classification tree uses interactions between variables \(\rightarrow\) the ultimate groups/leafs may depend on many variables
      • +
      • the classification tree uses interactions between variables \(\rightarrow\) the ultimate groups/leafs may depend on many variables
    • the result (tree) is easy to interpret, and generally performs better predictions than regression models when the relationships are non-linear
    • -
    • transformations less important \(\rightarrow\) monotone transformations (order unchanged, such as \(\log\)) will produce same splits
    • +
    • transformations less important \(\rightarrow\) monotone transformations (order unchanged, such as \(\log\)) will produce same splits
    • trees can be used for regression problems as well and use RMSE as measure of impurity
    • however, without proper cross-validation, the model can be over-fitted (especially with large number of variables) and results may be variable from one run to the next
        @@ -1474,56 +1551,56 @@

        Process

    Measures of Impurity (Reference)

    -

    \[\hat{p}_{mk} = \frac{\sum_{i}^m \mathbb{1}(y_i = k)}{N_m}\]

    +

    \[\hat{p}_{mk} = \frac{\sum_{i}^m \mathbb{1}(y_i = k)}{N_m}\]

      -
    • \(\hat p_mk\) is the probability of the objects in group \(m\) to take on the classification \(k\)
    • -
    • \(N_m\) is the size of the group

    • -
    • Misclassification Error \[ 1 - \hat{p}_{m~k(m)}\] where \(k(m)\) is the most common classification/group +
    • \(\hat p_mk\) is the probability of the objects in group \(m\) to take on the classification \(k\)
    • +
    • \(N_m\) is the size of the group

    • +
    • Misclassification Error \[ 1 - \hat{p}_{m~k(m)}\] where \(k(m)\) is the most common classification/group
      • 0 = perfect purity
      • 0.5 = no purity
          -
        • Note: it is not 1 here because when \(\hat{p}_{m~k(m)} < 0.5\) or there’s not predominant classification for the objects, it means the group should be further subdivided until there’s a majority
        • +
        • Note: it is not 1 here because when \(\hat{p}_{m~k(m)} < 0.5\) or there’s not predominant classification for the objects, it means the group should be further subdivided until there’s a majority
    • -
    • Gini Index\[ \sum_{k \neq k'} \hat{p}_{mk} \times \hat{p}_{mk'} = \sum_{k=1}^K \hat{p}_{mk}(1-\hat{p}_{mk}) = 1 - \sum_{k=1}^K p_{mk}^2\] +
    • Gini Index\[ \sum_{k \neq k'} \hat{p}_{mk} \times \hat{p}_{mk'} = \sum_{k=1}^K \hat{p}_{mk}(1-\hat{p}_{mk}) = 1 - \sum_{k=1}^K p_{mk}^2\]
      • 0 = perfect purity
      • 0.5 = no purity
    • -
    • Deviance \[ -\sum_{k=1}^K \hat{p}_{mk} \log_e\hat{p}_{mk} \] +
    • Deviance \[ -\sum_{k=1}^K \hat{p}_{mk} \log_e\hat{p}_{mk} \]
      • 0 = perfect purity
      • 1 = no purity
    • -
    • Information Gai \[ -\sum_{k=1}^K \hat{p}_{mk} \log_2\hat{p}_{mk} \] +
    • Information Gai \[ -\sum_{k=1}^K \hat{p}_{mk} \log_2\hat{p}_{mk} \]
      • 0 = perfect purity
      • 1 = no purity
    • example

    -
    # set margin and seed
    +
    # set margin and seed
     par(mar=c(1,1,1,1), mfrow = c(1, 2)); set.seed(1234);
     # simulate data
     x = rep(1:4,each=4); y = rep(1:4,4)
     # plot first scenario
     plot(x,y,xaxt="n",yaxt="n",cex=3,col=c(rep("blue",15),rep("red",1)),pch=19)
     # plot second scenario
    -plot(x,y,xaxt="n",yaxt="n",cex=3,col=c(rep("blue",8),rep("red",8)),pch=19)
    -

    +plot(x,y,xaxt="n",yaxt="n",cex=3,col=c(rep("blue",8),rep("red",8)),pch=19)
    +

    • left graph
        -
      • Misclassification: \(\frac{1}{16} = 0.06\)
      • -
      • Gini: \(1 - [(\frac{1}{16})^2 + (\frac{15}{16})^2] = 0.12\)
      • -
      • Information:\(-[\frac{1}{16} \times \log_2 (\frac{1}{16})+ \frac{15}{16} \times \log_2(\frac{1}{16})] = 0.34\)
      • +
      • Misclassification: \(\frac{1}{16} = 0.06\)
      • +
      • Gini: \(1 - [(\frac{1}{16})^2 + (\frac{15}{16})^2] = 0.12\)
      • +
      • Information:\(-[\frac{1}{16} \times \log_2 (\frac{1}{16})+ \frac{15}{16} \times \log_2(\frac{1}{16})] = 0.34\)
    • right graph
        -
      • Misclassification: \(\frac{8}{16} = 0.5\)
      • -
      • Gini: \(1 - [(\frac{8}{16})^2 + (\frac{8}{16})^2] = 0.5\)
      • -
      • Information:\(-[\frac{8}{16} \times \log_2 (\frac{8}{16})+ \frac{8}{16} \times \log_2(\frac{8}{16})] = 1\)
      • +
      • Misclassification: \(\frac{8}{16} = 0.5\)
      • +
      • Gini: \(1 - [(\frac{8}{16})^2 + (\frac{8}{16})^2] = 0.5\)
      • +
      • Information:\(-[\frac{8}{16} \times \log_2 (\frac{8}{16})+ \frac{8}{16} \times \log_2(\frac{8}{16})] = 1\)
    @@ -1540,12 +1617,12 @@

    Constructing Trees with caret Package

  • [rattle package] fancyRpartPlot(tree$finalModel) = produces more readable, better formatted classification tree diagrams
  • each split will have the condition/node in bold and the splits/leafs on the left and right sides following the “yes” or “no” indicators
      -
    • “yes” \(\rightarrow\) go left
    • -
    • “no” \(\rightarrow\) go right
    • +
    • “yes” \(\rightarrow\) go left
    • +
    • “no” \(\rightarrow\) go right
  • -
    # load iris data set
    +
    # load iris data set
     data(iris)
     # create test/train data sets
     inTrain <- createDataPartition(y=iris$Species,p=0.7, list=FALSE)
    @@ -1554,31 +1631,31 @@ 

    Constructing Trees with caret Package

    # fit classification tree as a model modFit <- train(Species ~ .,method="rpart",data=training) # print the classification tree -print(modFit$finalModel)
    -
    ## n= 105 
    -## 
    -## node), split, n, loss, yval, (yprob)
    -##       * denotes terminal node
    -## 
    -## 1) root 105 70 setosa (0.33333333 0.33333333 0.33333333)  
    -##   2) Petal.Length< 2.45 35  0 setosa (1.00000000 0.00000000 0.00000000) *
    -##   3) Petal.Length>=2.45 70 35 versicolor (0.00000000 0.50000000 0.50000000)  
    -##     6) Petal.Width< 1.65 34  1 versicolor (0.00000000 0.97058824 0.02941176) *
    -##     7) Petal.Width>=1.65 36  2 virginica (0.00000000 0.05555556 0.94444444) *
    -
    # plot the classification tree
    -rattle::fancyRpartPlot(modFit$finalModel)
    -

    -
    # predict on test values
    -predict(modFit,newdata=testing)
    -
    ##  [1] setosa     setosa     setosa     setosa     setosa     setosa    
    -##  [7] setosa     setosa     setosa     setosa     setosa     setosa    
    -## [13] setosa     setosa     setosa     versicolor versicolor versicolor
    -## [19] versicolor versicolor versicolor versicolor versicolor versicolor
    -## [25] versicolor versicolor versicolor versicolor versicolor versicolor
    -## [31] virginica  virginica  virginica  virginica  virginica  virginica 
    -## [37] versicolor virginica  virginica  versicolor versicolor virginica 
    -## [43] virginica  virginica  virginica 
    -## Levels: setosa versicolor virginica
    +print(modFit$finalModel)
    +
    n= 105 
    +
    +node), split, n, loss, yval, (yprob)
    +      * denotes terminal node
    +
    +1) root 105 70 setosa (0.33333333 0.33333333 0.33333333)  
    +  2) Petal.Length< 2.45 35  0 setosa (1.00000000 0.00000000 0.00000000) *
    +  3) Petal.Length>=2.45 70 35 versicolor (0.00000000 0.50000000 0.50000000)  
    +    6) Petal.Width< 1.65 34  1 versicolor (0.00000000 0.97058824 0.02941176) *
    +    7) Petal.Width>=1.65 36  2 virginica (0.00000000 0.05555556 0.94444444) *
    +
    # plot the classification tree
    +rattle::fancyRpartPlot(modFit$finalModel)
    +

    +
    # predict on test values
    +predict(modFit,newdata=testing)
    +
     [1] setosa     setosa     setosa     setosa     setosa     setosa    
    + [7] setosa     setosa     setosa     setosa     setosa     setosa    
    +[13] setosa     setosa     setosa     versicolor versicolor versicolor
    +[19] versicolor versicolor versicolor versicolor versicolor versicolor
    +[25] versicolor versicolor versicolor versicolor versicolor versicolor
    +[31] virginica  virginica  virginica  virginica  virginica  virginica 
    +[37] versicolor virginica  virginica  versicolor versicolor virginica 
    +[43] virginica  virginica  virginica 
    +Levels: setosa versicolor virginica
    @@ -1600,7 +1677,7 @@

    Bagging

    -
    # load data
    +
    # load data
     library(ElemStatLearn); data(ozone,package="ElemStatLearn")
     # reorder rows based on ozone variable
     ozone <- ozone[order(ozone$ozone),]
    @@ -1622,8 +1699,8 @@ 

    Bagging

    # plot each prediction model for(i in 1:10){lines(1:155,ll[i,],col="grey",lwd=2)} # plot the average in red -lines(1:155,apply(ll,2,mean),col="red",lwd=2)
    -

    +lines(1:155,apply(ll,2,mean),col="red",lwd=2)
    +

    Bagging Algorithms

      @@ -1648,7 +1725,7 @@

      Bagging Algorithms

  • example
  • -
    # load relevant package and data
    +
    # load relevant package and data
     library(party); data(ozone,package="ElemStatLearn")
     # reorder rows based on ozone variable
     ozone <- ozone[order(ozone$ozone),]
    @@ -1667,9 +1744,9 @@ 

    Bagging Algorithms

    # plot the first fit points(ozone$ozone,predict(treebag$fits[[1]]$fit,predictors),pch=19,col="red") # plot the aggregated predictions -points(ozone$ozone,predict(treebag,predictors),pch=19,col="blue")
    -

    -

    \(\pagebreak\)

    +points(ozone$ozone,predict(treebag,predictors),pch=19,col="blue")
    +

    +

    \(\pagebreak\)

    @@ -1680,7 +1757,7 @@

    Random Forest

  • one of the most used/accurate algorithms along with boosting
  • -

    +

    • process
        @@ -1707,7 +1784,7 @@

        R Commands and Examples

      • rf$finalModel$prox = returns matrix of proximities
    • ntree=500 = specify number of trees that should be constructed
    • -
    • do.trace=TRUE = prints logs as the trees are being built \(\rightarrow\) useful by indicating progress to user
    • +
    • do.trace=TRUE = prints logs as the trees are being built \(\rightarrow\) useful by indicating progress to user
    • Note: randomForest() function can be used to perform random forest algorithm (syntax is the same as train) and is much faster
  • getTree(rf$finalModel, k=2) = return specific tree from random forest model
  • @@ -1725,7 +1802,7 @@

    R Commands and Examples

  • example
  • -
    # load data
    +
    # load data
     data(iris)
     # create train/test data sets
     inTrain <- createDataPartition(y=iris$Species,p=0.7, list=FALSE)
    @@ -1734,38 +1811,38 @@ 

    R Commands and Examples

    # apply random forest modFit <- train(Species~ .,data=training,method="rf",prox=TRUE) # return the second tree (first 6 rows) -head(getTree(modFit$finalModel,k=2))
    -
    ##   left daughter right daughter split var split point status prediction
    -## 1             2              3         4        0.70      1          0
    -## 2             0              0         0        0.00     -1          1
    -## 3             4              5         4        1.75      1          0
    -## 4             6              7         3        5.30      1          0
    -## 5             0              0         0        0.00     -1          3
    -## 6             8              9         3        4.95      1          0
    -
    # compute cluster centers
    +head(getTree(modFit$finalModel,k=2))
    +
      left daughter right daughter split var split point status prediction
    +1             2              3         3        2.60      1          0
    +2             0              0         0        0.00     -1          1
    +3             4              5         4        1.65      1          0
    +4             6              7         3        5.25      1          0
    +5             8              9         3        4.85      1          0
    +6             0              0         0        0.00     -1          2
    +
    # compute cluster centers
     irisP <- classCenter(training[,c(3,4)], training$Species, modFit$finalModel$prox)
     # convert irisP to data frame and add Species column
     irisP <- as.data.frame(irisP); irisP$Species <- rownames(irisP)
     # plot data points
     p <- qplot(Petal.Width, Petal.Length, col=Species,data=training)
     # add the cluster centers
    -p + geom_point(aes(x=Petal.Width,y=Petal.Length,col=Species),size=5,shape=4,data=irisP)
    -

    -
    # predict outcome for test data set using the random forest model
    +p + geom_point(aes(x=Petal.Width,y=Petal.Length,col=Species),size=5,shape=4,data=irisP)
    +

    +
    # predict outcome for test data set using the random forest model
     pred <- predict(modFit,testing)
     # logic value for whether or not the rf algorithm predicted correctly
     testing$predRight <- pred==testing$Species
     # tabulate results
    -table(pred,testing$Species)
    -
    ##             
    -## pred         setosa versicolor virginica
    -##   setosa         15          0         0
    -##   versicolor      0         15         2
    -##   virginica       0          0        13
    -
    # plot data points with the incorrect classification highlighted
    -qplot(Petal.Width,Petal.Length,colour=predRight,data=testing,main="newdata Predictions")
    -

    -

    \(\pagebreak\)

    +table(pred,testing$Species)
    +
                
    +pred         setosa versicolor virginica
    +  setosa         15          0         0
    +  versicolor      0         14         1
    +  virginica       0          1        14
    +
    # plot data points with the incorrect classification highlighted
    +qplot(Petal.Width,Petal.Length,colour=predRight,data=testing,main="newdata Predictions")
    +

    +

    \(\pagebreak\)

    @@ -1774,28 +1851,28 @@

    Boosting

  • boosting = one of the most widely used and accurate prediction models, along with random forest
  • boosting can be done with any set of classifiers, and a well-known approach is gradient boosting
  • more detail tutorial can be found here

  • -
  • process: take a group of weak predictors \(\rightarrow\) weight them and add them up \(\rightarrow\) result in a stronger predictor +
  • process: take a group of weak predictors \(\rightarrow\) weight them and add them up \(\rightarrow\) result in a stronger predictor
      -
    • start with a set of classifiers \(h_1, \ldots, h_k\) +
    • start with a set of classifiers \(h_1, \ldots, h_k\)
      • examples: all possible trees, all possible regression models, all possible cutoffs (divide data into different parts)
    • -
    • calculate a weighted sum of classifiers as the prediction value \[f(x) = \sum_i \alpha_i h_i(x)\] where \(\alpha_i\) = coefficient/weight and \(h_i(x)\) = value of classifier +
    • calculate a weighted sum of classifiers as the prediction value \[f(x) = \sum_i \alpha_i h_i(x)\] where \(\alpha_i\) = coefficient/weight and \(h_i(x)\) = value of classifier
      • goal = minimize error (on training set)
      • -
      • select one \(h\) at each step (iterative)
      • +
      • select one \(h\) at each step (iterative)
      • calculate weights based on errors
      • -
      • up-weight missed classifications and select next \(h\)
      • +
      • up-weight missed classifications and select next \(h\)
  • example

  • -

    +

    • we start with space with blue + and red - and the goal is to classify all the object correctly
    • only straight lines will be used for classification
    -

    +

    • from the above, we can see that a group of weak predictors (lines in this case), can be combined and weighed to become a much stronger predictor
    @@ -1816,7 +1893,7 @@

    R Commands and Examples

  • predict function can be used to apply the model to test data, similar to the rest of the algorithms in caret package

  • example

  • -
    # load data
    +
    # load data
     data(Wage)
     # remove log wage variable (we are trying to predict wage)
     Wage <- subset(Wage,select=-c(logwage))
    @@ -1826,37 +1903,35 @@ 

    R Commands and Examples

    # run the gbm model modFit <- train(wage ~ ., method="gbm",data=training,verbose=FALSE) # print model summary -print(modFit)
    -
    ## Stochastic Gradient Boosting 
    -## 
    -## 2102 samples
    -##   10 predictor
    -## 
    -## No pre-processing
    -## Resampling: Bootstrapped (25 reps) 
    -## 
    -## Summary of sample sizes: 2102, 2102, 2102, 2102, 2102, 2102, ... 
    -## 
    -## Resampling results across tuning parameters:
    -## 
    -##   interaction.depth  n.trees  RMSE      Rsquared   RMSE SD   Rsquared SD
    -##   1                   50      35.64972  0.3317422  1.495165  0.02312183 
    -##   1                  100      34.95593  0.3429594  1.503223  0.02319651 
    -##   1                  150      34.84473  0.3451634  1.496016  0.02324176 
    -##   2                   50      34.91119  0.3462101  1.490004  0.02425481 
    -##   2                  100      34.74433  0.3487227  1.480423  0.02278780 
    -##   2                  150      34.74823  0.3487136  1.472941  0.02314265 
    -##   3                   50      34.83480  0.3467828  1.493846  0.02292988 
    -##   3                  100      34.85342  0.3449018  1.482881  0.02373242 
    -##   3                  150      34.99413  0.3401694  1.544378  0.02498133 
    -## 
    -## Tuning parameter 'shrinkage' was held constant at a value of 0.1
    -## 
    -## Tuning parameter 'n.minobsinnode' was held constant at a value of 10
    -## RMSE was used to select the optimal model using  the smallest value.
    -## The final values used for the model were n.trees = 100,
    -##  interaction.depth = 2, shrinkage = 0.1 and n.minobsinnode = 10.
    -

    \(\pagebreak\)

    +print(modFit)
    +
    Stochastic Gradient Boosting 
    +
    +2102 samples
    +  10 predictors
    +
    +No pre-processing
    +Resampling: Bootstrapped (25 reps) 
    +Summary of sample sizes: 2102, 2102, 2102, 2102, 2102, 2102, ... 
    +Resampling results across tuning parameters:
    +
    +  interaction.depth  n.trees  RMSE      Rsquared 
    +  1                   50      35.24716  0.3061446
    +  1                  100      34.74137  0.3165154
    +  1                  150      34.70108  0.3179425
    +  2                   50      34.67703  0.3199690
    +  2                  100      34.58430  0.3226554
    +  2                  150      34.67104  0.3202411
    +  3                   50      34.59842  0.3219575
    +  3                  100      34.73009  0.3183701
    +  3                  150      34.97108  0.3107605
    +
    +Tuning parameter 'shrinkage' was held constant at a value of 0.1
    +
    +Tuning parameter 'n.minobsinnode' was held constant at a value of 10
    +RMSE was used to select the optimal model using  the smallest value.
    +The final values used for the model were n.trees = 100,
    + interaction.depth = 2, shrinkage = 0.1 and n.minobsinnode = 10. 
    +

    \(\pagebreak\)

    @@ -1869,34 +1944,34 @@

    Model Based Prediction

  • can be reasonably accurate on real problems
  • this approach does make additional assumptions about the data, which can lead to model failure/reduced accuracy if they are too far off
  • -
  • goal = build parameter-based model (based on probabilities) for conditional distribution \(P(Y = k~|~X = x)\), or the probability of the outcome \(Y\) is equal to a particular value \(k\) given a specific set of predictor variables \(x\) +
  • goal = build parameter-based model (based on probabilities) for conditional distribution \(P(Y = k~|~X = x)\), or the probability of the outcome \(Y\) is equal to a particular value \(k\) given a specific set of predictor variables \(x\)
      -
    • Note: \(X\) is the data for the model (observations for all predictor variables), which is also known as the design matrix
    • +
    • Note: \(X\) is the data for the model (observations for all predictor variables), which is also known as the design matrix
  • typical approach/process
      -
    1. start with the quantity \(P(Y = k~|~X = x)\)
    2. -
    3. apply Bayes’ Theorem such that \[ P(Y = k ~|~ X=x) = \frac{P(X=x~|~Y=k)P(Y=k)}{\sum_{\ell=1}^K P(X=x ~|~Y = \ell) P(Y=\ell)}\] where the denominator is simply the sum of probabilities for the predictor variables are the set specified in \(x\) for all outcomes of \(Y\)
    4. -
    5. assume the term \(P(X=x~|~Y=k)\) in the numerator follows a parameter-based probability distribution, or \(f_k(x)\) +
    6. start with the quantity \(P(Y = k~|~X = x)\)
    7. +
    8. apply Bayes’ Theorem such that \[ P(Y = k ~|~ X=x) = \frac{P(X=x~|~Y=k)P(Y=k)}{\sum_{\ell=1}^K P(X=x ~|~Y = \ell) P(Y=\ell)}\] where the denominator is simply the sum of probabilities for the predictor variables are the set specified in \(x\) for all outcomes of \(Y\)
    9. +
    10. assume the term \(P(X=x~|~Y=k)\) in the numerator follows a parameter-based probability distribution, or \(f_k(x)\)
        -
      • common choice = Gaussian distribution \[f_k(x) = \frac{1}{\sigma_k \sqrt{2 \pi}}e^{-\frac{(x-\mu_k)^2}{2\sigma_k^2}}\]
      • +
      • common choice = Gaussian distribution \[f_k(x) = \frac{1}{\sigma_k \sqrt{2 \pi}}e^{-\frac{(x-\mu_k)^2}{2\sigma_k^2}}\]
    11. -
    12. assume the probability for the outcome \(Y\) to take on value of \(k\), or \(P(Y=k)\), is determined from the data to be some known quantity \(\pi_k\) +
    13. assume the probability for the outcome \(Y\) to take on value of \(k\), or \(P(Y=k)\), is determined from the data to be some known quantity \(\pi_k\)
    14. -
    15. so the quantity \(P(Y = k~|~X = x)\) can be rewritten as \[P(Y = k ~|~ X=x) = \frac{f_k(x) \pi_k}{\sum_{\ell = 1}^K f_{\ell}(x) \pi_{\ell}}\]
    16. -
    17. estimate the parameters (\(\mu_k\), \(\sigma_k^2\)) for the function \(f_k(x)\) from the data
    18. -
    19. calculate \(P(Y = k~|~X = x)\) using the parameters
    20. -
    21. the outcome \(Y\) is where the value of \(P(Y = k ~|~ X = x)\) is the highest
    22. +
    23. so the quantity \(P(Y = k~|~X = x)\) can be rewritten as \[P(Y = k ~|~ X=x) = \frac{f_k(x) \pi_k}{\sum_{\ell = 1}^K f_{\ell}(x) \pi_{\ell}}\]
    24. +
    25. estimate the parameters (\(\mu_k\), \(\sigma_k^2\)) for the function \(f_k(x)\) from the data
    26. +
    27. calculate \(P(Y = k~|~X = x)\) using the parameters
    28. +
    29. the outcome \(Y\) is where the value of \(P(Y = k ~|~ X = x)\) is the highest
  • prediction models that leverage this approach
      -
    • linear discriminant analysis = assumes \(f_k(x)\) is multivariate Gaussian distribution with same covariance for each predictor variables +
    • linear discriminant analysis = assumes \(f_k(x)\) is multivariate Gaussian distribution with same covariance for each predictor variables
      • effectively drawing lines through “covariate space”
    • -
    • quadratic discriminant analysis = assumes \(f_k(x)\) is multivariate Gaussian distribution with different covariance for predictor variables +
    • quadratic discriminant analysis = assumes \(f_k(x)\) is multivariate Gaussian distribution with different covariance for predictor variables
      • effectively drawing curves through “covariate space”
    • @@ -1910,35 +1985,36 @@

      Model Based Prediction

      Linear Discriminant Analysis

        -
      • to compare the probability for outcome \(Y = k\) versus probability for outcome \(Y = k\), we can look at the ratio of \[\frac{P(Y=k~|~X=x)}{P(Y=j~|~X=x)}\]
      • -
      • take the log of the ratio and apply Bayes’ Theorem, we get \[\log \frac{P(Y = k~|~X=x)}{P(Y = j~|~X=x)} = \log \frac{f_k(x)}{f_j(x)} + \log \frac{\pi_k}{\pi_j}\] which is effectively the log ratio of probability density functions plus the log ratio of prior probabilities +
      • to compare the probability for outcome \(Y = k\) versus probability for outcome \(Y = k\), we can look at the ratio of \[\frac{P(Y=k~|~X=x)}{P(Y=j~|~X=x)}\]
      • +
      • take the log of the ratio and apply Bayes’ Theorem, we get \[\log \frac{P(Y = k~|~X=x)}{P(Y = j~|~X=x)} = \log \frac{f_k(x)}{f_j(x)} + \log \frac{\pi_k}{\pi_j}\] which is effectively the log ratio of probability density functions plus the log ratio of prior probabilities
          -
        • Note: \(\log\) = monotone transformation, which means taking the \(\log\) of a quantity does not affect implication of the ratio since th \(\log(ratio)\) is directly correlated with ratio
        • +
        • Note: \(\log\) = monotone transformation, which means taking the \(\log\) of a quantity does not affect implication of the ratio since th \(\log(ratio)\) is directly correlated with ratio
      • -
      • if we substitute \(f_k(x)\) and \(f_l(x)\) with Gaussian probability density functions \[f(x) = \frac{1}{\sigma \sqrt{2 \pi}}e^{-\frac{(x-\mu)^2}{2\sigma^2}}\] so the ratio can be simplified to \[\log \frac{P(Y = k~|~X=x)}{P(Y = j~|~X=x)} = \log \frac{\pi_k}{\pi_j} - \frac{1}{2}(\mu_k + \mu_j)^T \Sigma^{-1}(\mu_k + \mu_j) + x^T \Sigma^{-1} (\mu_k - \mu_j)\] where \(\Sigma^{-1}\) = covariance matrix for the predictor variables, \(x^T\) = set of predictor variables, and \(\mu_k\) / \(\mu_j\) = mean of \(k\), \(j\) respectively
      • -
      • as annotated above, the log-ratio is effectively an equation of a line for a set of predictor variables \(x\) \(\rightarrow\) the first two terms are constants and the last term is in the form of \(X \beta\) +
      • if we substitute \(f_k(x)\) and \(f_l(x)\) with Gaussian probability density functions \[f(x) = \frac{1}{\sigma \sqrt{2 \pi}}e^{-\frac{(x-\mu)^2}{2\sigma^2}}\] so the ratio can be simplified to \[\log \frac{P(Y = k~|~X=x)}{P(Y = j~|~X=x)} = \log \frac{\pi_k}{\pi_j} - \frac{1}{2}(\mu_k + \mu_j)^T \Sigma^{-1}(\mu_k + \mu_j) + x^T \Sigma^{-1} (\mu_k - \mu_j)\] where \(\Sigma^{-1}\) = covariance matrix for the predictor variables, \(x^T\) = set of predictor variables, and \(\mu_k\) / \(\mu_j\) = mean of \(k\), \(j\) respectively
      • +
      • as annotated above, the log-ratio is effectively an equation of a line for a set of predictor variables \(x\) \(\rightarrow\) the first two terms are constants and the last term is in the form of \(X \beta\)
        • Note: the lines are also known as decision boundaries
      • -
      • therefore, we can classify values based on which side of the line the value is located (\(k\) vs \(j\))
      • -
      • discriminant functions are used to determine value of \(k\), the functions are in the form of \[\delta_k(x) = x^T \Sigma^{-1} \mu_k - \frac{1}{2}\mu_k \Sigma^{-1}\mu_k + log(\mu_k)\] +
      • therefore, we can classify values based on which side of the line the value is located (\(k\) vs \(j\))
      • +
      • discriminant functions are used to determine value of \(k\), the functions are in the form of \[\delta_k(x) = x^T \Sigma^{-1} \mu_k - \frac{1}{2}\mu_k \Sigma^{-1}\mu_k + log(\mu_k)\]
          -
        • plugging in the set of predictor variables, \(x^T\), into the discriminant function, we can find the value of \(k\) that maximizes the function \(\delta_k(x)\)
        • +
        • plugging in the set of predictor variables, \(x^T\), into the discriminant function, we can find the value of \(k\) that maximizes the function \(\delta_k(x)\)
        • the terms of the discriminant function can be estimated using maximum likelihood
      • -
      • the predicted value for the outcome is therefore \(\hat{Y}(x) = argmax_k \delta_k(x)\)

      • +
      • the predicted value for the outcome is therefore \(\hat{Y}(x) = argmax_k \delta_k(x)\)

      • example
          -
        • classify a group of values into 3 groups using 2 variables (\(x\), \(y\) coordinates)
        • +
        • classify a group of values into 3 groups using 2 variables (\(x\), \(y\) coordinates)
        • 3 lines are draw to split the data into 3 Gaussian distributions
            -
          • each line splits the data into two groups \(\rightarrow\) 1 vs 2, 2 vs 3, 1 vs 3
          • -
          • each side of the line represents a region where the probability of one group (1, 2, or 3) is the highest
          • +
          • each line splits the data into two groups \(\rightarrow\) 1 vs 2, 2 vs 3, 1 vs 3
          • +
          • each side of the line represents a region where the probability of one group (1, 2, or 3) is the highest
            +
        • the result is represented in the the following graph
      -

      +

      • R Commands
          @@ -1947,7 +2023,7 @@

          Linear Discriminant Analysis

        • example: caret package
      -
      # load data
      +
      # load data
       data(iris)
       # create training and test sets
       inTrain <- createDataPartition(y=iris$Species,p=0.7, list=FALSE)
      @@ -1958,27 +2034,27 @@ 

      Linear Discriminant Analysis

      # predict test outcomes using LDA model pred.lda <- predict(lda,testing) # print results -pred.lda
      -
      ##  [1] setosa     setosa     setosa     setosa     setosa     setosa    
      -##  [7] setosa     setosa     setosa     setosa     setosa     setosa    
      -## [13] setosa     setosa     setosa     versicolor versicolor versicolor
      -## [19] versicolor versicolor versicolor versicolor versicolor versicolor
      -## [25] versicolor versicolor versicolor versicolor versicolor versicolor
      -## [31] virginica  virginica  virginica  virginica  virginica  virginica 
      -## [37] virginica  virginica  virginica  virginica  virginica  virginica 
      -## [43] virginica  virginica  virginica 
      -## Levels: setosa versicolor virginica
      +pred.lda
      +
       [1] setosa     setosa     setosa     setosa     setosa     setosa    
      + [7] setosa     setosa     setosa     setosa     setosa     setosa    
      +[13] setosa     setosa     setosa     versicolor versicolor versicolor
      +[19] versicolor versicolor versicolor versicolor versicolor versicolor
      +[25] versicolor versicolor versicolor versicolor versicolor versicolor
      +[31] virginica  virginica  virginica  virginica  virginica  virginica 
      +[37] virginica  virginica  virginica  virginica  virginica  virginica 
      +[43] virginica  virginica  virginica 
      +Levels: setosa versicolor virginica
  • Naive Bayes

      -
    • for predictors \(X_1,\ldots,X_m\), we want to model \(P(Y = k ~|~ X_1,\ldots,X_m)\)
    • -
    • by applying Bayes’ Theorem, we get \[P(Y = k ~|~ X_1,\ldots,X_m) = \frac{\pi_k P(X_1,\ldots,X_m~|~ Y=k)}{\sum_{\ell = 1}^K P(X_1,\ldots,X_m ~|~ Y=k) \pi_{\ell}}\]
    • -
    • since the denominator is just a sum (constant), we can rewrite the quantity as \[P(Y = k ~|~ X_1,\ldots,X_m) \propto \pi_k P(X_1,\ldots,X_m~|~ Y=k)\] or the probability is proportional to the numerator +
    • for predictors \(X_1,\ldots,X_m\), we want to model \(P(Y = k ~|~ X_1,\ldots,X_m)\)
    • +
    • by applying Bayes’ Theorem, we get \[P(Y = k ~|~ X_1,\ldots,X_m) = \frac{\pi_k P(X_1,\ldots,X_m~|~ Y=k)}{\sum_{\ell = 1}^K P(X_1,\ldots,X_m ~|~ Y=k) \pi_{\ell}}\]
    • +
    • since the denominator is just a sum (constant), we can rewrite the quantity as \[P(Y = k ~|~ X_1,\ldots,X_m) \propto \pi_k P(X_1,\ldots,X_m~|~ Y=k)\] or the probability is proportional to the numerator
      • Note: maximizing the numerator is the same as maximizing the ratio
    • -
    • \(\pi_k P(X_1,\ldots,X_m~|~ Y=k)\) can be rewritten as \[\begin{aligned} +
    • \(\pi_k P(X_1,\ldots,X_m~|~ Y=k)\) can be rewritten as \[\begin{aligned} \pi_k P(X_1,\ldots,X_m~|~ Y=k) & = \pi_k P(X_1 ~|~ Y = k)P(X_2,\ldots,X_m ~|~ X_1,Y=k) \\ & = \pi_k P(X_1 ~|~ Y = k) P(X_2 ~|~ X_1, Y=k) P(X_3,\ldots,X_m ~|~ X_1,X_2, Y=k) \\ & = \pi_k P(X_1 ~|~ Y = k) P(X_2 ~|~ X_1, Y=k)\ldots P(X_m~|~X_1\ldots,X_{m-1},Y=k) \\ @@ -1986,7 +2062,7 @@

      Naive Bayes

      • this is effectively indicating that each of the predictors may be dependent on other predictors
    • -
    • however, if we make the assumption that all predictor variables are independent to each other, the quantity can be simplified to \[ \pi_k P(X_1,\ldots,X_m~|~ Y=k) \approx \pi_k P(X_1 ~|~ Y = k) P(X_2 ~|~ Y = k)\ldots P(X_m ~|~,Y=k)\] which is effectively the product of the prior probability for \(k\) and the probability of variables \(X_1,\ldots,X_m\) given that \(Y = k\) +
    • however, if we make the assumption that all predictor variables are independent to each other, the quantity can be simplified to \[ \pi_k P(X_1,\ldots,X_m~|~ Y=k) \approx \pi_k P(X_1 ~|~ Y = k) P(X_2 ~|~ Y = k)\ldots P(X_m ~|~,Y=k)\] which is effectively the product of the prior probability for \(k\) and the probability of variables \(X_1,\ldots,X_m\) given that \(Y = k\)
      • Note: the assumption is naive in that it is unlikely the predictors are completely independent of each other, but this model still produces useful results particularly with large number of binary/categorical variables
          @@ -2000,21 +2076,22 @@

          Naive Bayes

        • example: caret package
      -
      # using the same data from iris, run naive Bayes on training data
      +
      # package needed: klaR
      +# using the same data from iris, run naive Bayes on training data
       nb <- train(Species ~ ., data=training,method="nb")
       # predict test outcomes using naive Bayes model
       pred.nb <- predict(nb,testing)
       # print results
      -pred.nb
      -
      ##  [1] setosa     setosa     setosa     setosa     setosa     setosa    
      -##  [7] setosa     setosa     setosa     setosa     setosa     setosa    
      -## [13] setosa     setosa     setosa     versicolor versicolor versicolor
      -## [19] versicolor versicolor versicolor versicolor versicolor versicolor
      -## [25] versicolor versicolor versicolor versicolor versicolor versicolor
      -## [31] virginica  virginica  virginica  virginica  virginica  virginica 
      -## [37] virginica  virginica  virginica  virginica  virginica  virginica 
      -## [43] virginica  virginica  virginica 
      -## Levels: setosa versicolor virginica
      +pred.nb
    +
     [1] setosa     setosa     setosa     setosa     setosa     setosa    
    + [7] setosa     setosa     setosa     setosa     setosa     setosa    
    +[13] setosa     setosa     setosa     versicolor versicolor versicolor
    +[19] versicolor versicolor versicolor versicolor versicolor versicolor
    +[25] versicolor versicolor versicolor versicolor versicolor versicolor
    +[31] virginica  virginica  virginica  virginica  virginica  virginica 
    +[37] versicolor virginica  virginica  virginica  virginica  virginica 
    +[43] virginica  virginica  virginica 
    +Levels: setosa versicolor virginica

    Compare Results for LDA and Naive Bayes

    @@ -2022,22 +2099,22 @@

    Compare Results for LDA and Naive Bayes

  • linear discriminant analysis and naive Bayes generally produce similar results for small data sets
  • for our example data from iris data set, we can compare the prediction the results from the two models
  • -
    # tabulate the prediction results from LDA and naive Bayes
    -table(pred.lda,pred.nb)
    -
    ##             pred.nb
    -## pred.lda     setosa versicolor virginica
    -##   setosa         15          0         0
    -##   versicolor      0         15         0
    -##   virginica       0          0        15
    -
    # create logical variable that returns TRUE for when predictions from the two models match
    +
    # tabulate the prediction results from LDA and naive Bayes
    +table(pred.lda,pred.nb)
    +
                pred.nb
    +pred.lda     setosa versicolor virginica
    +  setosa         15          0         0
    +  versicolor      0         15         0
    +  virginica       0          1        14
    +
    # create logical variable that returns TRUE for when predictions from the two models match
     equalPredictions <- (pred.lda==pred.nb)
     # plot the comparison
    -qplot(Petal.Width,Sepal.Width,colour=equalPredictions,data=testing)
    -

    +qplot(Petal.Width,Sepal.Width,colour=equalPredictions,data=testing)
    +

    • as we can see from above, only one data point, which is located inbetween the two classes is predicted differently by the two models
    -

    \(\pagebreak\)

    +

    \(\pagebreak\)

    @@ -2049,7 +2126,7 @@

    Model Selection

  • the error for the prediction model on test set decreases first and then increases as number of predictors used approaches the total number of predictors available
  • -

    +

    • this is expected since as more predictors used, the model is more likely to overfit the training data
    • goal in selecting models = avoid overfitting on training data and minimize error on test data
    • @@ -2077,7 +2154,7 @@

      Example: Training vs Test Error for Combination of Predictors

    • to demonstrate the behavior of training and test errors, the prostate dataset from Elements of Statistical Learning is used
    • all combinations of predictors are used to produce prediction models, and Residual Squared Error (RSS) is calculated for all models on both the training and test sets
    -
    # load data and set seed
    +
    # load data and set seed
     data(prostate); set.seed(1)
     # define outcome y and predictors x
     covnames <- names(prostate[-(9:10)])
    @@ -2133,8 +2210,8 @@ 

    Example: Training vs Test Error for Combination of Predictors

    # plot line through the minimum test RSS data points in blue lines((1:p), minrss, col="red", lwd=1.7) # add legend -legend("topright", c("Train", "Test"), col=c("blue", "red"), pch=1)
    -

    +legend("topright", c("Train", "Test"), col=c("blue", "red"), pch=1)
    +

    • from the above, we can clearly that test RSS error approaches the minimum at around 3 predictors and increases slightly as more predictors are used
    @@ -2161,16 +2238,16 @@

    Split Samples

    Decompose Expected Prediction Error

      -
    • the outcome \(Y_i\) can be modeled by \[Y_i = f(X_i) + \epsilon_i\] where \(\epsilon_i\) = error term
    • -
    • the expected prediction error is defined as \[EPE(\lambda) = E\left[\left(Y - \hat{f}_{\lambda}(X)\right)^2\right]\] where \(\lambda\) = specific set of tuning parameters
    • -
    • estimates from the model constructed with training data can be denoted as \(\hat{f}_{\lambda}(x^*)\) where \(X = x^*\) is the new data point that we would like to predict at
    • -
    • the expected prediction error is as follows \[\begin{aligned} +
    • the outcome \(Y_i\) can be modeled by \[Y_i = f(X_i) + \epsilon_i\] where \(\epsilon_i\) = error term
    • +
    • the expected prediction error is defined as \[EPE(\lambda) = E\left[\left(Y - \hat{f}_{\lambda}(X)\right)^2\right]\] where \(\lambda\) = specific set of tuning parameters
    • +
    • estimates from the model constructed with training data can be denoted as \(\hat{f}_{\lambda}(x^*)\) where \(X = x^*\) is the new data point that we would like to predict at
    • +
    • the expected prediction error is as follows \[\begin{aligned} E\left[\left(Y - \hat{f}_{\lambda}(x^*)\right)^2\right] & = \sigma^2 + \left(E[\hat{f}_{\lambda}(x^*)] - f(x^*)\right)^2 + E\left[\hat{f}_{\lambda}(x^*) - E[\hat{f}_{\lambda}(x^*)]\right]^2\\ & = \mbox{Irreducible Error} + \mbox{Bias}^2 + \mbox{Variance}\\ \end{aligned} \]
      • goal of prediction model = minimize overall expected prediction error
      • -
      • irreducible error = noise inherent to the data collection process \(\rightarrow\) cannot be reduced
      • +
      • irreducible error = noise inherent to the data collection process \(\rightarrow\) cannot be reduced
      • bias/variance = can be traded in order to find optimal model (least error)
    @@ -2186,9 +2263,9 @@

    Hard Thresholding

  • hard thresholding can help estimate the coefficients/model by taking subsets of predictors and building models
  • process
      -
    • model the outcome as \[Y_i = f(X_i) + \epsilon_i\] where \(\epsilon_i\) = error term
    • -
    • assume the prediction estimate has a linear form \[\hat{f}_{\lambda}(x) = x'\beta\] where only \(\lambda\) coefficients for the set of predictors \(x\) are nonzero
    • -
    • after setting the value of \(\lambda\), compute models using all combinations of \(\lambda\) variables to find which variables’ coefficients should be set to be zero
    • +
    • model the outcome as \[Y_i = f(X_i) + \epsilon_i\] where \(\epsilon_i\) = error term
    • +
    • assume the prediction estimate has a linear form \[\hat{f}_{\lambda}(x) = x'\beta\] where only \(\lambda\) coefficients for the set of predictors \(x\) are nonzero
    • +
    • after setting the value of \(\lambda\), compute models using all combinations of \(\lambda\) variables to find which variables’ coefficients should be set to be zero
  • problem
      @@ -2199,43 +2276,43 @@

      Hard Thresholding

    • as we can see from the results below, some of the coefficients have values of NA
  • -
    # load prostate data
    +
    # load prostate data
     data(prostate)
     # create subset of observations with 10 variables
     small = prostate[1:5,]
     # print linear regression
    -lm(lpsa ~ .,data =small)
    -
    ## 
    -## Call:
    -## lm(formula = lpsa ~ ., data = small)
    -## 
    -## Coefficients:
    -## (Intercept)       lcavol      lweight          age         lbph  
    -##     9.60615      0.13901     -0.79142      0.09516           NA  
    -##         svi          lcp      gleason        pgg45    trainTRUE  
    -##          NA           NA     -2.08710           NA           NA
    +lm(lpsa ~ .,data =small)
    +
    
    +Call:
    +lm(formula = lpsa ~ ., data = small)
    +
    +Coefficients:
    +(Intercept)       lcavol      lweight          age         lbph  
    +    9.60615      0.13901     -0.79142      0.09516           NA  
    +        svi          lcp      gleason        pgg45    trainTRUE  
    +         NA           NA     -2.08710           NA           NA  

    Regularized Regression Concept (Resource)

    -

    +

    Regularized Regression - LASSO Regression

    • LASSO (least absolute shrinkage and selection operator) was introduced by Tibshirani (Journal of the Royal Statistical Society 1996)
    • similar to ridge, with slightly different penalty term
    • -
    • the penalized residual sum of squares (PRSS) takes the form of \[PRSS(\beta_j) = \sum_{i=1}^N \left(y_i - \beta_0 - \sum_{j=1}^p x_{ij}\beta_j \right)^2 + \lambda \sum_{j=1}^p |\beta_j|\]
    • -
    • this is equivalent to solving the equation \[PRSS(\beta_j) = \sum_{i=1}^N \left(y_i - \beta_0 - \sum_{j=1}^p x_{ij}\beta_j \right)^2\] subject to constraint \(\sum_{j=1}^p |\beta_j| \leq s\) where \(s\) is inversely proportional to \(\lambda\)
    • -
    • \(\lambda\) = tuning parameter +
    • the penalized residual sum of squares (PRSS) takes the form of \[PRSS(\beta_j) = \sum_{i=1}^N \left(y_i - \beta_0 - \sum_{j=1}^p x_{ij}\beta_j \right)^2 + \lambda \sum_{j=1}^p |\beta_j|\]
    • +
    • this is equivalent to solving the equation \[PRSS(\beta_j) = \sum_{i=1}^N \left(y_i - \beta_0 - \sum_{j=1}^p x_{ij}\beta_j \right)^2\] subject to constraint \(\sum_{j=1}^p |\beta_j| \leq s\) where \(s\) is inversely proportional to \(\lambda\)
    • +
    • \(\lambda\) = tuning parameter
      • controls size of coefficients or the amount of regularization
      • -
      • large values of \(\lambda\) will set some coefficient equal to zero +
      • large values of \(\lambda\) will set some coefficient equal to zero
        • Note: LASSO effectively performs model selection (choose subset of predictors) while shrinking other coefficients, where as Ridge only shrinks the coefficients
      • @@ -2323,7 +2400,7 @@

        Regularized Regression - LASSO Regression

      • Note: the predictors are centered and scaled first before the regression is run
      • as.matrix(x) = the predictors must be in matrix/dataframe format
      • trace=TRUE = prints progress of the lasso regression
      • -
      • lasso$lambda = return the \(\lambda\)s used for each step of the lasso regression
      • +
      • lasso$lambda = return the \(\lambda\)s used for each step of the lasso regression
      • plot(lasso) = prints plot that shows the progression of the coefficients as they are set to zero one by one
      • predit.lars(lasso, test) = use the lasso model to predict on test data
          @@ -2338,14 +2415,14 @@

          Regularized Regression - LASSO Regression

      • [enet package] lasso<-enet(predictors, outcome, lambda = 0) = perform elastic net regression on given predictors and outcome
          -
        • lambda=0 = default value for \(\lambda\) +
        • lambda=0 = default value for \(\lambda\)
          • Note: lasso regression is a special case of elastic net regression, and forcing lambda=0 tells the function to fit a lasso regression
        • plot(lasso) = prints plot that shows the progression of the coefficients as they are set to zero one by one
        • predict.ent(lasso, test)= use the lasso model to predict on test data
      • -
      • [caret package] train(outcome ~ predictors, data=training, method="lasso") = perform lasso regression with given outcome and predictors +
      • caret package train(outcome ~ predictors, data=training, method="lasso") = perform lasso regression with given outcome and predictors
        • Note: outcome and predictors must be in the same dataframe
        • preProcess=c("center", "scale") = centers and scales the predictors before the model is built @@ -2353,7 +2430,7 @@

          Regularized Regression - LASSO Regression

        • Note: this is generally a good idea for building lasso regressions
    • -
    • [caret package] train(outcome~predictors,data=train,method="relaxo",lambda=5,phi=0.3) = perform relaxed lasso regression on given predictors and outcome +
    • caret package train(outcome~predictors,data=train,method="relaxo",lambda=5,phi=0.3) = perform relaxed lasso regression on given predictors and outcome
      • lambda=5 = tuning parameter
      • phi=0.3 = relaxation parameter @@ -2362,11 +2439,11 @@

        Regularized Regression - LASSO Regression

      • phi=0 computes the OLS estimates on the set of variables selected by the Lasso
    -
  • [caret package] predict(model,test) = use the model to predict on test set \(\rightarrow\) similar to all other caret algorithms
  • +
  • caret package predict(model,test) = use the model to predict on test set \(\rightarrow\) similar to all other caret algorithms
  • example: lars package
  • -
    # load lars package
    +
    # load lars package
     library(lars)
     # perform lasso regression
     lasso.fit <- lars(as.matrix(x), y, type="lasso", trace=TRUE)
    @@ -2374,12 +2451,12 @@ 

    Regularized Regression - LASSO Regression

    plot(lasso.fit, breaks=FALSE, cex = 0.75) # add legend legend("topleft", covnames, pch=8, lty=1:length(covnames), - col=1:length(covnames), cex = 0.6)
    -

    -
    # plots the cross validation curve
    -lasso.cv <- cv.lars(as.matrix(x), y, K=10, type="lasso", trace=TRUE)
    -

    -

    \(\pagebreak\)

    + col=1:length(covnames), cex = 0.6)
    +

    +
    # plots the cross validation curve
    +lasso.cv <- cv.lars(as.matrix(x), y, K=10, type="lasso", trace=TRUE)
    +

    +

    \(\pagebreak\)

    @@ -2437,7 +2514,7 @@

    Example - Majority Vote

  • each has 70% accuracy
  • majority vote accuracy (mva) = probability of the majority of the models achieving 70% at the same time
  • -

    \[\begin{aligned} +

    \[\begin{aligned} \mbox{majority vote accuracy} & = p(3~correct,~2~wrong) + p(4~correct,~1~wrong) \\ &\qquad+ p(5~correct) \\ & = {5 \choose 3} \times(0.7)^3(0.3)^2 + {5 \choose 4} \times(0.7)^4(0.3)^1 - {5 \choose 5} (0.7)^5 \\ @@ -2450,7 +2527,7 @@

    Example - Majority Vote

    Example - Model Ensembling

    -
    # set up data
    +
    # set up data
     inBuild <- createDataPartition(y=Wage$wage,p=0.7, list=FALSE)
     validation <- Wage[-inBuild,]; buildData <- Wage[inBuild,]
     inTrain <- createDataPartition(y=buildData$wage,p=0.7, list=FALSE)
    @@ -2483,11 +2560,11 @@ 

    Example - Model Ensembling

    # validation data set RMSE Errors validation = c(sqrt(sum((glm.pred.val-validation$wage)^2)), sqrt(sum((rf.pred.val-validation$wage)^2)), - sqrt(sum((comb.pred.val-validation$wage)^2))))
    -
    ##                  glm        rf  combined
    -## test        858.7074  888.0702  849.3771
    -## validation 1061.0891 1086.2027 1057.8264
    -

    \(\pagebreak\)

    + sqrt(sum((comb.pred.val-validation$wage)^2))))
    +
                     glm        rf  combined
    +test        858.7074  888.5536  849.4654
    +validation 1061.0891 1085.4773 1056.5813
    +

    \(\pagebreak\)

    @@ -2530,8 +2607,8 @@

    Forecasting

  • approaches
      -
    • simple moving averages = prediction will be made for a time point by averaging together values from a number of prior periods \[ Y_{t}=\frac{1}{2*k+1}\sum_{j=-k}^k {y_{t+j}}\]

    • -
    • exponential smoothing/exponential moving average = weight time points that are closer to point of prediction than those that are further away \[\hat{y}_{t+1} = \alpha y_t + (1-\alpha)\hat{y}_{t-1}\] +
    • simple moving averages = prediction will be made for a time point by averaging together values from a number of prior periods \[ Y_{t}=\frac{1}{2*k+1}\sum_{j=-k}^k {y_{t+j}}\]

    • +
    • exponential smoothing/exponential moving average = weight time points that are closer to point of prediction than those that are further away \[\hat{y}_{t+1} = \alpha y_t + (1-\alpha)\hat{y}_{t-1}\]
      • Note: many different methods of exponential smoothing are available, more information can be found here
    • @@ -2568,7 +2645,7 @@

      R Commands and Examples

  • ts(googOpen, frequency=12) = convert data to a time series with frequency observations per time unit
      -
    • frequency=12 = number of observations per unit time (12 in this case because there are 12 months in each year \(\rightarrow\) converts data into yearly time series)
    • +
    • frequency=12 = number of observations per unit time (12 in this case because there are 12 months in each year \(\rightarrow\) converts data into yearly time series)
  • decompose(ts) = decomposes time series into trend, seasonal, and irregular components by using moving averages @@ -2607,25 +2684,25 @@

    R Commands and Examples

  • quandl package is also used for finance-related predictions
  • example: decomposed time series
  • -
    # load quantmod package
    +
    # load quantmod package
     library(quantmod);
     # specify to and from dates
     from.dat <- as.Date("01/01/00", format="%m/%d/%y")
     to.dat <- as.Date("3/2/15", format="%m/%d/%y")
     # get data for AAPL from Google Finance for the specified dates
    -getSymbols("AAPL", src="google", from = from.dat, to = to.dat)
    -
    ## [1] "AAPL"
    -
    # convert the retrieved daily data to monthly data
    +getSymbols("AAPL", src="google", from = from.dat, to = to.dat)
    +
    [1] "AAPL"
    +
    # convert the retrieved daily data to monthly data
     mAAPL <- to.monthly(AAPL)
     # extract the closing price and convert it to yearly time series (12 observations per year)
     ts <- ts(Cl(mAAPL), frequency = 12)
     # plot the decomposed parts of the time series
    -plot(decompose(ts),xlab="Years")
    -

    +plot(decompose(ts),xlab="Years")
    +

    • example: forecast
    -
    # load forecast library
    +
    # load forecast library
     library(forecast)
     # find the number of rows (years)
     rows <- ceiling(length(ts)/12)
    @@ -2636,24 +2713,24 @@ 

    R Commands and Examples

    # plot the training set plot(ts.train) # add the moving average in red -lines(ma(ts.train,order=3),col="red")
    -

    -
    # compute the exponential smoothing average
    +lines(ma(ts.train,order=3),col="red")
    +

    +
    # compute the exponential smoothing average
     ets <- ets(ts.train,model="MMM")
     # construct a forecasting model using the exponential smoothing function
     fcast <- forecast(ets)
     # plot forecast and add actual data in red
    -plot(fcast); lines(ts.test,col="red")
    -

    -
    # print the accuracy of the forecast model
    -accuracy(fcast,ts.test)
    -
    ##                      ME      RMSE       MAE       MPE     MAPE      MASE
    -## Training set  0.1188298  2.825883  1.646959  -0.61217 10.68901 0.1924329
    -## Test set     -7.8132889 16.736910 15.079222 -13.64900 20.31005 1.7618772
    -##                    ACF1 Theil's U
    -## Training set 0.09773823        NA
    -## Test set     0.84664431  3.360515
    -

    \(\pagebreak\)

    +plot(fcast); lines(ts.test,col="red")
    +

    +
    # print the accuracy of the forecast model
    +accuracy(fcast,ts.test)
    +
                        ME      RMSE       MAE        MPE     MAPE      MASE
    +Training set 0.3738734  2.576067  1.438703  0.2377946 10.38942 0.1759433
    +Test set     0.7292502 18.507031 15.347659 -3.6663982 19.01227 1.8769111
    +                  ACF1 Theil's U
    +Training set 0.1026281        NA
    +Test set     0.8661382  3.200942
    +

    \(\pagebreak\)

    @@ -2700,7 +2777,7 @@

    R Commands and Examples

  • cl_predict function in clue package provides similar functionality
  • -
    # load iris data
    +
    # load iris data
     data(iris)
     # create training and test sets
     inTrain <- createDataPartition(y=iris$Species,p=0.7, list=FALSE)
    @@ -2715,8 +2792,8 @@ 

    R Commands and Examples

    ggtitle("Clusters Classification") p2 <- qplot(Petal.Width,Petal.Length,colour=Species,data=training) + ggtitle("Species Classification (Truth)") -grid.arrange(p1, p2, ncol = 2)
    -

    +grid.arrange(p1, p2, ncol = 2)
    +

    • as we can see, there are three clear groups that emerge from the data
        @@ -2724,51 +2801,57 @@

        R Commands and Examples

      • we can compare the results from the clustering and Species classification by tabulating the values
    -
    # tabulate the results from clustering and actual species
    -table(kMeans1$cluster,training$Species)
    -
    ##    
    -##     setosa versicolor virginica
    -##   1     35          0         0
    -##   2      0          0        27
    -##   3      0         35         8
    +
    # tabulate the results from clustering and actual species
    +table(kMeans1$cluster,training$Species)
    +
       
    +    setosa versicolor virginica
    +  1      0          2        27
    +  2     35          0         0
    +  3      0         33         8
    • with the clusters determined, the training data can be trained on all predictors with the clusters from k-means as outcome
    -
    # build classification trees using the k-means cluster
    -clustering <- train(clusters ~.,data=subset(training,select=-c(Species)),method="rpart")
    +
    # build classification trees using the k-means cluster
    +clustering <- train(clusters ~.,data=subset(training,select=-c(Species)),method="rpart")
    • we can compare the prediction results on training set vs truth
    -
    # tabulate the prediction results on training set vs truth
    -table(predict(clustering,training),training$Species)
    -
    ##    
    -##     setosa versicolor virginica
    -##   1     35          0         0
    -##   2      0          0        29
    -##   3      0         35         6
    +
    # tabulate the prediction results on training set vs truth
    +table(predict(clustering,training),training$Species)
    +
       
    +    setosa versicolor virginica
    +  1      0          0        24
    +  2     35          0         0
    +  3      0         35        11
    • similarly, we can compare the prediction results on test set vs truth
    -
    # tabulate the prediction results on test set vs truth
    -table(predict(clustering,testing),testing$Species)
    -
    ##    
    -##     setosa versicolor virginica
    -##   1     15          0         0
    -##   2      0          1        12
    -##   3      0         14         3
    +
    # tabulate the prediction results on test set vs truth
    +table(predict(clustering,testing),testing$Species)
    +
       
    +    setosa versicolor virginica
    +  1      0          0        10
    +  2     15          0         0
    +  3      0         15         5
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deletions(-) diff --git a/8_PREDMACHLEARN/Practical Machine Learning Course Notes.Rmd b/8_PREDMACHLEARN/Practical Machine Learning Course Notes.Rmd index deec457..ace370a 100644 --- a/8_PREDMACHLEARN/Practical Machine Learning Course Notes.Rmd +++ b/8_PREDMACHLEARN/Practical Machine Learning Course Notes.Rmd @@ -2,13 +2,13 @@ title: "Practical Machine Learning Course Notes" author: "Xing Su" output: + pdf_document: + toc: yes + toc_depth: 3 html_document: highlight: pygments theme: spacelab toc: yes - pdf_document: - toc: yes - toc_depth: 3 header-includes: \usepackage{graphicx} \usepackage{mathtools} --- diff --git a/8_PREDMACHLEARN/Practical_Machine_Learning_Course_Notes.html b/8_PREDMACHLEARN/Practical_Machine_Learning_Course_Notes.html index 243f198..826225d 100644 --- a/8_PREDMACHLEARN/Practical_Machine_Learning_Course_Notes.html +++ b/8_PREDMACHLEARN/Practical_Machine_Learning_Course_Notes.html @@ -1644,7 +1644,7 @@

    Constructing Trees with caret Package

    7) Petal.Width>=1.65 36 2 virginica (0.00000000 0.05555556 0.94444444) *
    # plot the classification tree
     rattle::fancyRpartPlot(modFit$finalModel)
    -

    +

    # predict on test values
     predict(modFit,newdata=testing)
     [1] setosa     setosa     setosa     setosa     setosa     setosa    
    diff --git a/8_PREDMACHLEARN/Practical_Machine_Learning_Course_Notes.pdf b/8_PREDMACHLEARN/Practical_Machine_Learning_Course_Notes.pdf
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