Why statistics?

From Motulsky (2014) Chapter 1:

We tend to jump to conclusiong
We tend to be overconfident
We see patterns in random data
Coincidences are common
Incorrect intuition about probability
…

What is Probability?

Long-term Frequency — Frequentist

(# of head)/(# of coin toss)
predicted from model
estimated from data
probability of Higgs boson exists?
- We need to create the universe many times…

Strength of Belief — Bayesian

subjective probability
quantification of uncertainty/ignorance (Cox, 1946)
- sum rule: $prob(X) + prob(\bar{X}) = 1$
- product rule: $prob(X,Y) = prob(X) \times prob(Y)$

Probability and Statistics

\[\begin{matrix} & \mbox{Population (Model)}& \\ \mbox{Probability} & \downarrow \quad \uparrow & \mbox{Statistics} \\ & \mbox{Sample (Data)} & \end{matrix}\]

R Language

R was developed by R. Ihaka and R. Gentleman (1996) as a successor of S language (Chambers & Hastie, 1988).

Pros:
- Free and open-source, for Linux, Mac, and Windows
- Easy to start using
- Sample data sets
- Programming for custom use
- Many packages contributed by statisticians
Cons:
- Fortran legacy
- Object-oriented like, but not really
- Classis standard graphics

Running R

First, download R package for your computer from R Project site and install as instructed there.
The basic way is to run a command “r” on your computer terminal:

$ r

It is more convenient to use an integrated develpment environment (IDE) like RStudio
In RStudito, you can
- type in commands in the “Console” window
- run scripts in “Terminal” window
- run code chunks in R Markdown files within the editor window

R Markdown

R Markdown combines explanation, code, and result in a same file.

“File >New File >R Markdown…” menu to create a new file
You can embed LaTeX-like mathematical formula by Mathjax
-  or $ $ for inline, e.g., $\sum_{i=1}^n x_i$ or $\int_0^\infty f(x)dx$.
- \[ \] or $$ $$ for a separate line, e.g., \[\sum_{i=1}^n x_i\] or \[\int_0^\infty f(x)dx.\]
To embed your R code, “Option+Command+I” or “Insert” icon to make a code chunk

To run a code chunk:
- “Shift+Comman+Enter” or the triangle in the uppper-right corner
- “Comman+Enter” for each line
- “Run All” to run all the code chunks in the file
“File >Knit Document” menu, or “Knit” icon, to produce an html file.

R Notebook

R Notebook a new type of R Markdown for embedding R Markdown code in an html file. This file is an example of an R Notebook.

“File >New File >R Notebook” menu to create a new file, which includes
output: html_notebook
in the file header.
“Prevew >Preview Notebook” button to show a page preview.
Everytime you save your “file.Rmd”, R Studio creates “file.nb.html”.

R as a calculator

You can just type in numbers and operators in the Console window.

1+1

[1] 2

2*3

[1] 6

4/5

[1] 0.8

Either ** or ^ for power.

2**3

[1] 8

2^3

[1] 8

Many math functions and some constants are pre-defined.

exp(1)

[1] 2.718282

sin(pi/2)

[1] 1

Variables

You can use <- or = to assign a value to a variable.

x <- 1
y = 2
x+y

[1] 3

x/y

[1] 0.5

z = x+y
z

[1] 3

Getting information

ls() lists objects in your work space.
rm() removes objects.

ls()

 [1] "a"        "ai"       "alpha"   
 [4] "av"       "b"        "b0"      
 [7] "b0hat"    "b1"       "b1hat"   
[10] "c"        "CI"       "d2norm"  
[13] "dmn"      "eps"      "ess"     
[16] "f"        "F"        "fpbc"    
[19] "fpfd"     "fpt"      "fptt"    
[22] "g"        "i"        "interval"
[25] "j"        "k"        "L"       
[28] "lambda"   "m"        "M"       
[31] "miss"     "mu"       "n"       
[34] "nb"       "nu"       "ones"    
[37] "p"        "phat"     "pp"      
[40] "pval"     "q"        "r2"      
[43] "rss"      "s"        "S"       
[46] "SE"       "sig"      "sigma"   
[49] "t"        "T"        "t1"      
[52] "tss"      "tstar"    "tt"      
[55] "V"        "w"        "W"       
[58] "x"        "X"        "x1"      
[61] "X1"       "x2"       "X2"      
[64] "xbar"     "xq"       "XY"      
[67] "y"        "ybar"     "z"

rm(x,y)
ls()

 [1] "a"        "ai"       "alpha"   
 [4] "av"       "b"        "b0"      
 [7] "b0hat"    "b1"       "b1hat"   
[10] "c"        "CI"       "d2norm"  
[13] "dmn"      "eps"      "ess"     
[16] "f"        "F"        "fpbc"    
[19] "fpfd"     "fpt"      "fptt"    
[22] "g"        "i"        "interval"
[25] "j"        "k"        "L"       
[28] "lambda"   "m"        "M"       
[31] "miss"     "mu"       "n"       
[34] "nb"       "nu"       "ones"    
[37] "p"        "phat"     "pp"      
[40] "pval"     "q"        "r2"      
[43] "rss"      "s"        "S"       
[46] "SE"       "sig"      "sigma"   
[49] "t"        "T"        "t1"      
[52] "tss"      "tstar"    "tt"      
[55] "V"        "w"        "W"       
[58] "X"        "x1"       "X1"      
[61] "x2"       "X2"       "xbar"    
[64] "xq"       "XY"       "ybar"    
[67] "z"

getwd() to get or set the working directory

getwd()

[1] "/Users/doya/Dropbox (OIST)/R/StatisticalMethods"

help() or ? for help. ?? for related topics.

?sin
??sin

You can also use “Help” panel of RStudio.

Vectors

c( ) cancatenates objects to make a vector.

a = c(1,2,3)
a

[1] 1 2 3

You can access each component by [ ].

a[1]

[1] 1

a[c(2,3)]

[1] 2 3

Operators work component-wise. There are many functions that work with vectors.

a+a

[1] 2 4 6

3*a

[1] 3 6 9

a*a

[1] 1 4 9

mean(a)

[1] 2

exp(a)

[1]  2.718282  7.389056 20.085537

Regular vectors can be created by : or seq().

b = 1:5
b

[1] 1 2 3 4 5

c = seq(0,10,2)
c

[1]  0  2  4  6  8 10

You can also create an empty array and later assign values.

d = array(dim=10)
d[1] = 1
d

 [1]  1 NA NA NA NA NA NA NA NA NA

Matrix

You can create a new matrix by

A = matrix(0, 2, 3)
A

     [,1] [,2] [,3]
[1,]    0    0    0
[2,]    0    0    0

or by combining vectors

B = rbind(1:3, 4:6)
B

     [,1] [,2] [,3]
[1,]    1    2    3
[2,]    4    5    6

C = cbind(1:3, 4:6)
C

     [,1] [,2]
[1,]    1    4
[2,]    2    5
[3,]    3    6

You can reshaping a vector into a matrix.

D = 1:6
dim(D) = c(2, 3)
D

     [,1] [,2] [,3]
[1,]    1    3    5
[2,]    2    4    6

You can access the components by [ , ]

D[1,2]

[1] 3

D[1:2,2:3]

     [,1] [,2]
[1,]    3    5
[2,]    4    6

D[1,]

[1] 1 3 5

D[,1]

[1] 1 2

Most operators and functions work component-wise.

A = rbind(1:2, 3:4)
A

     [,1] [,2]
[1,]    1    2
[2,]    3    4

A+A

     [,1] [,2]
[1,]    2    4
[2,]    6    8

A*A

     [,1] [,2]
[1,]    1    4
[2,]    9   16

sin(A)

         [,1]       [,2]
[1,] 0.841471  0.9092974
[2,] 0.141120 -0.7568025

To perform matrix multiplication, we must use %*%

A %*% A

     [,1] [,2]
[1,]    7   10
[2,]   15   22

Inverse is given by solve() function

Ainv = solve(A)
Ainv

     [,1] [,2]
[1,] -2.0  1.0
[2,]  1.5 -0.5

A %*% Ainv

     [,1]         [,2]
[1,]    1 1.110223e-16
[2,]    0 1.000000e+00

Use t() for a transpose matrix.

t(A)

     [,1] [,2]
[1,]    1    3
[2,]    2    4

List

A list is a collection of components of arbitrary types, e.g., numbers, matrices, characters, or functions.

L = list(name="OIST", address=c("Tancha","Onna","Okinawa"), age=6)
L[1]

$name
[1] "OIST"

a component can be specified by its index or name

L[[1]]

[1] "OIST"

L[["name"]]

[1] "OIST"

L$name

[1] "OIST"

L$add

[1] "Tancha"  "Onna"    "Okinawa"

Script

You can store multiple lines of R commands as a script for repeated use.

Choose “File >New File >R Script” to make a new file. Type in, for example:

# bern.R: Bernoulli distribution of 0 and 1
p = 0.5  # probability of 1
if (runif(1) < p){  # sample a random number uniformly in 0 and 1
    x = 1  # 1 if that falls below p
  }else{ 
    x = 0  # 0 otherwise
  }
print(x)

# starts a comment, which is skipped by R but helpful for humans, including yourself, to understand what the code does and what the parameter means.

Choose “File >Save As…” and name it ‘bern.R’

Go to the directory where the file was saved, start R, and type in:

source("bern.R")

[1] 0

Defining your own function

For repeating a procedure with different parameters, you would define your own function, in the following convention:

bern <- function(p){  # p: probability of 1
  if (runif(1) < p){  # sample a uniform random number
    x = 1    # 1 if that falls below p
  }else{
    x = 0    # 0 otherwise
  }
  return(x)  
}

You can call it with any parameter:

bern(0.2)

[1] 0

bern(0.8)

[1] 1

Loops

Try making bionomial distribution by adding 1s of n bernoulli samples using for loop:

binom = function(n, p) { # n:samples, p:probability
  y = 0   # prepare a counter
  for (i in 1:n) {
    x = bern(p)   # 1 or 0
    # print(x)    # uncomment to check each x
    y = y + x
  }
  return(y)
}

You can test it with different parameters:

binom(5, 0.5)

[1] 2

binom(10, 0.3)

[1] 4

Plotting

Let us take samples from a binomial distribution

m = 1000  # number of samples
n = 10
p = 0.5
x = array(dim=m)   # prepare an empty array
for (i in 1:m) {
  x[i] = binom(n, p)  # between 0 to n
}

You can visualize the raw data by plot() function.

plot(x)

or make a histogram by hist().

hist(x)

You can plot any function by specifying x and y for plot() function.

X = seq(0, n)
plot(X, dbinom(X, n, p))

To overlay a curve on an existing plot, use lines() function.

hist(x, freq=FALSE)  # normalize to sum up to 1
lines(X, dbinom(X, n, p))

3D Plots

R has a simple 3D plot function persp().

x = seq(-5, 5, 0.1)
y = seq(0.5, 3, 0.1)
f = function(x,y){exp(-x**2/(2*y**2))/y}
z = outer(x, y, f)
persp(x, y, z, theta=45, phi=30)

There are fancier 3D packages like plot3D and rgl.

To be covered in later chapters

probability distributions
data frame
pre-loaded data sets
reading data from a file

---
title: "1. Introduction"
output: html_notebook
---

## Why statistics?

From Motulsky (2014) Chapter 1:

* We tend to jump to conclusiong  
* We tend to be overconfident  
* We see patterns in random data  
* Coincidences are common  
* Incorrect intuition about probability  
* ...  

## What is Probability?

### Long-term Frequency — Frequentist

* (# of head)/(# of coin toss)  
* predicted from model  
* estimated from data  
* probability of Higgs boson exists?  
    * We need to create the universe many times...
    
### Strength of Belief — Bayesian

* subjective probability  
* quantification of uncertainty/ignorance (Cox, 1946)
    + sum rule: $prob(X) + prob(\bar{X}) = 1$  
    + product rule: $prob(X,Y) = prob(X) \times prob(Y)$  

## Probability and Statistics

$$\begin{matrix} & \mbox{Population (Model)}& \\
\mbox{Probability} & \downarrow \quad \uparrow & \mbox{Statistics} \\
& \mbox{Sample (Data)} &
\end{matrix}$$

# R Language

R was developed by R. Ihaka and R. Gentleman (1996) as a successor of S language (Chambers & Hastie, 1988).

* Pros:
    * Free and open-source, for Linux, Mac, and Windows
    * Easy to start using
    * Sample data sets
    * Programming for custom use
    * Many packages contributed by statisticians
    
* Cons:
    * Fortran legacy
    * Object-oriented like, but not really 
    * Classis standard graphics
    
## Running R

* First, download R package for your computer from [R Project site](https://www.r-project.org)
and install as instructed there.

* The basic way is to run a command "r" on your computer terminal:
```
$ r
```

* It is more convenient to use an integrated develpment environment (IDE) like [RStudio](https://www.rstudio.com)

* In RStudito, you can
    * type in commands in the "Console" window
    * run scripts in "Terminal" window
    * run code chunks in R Markdown files within the editor window

## R Markdown

[R Markdown](https://rmarkdown.rstudio.com) combines explanation, code, and result in a same file.  

* "File >New File >R Markdown..." menu to create a new file

* You can embed LaTeX-like mathematical formula by [Mathjax](https://www.mathjax.org/)  

    * `\( \)` or `$ $` for inline, e.g., 
    \(\sum_{i=1}^n x_i\) or $\int_0^\infty f(x)dx$.   
    
    * `\[ \]` or `$$ $$` for a separate line, e.g., 
    \[\sum_{i=1}^n x_i\] or $$\int_0^\infty f(x)dx.$$   

* To embed your R code, "Option+Command+I" or "Insert" icon to make a *code chunk*

```{r}

```

* To run a code chunk:
    * "Shift+Comman+Enter" or the triangle in the uppper-right corner    
    * "Comman+Enter" for each line
    * "Run All" to run all the code chunks in the file

* "File >Knit Document" menu, or "Knit" icon, to produce an html file.

## R Notebook

R Notebook a new type of R Markdown for embedding R Markdown code in an html file. This file is an example of an R Notebook.

* "File >New File >R Notebook" menu to create a new file, which includes  
`output: html_notebook`  
in the file header.  

* "Prevew >Preview Notebook" button to show a page preview.

* Everytime you save your "file.Rmd", R Studio creates "file.nb.html".

## R as a calculator

You can just type in numbers and operators in the Console window.  


```{r}
1+1
2*3
4/5
```

Either `**` or `^` for power.

```{r}
2**3
2^3
```

Many math functions and some constants are pre-defined.

```{r}
exp(1)
sin(pi/2)
```

## Variables

You can use `<-` or `=` to assign a value to a variable.

```{r}
x <- 1
y = 2
x+y
x/y
z = x+y
z
```

## Getting information

`ls()` lists objects in your work space.  
`rm()` removes objects.  

```{r info}
ls()
rm(x,y)
ls()
```

`getwd()` to get or set the working directory

```{r}
getwd()
```

`help()` or `?` for help. `??` for related topics.

```{r}
?sin
??sin
```

You can also use "Help" panel of RStudio.

## Vectors

`c( )` cancatenates objects to make a vector.  

```{r vector}
a = c(1,2,3)
a
```

You can access each component by `[ ]`.  

```{r}
a[1]
a[c(2,3)]
```

Operators work component-wise. There are many functions that work with vectors.

```{r}
a+a
3*a
a*a
mean(a)
exp(a)
```

Regular vectors can be created by `:` or `seq()`.

```{r}
b = 1:5
b
c = seq(0,10,2)
c
```

You can also create an empty array and later assign values.

```{r}
d = array(dim=10)
d[1] = 1
d
```

## Matrix

You can create a new matrix by

```{r}
A = matrix(0, 2, 3)
A
```

or by combining vectors

```{r}
B = rbind(1:3, 4:6)
B
C = cbind(1:3, 4:6)
C
```

You can reshaping a vector into a matrix.

```{r}
D = 1:6
dim(D) = c(2, 3)
D
```

You can access the components by `[ , ]`

```{r}
D[1,2]
D[1:2,2:3]
D[1,]
D[,1]
```

Most operators and functions work component-wise.

```{r}
A = rbind(1:2, 3:4)
A
A+A
A*A
sin(A)
```

To perform matrix multiplication, we must use `%*%`  

```{r}
A %*% A
```

Inverse is given by `solve()` function

```{r}
Ainv = solve(A)
Ainv
A %*% Ainv
```

Use `t()` for a transpose matrix.

```{r}
t(A)
```

## List

* A `list` is a collection of components of arbitrary types, e.g., numbers, matrices, characters, or functions.  

    
```{r}
L = list(name="OIST", address=c("Tancha","Onna","Okinawa"), age=6)
L[1]
```

* a component can be specified by its index or name

```{r}
L[[1]]
L[["name"]]
L$name
L$add
```


## Script

You can store multiple lines of R commands as a *script* for repeated use.

* Choose "File >New File >R Script" to make a new file. Type in, for example:

```
# bern.R: Bernoulli distribution of 0 and 1
p = 0.5  # probability of 1
if (runif(1) < p){  # sample a random number uniformly in 0 and 1
    x = 1  # 1 if that falls below p
  }else{ 
    x = 0  # 0 otherwise
  }
print(x)
```

\# starts a comment, which is skipped by R but helpful for humans, including yourself, to understand what the code does and what the parameter means.

* Choose "File >Save As..." and name it 'bern.R'

Go to the directory where the file was saved, start R, and type in:

```{r script}
source("bern.R")
```

## Defining your own function

For repeating a procedure with different parameters, you would define your own function, in the following convention:

```{r function}
bern <- function(p){  # p: probability of 1
  if (runif(1) < p){  # sample a uniform random number
    x = 1    # 1 if that falls below p
  }else{
    x = 0    # 0 otherwise
  }
  return(x)  
}
```

You can call it with any parameter:

```{r bern}
bern(0.2)
bern(0.8)
```

## Loops

Try making bionomial distribution by adding 1s of n bernoulli samples using for loop:

```{r loop}
binom = function(n, p) { # n:samples, p:probability
  y = 0   # prepare a counter
  for (i in 1:n) {
    x = bern(p)   # 1 or 0
    # print(x)    # uncomment to check each x
    y = y + x
  }
  return(y)
}
```

You can test it with different parameters:

```{r binom}
binom(5, 0.5)
binom(10, 0.3)
```

## Plotting

Let us take samples from a binomial distribution

```{r binom_hist}
m = 1000  # number of samples
n = 10
p = 0.5
x = array(dim=m)   # prepare an empty array
for (i in 1:m) {
  x[i] = binom(n, p)  # between 0 to n
}
```

You can visualize the raw data by `plot()` function.

```{r}
plot(x)
```

or make a histogram by `hist()`.

```{r}
hist(x)
```

You can plot any function by specifying x and y for `plot()` function.

```{r}
X = seq(0, n)
plot(X, dbinom(X, n, p))
```

To overlay a curve on an existing plot, use `lines()` function.

```{r}
hist(x, freq=FALSE)  # normalize to sum up to 1
lines(X, dbinom(X, n, p))
```

## 3D Plots

R has a simple 3D plot function `persp()`.

```{r}
x = seq(-5, 5, 0.1)
y = seq(0.5, 3, 0.1)
f = function(x,y){exp(-x**2/(2*y**2))/y}
z = outer(x, y, f)
persp(x, y, z, theta=45, phi=30)
```

There are fancier 3D packages like `plot3D` and `rgl`.

## To be covered in later chapters

* probability distributions
* data frame
* pre-loaded data sets
* reading data from a file


1. Introduction