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

a side-by-side reference sheet

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

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:

# file suffix must be .txt, .dat, or .csv
# first row defines variable names:
# first row defines column names:
import csv

% first row defines variable names:
# first row defines variable names:
# first row defines column names:
set column separator df = readtable('/etc/passwd',
'Delimiter', ':',
sep=':',
comment.char='#')
# $grep -v '^#' /etc/passwd > /tmp/passwd df = pd.read_table('/tmp/passwd', sep=':', header=None) set column separator to whitespace df = read.delim('/path/to.txt', sep='') df = read_table('/path/to.txt', sep='\s+') set quote character # default quote character for both read.csv and read.delim # is double quotes. The quote character is escaped by doubling it. # use single quote as quote character: df = read.csv('/path/to/single-quote.csv', quote="'") # no quote character: df = read.csv('/path/to/no-quote.csv', quote="") Both read_table and read_csv use double quotes as the quote character and there is no way to change it. A double quote can be escaped by doubling it. import file w/o header # column names are V1, V2, … read.delim('/etc/passwd', sep=':', header=FALSE, comment.char='#') #$ grep -v '^#' /etc/passwd > /tmp/passwd
#
# column names are X0, X1, …

df.Properties.VariableNames = {'ht', 'wt', 'age'}
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 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
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 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 variables x1, x2, x3; minimize x1 + x2 + x3; subject to x1 + x2 >= 1; x2 + x3 >= 1; x1 + x3 >= 1; cvx_end; % argmin in x1, x2, x3 % minval in cvx_optval array of input variables 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; multidimensional array of input variables linear maximization variable constraints unbounded behavior infeasible behavior complex variables integer variables binary variables 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 x1*x2 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

# null hypothesis is true:
chisq.test(table(fair))

# alternative hypothesis is true:
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 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 + 3*y w = y + 0.1*rnorm(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 = [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])

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), '-')

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)

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')
__________________________________________________ __________________________________________________ __________________________________________________ __________________________________________________

# 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.

## 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.

r:

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 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 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.

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.

r:

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.

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 frame 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 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.

# 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.

## 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.

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)


## 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 aij = 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.

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.

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 !=.

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.

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.

# Optimization

A linear program is an optimization problem in which one tries to minimize or maximize a linear function called the objective function subject to linear inequality constraints.

The unknowns in the objective function and the constraints are called the decision variables.

## linear minimization

How to solve a linear program where the objective function is minimized.

If we have a maximization problem c * x, then we can find the solution by minimizing -c * x; that is, replacing c with -c.

matlab:

linprog is part of the Optimization toolbox.

The function linprog minimizes c * x subject to the constraint that A * x <= b. A * x and b are vectors, and when we say that A * x <= b, we mean that each component of A * x must be less than or equal to the corresponding component of b.

When formulating constraints in the form A * x <= b, we may find that we have some greater-than constraints of the form: ai1 x1 + … + ain xn >= bi. We convert each such constraint to a less-than constraint by multiplying it by -1. We use the values -ai1, …, -ain in A and -bi in b.

## linear minimization with bounds on decision variables

How to solve a linear program where the objective function is minimized and with extra constraints on the decision variables of the form Li <= xi <= Ui.

# Polynomials

## 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 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 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:

(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.

matlab:

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 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

(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}

## 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.

r:

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.

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 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

(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 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, …, 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 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

(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.

## empirical density function

How to construct a density function from a sample.

# 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.

# 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().

## 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)


r:

## independent two-sample t-test

Test whether two normal variables have same mean.

r:

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.

## 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.

# 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.

# 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:

(12)
\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.

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")


# Scatter Plots

## 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 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.

## 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.

How to add another line to a plot.

## function plot

How to plot a function.

# Chart Options

## 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")


## 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.

## 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.

r:

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

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

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 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/