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Wednesday 4th December
Elisa Schneider
We use the Hitters data set to predict a baseball player's Salary based on
Years: the number of years that he has played in the major leagues and
Hits: the number of hits that he made in the previous year.
Which model is better?
It depends on the problem at hand.
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If the relationship between the features and the response is well approximated by a linear model, then an approach such as linear regression will likely work well and will outperform a method such as a regression tree.
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If instead there is a highly non-linear and complex relationship between the features and the response as indicated by model, then decision trees may outperform classical approaches.
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The decision trees suffer from high variance: if we split the training data into two parts at random, and fit a decision tree to both halves, the results that we get could be quite different.
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A procedure with low variance will yield similar results; linear regression tends to have low variance.
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A natural way to reduce the variance and hence increase the prediction accuracy is to take many training sets from the population, build a separate prediction model using each training set, and average the resulting predictions.
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If we create six decision trees with sub-samples of the Boston housing data, we see that the top of the trees all have a very similar structure: Although there are 15 predictor variables, all six trees have both lstat and rm variables driving the first few splits.
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Tree correlation prevents variance reduction. The way Random Forest solve this is randomly choosing a subset of all the predictors that are used at each split.
class: small-code
library(readr)
library(randomForest)
library(MASS)
housing <- Boston
RFmodel <- randomForest(medv~., ntree=500, mtry=6, data=housing)
RFmodel
Call:
randomForest(formula = medv ~ ., data = housing, ntree = 500, mtry = 6)
Type of random forest: regression
Number of trees: 500
No. of variables tried at each split: 6
Mean of squared residuals: 9.72311
% Var explained: 88.48
class: small-code
plot(RFmodel)
varImpPlot(RFmodel)
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- ANN is an information processing model inspired by the biological neuron system.
- It is composed of a large number of interconnected processing elements: the neuron.
- ANN were designed to solve problems which are easy for humans and difficult for machines such as identifying patterns: distingushing pictures of cats and dogs or recognizing numbers in pictures
class: small-code
index <- sample(1:nrow(housing),round(0.75*nrow(housing)))
train <- housing[index,]
test <- housing[-index,] maxs <- apply(housing, 2, max)
mins <- apply(housing, 2, min)
scaled <- as.data.frame(scale(housing, center = mins, scale = maxs - mins))
train_ <- scaled[index,]
test_ <- scaled[-index,]class: small-code
library(neuralnet)
n <- names(train_)
f <- as.formula(paste("medv ~", paste(n[!n %in% "medv"], collapse = " + ")))
nn <- neuralnet(f, data=train_,hidden=c(5,3), linear.output=T)
plot (nn)
class: small-code
To test the model we can not use the same data used to fit the model. There are different strategies to test the model when we have only one data-set (this could be another hole MeetUp). One is what we did before, split the data set in two. An then calculate differen meassures to test the model:
- Regression
$$ RMSE = \sqrt\frac{\strut{\sum\limits_{i=1}^{n}{(\hat{y}{i}-y{i})^2}}}{n} $$
class: small-code
Classification (0,1)
library(verification)
verification::roc.area(obs, pred)Cassification (More than two categories)
table(obs, pred)
ggplot(longData, aes(x = Var2, y = Var1)) +
geom_raster(aes(fill=value))
library(caret)
caret::confusion.matrix(obs, pred)