Numerical Analysis & Statistics: MATLAB, R, NumPy, Julia

a side-by-side reference sheet

sheet two: tables | import and export | relational algebra | aggregation

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tables
	matlab	r	numpy	julia
construct from column arrays	sx = {'F' 'F' 'F' 'M' 'M' 'M'} ht = [69 64 67 68 72 71] wt = [148 132 142 149 167 165] cols = {'sx', 'ht', 'wt'} people = table(sx', ht', wt', 'VariableNames', cols)	# gender, height, weight of some people # in inches and lbs: sx = c("F", "F", "F", "M", "M", "M") ht = c(69, 64, 67, 68, 72, 71) wt = c(148, 132, 142, 149, 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 dictionaries			rows = [ {'sx': 'F', 'ht': 69, 'wt': 150}, {'sx': 'F', 'ht': 64, 'wt': 132}, {'sx': 'F', 'ht': 67, 'wt': 142}, {'sx': 'F', 'ht': 66, 'wt': 139}, {'sx': 'M', 'ht': 72, 'wt': 167}, {'sx': 'M', 'ht': 70, 'wt': 165}] people = pd.DataFrame(rows)
size	height(people) width(people)	nrow(people) ncol(people) # number of rows and cols in 2-element vector: dim(people)	len(people) len(people.columns)
column names as array	people.Properties.VariableNames	names(people) colnames(people)	returns Index object: people.columns
access column as array	people.ht people.(2)	# vectors: people$ht people[,2] people[['ht']] people[[2]] # 1 column data frame: people[2]	people['ht'] # if name does not conflict with any DataFrame attributes: people.ht
access row as tuple	people(1,:)	# 1 row data frame: people[1, ] # list: as.list(people[1, ])	people.ix[0]
access datum	% height of 1st person: people(1,2)	# height of 1st person: people[1,2]	people.get_value(0, 'ht')
order rows by column	sortrows(people, 'ht')	people[order(people$ht), ]	people.sort(['ht'])
order rows by multiple columns	sortrows(people, {'sx', 'ht'})	people[order(people$sx, people$ht), ]	people.sort(['sx', 'ht'])
order rows in descending order	sortrows(people, 'ht', 'descend')	people[order(-people$ht), ]	people.sort('ht', ascending=[False])
limit rows	people(1:3, :)	people[seq(1, 3), ]	people[0:3]
offset rows	people(4:6, :)	people[seq(4, 6), ]	people[3:]
reshape		people$couple = c(1, 2, 3, 1, 2, 3) reshape(people, idvar="couple", direction="wide", timevar="sx", v.names=c("ht", "wt"))
remove rows with null fields		sx = c('F', 'F', 'M', 'M') wt = c(120, NA, 150, 170) df = data.frame(sx, wt) df2 = na.omit(df)
attach columns		# put columns ht, wt, and sx # in variable name search path: attach(people) sum(ht) # alternative which doesn't put columns in # search path: with(people, sum(ht))	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
	matlab	r	numpy	julia
import tab delimited	# first row defines variable names: readtable('/tmp/password.txt', 'Delimiter', '\t') # file suffix must be .txt, .dat, or .csv	# first row defines variable names: df = read.delim('/path/to.tab', stringsAsFactors=F, quote=NULL)	# first row defines column names: df = pd.read_table('/path/to.tab')
import csv	% first row defines variable names: df = readtable('/path/to.csv')	# first row defines variable names: df = read.csv('/path/to.csv', stringsAsFactors=F)	# first row defines column names: df = pd.read_csv('/path/to.csv')
set column separator	df = readtable('/etc/passwd', 'Delimiter', ':', 'ReadVariableNames', 0, 'HeaderLines', 10)	df = read.delim('/etc/passwd', sep=':', header=FALSE, 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 escaped 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, … df = pd.read_table('/tmp/passwd', sep=':', header=None)
set column names	df = readtable('/path/to/no-header.csv', 'ReadVariableNames', 0) df.Properties.VariableNames = {'ht', 'wt', 'age'}	df = read.csv('/path/to/no-header.csv', header=FALSE, col.names=c('ht', 'wt', 'age'))	df = pd.read_csv('/path/to/no-header.csv', 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' # df = read.csv('/path/to/data.csv', colClasses=c('integer', 'numeric', 'character'))
recognize null values		df = read.csv('/path/to/data.csv', colClasses=c('integer', 'logical', 'character'), na.strings=c('nil'))	df = read_csv('/path/to/data.csv', 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=':', header=FALSE, comment.char='#')	none
skip rows	def = readtable('/path/to/data.csv', 'HeaderLines', 4)	df = read.csv('/path/to/data.csv', skip=4)	df = read_csv('/path/to/data.csv', skiprows=4) # rows to skip can be specified individually: df = read_csv('/path/to/data.csv', skiprows=range(0, 4))
max rows to read		df = read.csv('/path/to/data.csv', nrows=4)	df = read_csv('/path/to/data.csv', nrows=4)
index column		none	df = pd.read_csv('/path/to.csv', index_col='key_col') # hierarchical index: df = pd.read_csv('/path/to.csv', index_col=['col1', 'col2'])
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	julia
project columns by name	people(:, {'sx', 'ht'})	people[c('sx', 'ht')]	people[['sx', 'ht']]
project columns by position	people(:, [1 2])	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, :)	people[people$ht > 66, ]
rename columns		colnames(people) = c('gender', 'height', 'weight')
access sub data frame		# data frame of first 3 rows with # ht and wt columns reversed: people[1:3, c(1, 3, 2)]
select rows	people(people.ht > 66, :)	subset(people, ht > 66) people[people$ht > 66, ]	people[people['ht'] > 66]
select distinct rows	unique(people(:,{'sx'}))	unique(people[c('sx')])
split rows		# class(x) is list: x = split(people, people$sx == 'F') # data.frame only containing females: x$T
inner join		pw = read.delim('/etc/passwd', sep=':', header=F, comment.char='#', col.names=c('name', 'passwd', 'uid', 'gid', 'gecos', 'home', 'shell')) grp = read.delim('/etc/group', sep=':', header=F, 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']) grp = pd.read_table('/tmp/group', sep=':', header=None, names=['name', 'passwd', 'gid', 'members']) 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	julia
group by column			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
nulls and aggregation functions
vectors
	matlab	r	numpy	julia
vector literal	same as array	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	DimensionMismatch
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	3 * [1, 2, 3] [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)	dot([1, 1, 1], [2, 2, 2])
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)	cross([1, 0, 0], [0, 1, 0])
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)	v = [1, 2, 3] norm(v, 1) norm(v, 2) norm(v, Inf)
matrices
	matlab	r	numpy	julia
literal or constructor	% row-major order: A = [1, 2; 3, 4] B = [4 3 2 1]	# column-major order: A = matrix(c(1, 3, 2, 4), 2, 2) B = matrix(c(4, 2, 3, 1), nrow=2) # row-major order: A = matrix(c(1, 2, 3, 4), nrow=2, byrow=T)	# row-major order: A = np.matrix([[1, 2], [3, 4]]) B = np.matrix([[4, 3], [2, 1]])	A = [1 2; 3 4] B = [4 3; 2 1]
constant matrices all zeros, all ones	zeros(3, 3) or zeros(3) ones(3, 3) or ones(3)	matrix(0, 3, 3) matrix(1, 3, 3)	np.matrix(np.ones([3, 3])) np.matrix(np.zeros([3, 3]))	zeros(Float64, (3, 3)) ones(Float64, (3, 3))
diagonal matrices and identity	diag([1, 2, 3]) % 3x3 identity: eye(3)	diag(c(1, 2, 3) # 3x3 identity: diag(3)	np.diag([1, 2, 3]) np.identity(3)	diagm([1, 2, 3]) eye(3)
matrix by formula	i = ones(10, 1) * (1:10) j = (1:10)' * ones(1, 10) % use component-wise ops only: 1 ./ (i + j - 1)
dimensions	rows(A) columns(A)	dim(A)[1] dim(A)[2]	nrows, ncols = A.shape	nrows, ncols = size([1 2 3; 4 5 6])
element access	A(1, 1)	A[1, 1]	A[0, 0]	A[1, 1]
row access	A(1, 1:2)	A[1, ]	A[0]	A[1, :]
column access	A(1:2, 1)	A[, 1]	A[:, 0]	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]	A = np.matrix(range(1, 10)).reshape(3, 3) A[:2, :2]	reshape(1:9, 3, 3)[1:2, 1:2]
scalar multiplication	3 * A A * 3 also: 3 .* A A .* 3	3 * A A * 3	3 * A A * 3	3 * [1 2; 3 4] [1 2; 3 4] * 3
element-wise operators	.+ .- .* ./	+ - * /	+ - np.multiply() np.divide()	+ - .* ./
multiplication	A * B	A %*% B	np.dot(A, B)	A * B
power	A ^ 3 % power of each entry: A .^ 3		A ** 3	A ^ 3 # power of each entry: A .^ 3
kronecker product	kron(A, B)	kronecker(A, B)	np.kron(A, B)	kron(A, B)
comparison	all(all(A == B)) any(any(A ~= B))	all(A == B) any(A != B)	np.all(A == B) np.any(A != B)	A == B A != B
norms	norm(A, 1) norm(A, 2) norm(A, Inf) norm(A, 'fro')	norm(A, "1") ?? norm(A, "I") norm(A, "F")		norm(A, 1) norm(A, 2) norm(A, Inf) # Froebenius norm: vecnorm(A, 2)
transpose	transpose(A)	t(A)	A.transpose()	transpose([1 2; 3 4])
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()	[1im 2im; 3im 4im]' ctranspose([1im 2im; 3im 4im])
inverse	inv(A)	solve(A)	np.linalg.inv(A)	inv([1 2; 3 4])
pseudoinverse	A = [0 1; 0 0] pinv(A)	install.packages('corpcor') library(corpcor) A = matrix(c(0, 0, 1, 0), nrow=2) pseudoinverse(A)	A = np.matrix([[0, 1], [0, 0]]) np.linalg.pinv(A)	pinv([0 1; 0 0])
determinant	det(A)	det(A)	np.linalg.det(A)	det(1 2; 3 4])
trace	trace(A)	sum(diag(A))	A.trace()	trace([1 2; 3 4])
eigenvalues	eig(A)	eigen(A)$values	np.linalg.eigvals(A)	eigvals(A)
eigenvectors	[evec, eval] = eig(A) % each column of evec is an eigenvector % eval is a diagonal matrix of eigenvalues	eigen(A)$vectors	np.linalg.eig(A)[1]	eigvecs(A)
singular value decomposition	X = randn(10) [u, d, v] = svd(X)	X = matrix(rnorm(100), nrow=10) result = svd(X) # singular values: result$d # matrix of eigenvectors: result$u # unitary matrix: result$v	np.linalg.svd(np.random.randn(100).reshape(10, 10))	X = randn(10, 10) u, s, v = svds(X)
solve system of equations	A \ [2;3]	solve(A, c(2, 3))	np.linalg.solve(A, [2, 3])	[1 2; 3 4] \ [2; 3]
sparse matrices
	matlab	r	numpy	julia
sparse matrix construction	% 100x100 matrix with 5 at (1, 1) and 4 at (2, 2): X = sparse([1 2], [1 2], [5 4], 100, 100)	X = spMatrix(100, 100, c(1, 2), c(1, 2), c(5, 4))	import scipy.sparse as sparse row, col, val = [5, 4], [1, 2], [1, 2] X = sparse.coo_matrix((val, (row, col)), shape=(100, 100))
sparse matrix decomposition	[rows, cols, vals] = find(X) % just the values: nonzeros(X)
sparse identity matrix	% 100x100 identity: speye(100)		sparse.identity(100) # not square; ones on diagonal: sparse.eye(100, 200)
dense matrix to sparse matrix and back	X = sparse([1 0 0; 0 0 0; 0 0 0]) X2 = full(X)		imoprt scipy.sparse as sparse A = np.array([[1, 0, 0], [0, 0, 0], [0, 0, 0]]) X = sparse.coo_matrix(A) X2 = X.todense()
sparse matrix storage	% is storage sparse: issparse(X) % memory allocation in bytes: nzmax(X) % number of nonzero entries: nnz(X)	# memory allocation in bytes: object.size(X)	import scipy.sparse as sparse sparse.issparse(X)
optimization
	matlab	r	numpy	julia
linear minimization	% download and install cvx: cvx_begin variable x1; variable x2; variable x3; minimize x1 + x2 + x3; subject to x1 + x2 >= 1; x2 + x3 >= 1; x1 + x3 >= 1; cvx_end; % 'Solved' in cvx_status % argmin in x1, x2, x3 % minval in cvx_optval	# install.packages('lpSolve') require(lpSolve) obj = c(1, 1, 1) A = matrix(c(1, 1, 0, 0, 1, 1, 1, 0, 1), nrow=3, byrow=T) dir = c(">=", ">=", ">=") rhs = c(1, 1, 1) result = lp("min", obj, A, dir, rhs) # 0 in result$status # argmin in result$solution # minval in result$objval	# sudo pip install cvxopt from cvxopt.modeling import * x1 = variable(1, 'x1') x2 = variable(1, 'x2') x3 = variable(1, 'x3') c1 = (x1 + x2 >= 1) c2 = (x1 + x3 >= 1) c3 = (x2 + x3 >= 1) lp = op(x1 + x2 + x3, [c1, c2, c3]) lp.solve() # 'optimal' in lp.status # argmin in x1.value[0], x2.value[0], # x3.value[0] # minval in lp.objective.value()[0]
decision variable vector	cvx_begin variable x(3); minimize sum(x); subject to x(1) + x(2) >= 1; x(2) + x(3) >= 1; x(1) + x(3) >= 1; cvx_end;	# decision variables must be an array	# sudo pip install cvxopt from cvxopt.modeling import * x = variable(3, 'x') c1 = (x[0] + x[1] >= 1) c2 = (x[0] + x[2] >= 1) c3 = (x[1] + x[2] >= 1) lp = op(x[0] + x[1] + x[2], [c1, c2, c3]) lp.solve()
linear maximization	cvx_begin variable x(3); maximize sum(x); subject to x(1) + x(2) <= 1; x(2) + x(3) <= 1; x(1) + x(3) <= 1; cvx_end;	# install.packages('lpSolve') require(lpSolve) obj = c(1, 1, 1) A = matrix(c(1, 1, 0, 0, 1, 1, 1, 0, 1), nrow=3, byrow=T) dir = c("<=", "<=", "<=") rhs = c(1, 1, 1) result = lp("max", obj, A, dir, rhs)	# None; negate objective function before # solving; negate optimal value which # is found.
constraint in variable declaration	cvx_begin variable x(3) nonnegative; minimize 10x(1) + 5x(2) + 4*x(3); subject to x(1) + x(2) + x(3) >= 10; cvx_end	# none; but note that variables are assumed # to be nonnegative	# none
unbounded behavior	cvx_begin variable x(3); maximize sum(x); cvx_end % Inf in cvx_optval % 'Unbounded' in cvx_status	# install.packages('lpSolve') require(lpSolve) obj = c(1, 1, 1) A = matrix(c(1, 1, 1), nrow=1, byrow=T) dir = c(">=") rhs = c(1) result = lp("max", obj, A, dir, rhs) # result$status is 3	# sudo pip install cvxopt from cvxopt.modeling import * x = variable(3, 'x') c1 = (x[0] >= 0) c2 = (x[1] >= 0) c3 = (x[2] <= 0) lp = op(x[0] + x[1] + x[2], [c1, c2, c3]) lp.solve() # lp.status is 'dual infeasible'
infeasible behavior	cvx_begin variable x(3) nonnegative; maximize sum(x); subject to x(1) + x(2) + x(3) < -1; cvx_end % -Inf in cvx_optval % 'Infeasible' in cvx_status	# install.packages('lpSolve') require(lpSolve) obj = c(1, 1, 1) A = matrix(c(1, 1, 1), nrow=1, byrow=T) dir = c("<=") rhs = c(-1) result = lp("min", obj, A, dir, rhs) # result$status is 2	# sudo pip install cvxopt from cvxopt.modeling import * x = variable(3, 'x') c1 = (x[0] >= 0) c2 = (x[1] >= 0) c3 = (x[2] >= 0) c4 = (x[0] + x[1] + x[2] <= -1) lp = op(x[0] + x[1] + x[2], [c1, c2, c3, c4]) lp.solve() # lp.status is 'primal infeasible'
integer decision variable	% requires Optimization Toolbox: f = [1 1 1] A = [-1 -1 0; -1 0 -1; 0 -1 -1; -1 0 0; 0 -1 0; 0 0 -1] b = [-1 -1 -1 0 0 0] % 2nd arg indicates integer vars [x opt flag] = intlinprog(f, [1 1 1], A, b) % if solution found, flag is 1 % x is argmin % opt is optimal value	# install.packages('lpSolve') require(lpSolve) obj = c(1, 1, 1) A = matrix(c(1, 1, 0, 0, 1, 1, 1, 0, 1), nrow=3, byrow=T) dir = c(">=", ">=", ">=") rhs = c(1, 1, 1) result = lp("min", obj, A, dir, rhs, int.vec=c(1, 1, 1))
binary decision variable		# install.packages('lpSolve') require(lpSolve) obj = c(1, 1, 1) A = matrix(c(1, 1, 0, 0, 1, 1, 1, 0, 1), nrow=3, byrow=T) dir = c(">=", ">=", ">=") rhs = c(1, 1, 1) result = lp("min", obj, A, dir, rhs, binary.vec=c(1, 1, 1))	# integer solver not provided by default
polynomials
	matlab	r	numpy	julia
exact polynomial fit	x = [1 2 3 4] y = [3 9 2 1] % polynomial coefficient array: p = polyfit(x, y, 3) % plot polynomial: xx = -10:.1:10 yy = polyval(p, xx) plot(xx, yy)
exact polynomial fit with derivative values
piecewise polynomial fit
cubic spline	f = spline(1:20, normrnd(0, 1, 1, 20)) x = 1:.1:20 plot(x, ppval(f, x))	f = splinefun(rnorm(20)) x = seq(1, 20, .1) plot(x, f(x), type="l")
underdetermined polynomail fit
overdetermined polynomial fit
multivariate polynomial fit
descriptive statistics
	matlab	r	numpy	julia
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)	x = [1 2 3 8 12 19] sum(x) 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)	std(x) var(x)
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) iqr(x) quantile(x, .90)	min(x) median(x) max(x) IQR(x) quantile(x, prob=.90)	min(x) sp.median(x) max(x) ?? stats.scoreatpercentile(x, 90.0)
bivariate statistiscs correlation, covariance	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] ??
correlation matrix	x1 = randn(100, 1) x2 = 0.5 * x1 + randn(100, 1) x3 = 0.1 * x1 + 0.1 * x2 + 0.1 * randn(100, 1) corr([x1 x2 x3])	x1 = rnorm(100) x2 = x1 + 0.5 * rnorm(100) x3 = 0.3 * x1 + 0.1 * 2 + 0.1 * rnorm(100) cor(cbind(x1, x2, x3))
data set to frequency table		x = c(1,2,1,1,2,5,1,2,7) tab = table(x)
frequency table to data set		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 are (0, 3], (3, 6], and (6, 9]: bins = tapply(x, xf, length)
distributions
	matlab	r	numpy	julia
binomial density, cumulative, quantile, sample of 10	binopdf(x, n, p) binocdf(x, n, p) binoinv(y, n, p) binornd(n, p, 1, 10)	dbinom(x, n, p) pbinom(x, n, p) qbinom(y, n, p) rbinom(10, 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 density, cumulative, quantile, sample of 10	poisspdf(x, lambda) poisscdf(x, lambda) poissinv(y, lambda) poissrnd(lambda, 1, 10)	dpois(x, lambda) ppois(x, lambda) qpois(y, lambda) rpois(10, lambda)	stats.poisson.pmf(x, lambda) stats.poisson.cdf(x, lambda) stats.poisson.ppf(y, lambda) stats.poisson.rvs(lambda, size=1)
normal density, cumulative, quantile, sample of 10	normpdf(x, mu, sigma) normcdf(x, mu, sigma) norminv(y, mu, sigma) normrnd(mu, sigma, 1, 10)	dnorm(x, mu, sigma) pnorm(x, mu, sigma) qnorm(y, mu, sigma) rnorm(10, 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 density, cumulative, quantile, sample of 10	gampdf(x, k, theta) gamcdf(x, k, theta) gaminv(y, k, theta) gamrnd(k, theta, 1, 10)	dgamma(x, k, scale=theta) pgamma(x, k, scale=theta) qgamma(y, k, scale=theta) rgamma(10, 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 density, cumulative, quantile, sample of 10	exppdf(x, lambda) expcdf(x, lambda) expinv(y, lambda) exprnd(lambda, 1, 10)	dexp(x, lambda) pexp(x, lambda) qexp(y, lambda) rexp(10, 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 density, cumulative, quantile, sample of 10	chi2pdf(x, nu) chi2cdf(x, nu) chi2inv(y, nu) chi2rnd(nu, 1, 10)	dchisq(x, nu) pchisq(x, nu) qchisq(y, nu) rchisq(10, nu)	stats.chi2.pdf(x, nu) stats.chi2.cdf(x, nu) stats.chi2.ppf(y, nu) stats.chi2.rvs(nu)
beta density, cumulative, quantile, sample of 10	betapdf(x, alpha, beta) betacdf(x, alpha, beta) betainvf(y, alpha, beta) betarnd(alpha, beta, 1, 10)	dbeta(x, alpha, beta) pbeta(x, alpha, beta) qbeta(y, alpha, beta) rbeta(10, 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 density, cumulative, quantile, sample of 10	unifpdf(x, a, b) unifcdf(x, a, b) unifinv(y, a, b) unifrnd(a, b, 1, 10)	dunif(x, a, b) punif(x, a, b) qunif(y, a, b) runif(10, 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 density, cumulative, quantile, sample of 10	tpdf(x, nu) tcdf(x, nu) tinv(y, nu) trnd(nu, 1, 10)	dt(x, nu) pt(x, nu) qt(y, nu) rt(10, nu)	stats.t.pdf(x, nu) stats.t.cdf(x, nu) stats.t.ppf(y, nu) stats.t.rvs(nu)
Snedecor's F density, cumulative, quantile, sample of 10	fpdf(x, d1, d2) fcdf(x, d1, d2) finv(y, d1, d2) frnd(d1, d2, 1, 10)	df(x, d1, d2) pf(x, d1, d2) qf(y, d1, d2) rf(10, 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)
empirical density function	% $ apt-get install octave-econometrics x = (-3:.05:3)' y = kernel_density(x, normrnd(0, 1, 100, 1))	dfunc = density(rnorm(100)) dfunc$x dfunc$y
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)))
linear regression
	matlab	r	numpy	julia
simple linear regression coefficient, intercept, and residuals	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 = seq(10) y = 2 * x + 1 + rnorm(10) fit = lm(y ~ x) summary(fit) # yhat = ax + b: a = fit$coefficients[2] b = fit$coefficients[1] # y - (ax + b): fit$residuals	x = np.array([1,2,3]) y = np.array([2,4,7]) lsq = stats.linregress(x, y) a = lsq[0] b = lsq[1] y - (a*x+b)
no intercept		x = seq(10) y = 2 * x + 1 + rnorm(10) fit = lm(y ~ x + 0) summary(fit) # y = ax: a = fit$coefficients[1]
multiple linear regression		x1 = rnorm(100) x2 = rnorm(100) y = 2 * x2 + rnorm(100) fit = lm(y ~ x1 + x2) summary(fit)
interaction		x1 = rnorm(100) x2 = rnorm(100) y = 2 * x1 + x2 + 3 * x1 * x2 + rnorm(100) # x1, x2, and x1x2 as predictors: fit = lm(y ~ x1 x2) summary(fit) # just x1*x2 as predictor: fit2 = lm(Y ~ x1:x2)
logistic regression		y = round(runif(100)) x1 = round(runif(100)) x2 = y + rnorm(100) fit = glm(y ~ x1 + x2, family="binomial") summary(fit)
statistical tests
	matlab	r	numpy	julia
wilcoxon signed-rank test variable is symmetric around zero	x = unifrnd(-0.5, 0.5, 100, 1) % null hypothesis is true: wilcoxon_test(x, zeros(100, 1)) % alternative hypothesis is true: wilcoxon_test(x + 1.0, zeros(100, 1))	# null hypothesis is true: wilcox.test(runif(100) - 0.5) alternative hypothesis is true: wilcox.test(runif(100) + 0.5)	stats.wilcoxon()
kruskal-wallis rank sum test variables have same location parameter	x = unifrnd(0, 1, 200, 1) % null hypothesis is true: kruskal_wallis_test(randn(100, 1), randn(200, 1)) % alternative hypothesis is true: kruskal_wallis_test(randn(100, 1), x)	# null hypothesis is true: kruskal.test(list(rnorm(100), rnorm(200))) # alternative hypothesis is true: kruskal.test(list(rnorm(100), runif(200)))	stats.kruskal()
kolmogorov-smirnov test variables have same distribution	x = randn(100, 1) y1 = randn(100, 1) y2 = unifrnd(-0.5, 0.5, 100, 1) % null hypothesis is true: kolmogorov_smirnov_test_2(x, y1) % alternative hypothesis is true: kolmogorov_smirnov_test_2(x, y2)	# null hypothesis is true: ks.test(rnorm(100), rnorm(100)) # alternative hypothesis is true: ks.test(rnorm(100), runif(100) - 0.5)	stats.ks_2samp()
one-sample t-test mean of normal variable with unknown variance is zero	x1 = 3 * randn(100, 1) x2 = 3 * randn(100, 1) + 3 % null hypothesis is true: t_test(x1, 0) % alternative hypothesis is true: t_test(x2, 0)	# null hypothesis is true: t.test(rnorm(100, 0, 3)) # alternative hypothesis is true: t.test(rnorm(100, 3, 3))	stats.ttest_1samp()
independent two-sample t-test two normal variables have same mean	x = randn(100, 1) y1 = randn(100, 1) y2 = randn(100, 1) + 1.5 % null hypothesis is true: t_test_2(x, y1) % alternative hypothesis is true: t_test_2(x, y2)	# null hypothesis is true: t.test(rnorm(100), rnorm(100)) # alternative hypothesis is true: t.test(rnorm(100), rnorm(100, 3))	stats.ttest_ind()
one-sample binomial test binomial variable parameter is as given		n = 100 x = rbinom(1, n, 0.5) # null hypothesis that p=0.5 is true: binom.test(x, n) # alternative hypothesis is true: binom.test(x, n, p=0.3)	stats.binom_test()
two-sample binomial test parameters of two binomial variables are equal	prop_test_2()	n = 100 x1 = rbinom(1, n, 0.5) x2 = rbinom(1, n, 0.5) # null hypothesis that p=0.5 is true: prop.test(c(x1, x2), c(n, n)) y = rbinom(1, n, 0.3) # alternative hypothesis is true: prop.test(c(x1, y), c(n, n))
chi-squared test parameters of multinomial variable are all equal	chisquare_test_independence()	fair = floor(6 * runif(100)) + 1 loaded = floor(7 * runif(100)) + 1 loaded[which(loaded > 6)] = 6 # null hypothesis is true: chisq.test(table(fair)) # alternative hypothesis is true: chisq.test(table(loaded))	stats.chisquare()
poisson test parameter of poisson variable is as given		# null hypothesis is true: poisson.test(rpois(1, 100), r=100) # alternative test is true: poisson.test(rpois(1, 150), r=100)
F test ratio of variance of normal variables is as given	var_test()	x = rnorm(100) y = rnorm(100, 0, sd=sqrt(3)) # null hypothesis is true: var.test(y, x, ratio=3) # alternative hypothesis is true: var.test(y, x, ratio=1)
pearson product moment test normal variables are not correlated	cor_test()	x1 = rnorm(100) x2 = rnorm(100) y = x2 + rnorm(100) # null hypothesis is true: cor.test(y, x1) # alternative hypothesis is true: cor.test(y, x2)	stats.pearsonr()
shapiro-wilk test variable has normal distribution		# null hypothesis is true: shapiro.test(rnorm(1000)) # alternative hypothesis is true: shapiro.test(runif(1000))	stats.shapiro()
bartlett's test two or more normal variables have same variance	bartlett_test()	x = rnorm(100) y1 = rnorm(100) y2 = 0.1 * rnorm(100) # null hypothesis is true: bartlett.test(list(x, y1)) # alternative hypothesis is true: bartlett.test(list(x, y))	stats.bartlett()
levene's test two or more variables have same variance		install.packages('reshape', 'car') library(reshape) library(car) x = rnorm(100) y1 = rnorm(100) y2 = 0.1 * rnorm(100) # null hypothesis is true: df = melt(data.frame(x, y1)) leveneTest(df$value, df$variable) # alternative hypothesis is true: df = melt(data.frame(x, y2)) leveneTest(df$value, df$variable)	stats.levene()
one-way anova two or more normal variables have same mean	x1 = randn(100, 1) x2 = randn(100, 1) x3 = randn(100, 1) x = [x1; x2; x3] y = [x1; x2; x3 + 0.5] units = ones(100, 1) grp = [units; 2 * units; 3 * units] % null hypothesis is true: anova(x, grp) % alternative hypothesis is true: anova(y, grp)	install.packages('reshape') library(reshape) # null hypothesis that all means are the same # is true: x1 = rnorm(100) x2 = rnorm(100) x3 = rnorm(100) df = melt(data.frame(x1, x2, x3)) fit = lm(df$value ~ df$variable) anova(fit)	stats.f_oneway()
time series
	matlab	r	numpy	julia
time series		# first observation time is 1: y = ts(rnorm(100)) # first observation time is 0: y2 = ts(rnorm(100), start=0) plot(y)	# first observation time is 0: y = pd.Series(randn(100)) # first observation time is 1: y2 = pd.Series(randn(100), index=range(1,101)) y.plot()
monthly time series		# monthly observations 1993-1997: y = ts(rnorm(60), frequency=12, start=1993) # monthly observations from Oct 1993: y2 = ts(rnorm(60), frequency=12, start=c(1993, 10)) plot(y)	dt = pd.datetime(2013, 1, 1) idx = pd.date_range(dt, periods=60, freq='M') y = pd.Series(randn(60), index=idx) dt2 = pd.datetime(2013, 10, 1) idx2 = pd.date_range(dt2, periods=60, freq='M') y2 = pd.Series(randn(60), index=idx2)
lookup by time		start = tsp(y2)[1] end = tsp(y2)[2] freq = tsp(y2)[3] # value for Jan 1994: y2[(1994 - start) * freq + 1]	y2[pd.datetime(2014, 1, 31)]
lookup by position in series		for (i in 1:length(y)) { print(y[i]) }	for i in range(0, len(y)): y.ix[i]
aligned arithmetic		y = ts(rnorm(10), start=0) y2 = ts(rnorm(10), start=5) # time series with 5 data points: y3 = y + y2	y = pd.Series(randn(10)) y2 = pd.Series(randn(10), index=range(5, 15)) # time series with 15 data points; 10 of # which are NaN: # y3 = y + y2
lag operator		x = ts(rnorm(100)) y = x + lag(x, 1)	x = pd.Series(randn(100)) y = x + x.shift(-1)
lagged difference		delta = diff(y, lag=1)	delta = y.diff(1)
simple moving average		install.packages('TTR') library('TTR') ma = SMA(y, n=4) plot(y) lines(ma, col='red')	y = pd.Series(randn(50)) ma = pd.rolling_mean(y, 4) plot(y, 'k', ma, 'r')
weighted moving average		install.packages('TTR') library('TTR') ma = WMA(y, n=4, wts=c(1, 2, 3, 4)) plot(y) lines(ma, col='red')
exponential smoothing		x = rnorm(100) fit = HoltWinters(x, alpha=0.5, beta=F, gamma=F) values = fit$fitted plot(fit)	alpha = 0.5 span = (2 / alpha) - 1 fit = pd.ewma(y, span=span, adjust=False) fit.plot()
exponential smoothing with best least squares fit		x = rnorm(100) fit = HoltWinters(x, beta=F, gamma=F) alpha = fit$a plot(fit)
decompose into seasonal and trend		raw = seq(1,100) + rnorm(100) + rep(seq(1,10), 10) y = ts(raw, frequency=10) # additive model: t + s + r: yd = decompose(y) yd$trend yd$seasonal yd$random plot(yd) # multiplicative model: t * s * r: yd2 = decompose(y, type="multiplicative")
correlogram		x = rnorm(100) x2 = append(x[4:100], x[1:3]) acf(x, lag.max=20) acf(x + x2, lag.max=20)
test for stationarity
arma
arima
arima with automatic model selection
fast fourier transform
	matlab	r	numpy	julia
fft	x = 3 * sin(1:100) + sin(3 * (1:100)) + randn(1, 100) dft = fft(x)
inverse fft
shift constant component to center
two-dimensional fft
n-dimensional fft
clustering
	matlab	r	numpy	julia
distance matrix	pts = [1 1; 1 2; 2 1; 2 3; 3 4; 4 4] % value at (i, j) is distance between i-th % and j-th observation dm = squareform(pdist(pts, 'euclidean'))
distance options	'euclidean' 'seuclidian' 'cityblock' 'minkowski' 'chebychev' 'mahalanobis' 'cosine' 'correlation' 'spearman' 'hamming' 'jaccard'
hierarchical clusters
dendogram
silhouette plot
k-means
images
	matlab	r	numpy	julia
load from file	X = imread('cat.jpg');
display image	imshow(X)
image info	whos X imfinfo('cat.jpg')
write to file	imwrite(X, 'cat2.jpg')
sound
	matlab	r	numpy	julia
read from file	[y, fs] = audioread('speech.flac')
record clip	recObj = audiorecorder % record 5 seconds: recordblocking(recObj, 5) y = getaudiodata(recOjb);
write to file
clip info	info = audioinfo('speech.flac') info.NumChannels info.SampleRate info.TotalSamples info.Duration
play clip	sound(y, fs)
bar charts
	matlab	r	numpy	julia
vertical bar chart	bar([7 3 8 5 5]) set(gca, 'XTick', 1:5, ... 'XTickLabel', {'a', 'b', 'c', 'd', 'e'})	cnts = c(7,3,8,5,5) names(cnts) = c("a","b","c","d","e") barplot(cnts) # 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)	cnts = [7,3,8,5,5] plt.bar(range(0,len(cnts)), cnts)
bar chart with error bars
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)
grouped bar chart	d = [7 1; 3 2; 8 1; 5 3; 5 1] bar(d)	data = matrix(c(7,1,3,2,8,1,5,3,5,1), nrow=2) labels = c("a","b","c","d","e") barplot(data, names.arg=labels, beside=TRUE)
stacked bar chart	d = [7 1; 3 2; 8 1; 5 3; 5 1] bar(d, 'stacked')	data = matrix(c(7,1,3,2,8,1,5,3,5,1), nrow=2) labels = c("a","b","c","d","e") barplot(data, 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')
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)
histogram	hist(randn(1, 100), 10)	hist(rnorm(100), breaks=10) hist(rnorm(100), breaks=seq(-3, 3, 0.5)) # ggplot2: x = rnorm(50) binwidth = (max(x) - min(x)) / 10 qplot(x, geom="histogram", binwidth=binwidth)	plt.hist(sp.randn(100), bins=range(-5,5))
box plot	boxplot(randn(1, 100))	boxplot(rnorm(100))	plt.boxplot(sp.randn(100))
box plots side-by-side	boxplot([randn(1, 100) exprnd(1, 1, 100) unifrnd(0, 1, 1, 100)]')	boxplot(rnorm(100), rexp(100), runif(100))	plt.boxplot([sp.randn(100), np.random.uniform(size=100), np.random.exponential(size=100)])
scatter plots
	matlab	r	numpy	julia
strip chart	data = randn(1, 50) plot(data, zeros(size(data)), 'o')	stripchart(rnorm(50))
strip chart with jitter		stripchart(floor(50 * runif(20)), method="jitter")
scatter plot	plot(randn(1,50),randn(1,50),'+')	plot(rnorm(50), rnorm(50))	plt.scatter(sp.randn(50), sp.randn(50), marker='x')
additional point set	plot(randn(20), randn(20), '.k', randn(20), randn(20), '.r')	plot(rnorm(20), rnorm(20)) points(rnorm(20) + 1, rnorm(20) + 1, col='red')
point types	'.': point 'o': circle 'x': x-mark '+': plus '*': star 's': square 'd': diamond 'v': triangle (down) '^': triangle (up) '<': triangle (left) '>': traingle (right) 'p': pentagram 'h': hexagram	Integer values for pch parameter: 0: open square 1: open circle 2: open triangle, points up 3: cross 4: x 5: open diamond 6: open triangle, points down 15: solid square 16: solid circle 17: solid triangle, points up 18: solid diamond	marker parameter takes these string values: '.': point ',': pixel 'o': circle 'v': triangle_down '^': triangle_up '<': triangle_left '>': triangle_right '1': tri_down '2': tri_up '3': tri_left '4': tri_right '8': octagon 's': square 'p': pentagon '*': star 'h': hexagon1 'H': hexagon2 '+': plus 'x': x 'D': diamond 'd': thin_diamond '\|': vline '_': hline
point size		plot(rnorm(50), rnorm(50), cex=2)
scatter plot matrix		x = rnorm(20) y = rnorm(20) z = x + 3y w = y + 0.1rnorm(20) df = data.frame(x, y, z, w) pairs(df)
3d scatter plot		install.packages('scatterplot3d') library('scatterplot3d') scatterplot3d(rnorm(50), rnorm(50), rnorm(50), type="h")
bubble chart		install.packages('ggplot2') library('ggplot2') 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()
hexagonal bins		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', col='red')	x = range(0,20) err = sp.randn(20)10 y = [2i 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])
quantile-quantile plot		qqplot(runif(50), rnorm(50)) lines(c(-9,9), c(-9,9), col="red")
line charts
	matlab	r	numpy	julia
polygonal line plot	plot(1:20,randn(1,20))	plot(1:20, rnorm(20), type="l")	plt.plot(range(0,20), sp.randn(20), '-')
additional line	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")
line types	Optional 3rd argument to plot: '-': solid ':': dotted '-.': dashdot '—': dashed	Integer or string values for lty parameter: 0: 'blank' 1: 'solid' (default) 2: 'dashed' 3: 'dotted' 4: 'dotdash' 5: 'longdash' 6: 'twodash'	Optional 3rd argument to plot: '-': solid line '--': dashed line '-.': dash-dot line ':': dotted line '.': point ',': pixel 'o': circle 'v': triangle_down '^': triangle_up '<': triangle_left '>': triangle_right '1': tri_down '2': tri_up '3': tri_left '4': tri_right 's': square 'p': pentagon '*': star 'h': hexagon1 'H': hexagon2 '+': plus 'x': x 'D': diamond 'd': thin_diamond '\|': vline '_': hline
line thickness		plot(1:20, rnorm(20), type="l", lwd=5)
function plot	fplot(@sin, [-4 4])	x = seq(-4, 4, .01) plot(sin(x), type="l")	x = [i * .01 for i in range(-400, 400)] plt.plot(x, sin(x), '-')
stacked area chart		install.packages('ggplot2') library('ggplot2') 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		install.packages('ggplot2') library('ggplot2') 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()
surface charts
	matlab	r	numpy	julia
contour plot
heat map	i = ones(100, 1) * (1:100) j = (1:100)' * ones(1, 100) data = sin(.2 * i) .* sin(.2 * j) colormap(gray) imagesc(data)	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)
shaded surface plot
light source
mesh surface plot
view point
vector field plot
chart options
	matlab	r	numpy	julia
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')
axis labels	plot( 1:20, (1:20) .** 2) xlabel('x') ylabel('x squared')	plot(1:20, (1:20)^2, xlab="x", ylab="x squared")	x = range(0, 20) plt.plot(x, [i * i for i in x], '-') plt.xlabel('x') plt.ylabel('x squared')
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('topleft', c('first', 'second'), lty=c(1,1), lwd=c(2.5, 2.5), col=c('black', 'red'))
colors	Use color letters by themselves for colored lines. Use '.r' for red dots. 'b': blue 'g': green 'r': red 'c': cyan 'm': magenta 'y': yellow 'k': black 'w': white	# Use the col parameter to specify the color of # points and lines. # # The colors() function returns a list of # recognized names for colors. plot(rnorm(10), col='red') plot(rnorm(10), col='#FF0000')
axis limits	plot( 1:20, (1:20) .** 2) % [xmin, xmax, ymin, ymax]: axis([1 20 -200 500])	plot(1:20, (1:20)^2, xlim=c(0, 20), ylim=c(-200,500))	x = range(0, 20) plt.plot(x, [i * i for i in x], '-') plt.xlim([0, 20]) plt.ylim([-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])
superimposed plots with different y-axis scales		x <- 1:10 y <- rnorm(10) z <- rnorm(10) * 1000 par(mar = c(5, 4, 4, 4) + 0.3) plot(x, y, type='l') par(new=T) plot(x, z, col='red', type='l', axes=F, xlab='', ylab='') axis(side=4, col='red', col.axis='red', at=pretty(range(z))) mtext('z', side=4, line=3, col='red')
aspect ratio
ticks
grid lines
grid of subplots	% 3rd arg refers to the subplot; % subplots are numbered in row-major order. for i = 1:4 subplot(2, 2, i), hist(randn(50)) end	for (i in split.screen(c(2, 2))) { screen(n=i) hist(rnorm(100)) }	for i in [1, 2, 3, 4]: plt.subplot(2, 2, i) plt.hist(sp.randn(100), bins=range(-5,5))
open new plot window	open new plot figure open new plot	hist(rnorm(100)) dev.new() hist(rnorm(100))
close all plot windows	close all	graphics.off()
save plot as png	f = figure hist(randn(100)) print(f, '-dpng', 'histogram.png')	png('hist.png') hist(rnorm(100)) dev.off()	y = randn(50) plot(y) savefig('line-plot.png')
save plot as svg		svg('hist.svg') hist(rnorm(100)) dev.off()
	__________________________________________________	__________________________________________________	__________________________________________________	__________________________________________________

Tables

Tables are a data type which correspond to the tables of relational databases. In R this data type is called a data frame. The Python library Pandas provides a table data type called DataFrame.

A table is an array of tuples, each of the same length and type. If the type of the first element of the first type is integer, then all the tuples in the table must have first elements which are integers. The type of the tuples corresponds to the schema of a relational database table.

A table can also be

Pandas types: Series(), DataFrame(), Index()

construct from column arrays

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

octave:

Octave does not have the table data type.

size

How to get the number of columns and number of rows in a table.

construct from row tuples

column names as array

How to show the names of the columns.

access column as array

How to access a column in a data frame.

access row as tuple

How to access a row in a data frame.

people[1, ] returns the 1st row from the data frame people as a new data frame 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 frames have nearly the same behavior.

access datum

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

order rows by column

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

order rows by multiple columns

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.

Each column of the data frame 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 frame.

detach columns

How to remove attached column names from the current scope.

spreadsheet editor

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

Import and Export

import tab delimited file

Load a data frame from a tab delimited file.

By default strings are converted to factors. In older versions of R, this could reduce the amount of memory required to load the data frame; this is no longer true in newer versions.

import comma-separated values file

Load a data frame 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.

import file w/o header

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.

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.

skip rows

maximum rows to read

index column

export tab delimited file

export comma-separated values file

Save a data frame to a CSV file.

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 frame

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.

Aggregation

Vectors

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.

vector literal

element-wise arithmetic operators

scalar multiplication

dot product

cross product

norms

matlab:

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-major. If the elements of each column are adjacent to other elements of the same column we say the order is column-major.

matlab:

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.

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)

constant matrices

How to create a matrices with zeros for entries or with ones for entries.

diagonal matrices

How to create diagonal matrices including the identity matrix.

A matrix is diagonal if and only if a_ij = 0 for all i ≠ j.

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.

matlab:

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.

matlab:

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

~= is a synonym for !=.

== 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.

matlab:

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

Sparse Matrices

sparse matrix construction

How to construct a sparse matrix using coordinate format.

Coordinate format specifies a matrix with three arrays: the row indices, the the column indices, and the values.

sparse matrix decomposition

sparse identity matrix

dense matrix to sparse matrix

sparse matrix storage

Optimization

In an optimization problem one seeks the smallest or largest value assumed by an objective function. The inputs to the objective function are the decision variables. A set of equations or inequalities, the constraints, can be used to restrict the decision variables to a feasible region.

If the feasible region is empty, the problem is said to be infeasible. If a minimization problem does not have a lower bound on the feasible region, or if a maximization problem does not have an upper bound on the feasible region, the problem is said to be unbounded.

An optimization problem is linear if both its objective function and its constraints are linear. A constraint is linear if it can be written in the form ∑ aᵢ xᵢ ≤ b, ∑ aᵢ xᵢ ≥ b, or ∑ aᵢ xᵢ = b, where xᵢ are the decision variables.

An integer linear program is a linear optimization problem where the decision variables are constrained to assume integer values. Polynomial time algorithms exist for solving linear programs when the decision variables are real-valued, but solving integer linear programs is NP-hard. A mixed integer linear program has a mix of integer and real-valued decision variables. A special case of an integer linear program is a binary linear program where the decision variables assume the values 0 or 1.

linear minimization

An example of a linear minimization problem.

decision variable vector

How to declare a vector of decision variables.

linear maximization

An example of a linear maximization problem.

constraint in variable declaration

How to include a constraint on a decision variable in its declaration.

unbounded behavior

What happens when attempting to solve an unbounded optimization problem.

infeasible behavior

What happens when attempting to solve an infeasible optimization problem.

integer decision variable

How to declare a decision variable to be integer valued.

matlab:

The solvers which ship with CVX do not support integer programming.

binary decision variable

How to declare a decision variable to only take the values 0 or 1.

Polynomials

exact polynomial fit

cubic spline

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

Descriptive 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 x_i with mean μ is:

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

The nth central moment μ_n of the same population is:

\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 asymmetricality 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:

\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:

\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:

\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

\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.

matlab:

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

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 two middle 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 quartiles. The interquartile range (IQR) is the difference between the 3rd and 1st quartiles.

When the population is divided into ten parts the division markers are called deciles.

When the population is divided into a hundred parts the division markers are called percentiles.

If the population is divided into n parts the markers are called the 1st, 2nd, …, (n-1)th n-quantiles.

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

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

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

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

correlation matrix

data set to 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.

The table function returns an object of type table.

frequency table to data set

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.

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 constructed 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.

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

\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:

\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 Poisson 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 normal 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, …, X_n} are any independent and identically distributed random variables with mean μ and variance σ², then S_n := Σ X_i / 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 X_i are n independent random variables with Γ(k_i, θ) distribution, then Σ X_i has distribution Γ(Σ k_i, 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 the 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

\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.

empirical density function

How to construct a density function from a sample.

empirical cumulative distribution

empirical quantile function

Linear Regression

simple linear regression

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.

no intercept

multiple linear regression

interaction

logistic regression

Statistical Tests

A selection of statistical tests. For each test the null hypothesis of the test is stated in the left column.

In a null hypothesis test one considers the p-value, which is the chance of getting data which is as or more extreme than the observed data if the null hypothesis is true. The null hypothesis is usually a supposition that the data is drawn from a distribution with certain parameters.

The extremeness of the data is determined by comparing the expected value of a parameter according to the null hypothesis to the estimated value from the data. Usually the parameter is a mean or variance. In a one-tailed test the p-value is the chance the difference is greater than the observed amount; in a two-tailed test the p-value is the chance the absolute value of the difference is greater than the observed amount.

Octave and MATLAB have different names for the statistical test functions. The sheet shows the Octave functions; the corresponding MATLAB functions are:

octave	matlab
wilcoxon_test	ranksum
kruskal_wallis_test	kruskalwallis
kolmogorov_smirnov_test	kstest
kolmogorov_smirnov_test_2	kstest2
t_test	ttest
t_test_2	ttest2

wilcoxon signed-rank test

matlab

wilcoxon_test() is an Octave function. The MATLAB function is ranksum().

kruskal-wallis rank sum test

kolmogorov-smirnov test

Test whether two samples are drawn from the same distribution.

matlab:

kolmogorov_smirnov_test_2() and kolmogorov_smirnov_test() are Octave functions. The corresponding MATLAB functions are kstest2() and kstest().

kolmogorov_smirnov_test() is a one sample test; it tests whether a sample is drawn from one of the standard continuous distributions. A one sample KS test gives a repeatable p-value; generating a sample and using a two sample KS test does not.

x = randn(100, 1)

% null hypothesis is true:
kolmogorov_smirnov_test(x, "norm", 0, 1)

% alternative hypothesis is true:
kolmogorov_smirnov_test(x, "unif", -0.5, 0.5)

one-sample t-test

independent two-sample t-test

Test whether two normal variables have same mean.

If the normal variables are known to have the same variance, the variance can be pooled to estimate standard error:

t.test(x, y, var.equal=T)

If the variance cannot be pooled, then Welch's t-test is used. This uses a lower (often non-integral) degrees-of-freedom value, which in turn results in a higher p-value.

one-sample binomial test

two-sample binomial test

chi-squared test

poisson test

F test

pearson product moment test

shapiro-wilk test

bartlett's test

A test whether variables are drawn from normal distributions with the same variance.

levene's test

A test whether variables are drawn from distributions with the same variance.

one-way anova

Time Series

A time series is a sequence of data points collected repeatedly on a uniform time interval.

A time series can be represented by a dictionary which maps timestamps to the type of the data points. A more efficient implementation exploits the fact that the time interval is uniform and stores the data points in an array. To recover the timestamps of the data points, the timestamp of the first data point and the length of the time interval are also stored.

time series

How to create a time series from an array.

monthly time series

How to create a time series with one data point per month.

lookup by time

How to get to a data point in a time series by when the data point was collected.

lookup by position in series

How to get a data point in a time series by its ordinal position.

aligned arithmetic

lagged difference

simple moving average

weighted moving average

exponential smoothing

decompose into seasonal and trend

correlogram

arima

Fast Fourier Transform

[fft fft

inverse fft

shift constant component to center

two-dimensional fft

n-dimensional fft

Clustering

Images

Sound

Bar Charts

vertical bar chart

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

horizontal bar chart

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

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.

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.

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.

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:

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

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.

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")

Scatter Plots

scatter plot

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

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

additional point set

point types

hexagonal bins

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.

3d scatter plot

bubble chart

scatter plot matrix

linear regression line

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

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.

If the two data sets have the same number of elements, one can simply sort them and create the scatterplot.

If the number of elements is different, one generates a set of quantiles (such as percentiles) for each set. The quantile function of MATLAB and R is convenient for this. With Python, one can use scipy.stats.scoreatpercentile.

Line Charts

polygonal line plot

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

additional line

How to add another line to a plot.

line types

function plot

How to plot a function.

stacked area chart

overlapping area chart

Surface Charts

contour plot

Chart Options

chart title

How to set the chart title.

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

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

axis labels

How to label the x and y axes.

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.

logarithmic y-axis

colors

How to set the color of points and lines.

superimposed plots with different y-axis scales

How to superimpose two plots with different y-axis scales.

To minimize the risk that the reader will read off an incorrect y-value for a data point, the example uses the same color for the y-axis as it does for the corresponding data set.

legend

How to put a legend on a chart.

These strings can be used as the first argument to control the legend position:

"bottomright"
"bottom"
"bottomleft"
"left"
"topleft"
"top"
"topright"
"right"
"center"

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

Octave Manual
MATLAB Documentation
Differences between Octave and MATLAB
Octave-Forge Packages

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 results 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. 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.

R

An Introduction to R
Advanced R Programming
The Comprehensive R Archive Network

The primitive 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. 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), record (entries accessed by name), or even as a dictionary.

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 and SciPy Documentation
matplotlib intro
NumPy for Matlab Users
Pandas Documentation
Pandas Method/Attribute Index

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.

Julia

http://julialang.org/