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bellman.update <- function(action, state, values, gamma=1) {
    state.transition.prob <- transition[[action]]
    q <- rep(0, length(state.transition.prob))
    for(i in 1:length(state.transition.prob)) {        
        new.state <- act(names(state.transition.prob)[i], state) 
        q[i] <- (state.transition.prob[i] * (rewards[state["y"], state["x"]] + (gamma * values[new.state["y"], new.state["x"]])))
    }
    sum(q)
}

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actions <- c("N", "S", "E", "W")

x <- 1:4
y <- 1:3

rewards <- matrix(rep(0, 12), nrow=3)
rewards[2, 2] <- NA
rewards[1, 4] <- 1
rewards[2, 4] <- -1

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A = [1 1; 1 2; 1 3; 1 4]; b = [2 1 1 1]';
[m, n] = size(A);
C = A' * A; c = A' *b; bb = b'*b;
Cbar = [C c; c' bb];
Gbar = chol(Cbar)';
G = Gbar(1:2, 1:2); z = Gbar(3, 1:2)'; rho = Gbar(3, 3);
x = G'\z

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#
# Let's look at how the different models generalize between different datasets
#

n.training <- 10
n.test <- 100

error.function <- function(y, y.pred) sum((y.pred - y)^2) / 2

e.rms <- function(y, y.pred) sqrt(2 * error.function(y=y, y.pred=y.pred) / length(y))

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#
# Let's look at how the different models generalize between different datasets
#

n.training <- 10
n.test <- 100

error.function <- function(y, y.pred) sum((y.pred - y)^2) / 2

e.rms <- function(y, y.pred) sqrt(2 * error.function(y=y, y.pred=y.pred) / length(y))

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library(PolynomF)

n <- 10

f <- function(x) sin(2 * pi * x)

x <- seq(0, 1, length=n)
y <- f(x) + rnorm(n, sd=0.2)

plot(data.frame(x, y))

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# Plot the data
pairs(iris[1:4], main = "Anderson's Iris Data -- 3 species", pch = 21, bg = c("red", "green3", "blue")[unclass(iris$Species)])

# Use linear discriminant analysis
iris.lda <- lda(Species ~ ., data = iris)
summary(iris.lda)

# Use a multinomial logistic regression model
library(VGAM)
iris.vglm <- glm(Species ~ , family=multinomial, data=iris)

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num.iterations <- 1000

# Download South African heart disease data
sa.heart <- read.table("http://www-stat.stanford.edu/~tibs/ElemStatLearn/datasets/SAheart.data", sep=",",head=T,row.names=1)

x <- sa.heart[,c("age", "ldl")]
y <- sa.heart$chd
plot(x, pch=21, bg=c("red","green")[factor(y)])

# Function to standardize input values

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# Plot the sigmoid function
library(ggplot2)
qplot(-10:10, 1/(1 + exp(-(-10:10))), geom="line", xlab="z", ylab="sigmoid function")

# Download South African heart disease data
sa.heart <- read.table("http://www-stat.stanford.edu/~tibs/ElemStatLearn/datasets/SAheart.data", sep=",",head=T,row.names=1)

# Pretty plot
pairs(sa.heart[1:9],pch=21,bg=c("red","green")[factor(sa.heart$chd)])

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data <- read.csv("http://www.statalgo.com/wp-content/uploads/2011/10/housing.csv")
x <- as.matrix(cbind(intercept=rep(1, m), data[, c("area", "bedrooms")]))
theta <- solve(t(x) %*% x) %*% t(x) %*% y