Numerical Analysis Software: MATLAB, R, NumPy

a side-by-side reference sheet

matlab r numpy
version used Octave 3.2 2.6 Python 2.7
NumPy 1.6
SciPy 0.10
Pandas 0.9
Matplotlib 1.0
show version $octave --version$ r --version sys.version
np.__version__
sp.__version__
mpl.__version__
implicit prologue none install.packages('ggplot2')
library('ggplot2')
import sys, os, re, math
import numpy as np
import scipy as sp
import scipy.stats as stats
import pandas as pd
import matplotlib as mpl
import matplotlib.pyplot as plt
grammar and invocation
matlab r numpy
interpreter

$octave foo.m$ Rscript foo.r

$r -f foo.r$ python foo.py
repl

$octave$ r $python command line program$ octave --silent --eval 'printf("hi")' $Rscript -e 'print("hi")' python -c 'print("hi")' block delimiters function endfunction if elseif else endif while endwhile do until for endfor { } offside rule statement separator ; or newline ; or sometimes newline newline or ; newlines not separators inside (), [], {}, triple quote literals, or after backslash: \ end-of-line comment 1 + 1 % addition Octave only: 1 + 1 # addition 1 + 1 # addition 1 + 1 # addition variables and expressions matlab r numpy assignment i = 3 i = 3 i <- 3 3 -> i assign("i", 3) i = 3 compound assignment arithmetic, string, logical MATLAB has no compound assignment operators. Octave has these: += -= *= /= none none **= or ^= none none &= |= none none # do not return values: += -= *= /= //= %= **= += *= &= |= ^= increment and decrement ++x --x x++ x-- none none null only used in place of numeric values: NA NA NULL None null test isna(v) true for '', []: isnull(v) is.na(v) is.null(v) v == None v is None conditional expression none (if (x > 0) x else -x) ifelse(x > 0, x, -x) x if x > 0 else -x arithmetic and logic matlab r numpy true and false 1 0 true false TRUE FALSE T F True False falsehoods false 0 0.0 matrices evaluate to false unless nonempty and all entries evaluate to true FALSE F 0 0.0 matrices evaluate to value of first entry; string in boolean context causes error False None 0 0.0 '' [] {} logical operators ~true | (true & false) Optional negation operator in Octave: ! short-circuit operators: && || !TRUE | (TRUE & FALSE) short-circuit operators: && || & and | can operate on and return vectors, but && and || return scalars and or not relational operators == ~= > < >= <= Optional inequality operator in Octave: != == != > < >= <= == != > < >= <= arithmetic operators add, sub, mult, div, quot, rem + - * / none mod(n, divisor) + - * / ? %% + - * / // % integer division fix(13 / 5) as.integer(13 / 5) 13 // 5 integer division by zero Inf NaN or -Inf result of converting Inf or NaN to an integer with as.integer: NA raises ZeroDivisionError float division 13 / 5 13 / 5 float(13) / 5 float division by zero dividend is positive, zero, negative these values are literals: Inf NaN -Inf these values are literals: Inf NaN -Inf raises ZeroDivisionError power 2 ^ 16 % Octave only: 2 ** 16 2 ^ 16 2 ** 16 2 ** 16 sqrt sqrt(2) sqrt(2) math.sqrt(2) sqrt(-1) % returns 0 + 1i: sqrt(-1) # returns NaN: sqrt(-1) # returns 0+1i: sqrt(-1+0i) # raises ValueError: math.sqrt(-2) # returns 1.41421j: import cmath cmath.sqrt(-2) transcendental functions exp log sin cos tan asin acos atan atan2 exp log sin cos tan asin acos atan atan2 math.exp math.log math.sin math.cos math.tan math.asin math.acos math.atan math.atan2 transcendental constants pi e pi exp(1) math.pi math.e float truncation round towards zero, to nearest integer, down, up fix(x) round(x) floor(x) ceil as.integer(x) round(x) floor(x) ceiling(x) int(x) int(round(x)) math.floor(x) math.ceil(x) absolute value and signum abs sign abs sign abs(-3.7) math.copysign(1, -3.7) integer overflow becomes float; largest representable integer in the variable intmax becomes float; largest representable integer in the variable .Machine$integer.max becomes arbitrary length integer of type long
float overflow

Inf Inf raises OverflowError
float limits

eps
realmax
realmin
.Machine$double.eps .Machine$double.xmax
.Machine$double.xmin np.finfo(np.float64).eps np.finfo(np.float64).max np.finfo(np.float64).min complex construction 1 + 3i 1 + 3i 1 + 3j complex decomposition real imag abs arg conj Re Im abs Arg Conj import cmath z.real z.imag cmath.polar(z)[1] random number uniform integer, uniform float floor(100*rand) rand floor(100*runif(1)) runif(1) np.random.randint(0, 100) np.random.rand() random seed set, get, and restore rand('state', 17) sd = rand('state') rand('state', sd) set.seed(17) sd = .Random.seed none np.random.seed(17) sd = np.random.get_state() np.random.set_state(sd) bit operators bitshift(100, 3) bitshift(100, -3) bitand(1, 2) bitor(1, 2) bitxor(1, 2) % MATLAB: bitcmp(1, 'uint16') % Octave: bitcmp(1, 16) none 100 << 3 100 >> 3 1 & 2 1 | 2 1 ^ 2 ~1 strings matlab r numpy literal 'don''t say "no"' Octave also has double quoted strings: "don't say \"no\"" "don't say \"no\"" 'don\'t say "no"' 'don\'t say "no"' "don't say \"no\"" r"don't " r'say "no"' newline in literal no; use \n escape yes no literal escapes \\ \" \' \0 \a \b \f \n \r \t \v \\ \" \' \a \b \f \n \r \t \v \ooo single and double quoted: \newline \\ \' \" \a \b \f \n \r \t \v \ooo \xhh character access 'hello'(1) substr("hello", 1, 1) 'hello'[0] chr and ord char(65) toascii('A') intToUtf8(65) utf8ToInt("A") chr(65) ord('A') length length('hello') nchar("hello") len('hello') concatenate horzcat('one ', 'two ', 'three') paste("one ", "two ", "three") 'one ' + 'two ' + 'three' literals, but not variables, can be concatenated with juxtaposition: 'one ' "two " 'three' replicate hbar = repmat('-', 1, 80) hbar = paste(rep('-', 80), collapse='') hbar = '-' * 80 index of substring counts from one, returns zero if not found index('hello', 'el') counts from one, returns -1 if not found regexpr("el", "hello") counts from zero, raises ValueError if not found: 'hello'.index('el') extract substring substr('hello', 1, 4) substr("hello", 1, 4) 'hello'[0:4] split returns tuple: strsplit('foo,bar,baz',',') strsplit('foo,bar,baz', ',') 'foo,bar,baz'.split(',') join paste("foo", "bar", "baz", sep=",") paste(c('foo', 'bar', 'baz'), collapse=',') ','.join(['foo', 'bar', 'baz']) trim strtrim(' foo ') ?? deblank('foo ') gsub("(^[\n\t ]+|[\n\t ]+$)",
"",
" foo ")
sub("^[\n\t ]+", "", " foo")
sub("[\n\t ]+$", "", "foo ") ' foo '.strip() ' foo'.lstrip() 'foo '.rstrip() convert from string, to string 7 + str2num('12') 73.9 + str2num('.037') horzcat('value: ', num2str(8)) 7 + as.integer("12") 73.9 + as.double(".037") paste("value: ", toString("8")) 7 + int('12') 73.9 + float('.037') 'value: ' + str(8) case manipulation lower('FOO') upper('foo') tolower("FOO") toupper("foo") 'foo'.upper() 'FOO'.lower() 'foo'.capitalize() sprintf sprintf('%s: %.3f %d', 'foo', 2.2, 7) sprintf("%s: %.3f %d", "foo", 2.2, 7) '%s: %.3f %d' % ('foo', 2.2, 7) regular expressions matlab r numpy regex test regexp('hello', '^[a-z]+$')
regexp('hello', '^\S+$') regexpr("^[a-z]+$", "hello") > 0
regexpr('^\\S+$', "hello",perl=T) > 0 re.search('^[a-z]+$', 'hello')
re.search('^\S+$', 'hello') regex substitution regexprep('foo bar bar','bar','baz','once') regexprep('foo bar bar','bar','baz') sub('bar','baz','foo bar') gsub('bar','baz','foo bar bar') rx = re.compile('bar') s = rx.sub('baz', 'foo bar', 1) s2 = rx.sub('baz', 'foo bar bar') dates and time matlab r numpy current date/time t = now t = as.POSIXlt(Sys.time()) date/time type floating point number representing days since year 0 in the Gregorian calendar POSIXlt date/time difference type floating point number representing days a difftime object which behaves like a floating point number representing seconds get date parts datevec(t)(1) datevec(t)(2) datevec(t)(3) t$year + 1900
t$mon + 1 t$mday
get time parts datevec(t)(4)
datevec(t)(5)
datevec(t)(6)
t$hour t$min
t$sec build date/time from parts t = datenum([2011 9 20 23 1 2]) t = as.POSIXlt(Sys.time()) t$year = 2011 - 1900
t$mon = 9 - 1 t$mday = 20
t$hour = 23 t$min = 1
t$sec = 2 convert to string datestr(t) print(t) strptime t = datenum('2011-09-20 23:01:02', 'yyyy-mm-dd HH:MM:SS') t = strptime('2011-09-20 23:01:02', '%Y-%m-%d %H:%M:%S') strftime datestr(t, 'yyyy-mm-dd HH:MM:SS') format(t, format='%Y-%m-%d %H:%M:%S') tuples matlab r numpy tuple literal tup = {1.7, 'hello', [1 2 3]} tup = list(1.7, "hello", c(1, 2, 3)) tup = (1.7, "hello", [1,2,3]) tuple element access tup{1} tup[[1]] tup[0] tuple length length(tup) length(tup) len(tup) arrays matlab r numpy literal # arrays and vectors are same type a = c(1, 2, 3, 4) a = [1, 2, 3, 4] size length(a) len(a) empty test length(a) == 0 not a lookup # indices start at one a[1] # indices start at zero a[0] update a[1] = "lorem" a[0] = 'lorem' out-of-bounds behavior a = c() # evaluates as NA: a[10] # increases array size to 10: a[10] = "lorem" a = [] # raises IndexError: a[10] # raises IndexError: a[10] = 'lorem' index of element a = c('x', 'y', 'z', 'w', 'y') # c(2, 5): which(a == 'y') a = ['x', 'y', 'z', 'w', 'y'] a.index('y') # 1 a.rindex('y') # 4 slice by endpoints, by length a = c("a", "b", "c", "d", "e") # return c("c", "d"): a[seq(3, 4)] a[seq(3, 3 + 1)] a = ['a', 'b', 'c', 'd', 'e'] # return ['c', 'd']: a[2:4] a[2:2 + 2] slice to end tail(a, n=length(a) - 1) a[1:] manipulate back none a = [6, 7, 8] a.append(9) a.pop() manipulate front none a = [6, 7, 8] a.insert(0, 5) a.pop(0) concatenate a = c(1, 2, 3) a2 = append(a, c(4, 5, 6)) a = append(a, c(4, 5, 6)) a = [1, 2, 3] a2 = a + [4, 5, 6] a.extend([4, 5, 6]) replicate a = rep(NA, 10) a = [None] * 10 a = [None for i in range(0, 10)] copy address copy, shallow copy, deep copy # arrays cannot be elements of arrays a = [1, 2, 3, 4] none a2 = a import copy a = [1, 2, [3, 4]] a2 = a a3 = list(a) a4 = copy.deepcopy(a) arrays as function arguments modifying parameter will not modify original array parameter contains address copy; modifying parameter modifies original array iteration for i in [1, 2, 3]: print(i) indexed iteration a = ['do', 're', 'mi', 'fa'] for i, s in enumerate(a): print('%s at index %d' % (s, i)) reverse a = c(1, 2, 3) a2 = rev(a) a = rev(a) a = [1, 2, 3] a2 = a[::-1] a.reverse() sort a = c('b', 'A', 'a', 'B') a2 = sort(a) a = sort(a) a = ['b', 'A', 'a', 'B'] sorted(a) a.sort() a.sort(key=str.lower) dedupe a = c(1, 2, 2, 3) a2 = unique(a) a = unique(a) a = [1, 2, 2, 3] a2 = list(set(a)) a = list(set(a)) membership 7 %in% a is.element(7, a) 7 in a intersection intersect(c(1, 2), c(2, 3, 4)) {1, 2} & {2, 3, 4} union union(c(1, 2), c(2, 3, 4)) {1, 2} | {2, 3, 4} relative complement, symmetric difference setdiff(c(1, 2, 3), c(2)) union(setdiff(c(1, 2), c(2, 3, 4)), setdiff(c(2, 3, 4), c(1, 2))) {1, 2, 3} - {2} {1, 2} ^ {2, 3, 4} map map(lambda x: x * x, [1, 2, 3]) # or use list comprehension: [x * x for x in [1, 2, 3]] filter filter(lambda x: x > 1, [1, 2, 3]) # or use list comprehension: [x for x in [1, 2, 3] if x > 1] reduce reduce(lambda x, y: x + y, [1 ,2, 3], 0) universal and existential tests all(i % 2 == 0 for i in [1, 2, 3, 4]) any(i % 2 == 0 for i in [1, 2, 3, 4]) shuffle and sample from random import shuffle, sample a = [1, 2, 3, 4] shuffle(a) sample(a, 2) zip # array of 3 pairs: a = zip([1, 2, 3], ['a', 'b', 'c']) sequences matlab r numpy range 1:100 1:100 seq(1, 100) range(1, 101) iterate over range range replaces xrange in Python 3: for i in xrange(1, 1000001): code instantiate range as array a = range(1, 11) Python 3: a = list(range(1, 11)) arithmetic sequence of integers with difference 10 0:10:100 seq(0, 100, 10) range(0, 101, 10) arithmetic sequence of floats with difference 0.1 0:0.1:10 seq(0, 10, 0.1) [0.1 * x for x in range(0, 101)] 3rd arg is length of sequence, not step size: sp.linspace(0, 10, 100) multidimensional arrays matlab r numpy 1d array literal [1, 2, 3] commas are optional: [1 2 3] none none 2d array literal [1, 2; 3, 4] spaces and newlines can replace commas and semicolons: [1 2 3 4] none none 3d array from 2d arrays A = [1, 2; 3, 4] A(:,:,2) = [5, 6; 7, 8] 1d array from elements c(1, 2, 3) 1d 1d array from sequential data type array(c(1, 2, 3)) np.array([1, 2, 3]) np.array((1, 2, 3)) 2d array from sequential data type array(c(1,2,3,4),dim=c(2,2)) np.array([1, 2, 3, 4]).reshape(2, 2) 2d array from rows rbind(c(1, 2, 3), c(4, 5, 6)) 2d array from rows cbind(c(1, 4), c(2, 5), c(3, 6)) 3d array from sequential data type array(c(1,2,3,4,5,6,7,8),dim=c(2,2,2)) np.array([1, 2, 3, 4, 5, 6, 7, 8]).reshape(2, 2, 2) 2d array from nested sequential data types np.array([[1, 2], [3, 4]]) 3d array from nested sequential data types np.array([[[1, 2], [3, 4]], [[5, 6], [7, 8]]]) must arrays be homogeneous yes yes yes array data type always numeric class(c(1, 2, 3)) a = array(c(1, 2, 3)) class(c(a)) np.array([1, 2, 3]).dtype data types permitted in arrays numeric boolean, numeric, string np.bool, np.int64, np.float64, np.complex128, and others 1d array element access indices start at one: [1 2 3](1) indices start at one: c(1, 2, 3)[1] indices start at zero: a = np.array([1, 2, 3]) a[0] 2d array element access [1 2; 3 4](1, 1) a = array(c(1, 2, 3, 4), dim=c(2, 2) a[1, 1] a = np.array([[1, 2], [3, 4]]) a[0][0] or a[0, 0] 1d index access of higher dimensional array returns 4: [1 2; 3 4](4) a = array(c(1, 2, 3, 4), dim=c(2, 2)) returns 4: a[4] index of array element which(c(7,8,9)==9) array slice [1 2 3](1:2) c(1,2,3)[1:2] np.array([1,2,3])[0:2] assign to array slice integer array as index [1 2 3]([1,3,3]) c(1,2,3)[c(1,3,3)] np.array([1,2,3])[[0,2,2]] logical array as index [1 2 3]([true false true]) c(1,2,3)[c(T,F,T)] np.array([1,2,3])[[True,False,True]] array length length([1 2 3]) length(c(1,2,3)) len(np.array([1,2,3])) multidimensional array size length(dim(a)) dim(a) a.ndim a.shape array concatenation cat(2, [1 2 3], [4 5 6]) horzcat([1 2 3], [4 5 6]) append(c(1,2,3),c(4,5,6)) a1 = np.array([1,2,3]) a2 = np.array([4,5,6]) np.concatenate([a1,a2]) multidimensional array concatenation m = matrix(c(1, 2, 3, 4), nrow=2) m4_by_2 = rbind(m, m) m2_by_4 = cbind(m, m) array replication rep("a", 100) rep(c("a", "b", "c"), c(30, 50, 90)) sort a = [3 1 4 2] a = sort(a) a = c(3,1,4,2) a = sort(a) a = np.array([3,1,4,2]) a.sort() map arrayfun( @(x) x*x, [1 2 3]) sapply(c(1,2,3), function (x) { x * x}) a = np.array([1,2,3]) np.vectorize(lambda x: x*x)(a) filter v = [1 2 3] v(v > 2) v = c(1,2,3) v[v > 2] v = np.array([1,2,3]) a = [x for x in v if x > 2] np.array(a) sample w/o replacement x = c(3,7,5,12,19,8,4) sample(x, 3) from random import sample sample([3,7,5,12,19,8,4], 3) dictionaries matlab r numpy literal d = struct('n', 10, 'avg', 3.7, 'sd', 0.4) d = list(n=10, avg=3.7, sd=0.4) d = {'n': 10, 'avg': 3.7, 'sd': 0.4} size length(fieldnames(d)) length(d) len(d) lookup d.n d$n d['n']
update

d.var = d.sd**2 d$var = d$sd**2 d['var'] = d['sd']**2
out-of-bounds behavior

error NULL raises KeyError
is key present

isfield(d, 'var') is.null(d$var) 'var' in d delete d = rmfield(d, 'sd') d$sd = NULL del(d['sd'])
iterate for k, v in d.iteritems():
code
keys and values as arrays d.keys()
d.values()
merge d1 = list(a=1, b=2)
d2 = list(b=3, c=4)
values of first dictionary take precedence:
d3 = c(d1, d2)
d1 = {'a':1, 'b':2}
d2 = {'b':3, 'c':4}
d1.update(d2)
invert to_num = {'t':1, 'f':0}
to_let = {v:k for k, v
in to_num.items()}
sort by values from operator import itemgetter

pairs = sorted(d.iteritems(),
key=itemgetter(1))

for k, v in pairs:
print('{}: {}'.format(k, v))
functions
matlab r numpy
a+b
endfunction
add = function(a,b) {a + b}
invocation

return value how to declare a return variable:
retvar = a + b
endfunction

the return value is the value assigned to the return variable if one is defined; otherwise it's the last expression evaluated.
return argument or last expression evaluated. NULL if return called without an argument.
function value

anonymous function @(a,b) a+b function(a,b) {a+b}
missing argument raises error if code with the parameter that is missing an argument is executed raises error
extra argument

ignored raises error
default argument function mylog(x, base=10)
log(x) / log(base)
endfunction
mylog = function(x,base=10) {
log(x) / log(base)
}
variable number of arguments function s = add(varargin)
if nargin == 0
s = 0
else
s = varagin{1} + r
endif
endfunction
a = list(...)
if (length(a) == 0)
return(0)
s = 0
for(i in 1:length(a)) {
s = s + a[[i]]
}
return(s)
}
execution control
matlab r numpy
if if (x > 0)
printf('positive\n')
elseif (x < 0)
printf('negative\n')
else
printf('zero\n')
endif
if (x > 0) {
print('positive')
} else if (x < 0) {
print('negative')
} else {
print('zero')
}
if x > 0:
print('positive')
elif x < 0:
print('negative')
else:
print('zero')
while i = 0
while (i < 10)
i++
printf('%d\n', i)
endwhile
while (i < 10) {
i = i + 1
print(i)
}
while i < 10:
i += 1
print(i)
for for i = 1:10
printf('%d\n', i)
endfor
for (i in 1:10) {
print(i)
}
for i in range(1,11):
print(i)
break/continue

break continue break next break continue
raise exception

error('%s', 'failed') stop('failed') raise Exception('failed')
handle exception try
error('failed')
catch
printf('%s\n', lasterr())
end_try_catch
tryCatch(
stop('failed'),
error=function(e) print(message(e)))
try:
raise Exception('failed')
except Exception as e:
print(e)
finally block unwind_protect
if ( rand > 0.5 )
error('failed')
endif
unwind_protect_cleanup
printf('cleanup')
end_unwind_protect
risky = function() {
if (runif(1) > 0.5) {
stop('failed')
}
}

tryCatch(
risky(),
finally=print('cleanup'))
file handles
matlab r numpy
standard file handles stdin stdout stderr stdin() stdout() stderr() sys.stdin sys.stdout sys.stderr
write line to stdout puts("hello\n") cat("hello\n")

writeLines("hello")
print('hello')
write formatted string to stdout printf("%.2f\n", pi) cat(sprintf("%.2f\n", pi)) import math

print('%.2f' % math.pi)
open file for reading if ((f = fopen("/etc/hosts")) == -1)
error("failed to open file")
endif
f = file("/etc/hosts", "r") f = open('/etc/hosts')
open file for writing if ((f = fopen("/tmp/test", "w") == -1)
error("failed to open file")
endif
f = file("/tmp/test", "w") f = open('/tmp/test', 'w')
open file for appending if ((f = fopen("/tmp/err.log", "a") == -1)
error("failed to open file")
endif
f = file("/tmp/err.log", "a") f = open('/tmp/err.log', 'a')
close file fclose(f) close(f) f.close()
i/o errors fopen returns -1; fclose throws an error raise IOError exception
iterate over file by line while(!feof(f))
line = fgets(f)
puts(line)
endwhile
for line in f:
print(line)
write string fputs(f, "lorem ipsum") cat("lorem ipsum", file=f) f.write('lorem ipsum')
write line fputs(f, "lorem ipsum\n") writeLines("lorem ipsum", con=f) f.write('lorem ipsum\n')
flush file handle fflush(f) flush(f) f.flush()
file handle position
get, set
ftell(f)

% 3rd arg can be SEEK_CUR or SEEK_END
fseek(f, 0, SEEK_SET)
seek(f)

# sets seek point to 12 bytes after start;
# origin can also be "current" or "end"

seek(f, where=0, origin="start")
f.tell()

f.seek(0)
redirect stdout to file sink("foo.txt")
directories
matlab r numpy
working directory
get, set
pwd

cd("/tmp")
getwd()

setwd("/tmp")
os.path.abspath('.')

os.chdir('/tmp')
build pathname file.path("/etc", "hosts") os.path.join('/etc', 'hosts')
dirname and basename dirname("/etc/hosts")
basename("/etc/hosts")
os.path.dirname('/etc/hosts')
os.path.basename('/etc/hosts')
absolute pathname normalizePath("..") os.path.abspath('..')
processes and environment
matlab r numpy
command line arguments % does not include interpreter name:
argv()
# first arg is name of interpreter:
commandArgs()

# arguments after --args only:
commandArgs(TRUE)
sys.argv
environment variable
get, set
getenv("HOME")

setenv("PATH", "/bin")
Sys.getenv("HOME")

Sys.setenv(PATH="/bin")
os.getenv('HOME')

os.environ['PATH'] = '/bin'
exit

exit(0) quit(save="no", status=0) sys.exit(0)
external command if (shell_cmd("ls -l /tmp"))
error("ls failed")
endif
if (system("ls -l /tmp")) {
stop("ls failed")
}
if os.system('ls -l /tmp'):
raise Exception('ls failed')
libraries and namespaces
matlab r numpy
load library % if installed as Octave package:
require("foo") or
library("foo")
import foo
list loaded libraries none search() dir()
library search path path()
rmpath('~/foo')
.libPaths() sys.path
source file

source('foo.m') source("foo.r") none
% Octave-Forge in Octave:

pkg install foo-1.0.0.tar.gz
install.packages("ggplot2") $pip install scipy list installed packages pkg list library()$ pip freeze
reflection
matlab r numpy
data type class(x) class(x) type(x)
attributes if x is an object value:
x
attributes(x) [m for m in dir(x)
if not callable(getattr(o,m))]
methods note that most values are not objects:
methods(x)
none; objects are implemented by functions which dispatch based on type of first arg [m for m in dir(x)
if callable(getattr(o,m))]
variables in scope who() objects() dir()
undefine variable

clear('x') rm(v) del(x)
undefine all variables clear -a rm(list=objects())
eval

eval('1+1') eval(parse(text='1+1')) eval('1+1')
function documentation help tan help(tan)
?tan
math.tan.__doc__
list library functions none ls("package:moments") dir(stats)
search documentation not in Octave:
docsearch tan
??tan pydoc -k tan ordered dictionaries matlab r numpy constructor d = pd.Series([3, 5, 7], index=['a', 'b', 'c']) aligned arithmetic d1 = pd.Series([3, 5, 7], index=['a', 'b', 'c']) d2 = pd.Series([1, 2], index=['c', 'b') d3 = d1 + d2 values of d3 are: 'a': NaN, 'b': 7, 'c': 8 aligned arithmetic with fill value d1 = pd.Series([3, 5, 7], index=['a', 'b', 'c']) d2 = pd.Series([1, 2], index=['c', 'b') d3 = d1.add(d2, fill_value=0) values of d3 are: 'a': 3, 'b': 7, 'c': 8 reindex d = pd.Series([1, 2], index=['a', 'b']) d.reindex(['c', 'b', 'a']) values of d are: 'c': NaN, 'b': 2, 'a': 1 reindex with fill value data sets r numpy construct from column arrays gender, height, weight of some people in inches and lbs: sx = c("F","F","F","F","M","M") ht = c(69,64,67,66,72,70) wt = c(150,132,142,139,167,165) people = data.frame(sx, ht, wt) sx = ['F', 'F', 'F', 'F', 'M', 'M'] ht = [69, 64, 67, 66, 72, 70] wt = [150, 132, 142, 139, 167, 165] people = pd.DataFrame({'sx': sx, 'ht': ht, 'wt': wt}) construct from row tuples categorical variable column by default any column with character data is a factor index column column names as array names(people) colnames(people) returns Index object: people.columns access column as array vectors: peopleht
people[,2]
people[['ht']]
people[[2]]
1 column data set:
people[2]
people['ht']

if name does not conflict with any DataFrame attributes:
people.ht
access row as tuple 1 row data set:
people[1,]
list:
as.list(people[1,])
people.ix(0)
access datum datum in 1st row, 2nd column:
people[1,2]
people.get_value(0, 'ht')
order rows by column people[order(people$ht),] order rows by multiple columns people[order(sx, ht),] order rows in descending order people[order(-people$ht),]
limit rows people[seq(1, 3),]
offset rows people[seq(4, 6),]
attach columns copy columns into variables
named
sx, ht and wt:

attach(people)
none
detach columns detach(people) none
spreadsheet editor can edit data, in which case return value of edit must be saved
people = edit(people)
none
import and export
r numpy
import tab delimited # first row defines variable names:
# first row defines column names:
import csv

# first row defines variable names:
# first row defines column names:
set column separator df = read.delim('/etc/passwd',
sep=':',
comment.char='#')
# $grep -v '^#' /etc/passwd > /tmp/passwd df = pd.read_table('/tmp/passwd', sep=':', header=None) set column separator to whitespace df = read.delim('/path/to.txt', sep='') df = read_table('/path/to.txt', sep='\s+') set quote character default quote character for both read.csv and read.delim is double quotes. The quote character is escaped by doubling it. # use single quote as quote character: df = read.csv('/path/to/single-quote.csv', quote="'") # no quote character: df = read.csv('/path/to/no-quote.csv', quote="") Both read_table and read_csv use double quotes as the quote character and there is no way to change it. A double quote can be esacped by doubling it. import file w/o header # column names are V1, V2, … read.delim('/etc/passwd', sep=':', header=FALSE, comment.char='#') #$ grep -v '^#' /etc/passwd > /tmp/passwd
#
# column names are X0, X1, …
col.names=c('ht', 'wt', 'age'))
names=['ht', 'wt', 'age'])
set column types # possible values: NA, 'logical', 'integer', 'numeric',
# 'complex', 'character', 'raw', 'factor', 'Date',
# 'POSIXct'
#
# If type is set to NA, actual type will be inferred to be
# 'logical', 'integer', 'numeric', 'complex', or 'factor'
#

colClasses=c('integer', 'numeric', 'character'))
recognize null values df = read.csv('/path/to/data.csv',
colClasses=c('integer', 'logical', 'character'),
na.strings=c('nil'))
na_values=['nil'])
change decimal mark df = read.csv('/path/to.csv', dec=',')
recognize thousands separator none df = read_csv('/path/to.csv', thousands='.')
unequal row length behavior Missing fields will be set to NA unless fill is set to FALSE. If the column is of type character then the fill value is an empty string ''.

If there are extra fields they will be parsed as an extra row unless
flush is set to FALSE
skip comment lines df = read.delim('/etc/passwd',
sep=':',
comment.char='#')
none

# rows to skip can be specified individually:
index column none df = pd.read_csv('/path/to.csv', index_col='key_col')

# hierarchical index:
export tab delimited write.table(df, '/tmp/data.tab', sep='\t')
export csv

# first column contains row names unless row.names
# set to FALSE

write.csv(df, '/path/to.csv', row.names=F)
relational algebra
matlab r numpy
project columns by name people[c('sx', 'ht')] people[['sx', 'ht']]
project columns by position people[c(1, 2)]
project expression convert to cm and kg:
transform(people, ht=2.54*ht, wt=wt/2.2)
project all columns people[people$ht > 66,] rename columns colnames(people) = c('gender', 'height', 'weight') access sub data set data set of first 3 rows with ht and wt columns reversed people[1:3,c(1,3,2)] select rows subset(people, ht > 66) people[people$ht > 66,]
people[people['ht'] > 66]
select distinct rows
split rows
sep=':',
comment.char='#',
col.names=c('name', 'passwd', 'uid', 'gid', 'gecos',
'home', 'shell'))

sep=':',
comment.char='#',
col.names=c('name', 'passwd', 'gid', 'members'))

merge(pw, grp, by.x='gid', by.y='gid')
# $grep -v '^#' /etc/passwd > /tmp/passwd #$ grep -v '^#' /etc/group > /tmp/group

pw = pd.read_table('/tmp/passwd', sep=':', header=None, names=['name', 'passwd', 'uid', 'gid', 'gecos', 'home', 'shell'])

pd.merge(pw, grp, left_on='gid', right_on='gid')
nulls as join values
left join merge(pw, grp, by.x='gid', by.y='gid', all.x=T) pd.merge(pw, grp, left_on='gid', right_on='gid', how='left')
full join merge(pw, grp, by.x='gid', by.y='gid', all=T) pd.merge(pw, grp, left_on='gid', right_on='gid', how='outer')
antijoin pw[!(pw$gid %in% grp$gid), ]
cross join merge(pw, grp, by=c())
aggregation
matlab r numpy
row count length(pw[, 1]) len(pw)
group by column install.packages('data.table')
library('data.table')

# convert from data.frame to data.table:
people_dt = data.table(people)
people[,max(ht),by=sx]
grouped = people.groupby('sx')
grouped.aggregate(np.max)['ht']
multiple aggregated values grouped = people.groupby('sx')
grouped.aggregate(np.max)[['ht', 'wt']]
group by multiple columns
aggregation functions length sum min max mean sd
nulls and aggregation functions value of sum, min, max, mean, and sd is NA if any of the values is NA
rank
quantile
having
vectors
matlab r numpy
vector literal same as array same as array same as array
element-wise arithmetic operators + - .* ./ + - * / + - * /
result of vector length mismatch raises error values in shorter vector are recycled; warning if one vector is not a multiple length of the other raises ValueError
scalar multiplication 3 * [1, 2, 3]
[1, 2, 3] * 3
3 * c(1, 2, 3)
c(1, 2, 3) * 3
3 * np.array([1, 2, 3])
np.array([1, 2, 3]) * 3
dot product dot([1, 1, 1], [2, 2, 2]) c(1, 1, 1) %*% c(2, 2, 2) v1 = np.array([1, 1, 1])
v2 = np.array([2, 2, 2])
np.dot(v1, v2)
cross product cross([1, 0, 0], [0, 1, 0]) v1 = np.array([1, 0, 0])
v2 = np.array([0, 1, 0])
np.cross(v1, v2)
norms norm([1, 2, 3], 1)
norm([1, 2, 3], 2)
norm([1, 2, 3], Inf)
vnorm = function(x, t) {
norm(matrix(x, ncol=1), t)
}

vnorm(c(1, 2, 3), "1")
vnorm(c(1, 2, 3), "E")
vnorm(c(1, 2, 3), "I")
v = np.array([1, 2, 3])
np.linalg.norm(v, 1)
np.linalg.norm(v, 2)
np.linalg.norm(v, np.inf)
matrices
matlab r numpy
literal or constructor row contiguous:
A = [1, 2; 3, 4]
B = [4 3
2 1]
column contiguous:
A = matrix(c(1, 3, 2, 4), 2, 2)
B = matrix(c(4, 2, 3, 1), nrow=2)
row contiguous:
A = np.matrix([[1, 2], [3, 4]])
B = np.matrix([[4, 3], [2, 1]])
zero, identity, ones, diagonal matrix zeros(3, 3) or zeros(3)
eye(3)
ones(3, 3) or ones(3)
diag([1, 2, 3])
matrix(0, 3, 3)
diag(3)
matrix(1, 3, 3)
diag(c(1, 2, 3))
dimensions rows(A)
columns(A)
dim(A)[1]
dim(A)[2]
element access A(1, 1) A[1, 1] A[0, 0]
row access A(1, 1:2) A[1,] A[0]
column access A(1:2, 1) A[, 1]
submatrix access C = [1, 2, 3; 4, 5, 6; 7, 8, 9]
C(1:2, 1:2)
C = matrix(seq(1, 9), 3, 3, byrow=T)
C[1:2, 1:2]
scalar multiplication 3 * A
A * 3
also:
3 .* A
A .* 3
3 * A
A * 3
3 * A
A * 3
element-wise operators .+ .- .* ./ + - * / + - np.multiply() np.divide()
multiplication A * B A %*% B A * B
power A ** 3
kronecker product kron(A, B) kronecker(A, B) np.kron(A, B)
comparison all(all(A==B))
any(any(A!=A))
all(A==B)
any(A!=B)
norms norm(A, 1)
norm(A, 2)
norm(A, Inf)
norm(A, 'fro')
norm(A, "1")
??
norm(A, "I")
norm(A, "F")
transpose transpose(A)
A'
t(A) A.transpose()
conjugate transpose A = [1i, 2i; 3i, 4i]
A'
A = matrix(c(1i, 2i, 3i, 4i), nrow=2, byrow=T)
Conj(t(A))
A = np.matrix([[1j, 2j], [3j, 4j]])
A.conj().transpose()
inverse inv(A) solve(A) np.linalg.inv(A)
determinant det(A) det(A) np.linalg.det(A)
trace trace(A) sum(diag(A)) A.trace()
eigenvalues eig(A) eigen(A)$values np.linalg.eigvals(A) eigenvectors [evec, eval] = eig(A) evec(1:2) evec(3:4) eigen(A)$vectors np.linalg.eig(A)[1]
system of equations A \ [2;3] solve(A, c(2, 3)) np.linalg.solve(A, [2, 3])
statistics
matlab r numpy
first moment statistics x = [1 2 3 8 12 19]

sum(x)
mean(x)
x = c(1,2,3,8,12,19)

sum(x)
mean(x)
x = [1,2,3,8,12,19]

sp.sum(x)
sp.mean(x)
second moment statistics std(x, 1)
var(x, 1)
n = length(x)

sd(x) * sqrt((n-1)/n)
var(x) * (n-1)/n
sp.std(x)
sp.var(x)
second moment statistics for samples std(x)
var(x)
sd(x)
var(x)
n = float(len(x))

sp.std(x) * math.sqrt(n/(n-1))
sp.var(x) * n/(n-1)
skewness Octave uses sample standard deviation to compute skewness:
skewness(x)
install.packages('moments')
library('moments')

skewness(x)
stats.skew(x)
kurtosis Octave uses sample standard deviation to compute kurtosis:
kurtosis(x)
install.packages('moments')
library('moments')

kurtosis(x) - 3
stats.kurtosis(x)
nth moment and nth central moment n = 5

moment(x, n)
moment(x, n, "c")
install.packages('moments')
library('moments')

n = 5
moment(x, n)
moment(x, n, central=T)
n = 5

??
stats.moment(x, n)
mode mode([1 2 2 2 3 3 4]) samp = c(1,2,2,2,3,3,4)
names(sort(-table(samp)))[1]
stats.mode([1,2,2,2,3,3,4])[0][0]
quantile statistics min(x)
median(x)
max(x)
?
min(x)
median(x)
max(x)
quantile(x, prob=.90)
min(x)
sp.median(x)
max(x)
stats.scoreatpercentile(x, 90.0)
bivariate statistiscs x = [1 2 3]
y = [2 4 7]

cor(x, y)
cov(x, y)
x = c(1,2,3)
y = c(2,4,7)

cor(x, y)
cov(x, y)
x = [1,2,3]
y = [2,4,7]

stats.linregress(x, y)[2]
??
frequency table x = c(1,2,1,1,2,5,1,2,7)
tab = table(x)
invert frequency table rep(as.integer(names(tab)),
unname(tab))
bin x = c(1.1, 3.7, 8.9, 1.2, 1.9, 4.1)
xf = cut(x, breaks=c(0, 3, 6, 9))
bins = tapply(x, xf, length)
linear regression and curve fitting
matlab r numpy
linear regression y = ax + b x = [1 2 3]
y = [2 4 7]
[lsq, res] = polyfit(x, y, 1)

a = lsq(1)
b = lsq(2)
y - (a*x+b)
x = c(1,2,3)
y = c(2,4,7)
lsq = lm(y ~ x)

a = lsq$coefficients[2] b = lsq$coefficients[1]
lsq$residuals x = np.array([1,2,3]) y = np.array([2,4,7]) lsq = stats.linregress(x, y) a = lsq[0] b = lsq[1] y - (a*x+b) distributions matlab r numpy empirical density function empirical cumulative distribution F is a right-continuous step function: F = ecdf(rnorm(100)) empirical quantile function F = ecdf(rnorm(100)) Finv = ecdf(F(seq(0, 1, .01))) binomial binopdf(x, n, p) binocdf(x, n, p) binoinv(y, n, p) binornd(n, p) dbinom(x, n, p) pbinom(x, n, p) qbinom(y, n, p) rbinom(1, n, p) stats.binom.pmf(x, n, p) stats.binom.cdf(x, n, p) stats.binom.ppf(y, n, p) stats.binom.rvs(n, p) poisson poisspdf(x, lambda) poisscdf(x, lambda) poissinv(y, lambda) poissrnd(lambda) dpois(x, lambda) ppois(x, lambda) qpois(y, lambda) rpois(1, lambda) stats.poisson.pmf(x, lambda) stats.poisson.cdf(x, lambda) stats.poisson.ppf(y, lambda) stats.poisson.rvs(lambda, size=1) normal normpdf(x, mu, sigma) normcdf(x, mu, sigma) norminv(y, mu, sigma) normrnd(mu, sigma) dnorm(x, mu, sigma) pnorm(x, mu, sigma) qnorm(y, mu, sigma) rnorm(1, mu, sigma) stats.norm.pdf(x, mu, sigma) stats.norm.cdf(x, mu, sigma) stats.norm.ppf(y, mu, sigma) stats.norm.rvs(mu, sigma) gamma gampdf(x, k, theta) gamcdf(x, k, theta) gaminv(y, k, theta) gamrnd(k, theta) dgamma(x, k, scale=theta) pgamma(x, k, scale=theta) qgamma(y, k, scale=theta) rgamma(1, k, scale=theta) stats.gamma.pdf(x, k, scale=theta) stats.gamma.cdf(x, k, scale=theta) stats.gamma.ppf(y, k, scale=theta) stats.gamma.rvs(k, scale=theta) exponential exppdf(x, lambda) expcdf(x, lambda) expinv(y, lambda) exprnd(lambda) dexp(x, lambda) pexp(x, lambda) qexp(y, lambda) rexp(1, lambda) stats.expon.pdf(x, scale=1.0/lambda) stats.expon.cdf(x, scale=1.0/lambda) stats.expon.ppf(x, scale=1.0/lambda) stats.expon.rvs(scale=1.0/lambda) chi-squared chi2pdf(x, nu) chi2cdf(x, nu) chi2inv(y, nu) chi2rnd(nu) dchisq(x, nu) pchisq(x, nu) qchisq(y, nu) rchisq(1, nu) stats.chi2.pdf(x, nu) stats.chi2.cdf(x, nu) stats.chi2.ppf(y, nu) stats.chi2.rvs(nu) beta betapdf(x, alpha, beta) betacdf(x, alpha, beta) betainvf(y, alpha, beta) betarnd(alpha, beta) dbeta(x, alpha, beta) pbeta(x, alpha, beta) qbeta(y, alpha, beta) rbeta(1, alpha, beta) stats.beta.pdf(x, alpha, beta) stats.beta.cdf(x, alpha, beta) stats.beta.ppf(y, alpha, beta) stats.beta.pvs(alpha, beta) uniform unifpdf(x, a, b) unifcdf(x, a, b) unifinv(y, a, b) unifrnd(a, b) dunif(x, a, b) punif(x, a, b) qunif(y, a, b) runif(1, a, b) stats.uniform.pdf(x, a, b) stats.uniform.cdf(x, a, b) stats.uniform.ppf(y, a, b) stats.unifrom.rvs(a, b) Student's t dt(x, nu) pt(x, nu) qt(y, nu) rt(1, nu) stats.t.pdf(x, nu) stats.t.cdf(x, nu) stats.t.ppf(y, nu) stats.t.rvs(nu) Snedecor's F df(x, d1, d2) pf(x, d1, d2) qf(y, d1, d2) rf(1, d1, d2) stats.f.pdf(x, d1, d2) stats.f.cdf(x, d1, d2) stats.f.ppf(y, d1, d2) stats.f.rvs(d1, d2) univariate charts matlab r matplotlib vertical bar chart bar([7 3 8 5 5]) cnts = c(7,3,8,5,5) names(cnts) = c("a","b","c","d","e") barplot(cnts) x = floor(6*runif(100)) barplot(table(x)) cnts = [7,3,8,5,5] plt.bar(range(0,len(cnts)), cnts) horizontal bar chart barh([7 3 8 5 5]) cnts = c(7,3,8,5,5) names(cnts) = c("a","b","c","d","e") barplot(cnts, horiz=T) cnts = [7,3,8,5,5] plt.barh(range(0,len(cnts)), cnts) pie chart labels = {'a','b','c','d','e'} pie([7 3 8 5 5], labels) cnts = c(7,3,8,5,5) names(cnts) = c("a","b","c","d","e") pie(cnts) cnts = [7,3,8,5,5] labs = ['a','b','c','d','e'] plt.pie(cnts, labels=labs) dot plot stripchart(floor(10*runif(50)), method="stack", offset=1, pch=19) stem plot generates an ascii chart: stem(20*rnorm(100)) histogram hist(randn(1, 100), 10) hist(rnorm(100), breaks=10) plt.hist(sp.randn(100), bins=range(-5,5)) box plot boxplot(rnorm(100)) boxplot(rnorm(100), rexp(100), runif(100)) plt.boxplot(sp.randn(100)) plt.boxplot([sp.randn(100), np.random.uniform(size=100), np.random.exponential(size=100)]) chart title bar([7 3 8 5 5]) title('bar chart example') all chart functions except for stem accept a main parameter: boxplot(rnorm(100), main="boxplot example", sub="to illustrate options") plt.boxplot(sp.randn(100)) plt.title('boxplot example') bivariate charts matlab r matplotlib stacked bar chart d = [7 1; 3 2; 8 1; 5 3; 5 1] bar(d, 'stacked') d = matrix(c(7,1,3,2,8,1,5,3,5,1), nrow=2) labels = c("a","b","c","d","e") barplot(d,names.arg=labels) a1 = [7,3,8,5,5] a2 = [1,2,1,3,1] plt.bar(range(0,5), a1, color='r') plt.bar(range(0,5), a2, color='b') grouped bar chart d = [7 1; 3 2; 8 1; 5 3; 5 1] bar(d) d = matrix(c(7,1,3,2,8,1,5,3,5,1), nrow=2) labels = c("a","b","c","d","e") barplot(d,names.arg=labels,beside=TRUE) scatter plot plot(randn(1,50),randn(1,50),'+') plot(rnorm(50), rnorm(50)) plt.scatter(sp.randn(50), sp.randn(50)) hexagonal binning install.packages('hexbin') library('hexbin') plot(hexbin(rnorm(1000), rnorm(1000), xbins=12)) hexbin(randn(1000), randn(1000), gridsize=12) linear regression line x = 0:20 y = 2 * x + rnorm(21)*10 fit = lm(y ~ x) plot(y) lines(x, fit$fitted.values, type='l')
x = range(0,20)
err = sp.randn(20)*10
y = [2*i for i in x] + err

A = np.vstack([x,np.ones(len(x))]).T
m, c = np.linalg.lstsq(A, y)[0]

plt.scatter(x, y)
plt.plot(x, [m*i + c for i in x])

polygonal line plot
plot(1:20,randn(1,20)) plot(1:20, rnorm(20), type="l") plot(range(0,20), randn(20))

cubic spline
f = splinefun(rnorm(20))
x = seq(1, 20, .1)
plot(x, f(x), type="l")

function plot
fplot(@sin, [-4 4]) x = seq(-4, 4, .01)
plot(sin(x), type="l")

quantile-quantile plot
qqplot(runif(50),rnorm(50))
lines(c(-9,9), c(-9,9), col="red")
axis labels plot( 1:20, (1:20) .** 2)
xlabel('x')
ylabel('x squared')
plot(1:20, (1:20)^2,
xlab="x", ylab="x squared")
axis limits plot( 1:20, (1:20) .** 2)
axis([1 20 -200 500])
plot(1:20, (1:20)^2,
xlim=c(0, 20), ylim=c(-200,500))
logarithmic y-axis semilogy(x, x .** 2,
x, x .** 3,
x, x .** 4,
x, x .** 5)
x = 0:20
plot(x, x^2, log="y",type="l")
lines(x, x^3, col="blue")
lines(x, x^4, col="green")
lines(x, x^5, col="red")
x = range(0, 20)

for i in [2,3,4,5]:
y.append([j**i for j in x])

for i in [0,1,2,3]:
semilogy(x, y[i])
multivariate charts
matlab r matplotlib

plot(1:20, randn(1, 20),
1:20, randn(1, 20))

optional method:
plot(1:20, randn(1, 20))
hold on
plot(1:20, randn(1, 20))
plot(1:20, rnorm(20), type="l")
lines(1:20, rnorm(20), col="red")

legend
x = (1:20)
y = x + rnorm(20)
y2 = x - 2 + rnorm(20)

plot(x, y, type="l", col="black")
lines(x, y2, type="l", col="red")
legend(1, 15, c('first', 'second'),
lty=c(1,1), lwd=c(2.5, 2.5),
col=c('black', 'red'))

plot(rnorm(20), rnorm(20))
points(rnorm(20), rnorm(20),
col='red')

stacked area chart
x = rep(0:4, each=3)
y = round(5 * runif(15))
letter = rep(LETTERS[1:3], 5)
df = data.frame(x, y, letter)

p = ggplot(df, aes(x=x, y=y,
group=letter,
fill=letter))
p + geom_area(position='stack')

overlapping area chart
x = rep(0:4, each=3)
y = round(5 * runif(15))
letter = rep(LETTERS[1:3], 5)
df = data.frame(x, y, letter)
alpha = rep(I(2/10), each=15)

p = ggplot(df, aes(x=x, ymin=0, ymax=y,
group=letter, fill=letter,
alpha=alpha))
p + geom_ribbon()

3d scatter plot
install.packages('scatterplot3d')
library('scatterplot3d')

scatterplot3d(rnorm(50), rnorm(50),
rnorm(50), type="h")

bubble chart
df = data.frame(x=rnorm(20),
y=rnorm(20), z=rnorm(20))

p = ggplot(df, aes(x=x, y=y, size=z))
p + geom_point()

scatter plot matrix
x = rnorm(20)
y = rnorm(20)
z = x + 3*y
w = y + 0.1*rnorm(20)
df = data.frame(x, y, z, w)

pairs(df)

contour plot
m = matrix(0, 100, 100)
for (i in 2:100) {
for (j in 2:100) {
m[i,j] = (m[i-1,j] + m[i,j-1])/2 +
runif(1) - 0.5
}
}

filled.contour(1:100, 1:100, m)
_______________________________________________________ _______________________________________________________ _______________________________________________________

# General

## version used

The version of software used to check the examples in the reference sheet.

## show version

How to determine the version of an installation.

## implicit prologue

Code which examples in the sheet assume to have already been executed.

r:

The ggplot2 library must be installed and loaded to use the plotting functions qplot and ggplot.

# Grammar and Invocation

## interpreter

How to invoke the interpreter on a script.

## repl

How to launch a command line read-eval-print loop for the language.

r:

R installations come with a clickable GUI REPL.

## command line program

How to pass the code to be executed to the interpreter as a command line argument.

## environment variables

How to get and set an environment variable.

## block delimiters

Punctuation or keywwords which define blocks.

octave:

The list of keywords which define blocks is not exhaustive. Blocks are also defined by

• switch, case, otherwise, endswitch
• unwind_protect, unwind_protect_cleanup, end_unwind_protect
• try, catch, end_try_catch

## statement separator

How statements are separated.

octave:

Use a backslash to escape a newline and continue a statement on the following line. MATLAB, in contrast, uses three periods: '…' to continue a statement on the following line.

## end-of-line comment

Character used to start a comment that goes to the end of the line.

# Variables and Expressions

## assignment

r:

Traditionally <- was used in R for assignment. Using an = for assignment was introduced in version 1.4.0 sometime before 2002. -> can also be used for assignment:

3 -> x


## compound assignment

The compound assignment operators.

## increment and decrement operator

The operator for incrementing the value in a variable; the operator for decrementing the value in a variable.

## null

octave:

NA can be used for missing numerical values. Using a comparison operator on it always returns false, including NA == NA. Using a logical operator on NA raises an error.

r:

Relational operators return NA when one of the arguments is NA. In particular NA == NA is NA. When acting on values that might be NA, the logical operators observe the rules of ternary logic, treating NA is the unknown value.

## null test

How to test if a value is null.

## conditional expression

A conditional expression.

# Arithmetic and Logic

## true and false

The boolean literals.

octave:

true and false are functions which return matrices of ones and zeros of type logical. If no arguments are specified they return single entry matrices. If one argument is provided, a square matrix is returned. If two arguments are provided, they are the row and column dimensions.

## falsehoods

Values which evaluate to false in a conditional test.

octave:

When used in a conditional, matrices evaluate to false unless they are nonempty and all their entries evaluate to true. Because strings are matrices of characters, an empty string ('' or "") will evaluate to false. Most other strings will evaluate to true, but it is possible to create a nonempty string which evaluates to false by inserting a null character; e.g. "false\000".

r:

When used in a conditional, a vector evaluates to the boolean value of its first entry. Using a vector with more than one entry in a conditional results in a warning message. Using an empty vector in a conditional, c() or NULL, raises an error.

## logical operators

The boolean operators.

octave:

Note that MATLAB does not use the exclamation point '!' for negation.

&& and || are short circuit logical operators.

## relational operators

The relational operators.

octave:

Note that MATLAB does not use '!=' for an inequality test.

## arithmetic operators

The arithmetic operators: addition, subtraction, multiplication, division, quotient, remainder.

octave:

mod is a function and not an infix operator. mod returns a positive value if the first argument is positive, whereas rem returns a negative value.

## integer division

How to compute the quotient of two integers.

## integer division by zero

What happens when an integer is divided by zero.

## float division

How to perform float division, even if the arguments are integers.

## float division by zero

What happens when a float is divided by zero.

## power

octave:

^ is a synonym for **.

r:

^ is a synonym for **.

## sqrt

The square root function.

## sqrt(-1)

The result of taking the square root of a negative number.

## transcendental functions

The standard transcendental functions.

## transcendental constants

Constants for pi and e.

## float truncation

Ways of converting a float to a nearby integer.

## absolute value

The absolute value and signum of a number.

## integer overflow

What happens when an expression evaluates to an integer which is too big to be represented.

## float overflow

What happens when an expression evaluates to a float which is too big to be represented.

## float limits

The machine epsilon; the largest representable float and the smallest (i.e. closest to negative infinity) representable float.

## complex construction

Literals for complex numbers.

## complex decomposition

How to decompose a complex number into its real and imaginary parts; how to decompose a complex number into its absolute value and argument; how to get the complex conjugate.

## random number

How to generate a random integer from a uniform distribution; how to generate a random float from a uniform distribution.

## random seed

How to set, get, and restore the seed used by the random number generator.

octave:

At startup the random number generator is seeded using operating system entropy.

r:

At startup the random number generator is seeded using the current time.

numpy:

On Unix the random number generator is seeded at startup from /dev/random.

## bit operators

The bit operators left shift, right shift, and, or , xor, and negation.

matlab/octave:

bitshift takes a second argument which is positive for left shift and negative for right shift.

bitcmp takes a second argument which is the size in bits of the integer being operated on. Octave is not compatible with MATLAB in how the integer size is indicated.

r:

There is a library on CRAN called bitops which provides bit operators.

# Strings

## literal

The syntax for a string literal.

## newline in literal

Can a newline be included in a string literal? Equivalently, can a string literal span more than one line of source code?

## literal escapes

Escape sequences for including special characters in string literals.

## character access

How to get the character in a string at a given index.

## chr and ord

How to convert an ASCII code to a character; how to convert a character to its ASCII code.

## length

How to get the number of characters in a string.

## concatenate

How to concatenate strings.

## replicate

How to create a string which consists of a character of substring repeated a fixed number of times.

## index of substring

How to get the index of first occurrence of a substring.

## extract substring

How to get the substring at a given index.

## split

How to split a string into an array of substrings. In the original string the substrings must be separated by a character, string, or regex pattern which will not appear in the array of substrings.

The split operation can be used to extract the fields from a field delimited record of data.

## join

How to join an array of substrings into single string. The substrings can be separated by a specified character or string.

Joining is the inverse of splitting.

## trim

How to remove whitespace from the beginning and the end of a string.

Trimming is often performed on user provided input.

## convert from string, to string

How to convert strings to numbers and vice versa.

## case manipulation

How to put a string into all caps. How to put a string into all lower case letters. How to capitalize the first letter of a string.

## sprintf

How to create a string using a printf style format.

# Regular Expressions

## regex test

How to test whether a string matches a regular expression.

## regex substitution

How to replace all substring which match a pattern with a specified string; how to replace the first substring which matches a pattern with a specified string.

# Date and Time

## current date/time

How to get the current date and time.

r:

Sys.time() returns a value of type POSIXct.

## date/time type

The data type used to hold a combined date and time value.

## date/time difference type

The data type used to hold the difference between two date/time types.

## get date parts

How to get the year, the month as an integer from 1 through 12, and the day of the month from a date/time value.

## get time parts

How to get the hour as an integer from 0 through 23, the minute, and the second from a date/time value.

## build date/time from parts

How to build a date/time value from the year, month, day, hour, minute, and second as integers.

## convert to string

How to convert a date value to a string using the default format for the locale.

## strptime

How to parse a date/time value from a string in the manner of strptime from the C standard library.

## strftime

How to write a date/time value to a string in the manner of strftime from the C standard library.

# Tuples

homogeneous array vector tuple record map
NumPy list vector tuple dict dict
Octave rank 1 matrix rank 1 matrix cell array struct
R vector vector list list

## tuple literal

How to create a tuple, which we define as a fixed length, inhomogeneous list.

## tuple element access

How to access an element of a tuple.

## tuple length

How to get the number of elements in a tuple.

# Arrays

## copy

How to make an address copy, a shallow copy, and a deep copy of an array.

After an address copy is made, modifications to the copy also modify the original array.

After a shallow copy is made, the addition, removal, or replacement of elements in the copy does not modify of the original array. However, if elements in the copy are modified, those elements are also modified in the original array.

A deep copy is a recursive copy. The original array is copied and a deep copy is performed on all elements of the array. No change to the contents of the copy will modify the contents of the original array.

r:

R does not provide a way to perform an address copy.

Because arrays cannot be elements of arrays, there is no distinction between a shallow copy and a deep copy.

# Multidimensional Arrays

Arrays map integers to arbitrary values. The arrays supported by the languages in this reference sheet are homogeneous, which means that the values in the codomain of the array must all be of the same type.

The languages in this sheet all support multidimensional arrays. A multidimensional array maps tuples of integers to values. All tuples which can be used as indices in a multidimensional array are of the same length and this length is the dimension of the array.

Arrays use contiguous regions of memory to store their values. Thus, an array with an element at index 1 and index 10 must allocate space for elements at indices 2 through 9, even if values are not explicitly set or needed. The shape of a multidimensional array can be expressed by a tuple of positive integers with the same length as the dimension of the array.

Arrays provide constant time access when looking up values by their indices.

A vector is a one dimensional array which supports these operations:

• addition on vectors of the same length
• scalar multiplication
• a dot product
• a norm

The languages in this reference sheet provide the above operations for all one dimensional arrays which contain numeric values.

NumPy adds the homogeneous ndarray type to the native Python list. A Python list is nonhomogeneous and one dimensional, but because they can contain lists as values they can be used to hold multidimensional data. Python lists are described in the Python reference sheet.

## array literal

octave:

An array in Octave is in fact a 1 x n matrix.

r:

c(1,2,3) is a vector and array(c(1,2,3)) is a one dimensional array. The documentation says that some functions may treat the two objects differently. In the absence of knowing what those differences are it seems best to use the vector.

## must arrays be homogeneous

Can an array be created with elements of different type?

octave:

The array literal

[1,'foo',3]


will create an array with 5 elements of class char.

r:

The array literal

c(1,'foo',3)


will create an array of 3 elements of class character, which is the R string type.

## array data types

What data types are permitted in arrays.

octave:

Arrays in Octave can only contain numeric elements. This follows from the fact that Octave "arrays" are in fact 1 x n matrices.

Array literals can have a nested structure, but Octave will flatten them. The following literals create the same array:

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


Logical values can be put into an array because true and false are synonyms for 1 and 0. Thus the following literals create the same arrays:

[ true false false ]
[ 1 0 0 ]


If a string is encountered in an array literal, the string is treated as an array of ASCII values and it is concatenated with other ASCII values to produce as string. The following literals all create the same string:

[ 'foo', 98, 97, 114]
[ 'foo', 'bar' ]
'foobar'


If the other numeric values in an array literal that includes a string are not integer values that fit into a ASCII byte, then they are converted to byte sized values.

r:

Array literals can have a nested structure, but R will flatten them. The following literals produce the same array of 6 elements:

c(1,2,3,c(4,5,6))
c(1,2,3,4,5,6)


If an array literal contains a mixture of booleans and numbers, then the boolean literals will be converted to 1 (for TRUE and T) and 0 (for FALSE and F).

If an array literal contains strings and either booleans or numbers, then the booleans and numbers will be converted to their string representations. For the booleans the string representations are "TRUE'" and "FALSE".

# Dictionaries

## dictionary literal

The syntax for a dictionary literal.

## dictionary lookup

How to use a key to lookup a value in a dictionary.

# Data Sets

r pandas
ordered dictionary list() Series()
data set data frame DataFrame()
data set column type vector Series
row.names Index()
hierarchical index
factor (ordered and unordered)

## construct from column arrays

How to construct a data set from a set of arrays representing the columns.

## column names as array

How to show the names of the columns.

## access column as array

How to access a column in a data set.

## access row as tuple

How to access a row in a data set.

r:

people[1,] returns the 1st row from the data set people as a new data set with one row. This can be converted to a list using the function as.list. There is often no need because lists and one row data sets have nearly the same behavior.

## access datum

How to access a single datum in a data set; i.e. the value in a column of a single row.

## order rows by column

How to sort the rows in a data set according to the values in a specified column.

## order rows in descending order

How to sort the rows in descending order according to the values in a specified column.

## limit rows

How to select the first n rows according to some ordering.

## offset rows

How to select rows starting at offset n according to some ordering.

## attach columns

How to make column name a variable in the current scope which refers to the column as an array.

r:

Each column of the data set is copies into a variable named after the column containing the column as a vector. Modifying the data in the variable does not alter the original data set.

## detach columns

How to remove attached column names from the current scope.

How to view and edit the data set in a spreadsheet.

# Import and Export

## import tab delimited file

Load a data set from a tab delimited file.

## import comma-separated values file

Load a data set from a CSV file.

## set column separator

How to set the column separator when importing a delimited file.

## set quote character

How to change the quote character. Quoting is used when strings contain the column separator or the line terminator.

How to import a file that lacks a header.

## set column names

How to set the column names.

## set column types

How to indicate the type of the columns.

r:

If the column types are not set or if the type is set to NA or NULL, then the type will be set to logical, integer, numeric, complex, or factor.

## recognize null values

Specify the input values which should be converted to null values.

## unequal row length behavior

What happen when a row of input has less than or more than the expected number of columns.

## skip comment lines

How to skip comment lines.

## export comma-separated values file

Save a data set to a CSV file.

r:

If row.names is not set to F, the initial column will be the row number as a string starting from "1".

# Relational Algebra

## map data set

How to apply a mapping transformation to the rows of a data set.

## filter data set

How to select the rows of a data set that satisfy a predicate.

# Execution Control

## if

How to write a branch statement.

## while

How to write a conditional loop.

## for

How to write a C-style for statement.

## break/continue

How to break out of a loop. How to jump to the next iteration of a loop.

## raise exception

How to raise an exception.

## handle exception

How to handle an exception.

## finally block

How to write code that executes even if an exception is raised.

# File Handles

## standard file handles

Standard input, standard output, and standard error.

# Directories

## working directory

How to get and set the working directory.

# Processes and Environment

## command line arguments

How to get the command line arguments.

## environment variables

How to get and set and environment variable.

# Libraries and Namespaces

Show the list of libraries which have been loaded.

## library search path

The list of directories the interpreter will search looking for a library to load.

## source file

How to source a file.

r:

When sourcing a file, the suffix if any must be specified, unlike when loading library. Also, a library may contain a shared object, but a sourced file must consist of just R source code.

## install package

How to install a package.

## list installed packages

How to list the packages which have been installed.

# Reflection

## data type

How to get the data type of a value.

r:

For vectors class returns the mode of the vector which is the type of data contained in it. The possible modes are

• numeric
• complex
• logical
• character
• raw

Some of the more common class types for non-vector entities are:

• matrix
• array
• list
• factor
• data.frame

## attributes

How to get the attributes for an object.

r:

Arrays and vectors do not have attributes.

## methods

How to get the methods for an object.

## variables in scope

How to list the variables in scope.

## undefine variable

How to undefine a variable.

## undefine all variables

How to undefine all variables.

## eval

How to interpret a string as source code and execute it.

## function documentation

How to get the documentation for a function.

## list library functions

How to list the functions and other definitions in a library.

## search documentation

How to search the documentation by keyword.

# Vectors

## norms

octave:

The norm function returns the p-norm, where the second argument is p. If no second argument is provided, the 2-norm is returned.

# Matrices

## literal or constructor

Literal syntax or constructor for creating a matrix.

The elements of a matrix must be specified in a linear order. If the elements of each row of the matrix are adjacent to other elements of the same row in the linear order we say the order is row contiguous. If the elements of each column are adjacent to other elements of the same column we say the order is column contiguous.

octave:

Square brackets are used for matrix literals. Semicolons are used to separate rows, and commas separate row elements. Optionally, newlines can be used to separate rows and whitespace to separate row elements.

r:

Matrices are created by passing a vector containing all of the elements, as well as the number of rows and columns, to the matrix constructor.

If there are not enough elements in the data vector, the values will be recycled. If there are too many extra values will be ignored. However, the number of elements in the data vector must be a factor or a multiple of the number of elements in the final matrix or an error results.

When consuming the elements in the data vector, R will normally fill by column. To change this behavior pass a byrow=T argument to the matrix constructor:

A = matrix(c(1,2,3,4),nrow=2,byrow=T)


## dimensions

How to get the dimensions of a matrix.

## element access

How to access an element of a matrix. All languages described here follow the convention from mathematics of specifying the row index before the column index.

octave:

Rows and columns are indexed from one.

r:

Rows and columns are indexed from one.

## row access

How to access a row.

## column access

How to access a column.

## submatrix access

How to access a submatrix.

## scalar multiplication

How to multiply a matrix by a scalar.

## element-wise operators

Operators which act on two identically sized matrices element by element. Note that element-wise multiplication of two matrices is used less frequently in mathematics than matrix multiplication.

from numpy import array
matrix(array(A) * array(B))
matrix(array(A) / array(B))


## multiplication

How to multiply matrices. Matrix multiplication should not be confused with element-wise multiplication of matrices. Matrix multiplication in non-commutative and only requires that the number of columns of the matrix on the left match the number of rows of the matrix. Element-wise multiplication, by contrast, is commutative and requires that the dimensions of the two matrices be equal.

## kronecker product

The Kronecker product is a non-commutative operation defined on any two matrices. If A is m x n and B is p x q, then the Kronecker product is a matrix with dimensions mp x nq.

## comparison

How to test two matrices for equality.

octave:

== and != perform entry-wise comparison. The result of using either operator on two matrices is a matrix of boolean values.

~= is a synonym for !=.

r:

== and != perform entry-wise comparison. The result of using either operator on two matrices is a matrix of boolean values.

## norms

How to compute the 1-norm, the 2-norm, the infinity norm, and the frobenius norm.

octave:

norm(A) is the same as norm(A,2).

# Statistics

A statistic is a single number which summarizes a population of data. The most familiar example is the mean or average. Statistics defined for discrete populations can often be meaningfully extended to continuous distributions by replacing summations with integration.

An important class of statistics are the nth moments. The nth moment $\mu'_n$ of a population of k values xi with mean μ is:

(1)
\begin{align} \mu'_n = \sum_{i=1}^k x_i^n \end{align}

The nth central moment μn of the same population is:

(2)
\begin{align} \mu_n = \sum_{i=1}^k (x_i - \mu)^n \end{align}

## first moment statistics

The sum and the mean.

The mean is the first moment. It is one definition of the center of the population. The median and the mode are also used to define the center. In most populations they will be close to but not identical to the mean.

## second moment statistics

The variance and the standard deviation. The variance is the second central moment. It is a measure of the spread or width of the population.

The standard deviation is the square root of the variance. It is also a measurement of population spread. The standard deviation has the same units of measurement as the data in the population.

## second moment statistics for samples

The sample variance and sample standard deviation.

## skewness

The skewness of a population.

The skewness measures the asymmetrically of the population. The skewness will be negative, positive, or zero when the population is more spread out on the left, more spread out on the right, or similarly spread out on both sides, respectively.

The skewness can be calculated from the third moment and the standard deviation:

(3)
\begin{align} \gamma_1 = E\Big[\Big(\frac{x - \mu}{\sigma}\Big)^3\Big] = \frac{\mu_3}{\sigma^3} \end{align}

When estimating the population skewness from a sample a correction factor is often used, yielding the sample skewness:

(4)
\begin{align} \frac{(n(n-1))^{\frac{1}{2}}}{n-2} \gamma_1 \end{align}

octave and matlab:

Octave uses the sample standard deviation to compute skewness. This behavior is different from Matlab and should possibly be regarded as a bug.

Matlab, but not Octave, will take a flag as a second parameter. When set to zero Matlab returns the sample skewness:

skewness(x, 0)


numpy:

Set the named parameter bias to False to get the sample skewness:

stats.skew(x, bias=False)


## kurtosis

The kurtosis of a population.

The formula for kurtosis is:

(5)
\begin{align} \gamma_2 = \frac{\mu_4}{\sigma^4} - 3 \end{align}

When kurtosis is negative the sides of a distribution tend to be more convex than when the kurtosis is is positive. A negative kurtosis distribution tends to have a wide, flat peak and narrow tails. Such a distribution is called platykurtic. A positive kurtosis distribution tends to have a narrow, sharp peak and long tails. Such a distribution is called leptokurtic.

The fourth standardized moment is

(6)
\begin{align} \beta_2 = \frac{\mu_4}{\sigma^4} \end{align}

The fourth standardized moment is sometimes taken as the definition of kurtosis in older literature. The reason the modern definition is preferred is because it assigns the normal distribution a kurtosis of zero.

octave:

Octave uses the sample standard deviation when computing kurtosis. This should probably be regarded as a bug.

r:

R uses the older fourth standardized moment definition of kurtosis.

## nth moment and nth central moment

How to compute the nth moment (also called the nth absolute moment) and the nth central moment for arbitrary n.

## mode

The mode is the most common value in the sample.

The mode is a measure of central tendency like the mean and the median. A problem with the mean is that it can produce values not found in the data. For example the mean number of persons in an American household was 2.6 in 2009.

The mode might not be unique. If there are two modes the sample is said to be bimodal, and in general if there is more than one mode the sample is said to be multimodal.

## quantile statistics

If the data is sorted from smallest to largest, the minimum is the first value, the median is the middle value, and the maximum is the last value. If there are an even number of data points, the median is the average of the middle two points.

The median divides the population into two halves. When the population is divided into four parts the division markers are called the first, second, and third quartile. When the population is divided into a hundred the division markers are called percentiles. If the population is divided into nparts the markers are called the 1st, 2nd, …, (n-1)th n-quantile.

## bivariate statistics

The correlation and the covariance.

The correlation is a number from -1 to 1. It is a measure of the linearity of the data, with values of -1 and 1 representing indicating a perfectly linear relationship. When the correlation is positive the quantities tend to increase together and when the correlation is negative one quantity will tend to increase as the other decreases.

A variable can be completely dependent on another and yet the two variables can have zero correlation. This happens for Y = X2 where uniform X on the interval [-1, 1]. Anscombe's quartet gives four examples of data sets each with the same fairly high correlation 0.816 and yet which show significant qualitative differences when plotted.

The covariance is defined by

(7)
\begin{align} E[X -\mu_X)(Y- \mu_Y)] \end{align}

The correlation is the normalized version of the covariance. It is defined by

(8)
\begin{align} \frac{E[X -\mu_X)(Y- \mu_Y)]}{\sigma_X \sigma_Y} \end{align}

## frequency table

How to compute the frequency table for a data set. A frequency table counts how often each value occurs in the data set.

r:

The table function returns an object of type table.

## invert frequency table

How to convert a frequency table back into the original data set.

The order of the original data set is not preserved.

## bin

How to bin a data set. The result is a frequency table where each frequency represents the number of samples from the data set for an interval.

r:

The cut function returns a factor.

A labels parameter can be provided with a vector argument to assign the bins names. Otherwise bin names are contructed from the breaks using "[0.0,1.0)" style notation.

The hist function can be used to bin a data set:

x = c(1.1, 3.7, 8.9, 1.2, 1.9, 4.1)
hist(x, breaks=c(0, 3, 6, 9), plot=FALSE)


hist returns an object of type histogram. The counts are in the \$counts attribute.

# Linear Regression and Curve Fitting

## linear regression y = ax + b

How to get the slope a and intercept b for a line which best approximates the data. How to get the residuals.

If there are more than two data points, then the system is overdetermined and in general there is no solution for the slope and the intercept. Linear regression looks for line that fits the points as best as possible. The least squares solution is the line that minimizes the sum of the square of the distances of the points from the line.

The residuals are the difference between the actual values of y and the calculated values using ax + b. The norm of the residuals can be used as a measure of the goodness of fit.

# Distributions

A distribution density function f(x) is a non-negative function which, when integrated over its entire domain is equal to one. The distributions described in this sheet have as their domain the real numbers. The support of a distribution is the part of the domain on which the density function is non-zero.

A distribution density function can be used to describe the values one is likely to see when drawing an example from a population. Values in areas where the density function is large are more likely than values in areas where the density function is small. Values where there density function is zero do not occur. Thus it can be useful to plot the density function.

To derive probabilities from a density function one must integrate or use the associated cumulative density function

(9)
\begin{align} F(x) = \int_{-\infty}^x f(t) dt \end{align}

which gives the probability of seeing a value less than or equal to x. As probabilities are non-negative and no greater than one, F is a function from (-, ) to [0,1]. The inverse of F is called the inverse cumulative distribution function or the quantile function for the distribution.

For each distribution statistical software will generally provide four functions: the density, the cumulative distribution, the quantile, and a function which returns random numbers in frequencies that match the distribution. If the software does not provide a random number generating function for the distribution, the quantile function can be composed with the built-in random number generator that most languages have as long as it returns uniformly distributed floats from the interval [0, 1].

 density probability density probability mass cumulative density cumulative distribution distribution inverse cumulative density inverse cumulative distribution quantile percentile percent point random variate

Discrete distributions such as the binomial and the poisson do not have density functions in the normal sense. Instead they have probability mass functions which assign probabilities which sum up to one to the integers. In R warnings will be given if non integer values are provided to the mass functions dbinom and dpoiss.

The cumulative distribution function of a discrete distribution can still be defined on the reals. Such a function is constant except at the integers where it may have jump discontinuities.

Most well known distributions are in fact parametrized families of distributions. This table lists some of them with their parameters and properties.

The information entropy of a continuous distribution with density f(x) is defined as:

(10)
\begin{align} -\int_\mathbb{R} f(x) \; \log(f(x)) \; dx \end{align}

In Bayesian analysis the distribution with the greatest entropy, subject to the known facts about the distribution, is called the maximum entropy probability distribution. It is considered the best distribution for modeling the current state of knowledge.

## binomial

The probability mass, cumulative distribution, quantile, and random number generating functions for the binomial distribution.

The binomial distribution is a discrete distribution. It models the number of successful trails when n is the number of trials and p is the chance of success for each trial. An example is the number of heads when flipping a coin 100 times. If the coin is fair then p is 0.50.

numpy:

Random numbers in a binomial distribution can also be generated with:

np.random.binomial(n, p)


## poisson

The probability mass, cumulative distribution, quantile, and random number generating functions for the binomial distribution.

The poisson distribution is a discrete distribution. It is described by a parameter lam which is the mean value for the distribution. The poisson distribution is used to model events which happen at a specified average rate and independently of each other. Under these circumstances the time between successive events will be described by an exponential distribution and the events are said to be described by a poisson process.

numpy:

Random numbers in a poisson distribution can also be generated with:

np.random.poisson(lam, size=1)


## normal

The probability density, cumulative distribution, quantile, and random number generating functions for the uniform distribution.

The parameters are the mean μ and the standard deviation σ. The standard normal distribution has μ of 0 and σ of 1.

The normal distribution is the maximum entropy distribution for a given mean and variance. According to the central limit theorem, if {X1, …, Xn} are any independent and identically distributed random variables with mean μ and variance σ2, then Sn := Σ Xi / n converges to a normal distribution with mean μ and variance σ2/n.

numpy:

Random numbers in a normal distribution can also be generated with:

np.random.randn()


## gamma

The probability density, cumulative distribution, quantile, and random number generating functions for the gamma distribution.

The parameter k is called the shape parameter and θ is called the scale parameter. The rate of the distribution is β = 1/θ.

If Xi are n independent random variables with Γ(ki, θ) distribution, then Σ Xi has distribution Γ(Σ ki, θ).

If X has Γ(k, θ) distribution, then αX has Γ(k, αθ) distribution.

## exponential

The probability density, cumulative distribution, quantile, and random number generating functions for the exponential distribution.

## chi-squared

The probability density, cumulative distribution, quantile, and random number generating functions for the chi-squared distribution.

## beta

The probability density, cumulative distribution, quantile, and random number generating functions for the beta distribution.

## uniform

The probability density, cumulative distribution, quantile, and random number generating functions for the uniform distribution.

The uniform distribution is described by the parameters a and b which delimit the interval on which the density function is nonzero.

The uniform distribution is maximum entropy probability distribution with support [a, b].

Consider the uniform distribution on [0, b]. Suppose that we take k samples from it, and m is the largest of the samples. The minimum variance unbiased estimator for b is

(11)
\begin{align} \frac{k+1}{k}m \end{align}

octave, r, numpy:

a and b are optional parameters and default to 0 and 1 respectively.

## Student's t

The probability density, cumulative distribution, quantile, and random number generating functions for Student's t distribution.

## Snedecor's F

The probability density, cumulative distribution, quantile, and random number generating functions for Snedecor's F distribution.

# Univariate Charts

## vertical bar chart

A chart in which numerical values are represented by horizontal bars. The bars are aligned at the bottom.

r:

How to produce a bar chart using ggplot2:

cnts = c(7,3,8,5,5)
names = c("a","b","c","d","e")
df = data.frame(names, cnts)
qplot(names, data=df, geom="bar", weight=cnts)


## horizontal bar chart

A bar chart with horizontal bars which are aligned on the left.

## pie chart

A pie chart displays values using the areas of circular sectors or equivalently the lengths of the arcs of those sectors.

A pie chart implies that the values are percentages of a whole.

## dot plot

A chart which displays small, integral values with stacks of dots.

## stem plot

Also called a stem-and-leaf plot.

A stem plot is a concise way of storing a small set of numbers which makes their distribution visually evident.

The original set of numbers can be recovered with some loss of accuracy by appending the number on the left with each of the digits on the right. In the example below the original data set contained -43, -42, -41, -39, -38, -35, …, 35, 44, 46, 50, 58.

> stem(20*rnorm(100))

The decimal point is 1 digit(s) to the right of the |

-4 | 321
-2 | 98544054310
-0 | 8864333111009998776444332222110
0 | 0001122333333466667778899122334555666789
2 | 00023669025
4 | 4608


## histogram

A histogram is a bar chart where each bar represents a range of values that the data points can fall in. The data is tabulated to find out how often data points fall in each of the bins and in the final chart the length of the bars corresponds to the frequency.

A common method for choosing the number of bins using the number of data points is Sturges' formula:

(12)
\begin{align} \lceil \log_2{x} + 1 \rceil \end{align}

r:

How to make a histogram with the ggplot2 library:

qplot(rnorm(50), geom="histogram", binwidth=binwidth)
binwidth = (max(x)-min(x))/10
qplot(rnorm(50), geom="histogram", binwidth=binwidth)


## box plot

Also called a box-and-whisker plot.

The box shows the locations of the 1st quartile, median, and 3rd quartile. These are the same as the 25th percentile, 50th percentile, and 75th percentile.

The whiskers are sometimes used to show the maximum and minimum values of the data set. Outliers are sometimes shown explicitly with dots, in which case all remaining data points occur inside the whiskers.

r:

How to create a box plot with ggplot2:

qplot(x="rnorm", y=rnorm(50), geom="boxplot")

qplot(x=c("rnorm", "rexp", "runif"), y=c(rnorm(50), rexp(50), runif(50)), geom="boxplot")


## chart title

How to set the chart title.

r:

The qplot commands supports the main options for setting the title:

qplot(x="rnorm", y=rnorm(50), geom="boxplot", main="boxplot example")


# Bivariate Charts

## stacked bar chart

Two or more data sets with a common set of labels can be charted with a stacked bar chart. This makes the sum of the data sets for each label readily apparent.

## grouped bar chart

Optionally data sets with a common set of labels can be charted with a grouped bar chart which clusters the bars for each label. The grouped bar chart makes it easier to perform comparisons between labels for each data set.

## scatter plot

A scatter plot can be used to determine if two variables are correlated.

r:

How to make a scatter plot with ggplot:

x = rnorm(50)
y = rnorm(50)
p = ggplot(data.frame(x, y), aes(x, y))
p = p + layer(geom="point")
p


## hexagonal binning

A hexagonal binning is the two-dimensional analog of a histogram. The number of data points in each hexagon is tabulated, and then color or grayscale is used to show the frequency.

A hexagonal binning is superior to a scatter-plot when the number of data points is high because most scatter-plot software doesn't indicate when points are occur on top of each other.

## linear regression line

How to plot a line determined by linear regression on top of a scatter plot.

## polygonal line plot

How to connect the dots of a data set with a polygonal line.

## cubic spline

How to connect the dots of a data set with a line which has a continuous 2nd derivative.

## function plot

How to plot a function.

## quantile-quantile plot

Also called a Q-Q plot.

A quantile-quantile plot is a scatter plot created from two data sets. Each point depicts the quantile of the first data set with its x position and the corresponding quantile of the second data set with its y position.

If the data sets are drawn from the same distribution then most of the points should be close to the line y = x. If the data sets are drawn from distributions which have a linear relation then the Q-Q plot should also be close to linear.

## axis labels

How to label the x and y axes.

r:

How to label the axes with ggplot2:

x = rnorm(20)
y = x^2

p = ggplot(data.frame(x, y), aes(x, y))
p + layer(geom="point") + xlab('x') + ylab('x squared')


## axis limits

How to manually set the range of values displayed by an axis.

# Multivariate Charts

## legend

How to put a legend on a chart.

r:

The named parameter lwd is the line width. It is roughly the width in pixels, though the exact interpretation is device specific.

The named parameter lty specifies the line type. The value can be either an integer or a string:

number string
0 'blank'
1 'solid'
2 'dashed'
3 'dotted'
4 'dotdash'
5 'longdash'
6 'twodash'

# MATLAB

The basic data type of MATLAB is a matrix of floats. There is no distinction between a scalar and a 1x1 matrix, and functions that work on scalars typically work on matrices as well by performing the scalar function on each entry in the matrix and returning the resultings in a matrix with the same dimensions. Operators such as the logical operators ('&' '|' '!'), relational operators ('==', '!=', '<', '>'), and arithmetic operators ('+', '-') all work this way. However the multiplication '*' and division '/' operators perform matrix multiplication and matrix division, respectively. The '.*' and '.*' operators are available if entry-wise multiplication or division is desired.

Floats are by default double precision; single precision can be specified with the single constructor. MATLAB has convenient matrix literal notation: commas or spaces can be used to separate row entries, and semicolons or newlines can be used to separate rows.

Arrays and vectors are implemented as single-row (1xn) matrices. As a result an n-element vector must be transposed before it can be multiplied on the right of a mxn matrix.

Numeric literals that lack a decimal point such as 17 and -34 create floats, in contrast to most other programming languages. To create an integer, an integer constructor which specifies the size such as int8 and uint16 must be used. Matrices of integers are supported, but the entries in a given matrix must all have the same numeric type.

Strings are implemented as single-row (1xn) matrices of characters, and as a result matrices cannot contain strings. If a string is put in matrix literal, each character in the string becomes an entry in the resulting matrix. This is consistent with how matrices are treated if they are nested inside another matrix. The following literals all yield the same string or 1xn matrix of characters:

'foo'
[ 'f' 'o' 'o' ]
[ 'foo' ]
[ [ 'f' 'o' 'o' ] ]


true and false are functions which return matrices of ones and zeros. The ones and zeros have type logical instead of double, which is created by the literals 1 and 0. Other than having a different class, the 0 and 1 of type logical behave the same as the 0 and 1 of type double.

MATLAB has a tuple type (in MATLAB terminology, a cell array) which can be used to hold multiple strings. It can also hold values with different types.

Octave is a free, open source application for floating point and matrix computations which can interface with numerical routines implemented in C or Fortran. Octave implements the core MATLAB language, and as a result MATLAB scripts will usually run under Octave. Octave scripts are less likely to run under MATLAB because of extensions which Octave is made to the core language.. Octave's plotting functions use gnuplot.

# R

R is an application for statistical analysis. It is a free, open source implementation of the S programming language developed at Bell Labs.

The basic data types of R are vectors of floats, vectors of strings, and vectors of booleans. There is no distinction between a scalar and a vector with one entry in it, and functions and operators which accept a scalar argument will typically accept a vector argument, returning a vector of the same size with the scalar operation performed on each the entries of the original vector.

The scalars in a vector must all be of the same type, but R also provides a list data type which can be used as a tuple (entries accessed by index) or a record (entries accessed by name).

In addition R provides a data frame type which is a list (in R terminology) of vectors all of the same length. Data frames are equivalent to the data sets of other statistical analysis packages.

# NumPy

NumPy is a Python library which provides a data type called array. It differs from the Python list data type in the following ways:

• N-dimensional. Although the list type can be nested to hold higher dimension data, the array can hold higher dimension data in a space efficient manner without using indirection.
• homogeneous. The elements of an array are restricted to be of a specified type. The NumPy library introduces new primitive types not available in vanilla Python. However, the element type of an array can be object which permits storing anything in the array.

In the reference sheet the array section covers the vanilla Python list and the multidimensional array section covers the NumPy array.

List the NumPy primitive types

SciPy, Matplotlib, and Pandas are libraries which depend on Numpy.

page revision: 1455, last edited: 24 Apr 2013 13:50