lec01.rnw

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\title[BI476]{BI476: Biostatistics - Case Studies}
\author[Maoying Wu]{Maoying, Wu\\{\small ricket.woo@gmail.com}}
\institute[CBB] % (optional, but mostly needed)
{
  \inst{}
  Dept. of Bioinformatics \& Biostatistics\\
  Shanghai Jiao Tong University
}
\date{Spring, 2018}

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\section{Syllabus}

\begin{frame}
\frametitle{BI476: Syllabus}
\begin{itemize}
	\item This is a one-semester course for undergraduate students majored in biostatistics 
		or bioinformatics.  
	\item Topics will cover experiment design, intuitive hypothesis testing, (generalized 
		linear models (including generalized estimating equations), survival analysis and 
		multivariate statistics methods. 
	\item Advanced topics will be included, such as penalized regression or hierarchical 
		/mixed-effects linear models or Bayesian statistics,  if time permits.
	\item Estimation, interpretation, and diagnostic approaches will be discussed.  
	\item Software instruction will be provided in class in R.  
	\item Performance will be evaluated based on homeworks (35\%), two exams (30\%), 
		lab assignments (20\%) and projects (15\%).
	\item This is a two-credit course.
\end{itemize}
\end{frame}


\begin{frame}
\frametitle{BI476: Syllabus}
\begin{itemize}
	\item Textbooks: No textbook is required. We will provide the readings and 
		related materials on the website.
	\item Prerequisites:  Linear algebra, Probability, Biostatistics

	\item Course Objectives:
	Upon successful completion of the course, the student will be able to
	\begin{itemize}
		\item Design the experiment and analyze the experiment data 
		\item Apply, interpret and diagnose linear regression models
		\item Apply, interpret and diagnose logistic, poisson and Cox regresssion models
		\item Apply and interpret the multivariate analysis methods.
	\end{itemize}
\end{itemize}
\end{frame}


\begin{frame}[t]
\frametitle{BI476: Syllabus}
\begin{figure}[ht]
\begin{tabular}{ll}
\textbf{Instructor}:	& Maoying Wu (ricket.woo@gmail.com)\\
\textbf{Website}:	& \url{http://cbb.sjtu.edu.cn/~mywu/bi476}\\
\textbf{Github}: & \url{https://github.com/ricket.sjtu/bi476}\\
\textbf{Office}:	 & Biopharmaceutics Building, Rm 4A-223\\
\textbf{Time}:	 & Mondays, 14:00-15:40\\
\textbf{Location}:	& East Lower Hall, Rm 403\\
\textbf{Office Hours}:	 & Monday-Friday, 8:30 – 17:00\\
\end{tabular}
\end{figure}
\end{frame}


\begin{frame}
\frametitle{BI476: Schedule}
\begin{itemize}
	\item Lecture 1: Recaps of Mathematical Knowledge for Biostatistics (1-2)
	\item Lecture 2: Observational Studies and Analysis (3-4)
	\item Lecture 3: Randomized Clinical Trials and Analysis (5-6)
	\item Lecture 4: Linear Regression Models and Extensions (7-8)
	\item Midterm (9)
	\item Lecture 5: Panel Data Analysis (10-11)
	\item Lecture 6: Survival Analysis and Competing Risks (12-13) 
	\item Lecture 7: Multivariate Statistcal Analysis (14-15)
	\item Projects (16)
	\item Final (17-18)
\end{itemize}
\end{frame}


\begin{frame}
\frametitle{List of Symbols}
\begin{itemize}
	\item $\mathcal{X}$: A set
	\item $\mathbb{R}$: Real set
	\item $X$: A random variable
	\item $\mathbf{X} \in \mathbb{R}^{m \times n}$: A $m$-by-$n$ real matrix
	\item $\mathbf{x} \in \mathbb{R}^{n}$: A real vector of length $n$
	\item $x \in \mathbb{R}$: A real number
	\item $\Phi(x) = P(X \le x)$: Cumulative distribution function
	\item $f(x) = f(X=x)$： Probability density function  
\end{itemize}
\end{frame}

\begin{frame}[t]
\frametitle{Lecture 1: Mathematics for Biostatistics}
\begin{itemize}
	\item Calculus (微积分)
	\begin{itemize}
		\item Limits (极限)
		\item Derivatives (导数): First-order and second-order
		\item Integration (积分)
		\item Gradient (梯度)： Jacobian, Hessian
		\item Convex function (凸函数) and Jensen's inequality (简森不等式)
		\item Taylor's expansion (泰勒展开)
	\end{itemize}
	\item Linear algebra (线性代数)
	\begin{itemize}
		\item Vector (向量), matrix (矩阵)
		\item Norm (范数)
		\item Rank (秩), determinant (行列式), trace (迹)
		\item Matrix multiplication (矩阵乘积)
		\item Eigendecomposition (正交分解)
		\item Singular value decomposition (SVD，奇异值分解)
	\end{itemize}
\end{itemize}
\end{frame}

\begin{frame}[t]
\frametitle{Lecture 1: Mathematics for Biostatistics}
\begin{itemize}
	\item Probability (概率论)
	\begin{itemize}
		\item Probability density function (pdf, 概率密度函数)
		\item Probability mass function (pmf, 概率质量函数)
		\item Cumulative distribution function (cdf, 累积分布函数)
		\item Moment generating function (mgf, 矩母函数) 
		\item Joint probability distribution (联合概率分布)
		\item Conditional probability distribution (条件概率分布)
		\item Marginal distribution (边缘概率分布)
		\item Bayes' Equation/Theorem (贝叶斯公式/定理)
		\item Continuous distributions (连续概率分布)
		\item Discrete distribution (离散概率分布)
	\end{itemize}
	\item Numerical Optimization (数值优化方法)
	\begin{itemize}
		\item Convex set (凸集)
		\item Convex function (凸函数), Concave function (凹函数)
		\item Gradient descent (梯度下降), gradient ascent (梯度上升)
		\item Newton's method (牛顿法)
		\item Quasi-Newton's methods (拟牛顿法)
		\item Method of multiplers (乘子法)
		\item Laglangian method of multiplier (拉格朗日乘子法)
	\end{itemize}
\end{itemize}
\end{frame}

\section{Calculus}

\begin{frame}[t]
\frametitle{Limit (极限)}
Compute the limit
$$
\lim_{x \to 0} \frac{3 \sin^2 x}{4 x^2}
$$
\end{frame}


\begin{frame}[t]
\frametitle{Derivative (导数)}
$$
f^\prime(x)=\frac{d}{dx} f(x)=\lim_{h\rightarrow 0} \frac{f(x+h)-f(x)}{h}
$$
The derivative is the slope of the tangent line to the graph of $f(x)$, 
assuming the tangent line exists.
\end{frame}

\begin{frame}[t]
\frametitle{Common Derivatives and Rules}
\begin{itemize}
	\item $\frac{d}{dx}x^n = n x^{n-1}$
	\item $\frac{d}{dx}\ln = \frac{1}{x}$
	\item $\frac{d}{dx}a^x = a^x \ln a$
	\item $\frac{d}{dx}e^x = e^x$
\end{itemize}
\begin{block}{\center Product rule}
$$
[f(x)g(x)]^{\prime} = f^{\prime}(x)g(x) + f(x)g^{\prime}(x)
$$
\end{block}
\begin{block}{\center Quotient rule}
$$
\left[ \frac{f(x)}{g(x)}\right]^{\prime} = \frac{g(x)f^{\prime}(x) - f(x)g^{\prime}(x)}{g^2(x)}
$$
\end{block}
\begin{block}{\center Chain rule}
Let $y = f(g(x))$,
$$
y^{\prime} = f^{\prime}(g(x)) g^{\prime}(x)
$$
\end{block}
\end{frame}


\begin{frame}[t]
\frametitle{Integrals (积分)}
For a function $f(x)$, its indefinite integral is:
$$
\int f(x)dx = F(x) + C, \mbox{where } F'(x) = f(x)
$$
\end{frame}


\begin{frame}[t]
\frametitle{Common integrals and rules}
\begin{enumerate}[(a)]
	\item $\int_a^a f(x)dx = 0$
	\item $\int_a^b f(x)dx = -\int_b^a f(x)dx$
	\item $\int x^r dx = \frac{1}{r+1}x^{r+1} + C$
	\item $\int_a^b f(x)dx = F(b) - F(a)$
	\item $\int x^n dx = \frac{1}{n+1} x^{n+1} + C, n \neq -1$
	\item $\int \frac{1}{x} dx = \ln |x| + C$
	\item $\int e^x dx = e^x + C$
	\item $\int_a^b f(g(x)) g'(x) = \int_{g(a)}^{g(b)} f(u)du, \mbox{where } u= g(x)$
	\item Integration by parts (分部积分): $\int_a^b u dv = uv|_a^b - \int_a^b v du$
\end{enumerate}
\end{frame}

\begin{frame}
\frametitle{Integrals: Exercises}
\begin{itemize}
	\item $\int_0^5 x^2 e^{-x} dx$
	\item $\int x \ln x$
	\item $\int_0^t \frac{2}{1000^2} x e^{-(x/1000)^2}$
	\item $\int_{-\infty}^{\infty} \frac{1}{2} e^{-|y| + ty}$
\end{itemize}
\end{frame}


\begin{frame}[t]
\frametitle{Taylor's Expansion (泰勒展开)}
$$
f(x) = f(x_0) + \frac{f'(x_0)}{1!}(x-x_0) + \frac{f''(x_0)}{2!}(x-x_0)^2 + \dots = \sum_{k=0}^{\infty} \frac{f^{(k)}(x_0)}{k!}(x-x_0)^k
$$
\begin{block}{Exercise}
$$
1 +  \frac{1}{2!} + \frac{1}{3!} + \dots
$$
\end{block}
\end{frame}


\begin{frame}
\frametitle{Multivariate Calculus}
\begin{itemize}
	\item $f_x(x, y) = \frac{\partial}{\partial x} f(x, y) = \frac{\partial f}{\partial x}$
	\item $\int_a^b \left[ \int_c^d f(x, y)dx\right]dy = \int_c^d \left[ \int_a^b f(x, y)dy\right]dx$
\end{itemize}
\end{frame}

\section{Linear algebra}

\begin{frame}[t]
\frametitle{Linear vector space (线性向量空间)}
A linear vector space $\mathcal{X}$ is a collection of elements 
satisfying the following properties:
\begin{itemize} 
	\item Rule of addition (加法律): $\forall x, y, z \in \mathcal{X}$,
	\begin{enumerate}
		\item $x + y \in \mathcal{X}$
		\item $x + y = y + x$
		\item $(x + y) + z = x + (y + z)$
		\item $\exists 0 \in \mathcal{X}$, such that $x + 0 = x$
		\item $\forall x \in \mathcal{X}$, $\exists -x \in \mathcal{X}$ such that $x + (-x) = 0$
	\end{enumerate}
	\item Rule of multiplication (乘法律): $\forall x, y \in \mathcal{X}$ and $a, b \in R$,
	\begin{enumerate}
		\item $ax \in \mathcal{X}$
		\item $a(bx) = (ab) x$
		\item $1x = x, 0x = 0$
		\item $a(x + y) = ax + ay$
	\end{enumerate}
\end{itemize}
\begin{block}{Example: $\mathbb{R}^n$}
The $n$-dimensional Euclidean $\mathbb{R}^n$, is a linear vector space.
\end{block}
\end{frame}

\begin{frame}[t]
\frametitle{Inner product (向量内积)}
An \textbf{inner product} is a mapping: $\mathcal{X} \times \mathcal{X} \mapsto \mathbb{R}$.

The inner product between any $x, y \in \mathcal{X}$ is denoted by $\langle x, y \rangle$ and it satisfies
the following properties for all $x, y, z \in \mathcal{X}$:
\begin{enumerate}[(1)]
	\item $\langle x, y \rangle = \langle y, x \rangle$
	\item $\langle ax, y \rangle = a \langle x, y \rangle$ for all scalars $a$
	\item $\langle x + y, z \rangle = \langle x, z \rangle + \langle y, z \rangle$
	\item $\langle x, x \rangle \ge 0$ and $\langle x, x \rangle = 0 \Rightarrow x = 0$
\end{enumerate}
A space $\mathcal{X}$ equipped with an inner product is called an \textbf{inner product space}.
\end{frame}

\begin{frame}[t]
\frametitle{Orthongonal Vectors (正交向量)}

$\mathbf{x}$ and $\mathbf{y}$ are \textbf{orthogonal vectors} if:
$$
\langle \mathbf{x}, \mathbf{y} \rangle = 0
$$

Let $\mathcal{X} = \mathbb{R}^n$, then
$$
\langle \mathbf{x}, \mathbf{y} \rangle := \mathbf{x}^T \mathbf{y} = \sum_{i=1}^n x_i y_i
$$
\end{frame}

\begin{frame}
\frametitle{Norms (向量范数)}

The inner product induces the defintion of $\ell_2$-norm:
$$
\lVert \mathbf{x} \rVert_2 = \sqrt{\langle \mathbf{x}, \mathbf{x} \rangle}
$$
here the norm measure the size (length) of $\mathbf{x}$.

The inner product can be written into the following form with norms:
$$
\langle \mathbf{x}, \mathbf{y} \rangle = \Vert \mathbf{x} \Vert \Vert \mathbf{y} \Vert \cos \theta
$$
where $\theta$ is the angle between vectors $\mathbf{x}$ and $\mathbf{y}$.

The general $\ell_p$-norm for $\mathbf{x}$ is:
$$
\lVert \mathbf{x} \rVert_p = (\sum_i x_i^p)^{1/p}, p=0,1,2,\dots, \infty
$$

We have $\ell_0$ and $\ell_1$ norms:
$$
\lVert \mathbf{x} \rVert_0 = \sum_i I(x_i \neq 0), \lVert \mathbf{x} \rVert_1 = \sum_i |x_i|
$$
\end{frame} 

\begin{frame}[t]
\frametitle{Cauchy-Schwartz inequality (柯西-施瓦茨不等式)}
$$
\langle x, y \rangle \le \Vert x \Vert \Vert y \Vert
$$
\textcolor{red}{Q: When does the equation hold?}
\end{frame}


\begin{frame}[t]
\frametitle{Triangle inequality (三角不等式)}
$$
\Vert x - y \Vert \le \Vert x \Vert + \Vert y \Vert
$$
\textcolor{red}{Q: When does the equation hold?}
\end{frame}

\begin{frame}[t]
\frametitle{Linearly independent (线性无关)}

For a set of vectors
$$
x_1, x_2, \ldots, x_p \in \mathcal{X},
$$

if there exists a set of scalars $a_1, a_2, \ldots, a_p \in \mathbb{R}$ such that not all $a_i = 0$ and 
$$
\sum_{i=1}^p a_i x_i = 0
$$

we say that $x_1, x_2, \ldots, x_p$ are \textbf{linearly dependent（线性相关)}.

If equation only holds in the case $a_1=a_2=\ldots=a_p=0$, then we say that the vectors are \textbf{linearly independent (线性无关)}.
\end{frame}


\begin{frame}[t]
\frametitle{Basis (基)}
A set of vectors $\{\phi_i\}(i=1, \dots, n)$ is a \textbf{basis (基)} for $\mathcal{X}$ if an 
arbitrary vector $x \in \mathcal{X}$ can be expressed as the linear combination of 
$\{\phi_i\}(i=1,\dots,n)$. That is, there exists a set of scalars $\{\theta_i\}(i=1,\dots,n)$, 
such that
$$
x = \sum_{i=1}^n \theta_i \phi_i
$$
\begin{block}{Orthonormal basis (正交基)}
The bases $\{\phi_i\}_{i=1}^n$ are orthonormal if 
$$
\phi_i^T \phi_j = \left\{
\begin{array}{ll}
0, & i \neq j\\
1, & i = j
\end{array}
\right.
$$
\end{block}
\end{frame}


\begin{frame}
\frametitle{Orthobasis of Hilbert space (希尔伯特空间的正交基)}

Every $x \in \mathcal{H}$ can be represented in terms of an orthonormal basis 
$\{\phi_i\}_{i \ge 1}$ (or "orthobasis" for short) according to:
$$
x = \sum_{i \ge 1} \langle x, \phi_i \rangle \phi_i
$$

This is easy to see as follows. Suppose x has a representation $\sum_i \theta_i \phi_i$. Then
$$
\theta_i = \langle x, \phi_i \rangle
$$
\begin{block}{Example: Orthonormal basis for $\mathbb{R}^n$}
$$
\phi_k = \left[0, \cdots, 1, \cdots, 0 \right]^{-1}
$$
where
$$
\phi_{k, i} = \left\{
\begin{array}{ll}
0, & i \neq k\\
1, & i = k
\end{array}
\right.
$$
\end{block}
\end{frame}

\begin{frame}
\frametitle{Subspace (子空间)}

Consider a set of vectors $\{x_i\}_{i=1}^p \in \mathcal{X}$. The \textbf{span} of these vectors 
is the set of all vectors $x \in \mathcal{X}$ that can be generated from linear combinations of 
the set
$$
\operatorname{span}(\left\{xi\right\}_{i=1}^p) := \left\{ y: y = \sum_{i=1}^p a_i x_i, a_1, \ldots, a_p \in \mathbb{R} \right\}
$$
This set is also called a subspace of $\mathcal{X}$.

A subset $\mathcal{M} \subset \mathcal{X}$ is a subspace if $x, y \in \mathcal{M}$, we have
$$
ax + by \subset \mathcal{M}
$$

注：如果$\phi_1, \cdots, \phi_p$是子空间$\mathcal{M} \subset \mathbb{R}^n$的一组正交基，
则该子空间中的任意向量$x \in \mathcal{M}$可以写成：
$$
x = \sum_{i=1}^p \theta_i \phi_i
$$

这样虽然$x \in \mathbb{R}^n$，但由于其是$\mathcal{M}$中的向量，所以可以写成$p$个自由参数的线性组合，也就是说其自由度为$p$。

\end{frame}

\begin{frame}
\frametitle{Orthogonal projection (正交投影)}

Let $\mathcal{H}$ be a \textbf{Hilbert space} and let $\mathcal{M} \subset \mathcal{H}$ be a subspace. 
Every $x \in \mathcal{H}$ can be written as
$$
x = y + z
$$
where $y \in \mathcal{M}$ and $z \perp \mathcal{M}$, which is shorthand for $z$ orthogonal to 
$\mathcal{M}$; that is
$$
\forall v \in \mathcal{M}, \langle v, z \rangle = 0
$$

The vector $y$ is the optimal approximation to $x$ in terms of vectors in $M$ in the following sense:
$$
y = \operatorname{argmin}_{v \in \mathcal{M}} \| x - v \|
$$
The vector $y$ is called the \textbf{orthogonal projection} of $x$ onto $\mathcal{M}$.
\end{frame}

\begin{frame}
\frametitle{Orthogonal subspace projection (正交子空间投影)}

Let $\mathcal{M} \subset \mathcal{H}$ and let $\{\phi_i\}_{i=1}^r$ be an orthobasis for $\mathcal{M}$. For any
$x \in \mathcal{H}$, the projection of $x$ onto $\mathcal{M}$ is given by
$$
y = \sum_{i=1}^r \langle \phi_i, x \rangle \phi_i
$$
and this projection can be viewed as a sort of filter that \textbf{removes all components of the signal $x$ 
that are orthogonal to $\mathcal{M}$}.

\begin{block}{Example}
Let $\mathcal{H} = \mathbb{R}^2$. Consider the canonical coordinate system $\phi_1 = \left[\begin{array}{c}
1\\
0
\end{array}\right]$ and $\phi_2 = \left[\begin{array}{c}
0\\
1
\end{array}\right]$. Let $\mathcal{M}$ be the subspace spanned by $\phi_1$. The projection of any 
$x = [x_1\; x_2]^T \in \mathbb{R}^2$ onto $\mathcal{M}$ is

$$
\begin{array}{lcl}
P_1x &=& \langle x, \phi_1 \rangle \phi_1\\
& = & \left[ \begin{array}{cc}
x_1 & x_2
\end{array}\right] \left[ \begin{array}{c}
1\\
0
\end{array}\right] \left[ \begin{array}{c}
1\\
0
\end{array}\right] \\
& = & \left[ \begin{array}{c}
x_1\\
0
\end{array}\right]
\end{array}
$$


The projection operator $P_1$ is just a matrix and it is given by
$$
\begin{array}{lcl}
P_1 &:=& \phi_1 \phi_1^T \\
& = & \left[ \begin{array}{c}
1\\
0
\end{array}\right] \left[ \begin{array}{cc}
1 & 0
\end{array}\right]\\
& = & \left[ \begin{array}{cc}
1 & 0\\
0 & 0
\end{array}\right]
\end{array}
$$
\end{block}
\end{frame}

\begin{frame}
\frametitle{Orthogonal projections in Euclidean subspaces}

More generally suppose we are considering  $\mathbb{R}^n$ and we have a orthonormal 
basis $\{\phi_i\}_{i=1}^r$ for some $r$-dimensional ($r < n$) subspace $\mathcal{M}$ 
of $\mathbb{R}^n$. Then the projection matrix is given by
$$
P_{\mathcal{M}} = \sum_{i=1}^r \phi_i \phi_i^T
$$

Moreover, if $\{\phi_i\}_{i=1}^r$ is a basis for $\mathcal{M}$, but not necessarily orthonormal, then
$$
P_{\mathcal{M}} = \Phi(\Phi^T \Phi)^{-1}\Phi^T
$$
where $\Phi = [\phi_1, \ldots, \phi_r]$, a matrix whose columns are the basis vectors.

注：这被用在线性回归模型$y=X\beta$的求解上，其最小二乘解析解就是$y$到$X$张成的$p$-维子空间的正交投影：
$$
\hat{\beta} = (X^T X)^{-1} X^T y
$$
\end{frame}

\begin{frame}
\frametitle{Eigendecomposition of a symmetric matrix (对称阵的特征分解)}

Let $C \in \mathbb{R}^{n \times n}$ is a real, symmetric matrix ($C^T = C$). 
$v \in \mathbb{R}^n$ is the \textbf{eigenvector (特征向量)} of $C$ such that:
$$
C v = \lambda v
$$
where $\lambda$ is the eigenvalue (特征值) of $C$ corresponding to $v$.

There are $n$ orthonormal eigenvectors for $C$ such that
$$
\langle v_i, v_j \rangle = \delta_{ij}
$$

Let $V = [v_1, \ldots, v_n]$, then
$$
C = V \Lambda V^T
$$
where $\Lambda = \operatorname{diag}(\lambda_1, \ldots, \lambda_n)$.
\end{frame}


\begin{frame}
\frametitle{Singular value decomposition (SVD, 奇异值分解)}

The SVD of an $n \times p$ matrix $H$ is written as
$$
H = \underbrace{U}_{n \times p}\quad \underbrace{\Sigma}_{p \times p}\quad \underbrace{V^T}_{p \times p}
$$

\begin{itemize}
	\item $U=[u_1, \cdots, u_p]$ where $\{u_i\}_{i=1}^p$ are real $n$-dimensional 
		vectors, and called the \textbf{left singular vectors} of $H$. $U^TU = I_p$.
	\item $\Sigma = \operatorname{diag}(\sigma_1, \cdots, \sigma_p), \sigma_1 \ge 
		\sigma_2 \ge \cdots \ge \sigma_p \ge 0$. And $\{\sigma_i\}_{i=1}^p$ are 
		called the \textbf{singular values} of $H$.
	\item $V=[v_1, \cdots, v_p]$ where $\{v_i\}_{i=1}^p$ are $p$-dimensional vectors, 
		and called the \textbf{right singular vectors} of $H$. $V^T V = I_p$.
\end{itemize}
\end{frame}

\begin{frame}[t]
\frametitle{SVD}
Also note that:
$$
\begin{array}{lcl}
H^T H & = & (U \Sigma V^T)^T U \Sigma V^T \\
& = & V \Sigma U^T U \Sigma V^T \\
& = & V \Sigma^2 V^T\\
HH^T & = & U \Sigma^2 U^T
\end{array}
$$

Therefore, 
\begin{itemize}
	\item $\{\sigma_1^2, \cdots, \sigma_p^2\}$ are the eigenvalues of $H^TH$ and 
		$\{v_1, \cdots, v_p\}$ are the corresponding eigenvectors.
	\item $\{\sigma_1^2, \cdots, \sigma_p^2\}$ are the $p$-first eigenvalues of 
		$HH^T$ (the remaining $n-p$ eigenvalues are all zeros) and $\{u_1, \cdots, u_p\}$ 
		are the associated eigenvectors.
\end{itemize}
\end{frame}


\begin{frame}
\frametitle{Application of SVD}

Say we want to solve an \textbf{over-determined} linear equations:
$$
\underbrace{y}_{n \times 1} = \underbrace{X}_{n \times p}\; \underbrace{\beta}_{p \times 1}
$$

\begin{itemize}
	\item If $n=p$ and $X=U \Sigma V^T$ with $\sigma_1 \ge \cdots \ge \sigma_p > 0$, 
		we say $X$ is \textbf{square and non-singular}, $\beta = X^{-1}y$
	\item If $n>p$ and $X=U \Sigma V^T$ with $\sigma_1 \ge \cdots \ge \sigma_p > 0$, 
		we say $X$ is \textbf{non-square and non-singular}, $\beta = (X^TX)^{-1}X^Ty$. 
		This is called the least squares solution to the over-determined linear equations.
	\item When $n<p$, this is an \textbf{under-determined} linear equations, and can be 
		solved using \textbf{penalized regression}.
\end{itemize}
\end{frame}

\begin{frame}[t]
\frametitle{Assignment 1}
\begin{enumerate}
	\item Can you extend the derivatives to vector/matrix form, say, $\frac{d x^T y}{dx}$, 
		$\frac{d x^T y}{d x^T}$, $\frac{d x^T A y}{dx}$?
	\item What is least squares fitting of $\mathbf{X\beta = y}$? Can you use the above matrix 
		derivatives to reach the normal equation?
	\item What is QR-decomposition and Cholesky decomposition? Can you give some comments 
		on the application of the two decomposition techniques?
	\item \texttt{base::qr()} and \texttt{chol()} can be used to compute the two kinds of 
		decompositions. Give an example to illustrate the usage of the decompositions. 
	\item What kinds of matrices are positive-definite? positive semi-definite?
\end{enumerate}
\end{frame}

\section{Optimization}

\begin{frame}[t]
\frametitle{Mathematical Optimization}
\begin{itemize}
  \item Optimization uses a rigorous mathematical model to determine the 
      most efficient solution to a described problem.
  \item You should identify an objective function
  \begin{itemize}
    \item Objective is a quantitative measure of the performance
    \item Objective is usually a single number
  \end{itemize}
\end{itemize}
\end{frame}


\begin{frame}[t]
\frametitle{Classification of optimization problems}
Optimization problem can be constrained or unconstrained.
\begin{block}{Common groups}
\begin{itemize}
  \item Linear programming (LP)
  \begin{itemize}
    \item Objective function and constraints are both linear
    \item $\min_{\bf x} {\bf c^T x}$ s.t. ${\bf Ax \le b}$ and ${\bf x} \ge 0$
  \end{itemize}
  \item Quadratic programming (QP)
  \begin{itemize}
    \item Objective function is quadratic and constraints are linear
    \item $\min_{\bf x} {\bf x^TQx + c^T x}$ s.t. ${\bf Ax \le b}$ and ${\bf x} \ge 0$
  \end{itemize}
  \item Nonlinear programming (NLP)
  \begin{itemize}
    \item Objective function or at least one constraint is nonlinear
  \end{itemize}
\end{itemize}
\end{block}
\end{frame}


\begin{frame}[t]
\frametitle{Nonlinear optimization}
\scriptsize
\begin{figure}[]
\renewcommand\arraystretch{2.5}
\begin{tabular}{l|p{1.5cm}|p{1.5cm}p{1.5cm}p{1.5cm}}
\hline
\textbf{Dimensionality} & \multicolumn{1}{|c|}{One-dimensional} & \multicolumn{3}{c}{Multi-dimensional}\\
\hline
\textbf{Category} & Non-gradient based & Gradient-based & Hessian-based & Non-gradient based \\
\hline
\textbf{Algorithms} & Golden Section Search & Gradient descent & Newton/Quasi-Newton (L-BFGS, BFGS) & Nelder-Mead\\
\hline
\end{tabular}
\end{figure}
\end{frame}

\begin{frame}[t,fragile,containsverbatim]
\frametitle{One-dimensional nonlinear programming}
\begin{itemize}
  \item Golden section search
  \item Basic steps:
  \begin{enumerate}
    \item Golden ratio: $\phi = (\sqrt{5} - 1)/2 = .618$
    \item Pick an interval $[a,b]$ containing the optimum
    \item Evaluate $f(x_1)$ at $x_1 = a + (1-\phi)(b-a)$ and compare with 
      $f(x_2)$ at $x_2 = a + \phi(b-a)$
    \item if $f(x_1) < f(x_2)$, continue the search in the interval $[a, x_1]$, else $[x_2, b]$
  \end{enumerate}
  \item R command \texttt{stats::optimize()}
  \begin{lstlisting}[language=R]
optimize(f=, interval=, ...,
        tol = .Machine$double.eps^0.25)
  \end{lstlisting}
\end{itemize}
\end{frame}

\begin{frame}[t]
\frametitle{Nondifferentiable function}
\begin{itemize}
  \item $f(x) = |x - 2| + 2|x - 1|$
  \item How to solve it?
\end{itemize}
<<echo=FALSE,fig=TRUE>>=
f <- function(x) abs(x-2) + 2*abs(x-1)
curve(f, 0, 3)
@
\end{frame}



\begin{frame}[t]
\frametitle{Newton-Raphson}
\begin{itemize}
  \item Newton method is often used to find the zeros of a function
  \item Minima fulfill the conditions $f^{\prime}(x^*)=0$ and $f^{\prime\prime}(x^*)>0$, so 
      Newton can be used to find the zeros of the first derivative
  \item Basic steps
  \begin{enumerate}
    \item Approximate the function at the starting point with a linear 
      tangent (e.g., second-order Taylor expansion $t(x) \approx f^{\prime}(x_0) + (x-x_0)f^{\prime\prime}(x_0)$)
    \item Find the intersect $t(x_i)=0$ as an approximation to $f^{\prime}(x^*)=0$
    \item Use the intersect as the new starting point
    \item Finally, the algorithm $x_{n+1} = x_n - \frac{f^{\prime}(x_n)}{f^{\prime\prime}(x_n)}$ 
      is repeated until $f^{\prime}(x_n)$ is close enough to 0.
  \end{enumerate}
\end{itemize}
\end{frame}


\begin{frame}[t]
\frametitle{Hessian-based: BFGS and L-BFGS-B}
\begin{itemize}
  \item Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm builds on 
    the idea of Newton method to take gradient information into 
    account
  \item Gradient information comes from an approximation of the Hessian 
    matrix
  \item No guaranteed conversion; especially problematic if Taylor 
    expansion does not fit well
  \item L-BFGS-B stands for limited-memory-BFGS-box:
  \begin{itemize}
    \item Extension of BFGS
    \item Memory-efficient implementation
    \item Additional handles box constraints
  \end{itemize}
\end{itemize}
\end{frame}


\begin{frame}[t]
\frametitle{Himmelblau's function}
$$
f(x, y) = (x^2 + y - 11)^2 + (x + y^2 - 7)^2
$$
\begin{itemize}
	\item Himmelblau's function is a popular multi-modal function to 
		benchmark optimization algorithms.
	\item For equivalent minima are located at $f(-3.78,-3.28)=0$, 
		$f(-2.80,3.13)=0$, $f(3,2)=0$, $f(3.58,-1.85)=0$
\end{itemize}
<<echo=FALSE,fig=TRUE>>=
fn <- function(para) {
	matrix.A <- matrix(para, ncol=2)
	x <- matrix.A[,1]
	y <- matrix.A[,2]
	f.x <- (x^2+y-11)^2 + (x+y^2-7)^2
	return(f.x)
}
par <- c(1,1)
xy <- as.matrix(expand.grid(seq(-5,5, length=101),
					seq(-5,5,length=101)))
colnames(xy) <- c('x', 'y')
df <- data.frame(fnxy = fn(xy), xy)
library(lattice)
wireframe(fnxy ~ x*y, data=df, shade=TRUE, drape=FALSE,
			scales=list(arrows=FALSE),
			screen=list(z=-240, x=-70, y=0))
@
\end{frame}

\begin{frame}[t]
\frametitle{Gradient-free method: Nelder-Mead}
\begin{itemize}
	\item Nelder-Mead sovles multi-dimensional equations using function values
	\item Works with non-differentiable functions
	\item Basic steps:
	\begin{enumerate}
		\item Choose a simplex consisting of $n+1$ points $p_1, p-2, \dots, p_{n+1}$ 
			are chosen with $n$ being the number of variables
		\item Calculate $f(p_i)$ and sort by value, e.g. $f(p_1) \le f(p_2) \le \dots 
			\le f(p_{n+1})$
		\item Check if the best value is good enough, if so, stop
		\item Drop the point with hightest $f(p_i)$ from the simplex
		\item Choose a new point to be added to the simplex
		\item Continue with step 2
 	\end{enumerate}
\end{itemize}
\end{frame}

\end{CJK*}
\end{document}