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Unit 7: Multivariate AnalysisStatistics for Linguists with R – A SIGIL Course
Designed by Stefan Evert1 and Marco Baroni2
1Computational Corpus Linguistics GroupFriedrich-Alexander-Universität Erlangen-Nürnberg, Germany
2Center for Mind/Brain Sciences (CIMeC)University of Trento, Italy
SIGIL (Evert & Baroni) 7. Multivariate Analysis sigil.r-forge.r-project.org 1 / 29
Outline
Outline
IntroductionMultivariate analysisSetting up
Mathematical backgroundFeature matrixDistance metricOrthogonal projection
SIGIL (Evert & Baroni) 7. Multivariate Analysis sigil.r-forge.r-project.org 2 / 29
Introduction Multivariate analysis
Outline
IntroductionMultivariate analysisSetting up
Mathematical backgroundFeature matrixDistance metricOrthogonal projection
SIGIL (Evert & Baroni) 7. Multivariate Analysis sigil.r-forge.r-project.org 3 / 29
Introduction Multivariate analysis
What is multivariate analysis?
I Univariate statisticsI focus on a single variable of interest (at a time)I estimate population parameters (π, µ, σ2, . . . )I comparison of two or more groups
I Bivariate statisticsI focus on interdependencies of two variablesI correlation & co-occurrence
I Regression modellingI predict single target variable (“dependent”)I based on multiple other variables (“independent”)
I Multivariate statisticsI combined effects of many variablesI correlations & distribution patternsI often “unsupervised”: no target variable or comparison groups
SIGIL (Evert & Baroni) 7. Multivariate Analysis sigil.r-forge.r-project.org 4 / 29
Introduction Multivariate analysis
What is multivariate analysis?
I Univariate statisticsI focus on a single variable of interest (at a time)I estimate population parameters (π, µ, σ2, . . . )I comparison of two or more groups
I Bivariate statisticsI focus on interdependencies of two variablesI correlation & co-occurrence
I Regression modellingI predict single target variable (“dependent”)I based on multiple other variables (“independent”)
I Multivariate statisticsI combined effects of many variablesI correlations & distribution patternsI often “unsupervised”: no target variable or comparison groups
SIGIL (Evert & Baroni) 7. Multivariate Analysis sigil.r-forge.r-project.org 4 / 29
Introduction Multivariate analysis
What is multivariate analysis?
I Univariate statisticsI focus on a single variable of interest (at a time)I estimate population parameters (π, µ, σ2, . . . )I comparison of two or more groups
I Bivariate statisticsI focus on interdependencies of two variablesI correlation & co-occurrence
I Regression modellingI predict single target variable (“dependent”)I based on multiple other variables (“independent”)
I Multivariate statisticsI combined effects of many variablesI correlations & distribution patternsI often “unsupervised”: no target variable or comparison groups
SIGIL (Evert & Baroni) 7. Multivariate Analysis sigil.r-forge.r-project.org 4 / 29
Introduction Multivariate analysis
What is multivariate analysis?
I Univariate statisticsI focus on a single variable of interest (at a time)I estimate population parameters (π, µ, σ2, . . . )I comparison of two or more groups
I Bivariate statisticsI focus on interdependencies of two variablesI correlation & co-occurrence
I Regression modellingI predict single target variable (“dependent”)I based on multiple other variables (“independent”)
I Multivariate statisticsI combined effects of many variablesI correlations & distribution patternsI often “unsupervised”: no target variable or comparison groups
SIGIL (Evert & Baroni) 7. Multivariate Analysis sigil.r-forge.r-project.org 4 / 29
Introduction Multivariate analysis
Application examples
I Register variation (Biber 1988, 1993)I Translation studies
(Evert & Neumann 2017; De Sutter et al. 2012)I Stylometry: authorshop attribution (Evert et al. 2017)I Dialectology (Speelman et al. 2003)I Historical linguistics (Sagi et al. 2009; Perek 2018)I Identification of confounding variables (Tummers et al. 2014)I Linguistic productivity (Jenset & McGillivray 2012)I Correspondence analysis (Greenacre 2007)I Distributional semantics (see ESSLLI course)
SIGIL (Evert & Baroni) 7. Multivariate Analysis sigil.r-forge.r-project.org 5 / 29
Introduction Setting up
Outline
IntroductionMultivariate analysisSetting up
Mathematical backgroundFeature matrixDistance metricOrthogonal projection
SIGIL (Evert & Baroni) 7. Multivariate Analysis sigil.r-forge.r-project.org 6 / 29
Introduction Setting up
R packages
Required R packages:I corpora (≥ 0.5)I wordspace (≥ 0.2)
Recommended packages:I ggplot2, reshape2 . . . for plotting feature weightsI rgl . . . for interactive 3-d visualizationI Hotelling, ellipse . . . for significance testingI e1071 . . . for machine learning (SVM)I Rtsne . . . for low-dimensional mapsI ca . . . for correspondence analysis
+ install with package manager in RStudio or R GUI
SIGIL (Evert & Baroni) 7. Multivariate Analysis sigil.r-forge.r-project.org 7 / 29
Introduction Setting up
Code & data sets
Download additional code & data sets from SIGIL homepage:I multivar_utils.RI unit7_data.rda
+ put all files in RStudio project directory (or working directory)
> library(corpora) # basic utilities and some data sets> library(wordspace) # for large and sparse matrices
> source("multivar_utils.R") # additional functions
> load("unit7_data.rda", verbose=TRUE) # further data sets
SIGIL (Evert & Baroni) 7. Multivariate Analysis sigil.r-forge.r-project.org 8 / 29
Introduction Setting up
Overview of data sets
I 65 Biber features for British National CorpusI BNCbiber = 4048× 65 feature matrixI BNCmeta = complete metadata tableI extensive documentation with ?BNCbiber, ?BNCmeta
I 67 Biber features for Brown Family corporaI BrownBiber_Matrix = 3500x67 feature matrixI BrownBiber_Meta = metadata tableI features are Biber-scaled z-scores obtained with MAT v1.3
http://sites.google.com/site/multidimensionaltagger/I see tagger manual for feature definitions
SIGIL (Evert & Baroni) 7. Multivariate Analysis sigil.r-forge.r-project.org 9 / 29
Introduction Setting up
Overview of data sets
I 27 SFL-inspired features for translation pairs (CroCo corpus)I CroCo_Matrix = 452× 27 feature matrixI CroCo_Meta = metadata tableI CroCo_orig2trans = row numbers of translation pairsI data from Evert & Neumann (2017)
I Literary authorship attribution with ∆ measuresI data: sparse document-term matrices for 20,000 most frequent
words (mfw) as wordspace DSM objectsI Delta$DE = 75× 20000 matrix (German novels, 25 authors)I Delta$EN = 75× 20000 matrix (English novels, 25 authors)I Delta$FR = 75× 20000 matrix (French novels, 25 authors)I Delta$DE$rows, Delta$EN$rows, . . . = metadata tablesI DeltaLemma = lemmatized versionI data from Jannidis et al. (2015); Evert et al. (2017)
SIGIL (Evert & Baroni) 7. Multivariate Analysis sigil.r-forge.r-project.org 10 / 29
Introduction Setting up
Overview of data sets
I 19 type-token complexity measures for ∆ corpusI complexity scores for 10,000-token text slices from 75 novelsI DeltaComplexity$DE$Matrix = 996× 19 matrix (German)I DeltaComplexity$EN$Matrix = 1147× 19 matrix (English)I DeltaComplexity$FR$Matrix = 679× 19 matrix (French)I DeltaComplexity$DE$Meta, . . . = metadata tablesI can be used to study correlational patterns between measures
I 7 syntactic complexity measures for 969 German novelsI SyntacticComplexity_Matrix = 969× 7 feature matrixI SyntacticComplexity_Meta = metadata tablesI can be used to compare high-brow against low-brow literature
SIGIL (Evert & Baroni) 7. Multivariate Analysis sigil.r-forge.r-project.org 11 / 29
Mathematical background Feature matrix
Outline
IntroductionMultivariate analysisSetting up
Mathematical backgroundFeature matrixDistance metricOrthogonal projection
SIGIL (Evert & Baroni) 7. Multivariate Analysis sigil.r-forge.r-project.org 12 / 29
Mathematical background Feature matrix
Feature matrix
Feature matrix records quantitative features for each text
M =
· · · m1 · · ·· · · m2 · · ·
...
...· · · mk · · ·
nomina
l
pass
prep
subord
ttr
orig1 1.205 5.013 6.883 4.483 1.285orig2 0.738 2.537 6.486 6.157 1.714orig3 1.252 4.462 8.463 4.785 2.476orig4 1.105 2.899 8.119 3.966 1.519orig5 1.764 4.268 7.167 3.947 1.792orig8 1.545 7.268 7.461 5.455 1.572
trans1 0.463 2.208 6.297 6.089 2.339trans2 1.131 2.597 6.307 4.844 1.810trans4 0.935 1.744 7.098 4.012 1.403trans5 0.867 3.604 7.511 5.154 1.902trans7 1.387 4.290 8.211 3.998 1.822
> M <- MultiVar_Matrix> MSIGIL (Evert & Baroni) 7. Multivariate Analysis sigil.r-forge.r-project.org 13 / 29
Mathematical background Distance metric
Outline
IntroductionMultivariate analysisSetting up
Mathematical backgroundFeature matrixDistance metricOrthogonal projection
SIGIL (Evert & Baroni) 7. Multivariate Analysis sigil.r-forge.r-project.org 14 / 29
Mathematical background Distance metric
Geometric distance = metric
I Distance between vectorsu, v ∈ Rn Ü (dis)similarity
I u = (u1, . . . , un)I v = (v1, . . . , vn)
I Euclidean distance d2 (u, v)
I “City block” Manhattandistance d1 (u, v)
I Both are special cases of theMinkowski p-distance dp (u, v)(for p ∈ [1,∞])
x1
v
x2
1 2 3 4 5
1
2
3
4
5
6
6 u
d2 (!u,!v) = 3.6
d1 (!u,!v) = 5
SIGIL (Evert & Baroni) 7. Multivariate Analysis sigil.r-forge.r-project.org 15 / 29
Mathematical background Distance metric
Geometric distance = metric
I Distance between vectorsu, v ∈ Rn Ü (dis)similarity
I u = (u1, . . . , un)I v = (v1, . . . , vn)
I Euclidean distance d2 (u, v)
I “City block” Manhattandistance d1 (u, v)
I Both are special cases of theMinkowski p-distance dp (u, v)(for p ∈ [1,∞])
x1
v
x2
1 2 3 4 5
1
2
3
4
5
6
6 u
d2 (!u,!v) = 3.6
d1 (!u,!v) = 5
d2 (u, v) :=√
(u1 − v1)2 + · · ·+ (un − vn)2
SIGIL (Evert & Baroni) 7. Multivariate Analysis sigil.r-forge.r-project.org 15 / 29
Mathematical background Distance metric
Geometric distance = metric
I Distance between vectorsu, v ∈ Rn Ü (dis)similarity
I u = (u1, . . . , un)I v = (v1, . . . , vn)
I Euclidean distance d2 (u, v)
I “City block” Manhattandistance d1 (u, v)
I Both are special cases of theMinkowski p-distance dp (u, v)(for p ∈ [1,∞])
x1
v
x2
1 2 3 4 5
1
2
3
4
5
6
6 u
d2 (!u,!v) = 3.6
d1 (!u,!v) = 5
d1 (u, v) := |u1 − v1|+ · · ·+ |un − vn|
SIGIL (Evert & Baroni) 7. Multivariate Analysis sigil.r-forge.r-project.org 15 / 29
Mathematical background Distance metric
Geometric distance = metric
I Distance between vectorsu, v ∈ Rn Ü (dis)similarity
I u = (u1, . . . , un)I v = (v1, . . . , vn)
I Euclidean distance d2 (u, v)
I “City block” Manhattandistance d1 (u, v)
I Both are special cases of theMinkowski p-distance dp (u, v)(for p ∈ [1,∞])
x1
v
x2
1 2 3 4 5
1
2
3
4
5
6
6 u
d2 (!u,!v) = 3.6
d1 (!u,!v) = 5
dp (u, v) :=(|u1 − v1|p + · · ·+ |un − vn|p
)1/p
SIGIL (Evert & Baroni) 7. Multivariate Analysis sigil.r-forge.r-project.org 15 / 29
Mathematical background Distance metric
Geometric distance = metric
I Distance between vectorsu, v ∈ Rn Ü (dis)similarity
I u = (u1, . . . , un)I v = (v1, . . . , vn)
I Euclidean distance d2 (u, v)
I “City block” Manhattandistance d1 (u, v)
I Both are special cases of theMinkowski p-distance dp (u, v)(for p ∈ [1,∞])
x1
v
x2
1 2 3 4 5
1
2
3
4
5
6
6 u
d2 (!u,!v) = 3.6
d1 (!u,!v) = 5
dp (u, v) :=(|u1 − v1|p + · · ·+ |un − vn|p
)1/pd∞ (u, v) = max
{|u1 − v1|, . . . , |un − vn|
}SIGIL (Evert & Baroni) 7. Multivariate Analysis sigil.r-forge.r-project.org 15 / 29
Mathematical background Distance metric
Geometric distance = metric
I Distance between vectorsu, v ∈ Rn Ü (dis)similarity
I u = (u1, . . . , un)I v = (v1, . . . , vn)
I Euclidean distance d2 (u, v)
I “City block” Manhattandistance d1 (u, v)
I Extension of p-distance dp (u, v)(for 0 ≤ p ≤ 1)
x1
v
x2
1 2 3 4 5
1
2
3
4
5
6
6 u
d2 (!u,!v) = 3.6
d1 (!u,!v) = 5
dp (u, v) := |u1 − v1|p + · · ·+ |un − vn|p
d0 (u, v) = #{i∣∣ ui 6= vi
}SIGIL (Evert & Baroni) 7. Multivariate Analysis sigil.r-forge.r-project.org 16 / 29
Mathematical background Distance metric
Distance and vector length = norm
I Intuitively, distanced (u, v) should correspondto length ‖u− v‖ ofdisplacement vector u− v
I d (u, v) is a metricI ‖u− v‖ is a normI ‖u‖ = d
(u, 0)
I Any norm-induced metricis translation-invariant
I dp (u, v) = ‖u− v‖p
x1
origin
v
x2
1 2 3 4 5
1
2
3
4
5
6
6 u!!u! = d!!u,!0
"
d (!u,!v) = !!u " !v!
!!v! = d!!v,!0
"
I Minkowski p-norm for p ∈ [1,∞] (not p < 1):
‖u‖p :=(|u1|p + · · ·+ |un|p
)1/p
SIGIL (Evert & Baroni) 7. Multivariate Analysis sigil.r-forge.r-project.org 17 / 29
Mathematical background Distance metric
Distance and vector length = norm
I Intuitively, distanced (u, v) should correspondto length ‖u− v‖ ofdisplacement vector u− v
I d (u, v) is a metricI ‖u− v‖ is a normI ‖u‖ = d
(u, 0)
I Any norm-induced metricis translation-invariant
I dp (u, v) = ‖u− v‖p
x1
origin
v
x2
1 2 3 4 5
1
2
3
4
5
6
6 u!!u! = d!!u,!0
"
d (!u,!v) = !!u " !v!
!!v! = d!!v,!0
"
I Minkowski p-norm for p ∈ [1,∞] (not p < 1):
‖u‖p :=(|u1|p + · · ·+ |un|p
)1/p
SIGIL (Evert & Baroni) 7. Multivariate Analysis sigil.r-forge.r-project.org 17 / 29
Mathematical background Distance metric
Distance and vector length = norm
I Intuitively, distanced (u, v) should correspondto length ‖u− v‖ ofdisplacement vector u− v
I d (u, v) is a metricI ‖u− v‖ is a normI ‖u‖ = d
(u, 0)
I Any norm-induced metricis translation-invariant
I dp (u, v) = ‖u− v‖p
x1
origin
v
x2
1 2 3 4 5
1
2
3
4
5
6
6 u!!u! = d!!u,!0
"
d (!u,!v) = !!u " !v!
!!v! = d!!v,!0
"
I Minkowski p-norm for p ∈ [1,∞] (not p < 1):
‖u‖p :=(|u1|p + · · ·+ |un|p
)1/pSIGIL (Evert & Baroni) 7. Multivariate Analysis sigil.r-forge.r-project.org 17 / 29
Mathematical background Distance metric
Computing distances
Compute distances between all pairs of texts:
> round(dist(M), 2) # returns a triangular ‘dist’ object> round(dist(M, method="manhattan"), 2) # Manhattan metric
Use wordspace function for additional metrics:
> dist.matrix(M, method="mink", p=0.5) # full matrix> dist.matrix(M, method="mink", p=0.5, as.dist=TRUE)
Standardize features for equal contribution to Euclidean metric:
> Z <- scale(M) # matrix of z-scores> round(dist(Z), 2) # default: Euclidean metric
SIGIL (Evert & Baroni) 7. Multivariate Analysis sigil.r-forge.r-project.org 18 / 29
Mathematical background Distance metric
Computing distances
Compute distances between all pairs of texts:
> round(dist(M), 2) # returns a triangular ‘dist’ object> round(dist(M, method="manhattan"), 2) # Manhattan metric
Use wordspace function for additional metrics:
> dist.matrix(M, method="mink", p=0.5) # full matrix> dist.matrix(M, method="mink", p=0.5, as.dist=TRUE)
Standardize features for equal contribution to Euclidean metric:
> Z <- scale(M) # matrix of z-scores> round(dist(Z), 2) # default: Euclidean metric
SIGIL (Evert & Baroni) 7. Multivariate Analysis sigil.r-forge.r-project.org 18 / 29
Mathematical background Distance metric
Computing distances
Compute distances between all pairs of texts:
> round(dist(M), 2) # returns a triangular ‘dist’ object> round(dist(M, method="manhattan"), 2) # Manhattan metric
Use wordspace function for additional metrics:
> dist.matrix(M, method="mink", p=0.5) # full matrix> dist.matrix(M, method="mink", p=0.5, as.dist=TRUE)
Standardize features for equal contribution to Euclidean metric:
> Z <- scale(M) # matrix of z-scores> round(dist(Z), 2) # default: Euclidean metric
SIGIL (Evert & Baroni) 7. Multivariate Analysis sigil.r-forge.r-project.org 18 / 29
Mathematical background Orthogonal projection
Outline
IntroductionMultivariate analysisSetting up
Mathematical backgroundFeature matrixDistance metricOrthogonal projection
SIGIL (Evert & Baroni) 7. Multivariate Analysis sigil.r-forge.r-project.org 19 / 29
Mathematical background Orthogonal projection
Linear subspace & basis
I A linear subspace B ⊆ Rn of rank r ≤ n is spanned by a setof r linearly independent basis vectors
B = {b1, . . . ,br}
I Every point u in the subspaceis a unique linear combinationof the basis vectors
u = x1b1 + . . .+ xrbr
I Coordinate vector x ∈ Rr
with respect to the basis
1 2 3 4 5
1
2
3
4
5
6
6 u = (4,5)
b(2)
b(1)
x = (3,2)
SIGIL (Evert & Baroni) 7. Multivariate Analysis sigil.r-forge.r-project.org 20 / 29
Mathematical background Orthogonal projection
Linear subspace & basis
I A linear subspace B ⊆ Rn of rank r ≤ n is spanned by a setof r linearly independent basis vectors
B = {b1, . . . ,br}
I Every point u in the subspaceis a unique linear combinationof the basis vectors
u = x1b1 + . . .+ xrbr
I Coordinate vector x ∈ Rr
with respect to the basis1 2 3 4 5
1
2
3
4
5
6
6 u = (4,5)
b(2)
b(1)
x = (3,2)
SIGIL (Evert & Baroni) 7. Multivariate Analysis sigil.r-forge.r-project.org 20 / 29
Mathematical background Orthogonal projection
Linear subspace & basis
I Basis matrix V ∈ Rn×r with column vectors bi :
u = x1b1 + . . .+ xrbr = Vx
x1b11 + . . .+ xrb1rx1b21 + . . .+ xrb2r
...x1bn1 + . . .+ xrbnr
=
b11 · · · b1rb21 · · · b2r...
...bn1 · · · bnr
·
x1...xr
u = V · x
(n × 1) (n × r) (r × 1)
SIGIL (Evert & Baroni) 7. Multivariate Analysis sigil.r-forge.r-project.org 21 / 29
Mathematical background Orthogonal projection
An aside: Matrix multiplication
aij =
bi1 · · · bin ·
c1j......cnj
A = B · C
(k ×m) (k × n) (n ×m)
I B and C must be conformable (in dimension n)I Element aij is the inner product of the i-th row of B and the
j-th column of C
aij = bi1c1j + . . .+ bincnj =n∑
t=1
bi tctj
SIGIL (Evert & Baroni) 7. Multivariate Analysis sigil.r-forge.r-project.org 22 / 29
Mathematical background Orthogonal projection
An aside: Matrix multiplication
aij =
bi1 · · · bin ·
c1j......cnj
A = B · C
(k ×m) (k × n) (n ×m)
I B and C must be conformable (in dimension n)I Element aij is the inner product of the i-th row of B and the
j-th column of C
aij = bi1c1j + . . .+ bincnj =n∑
t=1
bi tctj
SIGIL (Evert & Baroni) 7. Multivariate Analysis sigil.r-forge.r-project.org 22 / 29
Mathematical background Orthogonal projection
An aside: Matrix multiplication
aij =
bi1 · · · bin
·
c1j......cnj
A = B · C
(k ×m) (k × n) (n ×m)
I B and C must be conformable (in dimension n)I Element aij is the inner product of the i-th row of B and the
j-th column of C
aij = bi1c1j + . . .+ bincnj =n∑
t=1
bi tctj
SIGIL (Evert & Baroni) 7. Multivariate Analysis sigil.r-forge.r-project.org 22 / 29
Mathematical background Orthogonal projection
Orthonormal basis
I Particularly convenient with orthonormal basis:
‖bi‖2 = 1
bTi bj = 0 for i 6= j
I Corresponding basis matrix V is (column)-orthogonal
VTV = Ir
and defines a Cartesian coordinate system in the subspace
+ From now on always assume orthonormal basis
SIGIL (Evert & Baroni) 7. Multivariate Analysis sigil.r-forge.r-project.org 23 / 29
Mathematical background Orthogonal projection
Orthonormal basis
I Particularly convenient with orthonormal basis:
‖bi‖2 = 1
bTi bj = 0 for i 6= j
I Corresponding basis matrix V is (column)-orthogonal
VTV = Ir
and defines a Cartesian coordinate system in the subspace
+ From now on always assume orthonormal basis
SIGIL (Evert & Baroni) 7. Multivariate Analysis sigil.r-forge.r-project.org 23 / 29
Mathematical background Orthogonal projection
The mathematics of projections
I 1-d subspace spanned bybasis vector ‖b‖2 = 1
I For any point u, we have
cosϕ =bTu
‖b‖2 · ‖u‖2=
bTu‖u‖2
I Trigonometry: coordinateof point on the line isx = ‖u‖2 · cosϕ = bTu
.
u
kbk2 = 1
'
u0 =u
kuk2
Pu = b(bTu)
x
I The projected point in original space is then given by
b · x = b(bTu) = (bbT )u = Pu
where P is a projection matrix of rank 1
SIGIL (Evert & Baroni) 7. Multivariate Analysis sigil.r-forge.r-project.org 24 / 29
Mathematical background Orthogonal projection
The mathematics of projections
I 1-d subspace spanned bybasis vector ‖b‖2 = 1
I For any point u, we have
cosϕ =bTu
‖b‖2 · ‖u‖2=
bTu‖u‖2
I Trigonometry: coordinateof point on the line isx = ‖u‖2 · cosϕ = bTu
.
u
kbk2 = 1
'
u0 =u
kuk2
Pu = b(bTu)
x
I The projected point in original space is then given by
b · x = b(bTu) = (bbT )u = Pu
where P is a projection matrix of rank 1
SIGIL (Evert & Baroni) 7. Multivariate Analysis sigil.r-forge.r-project.org 24 / 29
Mathematical background Orthogonal projection
The mathematics of projections
I 1-d subspace spanned bybasis vector ‖b‖2 = 1
I For any point u, we have
cosϕ =bTu
‖b‖2 · ‖u‖2=
bTu‖u‖2
I Trigonometry: coordinateof point on the line isx = ‖u‖2 · cosϕ = bTu
.
u
kbk2 = 1
'
u0 =u
kuk2
Pu = b(bTu)
x
I The projected point in original space is then given by
b · x = b(bTu) = (bbT )u = Pu
where P is a projection matrix of rank 1
SIGIL (Evert & Baroni) 7. Multivariate Analysis sigil.r-forge.r-project.org 24 / 29
Mathematical background Orthogonal projection
The mathematics of projections
I For an orthogonal basis matrix V with columns b1, . . . ,br , theprojection into the rank-r subspace B is given by
Pu =
(r∑
i=1
bibTi
)u = VVTu
and its subspace coordinates are x = VTu
I Projection can be seen as decomposition into the projectedvector and its orthogonal complement
u = Pu + (u− Pu) = Pu + (I− P)u = Pu + Qu
I Because of orthogonality, this also applies to the squaredEuclidean norm (according to the Pythagorean theorem)
‖u‖2 = ‖Pu‖2 + ‖Qu‖2
SIGIL (Evert & Baroni) 7. Multivariate Analysis sigil.r-forge.r-project.org 25 / 29
Mathematical background Orthogonal projection
The mathematics of projections
I For an orthogonal basis matrix V with columns b1, . . . ,br , theprojection into the rank-r subspace B is given by
Pu =
(r∑
i=1
bibTi
)u = VVTu
and its subspace coordinates are x = VTuI Projection can be seen as decomposition into the projected
vector and its orthogonal complement
u = Pu + (u− Pu) = Pu + (I− P)u = Pu + Qu
I Because of orthogonality, this also applies to the squaredEuclidean norm (according to the Pythagorean theorem)
‖u‖2 = ‖Pu‖2 + ‖Qu‖2
SIGIL (Evert & Baroni) 7. Multivariate Analysis sigil.r-forge.r-project.org 25 / 29
Mathematical background Orthogonal projection
The mathematics of projections
I For an orthogonal basis matrix V with columns b1, . . . ,br , theprojection into the rank-r subspace B is given by
Pu =
(r∑
i=1
bibTi
)u = VVTu
and its subspace coordinates are x = VTuI Projection can be seen as decomposition into the projected
vector and its orthogonal complement
u = Pu + (u− Pu) = Pu + (I− P)u = Pu + Qu
I Because of orthogonality, this also applies to the squaredEuclidean norm (according to the Pythagorean theorem)
‖u‖2 = ‖Pu‖2 + ‖Qu‖2
SIGIL (Evert & Baroni) 7. Multivariate Analysis sigil.r-forge.r-project.org 25 / 29
Mathematical background Orthogonal projection
Optimal projections and subspaces
I Orthogonal decomposition of squared distances btw. vectors
‖u− v‖2 = ‖Pu− Pv‖2 + ‖Qu−Qv‖2
I Define projection loss asdifference btw. squared distances∣∣ ‖P(u− v)‖2 − ‖u− v‖2
∣∣= ‖u− v‖2 − ‖P(u− v)‖2
= ‖Q(u− v)‖2
I Projection quality measure:
R2 =‖P(u− v)‖2
‖u− v‖2
.
.
.
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Q(u � v)
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SIGIL (Evert & Baroni) 7. Multivariate Analysis sigil.r-forge.r-project.org 26 / 29
Mathematical background Orthogonal projection
Optimal projections and subspaces
I Orthogonal decomposition of squared distances btw. vectors
‖u− v‖2 = ‖Pu− Pv‖2 + ‖Qu−Qv‖2
I Define projection loss asdifference btw. squared distances∣∣ ‖P(u− v)‖2 − ‖u− v‖2
∣∣= ‖u− v‖2 − ‖P(u− v)‖2
= ‖Q(u− v)‖2
I Projection quality measure:
R2 =‖P(u− v)‖2
‖u− v‖2
.
.
.
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Q(u � v)
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SIGIL (Evert & Baroni) 7. Multivariate Analysis sigil.r-forge.r-project.org 26 / 29
Mathematical background Orthogonal projection
Optimal projections and subspaces
I Orthogonal decomposition of squared distances btw. vectors
‖u− v‖2 = ‖Pu− Pv‖2 + ‖Qu−Qv‖2
I Define projection loss asdifference btw. squared distances∣∣ ‖P(u− v)‖2 − ‖u− v‖2
∣∣= ‖u− v‖2 − ‖P(u− v)‖2
= ‖Q(u− v)‖2
I Projection quality measure:
R2 =‖P(u− v)‖2
‖u− v‖2
.
.
.
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Q(u � v)
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SIGIL (Evert & Baroni) 7. Multivariate Analysis sigil.r-forge.r-project.org 26 / 29
Mathematical background Orthogonal projection
References I
Biber, Douglas (1988). Variation Across Speech and Writing. Cambridge UniversityPress, Cambridge.
Biber, Douglas (1993). The multi-dimensional approach to linguistic analyses of genrevariation: An overview of methodology and findings. Computers and theHumanities, 26, 331–345.
De Sutter, Gert; Delaere, Isabelle; Plevoets, Koen (2012). Lexical lectometry incorpus-based translation studies: combining profile-based correspondence analysisand logistic regression modeling. In M. P. Oakes and J. Meng (eds.), Quantitativemethods in corpus-based translation studies: a practical guide to descriptivetranslation research, volume 51 of Studies in Corpus Linguistics, pages 325–345.John Benjamins.
Evert, Stefan and Neumann, Stella (2017). The impact of translation direction oncharacteristics of translated texts. A multivariate analysis for English and German.In G. De Sutter, M.-A. Lefer, and I. Delaere (eds.), Empirical Translation Studies.New Theoretical and Methodological Traditions, number 300 in Trends inLinguistics. Studies and Monographs (TiLSM), pages 47–80. Mouton de Gruyter,Berlin.
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Mathematical background Orthogonal projection
References II
Evert, Stefan; Proisl, Thomas; Jannidis, Fotis; Reger, Isabella; Pielström, Steffen;Schöch, Christof; Vitt, Thorsten (2017). Understanding and explaining Deltameasures for authorship attribution. Digital Scholarship in the Humanities,22(suppl_2), ii4–ii16.
Greenacre, Michael (2007). Correspondence Analysis in Practice. InterdisciplinaryStatistics Series. Chapman & Hall, CRC, 2nd edition.
Jannidis, Fotis; Pielström, Steffen; Schöch, Christof; Vitt, Thorsten (2015). ImprovingBurrows’ Delta. An empirical evaluation of text distance measures. In Proceedingsof the Digital Humanities Conference 2015, Sydney, Australia.
Jenset, Gard B. and McGillivray, Barbara (2012). Multivariate analyses of affixproductivity in translated english. In M. P. Oakes and J. Meng (eds.), Quantitativemethods in corpus-based translation studies: a practical guide to descriptivetranslation research, volume 51 of Studies in Corpus Linguistics, pages 301–324.John Benjamins.
Perek, Florent (2018). Recent change in the productivity and schematicity of theway-construction: A distributional semantic analysis. Corpus Linguistics andLinguistic Theory, 14(1), 65–97.
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Mathematical background Orthogonal projection
References III
Sagi, Eyal; Kaufmann, Stefan; Clark, Brady (2009). Semantic density analysis:Comparing word meaning across time and phonetic space. In Proceedings of theWorkshop on Geometrical Models of Natural Language Semantics (GEMS), pages104–111, Athens, Greece.
Speelman, Dirk; Grondelaers, Stefan; Geeraerts, Dirk (2003). Profile-based linguisticuniformity as a generic method for comparing language varieties. Computers andthe Humanities, 37, 317–337.
Tummers, José; Speelman, Dirk; Geeraerts, Dirk (2014). Spurious effects invariational corpus linguistics: Identification and implications of confounding.International Journal of Corpus Linguistics, 19(4), 478–504.
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