- vectors
- you always need to know
- stat functions
- conditional calcs
- полезные функции
- types
- input / output
- read binary data
- write binary data
- factor
- dataframe
- plotting
- distributions
- binom test
- Shapiro
- Student
- Wilcoxon
- regression - linear model
- analysis of variance
- regression - logistic model
- regression - loglinear model
- Fisher
- Kolmogorov
- Bayesian model
- notes: scripts, packets and so on ...

`set.seed(k=integer)`

is the recommended way to specify seeds in R-soft

if you add two vectors of the same length, you get another vector of the same length, where each entry is the sum of that entry in the other two vectors

x = c (3, 1, 9)

y = c (2, 5, 6)

each entry adds

x + y [1] 5 6 15

we can index into vectors with square brackets []. for example, we can pull out the second entry of a vector:

x = c(3, 1, 9) #print 2nd element x[2] [1] 1

the next bit of code

s = rep (NA, 100)

NA is short for ‘not available.’ R cannot handle NA when doing calculations; for example, we couldn’t take the mean of a vector with an NA in it. this is why it is good practice to fill a vector with NA before filling it with our actual data; if we make a mistake and accidentally don’t fill a specific entry in the vector, R will let us know because we can’t even take a mean

to create a vector from 1 to 10, increment by 1

seq (from = 1, to = 10, by = 1) [1] 1 2 3 4 5 6 7 8 9 10

- mean (x)
- median (x)
- min (x)
- max (x)
- var (x)
- sd (x)

> mean (x$Field_Name) ; median (x$Field_Name) [1] 0.7649074 [1] 0.72 > quantile (x$Field_Name) 0% 25% 50% 75% 100% 0.1300 0.4800 0.7200 1.0075 1.7600 > min (x$Field_Name) ; max (x$Field_Name) [1] 0.13 [1] 1.76 > var (x$Field_Name) ; sd (x$Field_Name) [1] 0.1429382 [1] 0.3780717

- summary (x)
- min, 25th quantile, mean, median, 75th quantile, max
- fivenum (x)
- min, lower-hinge, median, upper-hinge, max
- quantile (x)

quantile (x, grades)

> z = rnorm (1000) > mean (z) [1] -0.02373456 > quantile (z, c (.1, .3, .7, .9)) 10% 30% 70% 90% -1.3458127 -0.5073407 0.5294266 1.2001054

- mad (xs)

mad (xs, center, const) - const * median (abs (xi - center))

by default: center = median and const = 1.4826 - for asymptotically normal consistency - cov (xs, ys)
- covariance between two series

cov = mean (xs * ys) - mean (xs) * mean (ys) - cor (xs, ys)
- correlation between two series

cor = cov²(xs,ys) / ( σ²xs * σ²ys)

- the range of the values in your sample data
- how these values are distributed
- how values in different variables relate to each other

> x = 1:10 > range (x) [1] 1 10 > quantile (x) 0% 25% 50% 75% 100% 1.00 3.25 5.50 7.75 10.00

the greater the variability in the data, the greater will be your uncertainty and lower your ability to distinguish between competing hypotheses

two populations can have different means but the same variance

> x = c (1, 3, 5) > z = c (2, 4, 6) > mean (z) ; mean (x) [1] 3 [1] 4 > var (z) ; var (x) [1] 4 [1] 4

two populations can have the same mean but different variances

> x = c (1, 3, 5) > y = c (0, 3, 6) > mean (y) ; mean (x) [1] 3 [1] 3 > var (x) ; var (y) [1] 4 [1] 9

comparing **means**
when the **variances**
are different is an extremely bad idea

in order to be reasonably confident that your inferences are correct, you need to establish some facts about the **distribution**
of the data:

- are the values normally distributed or not?
- are there outliers in the data?
- if data were collected over a period of time, is there evidence for correlation?

this rounds a value (the first argument) to a specific decimal point (the second argument). this can be useful because R generates random values to many decimal places

round (rnorm (1), 2) [1] 0.97

> choose(3,2) [1] 3

number

> numbers <- 30:1 > numbers [1] 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 [18] 13 12 11 10 9 8 7 6 5 4 3 2 1 > numbers[5] [1] 26 > numbers[c(5,11,3)] [1] 26 20 28 > indices <- c(5,11,3) > numbers[indices] [1] 26 20 28

string

logical T[RUE] | F[ALSE]

array

list

vector

> x <- c (10, 20, 30, 40, 50, 60) > y <- c (1, 2, 3, 4, 5, 6, 7, 8, 9, 10) > x[4] > y[3:5] > sum (x) > sum (sort (y)[1:3]) > prod (rev (x)) > rep (5.3, 17) > rep (1:6, 2)

seq (from, to, lenght.out = ..)

seq (from, by, along x)

seq (from, by, along = 1:20)

matrix

> y <- c (3, 4, 7, 2, 8, 3, 4, 7, 1, 6, 7, 8, 9, 3, 7) > m <- matrix (y, nrow = 5) > n <- matrix (y, ncol = 3) > m [,1] [,2] [,3] [1,] 3 3 7 [2,] 4 4 8 [3,] 7 7 9 [4,] 2 1 3 [5,] 8 6 7 > n [,1] [,2] [,3] [1,] 3 3 7 [2,] 4 4 8 [3,] 7 7 9 [4,] 2 1 3 [5,] 8 6 7

basic functions which support complex arithmetic, in addition to the arithmetic operators +, -, *, /, and ^

- complex (length.out = 0, real = numeric(), imaginary = numeric(), modulus = 1, argument = 0)
length.out numeric. desired length of the output vector, inputs being recycled as needed real numeric vector imaginary numeric vector modulus numeric vector argument numeric vector

> cs1 = complex(2, c(1,2,3)) > cs1 [1] 1+0i 2+0i 3+0i > cs2 = complex(7, c(1,2,3)) > cs2 [1] 1+0i 2+0i 3+0i 1+0i 2+0i 3+0i 1+0i

- as.complex (x, ...)
- x is an object, probably of mode complex
- is.complex (x)
- x is an object, probably of mode complex

return value is TRUE|FALSE - Re (z)
- Im (z)
- Mod (z)
- Arg (z)
- Conj (z)

useful functions:

z an object of mode complex

две рабочие лошадки: ` readline `

и ` print `

> x = as.numeric (readline (prompt="your num: ")) > print (x)

> x = scan () 1: 123 2: 567 890 34 5: 34.6 6: Read 5 items > x [1] 123.0 567.0 890.0 34.0 34.6 > print (x) [1] 123.0 567.0 890.0 34.0 34.6 > y <- scan (what = " ") 1: old young bible 4: stout 5: Read 4 items > y [1] "old" "young" "bible" "stout" > print (y) [1] "old" "young" "bible" "stout"

- file (fileName, open)
- this function will create file if it does not exist
open :
> fd = file ("aaa.dat", "w") > close (fd)

"r" or "rt" reading in text mode "w" or "wt" writing in text mode "a" or "at" appending in text mode "rb" reading in binary mode "wb" writing in binary mode "ab" appending in binary mode "r+" or "r+b" reading and writing "w+" or "w+b" reading and writing, truncating file initially "a+" or "a+b" reading and appending

- scan (filename, what, sep, n, nlines, fileEncoding)
- what: ‘logical’, ‘integer’, ‘numeric’, ‘complex’, ‘character’, ‘raw’, ‘list’

n: integer: the maximum number of data values to be read, defaulting to no limit

nlines: if positive, the maximum number of lines of data to be read

sep: by default ‘white-space’ delimited input fields - read.csv (filename, head, sep)

read.table (filename, head, sep) - the second argument indicates whether or not the first row is a set of labels

the third argument indicates sepatate sign between each number of each line - write.csv (obj, filename)

write.table (obj, filename) - save (obj, file=filename)
- load (filename)
- save.image()
- all your variables from current session
- unlink (filename)
- unix-way to delete file

file "a.dat":

1 2 3 4 5 6 7 8 9

> y = scan ("a.dat") Read 9 items > y [1] 1 2 3 4 5 6 7 8 9 > is.numeric(y) [1] TRUE

> y = rnorm (50, 7.0, 0.1) > write (file="filename.dat", y) > z = read.csv (file="bbb", header=0, sep=" ") > is.list(z) [1] TRUE

> z <- read.fwf ("http://www.ats.ucla.edu/stat/data/test_fixed.txt", width = c (8, 1, 3, 1, 1, 1))

you use the **width** argument to indicate the number of signs of each variable. in a fixed format file you do not have the names of the variables on the first line, and therefore they must be added after you have read in the data

> z V1 V2 V3 V4 V5 V6 1 general 0 70 4 1 1 2 vocati 1 121 4 2 1 3 general 0 86 4 3 1 4 vocati 0 141 4 3 1 5 academic 0 172 4 2 1 6 academic 0 113 4 2 1 7 general 0 50 3 2 1 8 academic 0 11 1 2 1 > s <- scan ("http://www.ats.ucla.edu/stat/data/names.txt", what = character ()) Read 6 items > s [1] "prgtyp" "gender" "id" "ses" "schtyp" "level"

in the binary data file, information is stored in groups of binary digits. each binary digit is a zero or one (and eight binary digits grouped together is a byte)

in order to successfully read binary data, you must know how pieces of information have been parsed into binary

for example, if your data consists of integers, how may bytes should you interpret as representative of one integer in your data?

or if your data contains both positive and negative numbers, how can you distinguish the two?

how many pieces of information do you expect to find in the binary data?

ideally, you know the answers to these questions before starting to read in the binary file

to get started, we establish a connection to a file and indicate that we will be using the connection to read in binary data. we do this with the ```
file
```

command, providing first the pathname, and the ```
rb
```

for “reading binary”

> to.read = file (“https://stats.idre.ucla.edu/stat/r/faq/bintest.dat”, “rb”)

next, we use the ` readBin `

command to begin. if we think the file contains integers, we can start by reading in the first integer and hoping that the size of the integer does not require further specifications. different platforms store binary data in different ways, and which end of a string of binary values represents the greatest values or smallest values is a difference that can yield very different results from the same set of binary values. this characteristic is called the “endian”. the binary files in the examples on this page were written using a PC, which suggests they are little-endian. when reading in binary data that may or may not have been written on a different platform, indicating an "endian" can be crucial. for example, without adding endian = “little” to the command below while running R on a Mac, the command reads the first integer as 16777216

> readBin (to.read, integer(), endian = “little”) [1] 1

thus, it looks like the first integer in the file is 1. as we repeatedly use ` readBin `

commands, we will work our way through the binary file until we hit the end. we can read in multiple integers at once by adding an ` n= `

option to our command. if the n you specify is greater than the number of integers you specified, ` readBin `

will read and display as much as is available, so there is no danger of guessing too large an n. since we have already read in the first integer, this command will begin at the second

> readBin (to.read, integer(), n = 4, endian = “little”) [1] 2 3 4 5

if you know have additional information about what is in your file, you should incorporate that into the ` readBin `

command. for example, if you know that you wish to read in integers stored on 4 bytes each, you can indicate this with the ` size `

option:

> readBin (to.read, integer(), n = 2, size = 4, endian = “little”) [1] 6 7

similarly, if you know that your file contains characters, complex numbers, or some other type of information, you would adjust the ` readBin `

command accordingly, changing integer() to character() or complex()

since you will likely want to do more than just look at what is contained in the binary file, you will need some strategies for formatting data as you read it in

for example, suppose you are given a binary file with the following description:

- three numeric variables collected from 200 subjects,

- the three variable names appear first in the file,

- the numeric values are integers store on two bytes each, and

- all of the values for the first variables are followed by all the values for the second and then all of the values for the third (as if they have be read in as columns, not rows)

first, open a connection to the data

> newdata = file (“https://stats.idre.ucla.edu/stat/r/faq/bindata.dat”, “rb”)

next, let’s read in the variable names and save them to a vector in R

> varnames = readBin (newdata, character(), n=3) > varnames [1] “read” “write” “math”

to read in the integer values, we can opt to read all 600 onto one vector, and then separate it out into the three variables:

> datavals = readBin (newdata, integer(), size = 4, n = 600, endian = “little”) > readvals = datavals[1:200] > writevals = datavals[201:400] > mathvals = datavals[401:600]

or we can read in each variable’s values with a separate ` readBin `

command:

> readvals = readBin (newdata, integer(), size = 4, n = 200, endian = “little”) > writevals = readBin (newdata, integer(), size = 4, n = 200, endian = “little”) > mathvals = readBin (newdata, integer(), size = 4, n = 200, endian = “little”)

then, we can combine our three value vectors into one data frame with the variable names as our column names:

> rdata = cbind (readvals, writevals, mathvals) > colnames (rdata) = varnames > rdata[1:5,] read write math [1,] 57 52 41 [2,] 68 59 53 [3,] 44 33 54 [4,] 63 44 47 [5,] 47 52 57

lastly, since we have finished reading data from the binary file, we can close the connection:

> close (newdata)

> xs = c(1,2,4,5) > ys = as.integer (xs) > fd = file ("mybin.dat","wb") > writeBin (ys, fd) > close (fd) > fd = file("mybin.dat", "rb") > readBin (fd, integer(), 4) [1] 1 2 4 5

there is a way to tell R to treat the some column as a set of factors. you specify that a variable is a factor using the `factor`

command. in the following example you convert column "x$month" (which can contain month's names) into a factor:

> x$month <- factor (x$month)

once a vector is converted into a set of factors then R treats it DIFFERENTLY - a set of factors have a DISCRETE SET of possible values, and it does not make sense to try to find averages or other NUMERICAL descriptions

> meteo$Month = factor (meteo$Month, ordered = T, levels = c ("Jan","Feb","Mar","Apr", "May","Jun","Jul","Aug", "Sep","Oct","Nov","Dec")) > is.factor (month) [1] TRUE > plot (meteo$Month, meteo$MeanTemp) # or > boxplot (meteo$MeanTemp ~ meteo$Month, col = "orange") > dev.off () > meteo$MeanTemp[meteo$Month == "Jan"]

dataframe

a dataframe is an object with rows and columns

the rows contain different observations/measurements from your experiment

the columns contain different variables

the values in the body of the dataframe can be numbers, but they could also be text; they could be calendar dates (like 23/5/04); or they could be logical variables

> d <- c (7, 4, 6, 8, 9, 1, 0, 3, 2, 5, 0) > r <- rank (d) > s <- sort (d) > o <- order (d) > v <- data.frame (d, r, s, o) > v d r s o 1 7 9.0 0 7 2 4 6.0 0 11 3 6 8.0 1 6 4 8 10.0 2 9 5 9 11.0 3 8 6 1 3.0 4 2 7 0 1.5 5 10 8 3 5.0 6 3 9 2 4.0 7 1 10 5 7.0 8 4 11 0 1.5 9 5

> w <- read.table ("some.txt", head = 1) > attach (w) > month <- factor (month) > h <- read.csv (file = "simple.csv", head = TRUE, sep = ", ")

> names(v1) <- c("x","y","sum")

**read.table** would fail if there were any spaces in any of the variable names in row 1 of the dataframe (the header row) or between any of the words within the same factor level

you can get a quick summary of the data by calculating a frequency table. a frequency table is a table that represents the number of occurrences of every unique value in the variable

- lapply (x, function, ...)
- apply a function over a List or Vector

returns a list of the same length as x, each element of which is the result of applying FUN to the corresponding element of x - sapply (x, function, ..., simplify=TRUE, USE.NAMES=TRUE)
- apply a function over a List or Vector

is a user-friendly version and wrapper of lapply - vapply (x, function, function.VALUE, ..., USE.NAMES=TRUE)
- apply a function over a List or Vector

is similar to sapply, but has a pre-specified type of return value, so it can be safer - apply (x, margin, function, ...)
- Apply Functions Over Array Margins

Returns a vector or array or list of values obtained by applying a function to margins of an array or matrix

MARGIN: a vector giving the subscripts which the function will be applied over

for a matrix ‘1’ indicates rows, ‘2’ indicates columns, ‘c(1, 2)’ indicates rows and columns.

Where x has named dimnames, it can be a character vector selecting dimension names - mapply (column, factor, function)
- Apply a Function to Multiple List or Vector Arguments

is a multivariate version of sapply. mapply applies FUN to the first elements of each ... argument, the second elements, the third elements, and so on - tapply (column, factor, function)
- apply a function to each (non-empty) group of values given by a unique combination of the levels of certain factors
- aggregate (column, list (title=factor ,...), function)
- by (column, list (title=factor,...), function)

> tapply (meteo$MeanTemp, meteo$Year, mean) > aggregate (meteo$MeanTemp, list (period = meteo$Year), mean) > by (meteo$MeanTemp, list (period = meteo$Year), mean) # exclude variables field1, field2, field3 m1 <- names (mydata) %in% c ("field1", "field2", "field3") n1 <- mydata[!m1] # exclude 3rd and 5th field n2 <- mydata[c (-3,-5)] # take first five observations n3 <- mydata[1:5,] # sql-style n4 <- mydata[ which (mydata$gender == 'F' & mydata$age > 65), ] # or attach (newdata) newdata <- mydata[ which (gender == 'F' & age > 65),] detach (newdata) # using subset function newdata <- subset (mydata, age > = 20 | age < 10, select = c (ID, Weight))

> p = read.csv ("temperature.csv") > p$Month = factor (p$Month, ordered = T, levels = c ("Jan","Feb","Mar","Apr", "May","Jun","Jul","Aug", "Sep","Oct","Nov","Dec")) > boxplot (p$Month, p$Temp, col="orange", main="raw data")

> mm = tapply (p$Temp, p$Month, median) > mm Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec -5.8 -5.8 -0.7 6.9 12.3 18.2 21.0 19.9 15.7 9.5 3.4 -2.8 > as.numeric (mm) [1] -5.8 -5.8 -0.7 6.9 12.3 18.2 21.0 19.9 15.7 9.5 3.4 -2.8 > boxplot (p$Month, p$Temp - as.numeric (mm), col="orange", main="deseasoned data") > dev.off()

> x = seq (1,7,1) > x [1] 1 2 3 4 5 6 7 > y = rep (1,7) > y [1] 1 1 1 1 1 1 1 > sum (y) [1] 7 > sum (y[x>3]) [1] 4 > sum (y[x==2]) [1] 1

> w1 <- read.table ("filename", header = T) > stripchart (w1$vals) > stripchart (w1$vals, method = "stack")

the histogram graphically shows the following:

- center (i.e., the location) of the data
- spread (i.e., the scale) of the data
- skewness of the data
- presence of outliers and
- presence of multiple modes in the data

> hist (w1$vals, col = 'grey', breaks = 12, xlim = c (0.9, 1.3))

box plots are an excellent tool for conveying location and variation information in data sets, particularly for detecting and illustrating location and variation changes between different groups of data

> boxplot (w1$vals, main = 'Main Title of the Plot', xlab = 'x axe label', horizontal = TRUE) > boxplot (e$MeanTemp~e$Month, col = "orange")

> plot (x, y, type = "l", pch = 3)

‘type’ possibilities:

the default plotting character (pch = 1) is ο

if you want Δ, use pch = 2

if you want + (plus signs), use pch = 3

if you want x use pch = 4

if you want ♦ use pch = 5

to draw the regression line through the data, you employ the straight line drawing directive

> abline (intercept, slope)

> abline (lm (y~x))

to plot four graphics (two-in-rows) use the command:

> par (mfrow = c (2, 2))

to plot two different funcs on the same plot:

> plot (t, p1, ylim = c (-6,4), type = "l", col = "red" ) > par (new = TRUE) > plot (t, p2, ylim = c (-6,4), type = "l", col = "green" )

> p1 = matrix (p1) > p2 = matrix (p2) > matplot (t, cbind (p1, p2), pch = 19)

> plot (t, p1, ylim = c (-6,5)) > points (t, p2)

> plot (t, p1, ylim = c (-6,5)) > lines (t, p2)

- par ()
- to look at current graphical params
- par (col.lab="red")
- to set the parameter

a second way to specify graphical parameters is by providing the **optionname=value** pairs directly to a plotting function

> hist (mtcars$mpg, col.lab="red")

function | output to |
---|---|

pdf ("mygraph.pdf") | pdf file |

png ("mygraph.png") | png file |

jpeg ("mygraph.jpg") | jpeg file |

bmp ("mygraph.bmp") | bmp file |

postscript ("mygraph.ps") | postscript file |

dev.off () | returns output to the terminal |

so to save a jpg file called "plot01.jpg" containing a plot of x and y:

> jpeg ('plot01.jpg') > plot (x, y) > dev.off ()

runif

generates a set number of random draws, n, from an interval with a specified lower bound, min, and upper bound, max (the r stands for random, and the unif means uniform, obviously)

rnorm

can generate random values from a normal distribution

runif(n = 10, min = 0, max = 5)

generates 10 random draws from 0 to 5. but if you don’t include the min and

max arguments…

runif(n = 10)

## [1] 0.2836422 0.7407605 0.3658010 0.2006554 0.8325042 0.9261639 0.7242986

0.1317093

## [9] 0.2777294 0.5944356

…the function defaults to the standard uniform with min of 0 and max of 1

#normal

?rnorm()

#binomial/bernoulli

?rbinom()

#geometric

?rgeom()

#exponential

?rexp()

#beta

?rbeta()

#gamma

?rgamma()

#poisson

?rpois()

#uniform

?runif()

n: number of observations

mean: vector of means

sd: vector of standard deviations

lambda: vector of (non-negative) means

x: number of successes (or a vector of length 2 giving the numbers of successes and failures, respectively)

n: number of trials (ignored if ‘x’ has length 2)

p: hypothesized probability of success

many statistical tests are based on the assumption of normality. the assumption of normality often leads to tests that are simple and powerful compared to tests that do not make the normality assumption. unfortunately, many real data sets are in fact not approximately normal

use Student' t-test when the means are independent and the errors are normally distributed

non-normality, outliers and serial correlation can all invalidate inferences made by Student' t-test. much better in such cases to use a non-parametric technique - Wilcoxon' signed-rank test. use Wilcoxon' rank sum test when the means are independent but errors are not normally distributed

neither the t-test nor the w-test can cope properly with situations where the variances are different, but the means are the same. this draws attention to a very general point: scientific importance and statistical significance are not the same thing

to be sure that your data sampling x is close to normal distribution use **Shapiro test**

**t-values** associated with different levels of confidence available in R:

the first argument is the probability and

the second is the degrees of freedom of your population

confidence intervals are always 2-tailed. thus, if you want to establish a 95% confidence interval you need to calculate (or look up) **t-value**

> qt (.025, 9) [1] -2.262157

> qt (.975, 9) [1] 2.262157

this says that

so, finally, you can write down the formula for the confidence interval of a mean based on a small sample:

> x = c(3, 5, 7) > qt (.975, 2) * var (x) [1] 17.21061 > qt (.025, 2) * var (x) [1] -17.21061

there is a built-in function **t.test**

> t.test (A, B) Welch Two Sample t-test data: A and B t = -3.873, df = 18, p-value = 0.001115 alternative hypothesis: true difference in means is not equal to 0 95 percent confidence interval: -3.0849115 -0.9150885 sample estimates: mean of x mean of y 3 5

you typically use 5% as the chance of rejecting the null hypothesis

when the covariance of A and B is positive, this is a great help because it reduces the variance of the difference, and should make it easier to detect significant differences between the means.

pairing is not always effective, because the correlation between A and B may be weak

> x <- c (20, 15, 10, 5, 20, 15, 10, 5, 20, 15, 10, 5, 20, 15, 10, 5) > y <- c (23, 16, 10, 4, 22, 15, 12, 7, 21, 16, 11, 5, 22, 14, 10, 6) > t.test (x, y) > t.test (x, y, paired = T)

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

78 | 24 | 64 | 45 | 64 | 52 | 30 | 50 | 64 | 50 | 78 | 22 | 84 | 40 | 90 | 72 |

78 | 24 | 62 | 48 | 68 | 56 | 25 | 44 | 56 | 40 | 68 | 36 | 68 | 20 | 58 | 32 |

> t = read.table ("marks.dat", header = T) > x = as.numeric (t[, 2]); x1 = x[1:16] > y = as.numeric (t[, 3]); y1 = y[1:16] > t.test (x1, y1, paired = T) Paired t-test data: x1 and y1 t = 2.2353, df = 15, p-value = 0.04103 alternative hypothesis: true difference in means is not equal to 0 95 percent confidence interval: 0.3600356 15.1399644 sample estimates: mean of the differences 7.75

Wilcoxon non-parametric test is used if the errors looked to be non-normal

this non-parametric test can be more powerful than the t-test if the distribution is strongly skewed

the Kolmogorov-Smirnov test is used to decide if a sample comes from a population with a specific distribution. the test is based on the empirical cumulative distribution function (ECDF)

ks.test (x, 'pnorm')

if ‘y’ is numeric, a two-sample test of the null hypothesis that ‘x’ and ‘y’ were drawn from the **same _continuous_ distribution** is performed

‘y’ can be a character string naming a __continuous__ CDF, or such a function. in this case, a one-sample test is carried out of the null that the distribution function which generated ‘x’ is distribution ‘y’

the K-S test has several important limitations:

- applies to continuous distributions only
- more sensitive near the center of the distribution than at the tails
- the distribution must be fully specified

suppose you are interested in investigating the assosiation between x and y

isn't just enought to caclulate the correlation ρ between x and y?

perhaps for this dataset (ρ=0.83) it is enought

what about this dataset? ρ=0.5

and this? ρ=-0.8

one of the best methods to describe some data is by fitting a statistical model. the model parameters tell you much more about the relationship between x and y than correlation coefficient. in statical modeling you are inerested in estimating the unknown parameters from your data

models have parameters some of which are unknown. you are intrested in the inferring the unknown parameters from your data. parameter's estimation needs be done in formal way

- is there a trend in the data?
- what is the slope of the trend (positive or negative)?
- is the trend linear or curved?
- is there any pattern to the scatter around the trend?

> x [1] 1 2 3 4 5 6 7 8 9 10 > y [1] 0.9252341 0.8546417 3.1241370 6.0860490 6.2729448 8.7434938 [7] 5.1923348 9.1469443 4.9776600 7.9872857

you begin with the simplest possible linear model; the straight line: y = a * x + b

b - interseption a - slope

- - the intercept a is greater than zero?
- - the slope b is negative?
- - the variance in y is constant?

> z = lm (x~y) > summary (z)

the first step is to fit a horizontal line though the data, using **abline (intersept,slope)**, showing the average value of y:

> plot (x, y) > abline (lm (x ~ y))

TODO

TODO

TODO

TODO

the likelihood function, when evaluated in certain point (args of the function), gives the probability of observing the data

the data is treated as fixed quantity and model's parameters treated as random variables

priors should be choosen before we see the data. if you know nothing about the parameter you should assign to it so-called **uninformative** prior

sayHello <- function () { print ('hello') } sayHello ()

how can you run this via command-line? if you want the output to print to the terminal it is best to use Rscript

$> Rscript a.R

note that when using R CMD BATCH a.R that instead of redirecting output to standard out and displaying on the terminal a new file called a.Rout will be created

>R CMD BATCH a.R

>cat a.Rout

if you really want to use the *./a.R* way of calling the script you could add an appropriate *#!* to the top of the script

if you can use vectorized functions then *loops* should be a last resort

you need to use them when you do something DIFFERENT to each element of an object

the `break`

statement can be used to terminate any loop, possibly abnormally. this is __the only way__ to terminate `repeat`

loops

#!/usr/local/bin/Rscript #binomial distribution args <- commandArgs (TRUE) z = as.numeric (args[1]) p = as.numeric (args[2]) g <- function (x) { choose (z, x) * p^x * (1-p)^(z-x) } x = 1:z plot (x, g (x), type = "l")

$> Rscript bintrial.r 100 0.5 $> display Rplots.pdf

#!/usr/local/bin/Rscript #erlang distribution args <- commandArgs(TRUE) k = as.numeric(args[1]) λ = as.numeric(args[2]) erlang <- function (t) { λ^k * t^(k - 1) * exp (- λ * t) / factorial (k - 1) } t = seq (from = 0, by = .1, to = 100) plot (t, erlang (t), type = "l")

$> Rscript erltrial.r 3 .07 $> display Rplots.pdf

> search () > library () > installed.packages () > list.of.packages <- c ("dlm", "hawkes") > new.packages <- list.of.packages[!(list.of.packages %in% installed.packages()[,"Package"])] > if (length (new.packages)) install.packages (new.packages)

and after that:

> require (dlm)