GibberLink is an acoustic data transmission project, with an open-source client available on GitHub, in which two conversational AI agents switch from speaking to one another in a Human-listenable language (such as English) to their own unique language that consists of a sound-level protocol after confirming they are both AI agents. The project was created by Anton Pidkuiko and Boris Starkov. == Reception == The project won the global top prize at the ElevenLabs Worldwide Hackathon. It has also been cited as raising questions around AI ethics and oversight. On February 23, 2025, a YouTube video of two independent conversational ElevenLabs AI agents being prompted to chat about booking a hotel (one as a caller, one as a receptionist) received coverage for going viral. In this video, both agents are prompted to switch to ggwave data-over-sound protocol when they identify the other side as AI, and keep speaking in English otherwise.
ZipBooks
ZipBooks is a free online accounting software company based in American Fork, Utah. The cloud-based software is an accounting and bookkeeping tool that helps business owners process credit cards, track finances, and send invoices, among other features. == History == ZipBooks was founded by Tim Chaves in June 2015, backed by venture capital firm Peak Ventures. The company secured an additional $2 million of funding in July 2016, and in 2017 it was awarded a $100,000 economic grant by the Utah Governor's Office of Economic Development Technology Commercialization and Innovation Program. == Products == ZipBooks' core modules are invoicing, transactions, bills, reporting, time tracking, contacts, and payroll. Accrual accounting was added in 2017. The application is available on G Suite, iOS, Slack, and as a web application. == Reception == Computerworld compared ZipBooks favorably with other accounting software. PC Magazine praised its user experience, but stated it lacked "a lot of features that competing sites offer".
Correlation clustering
Clustering is the problem of partitioning data points into groups based on similarity or dissimilarity. Correlation clustering is a clustering framework in which a set of objects is partitioned into clusters based on pairwise similarity and dissimilarity information, without requiring the number of clusters to be specified in advance. == Description of the problem == In machine learning, correlation clustering (also known as cluster editing) considers settings in which pairwise similarity or dissimilarity relationships between objects are known. A standard formulation models the input as an unweighted complete graph G = ( V , E ) {\displaystyle G=(V,E)} , where each edge is labeled either + {\displaystyle +} or − {\displaystyle -} (that is, the graph is a signed graph), indicating whether the corresponding endpoints are similar or dissimilar. The goal is to find a clustering (that is, a partition of V {\displaystyle V} ) that either maximizes the number of agreements—the sum of positive edges whose endpoints lie in the same cluster and negative edges whose endpoints lie in different clusters—or minimizes the number of disagreements—the sum of positive edges whose endpoints are separated and negative edges whose endpoints lie in the same cluster. Unlike other clustering methods such as k-means, correlation clustering does not require choosing the number of clusters k {\displaystyle k} in advance. It is not always possible to find a clustering with zero disagreements. For example, consider a triangle graph containing two positive edges and one negative edge. In this case, every clustering incurs at least one disagreement. Such configurations are referred to in the literature as bad triangles. From a computational perspective, optimizing the correlation clustering objective is challenging. The (decision version of the) problem is NP-complete. A large body of subsequent work has developed approximation algorithms for correlation clustering under various assumptions, including complete or general graphs and unweighted or weighted graphs, for both minimization and maximization objectives. This problem is considered one of the fundamental combinatorial optimization problems, and many algorithmic techniques have been developed to address it. The problem has also been studied extensively across multiple disciplines. A comprehensive literature review of early correlation clustering research is provided by Wahid and Hassini. == Formal Definitions == Let G = ( V , E ) {\displaystyle G=(V,E)} be a graph with nodes V {\displaystyle V} and edges E {\displaystyle E} . A clustering of G {\displaystyle G} is a partition of its node set Π = { π 1 , … , π k } {\displaystyle \Pi =\{\pi _{1},\dots ,\pi _{k}\}} with V = π 1 ∪ ⋯ ∪ π k {\displaystyle V=\pi _{1}\cup \dots \cup \pi _{k}} and π i ∩ π j = ∅ {\displaystyle \pi _{i}\cap \pi _{j}=\emptyset } for i ≠ j {\displaystyle i\neq j} . For a given clustering Π {\displaystyle \Pi } , let δ ( Π ) = { { u , v } ∈ E ∣ { u , v } ⊈ π ∀ π ∈ Π } {\displaystyle \delta (\Pi )=\{\{u,v\}\in E\mid \{u,v\}\not \subseteq \pi \;\forall \pi \in \Pi \}} denote the subset of edges of G {\displaystyle G} whose endpoints are in different subsets of the clustering Π {\displaystyle \Pi } . Now, let w : E → R ≥ 0 {\displaystyle w\colon E\to \mathbb {R} _{\geq 0}} be a function that assigns a non-negative weight to each edge of the graph and let E = E + ∪ E − {\displaystyle E=E^{+}\cup E^{-}} be a partition of the edges into attractive ( E + {\displaystyle E^{+}} ) and repulsive ( E − {\displaystyle E^{-}} ) edges; that is, the edges are signed. The minimum disagreement correlation clustering problem is the following optimization problem: minimize Π ∑ e ∈ E + ∩ δ ( Π ) w e + ∑ e ∈ E − ∖ δ ( Π ) w e . {\displaystyle {\begin{aligned}&{\underset {\Pi }{\operatorname {minimize} }}&&\sum _{e\in E^{+}\cap \delta (\Pi )}w_{e}+\sum _{e\in E^{-}\setminus \delta (\Pi )}w_{e}\;.\end{aligned}}} Here, the set E + ∩ δ ( Π ) {\displaystyle E^{+}\cap \delta (\Pi )} contains the attractive edges whose endpoints are in different components with respect to the clustering Π {\displaystyle \Pi } and the set E − ∖ δ ( Π ) {\displaystyle E^{-}\setminus \delta (\Pi )} contains the repulsive edges whose endpoints are in the same component with respect to the clustering Π {\displaystyle \Pi } . Together these two sets contain all edges that disagree with the clustering Π {\displaystyle \Pi } . Similarly to the minimum disagreement correlation clustering problem, the maximum agreement correlation clustering problem is defined as maximize Π ∑ e ∈ E + ∖ δ ( Π ) w e + ∑ e ∈ E − ∩ δ ( Π ) w e . {\displaystyle {\begin{aligned}&{\underset {\Pi }{\operatorname {maximize} }}&&\sum _{e\in E^{+}\setminus \delta (\Pi )}w_{e}+\sum _{e\in E^{-}\cap \delta (\Pi )}w_{e}\;.\end{aligned}}} Here, the set E + ∖ δ ( Π ) {\displaystyle E^{+}\setminus \delta (\Pi )} contains the attractive edges whose endpoints are in the same component with respect to the clustering Π {\displaystyle \Pi } and the set E − ∩ δ ( Π ) {\displaystyle E^{-}\cap \delta (\Pi )} contains the repulsive edges whose endpoints are in different components with respect to the clustering Π {\displaystyle \Pi } . Together these two sets contain all edges that agree with the clustering Π {\displaystyle \Pi } . Instead of formulating the correlation clustering problem in terms of non-negative edge weights and a partition of the edges into attractive and repulsive edges the problem is also formulated in terms of positive and negative edge costs without partitioning the set of edges explicitly. For given weights w : E → R ≥ 0 {\displaystyle w\colon E\to \mathbb {R} _{\geq 0}} and a given partition E = E + ∪ E − {\displaystyle E=E^{+}\cup E^{-}} of the edges into attractive and repulsive edges, the edge costs can be defined by c e = { w e if e ∈ E + − w e if e ∈ E − {\displaystyle {\begin{aligned}c_{e}={\begin{cases}\;\;w_{e}&{\text{if }}e\in E^{+}\\-w_{e}&{\text{if }}e\in E^{-}\end{cases}}\end{aligned}}} for all e ∈ E {\displaystyle e\in E} . An edge whose endpoints are in different clusters is said to be cut. The set δ ( Π ) {\displaystyle \delta (\Pi )} of all edges that are cut is often called a multicut of G {\displaystyle G} . The minimum cost multicut problem is the problem of finding a clustering Π {\displaystyle \Pi } of G {\displaystyle G} such that the sum of the costs of the edges whose endpoints are in different clusters is minimal: minimize Π ∑ e ∈ δ ( Π ) c e . {\displaystyle {\begin{aligned}&{\underset {\Pi }{\operatorname {minimize} }}&&\sum _{e\in \delta (\Pi )}c_{e}\;.\end{aligned}}} Similar to the minimum cost multicut problem, coalition structure generation in weighted graph games is the problem of finding a clustering such that the sum of the costs of the edges that are not cut is maximal: maximize Π ∑ e ∈ E ∖ δ ( Π ) c e . {\displaystyle {\begin{aligned}&{\underset {\Pi }{\operatorname {maximize} }}&&\sum _{e\in E\setminus \delta (\Pi )}c_{e}\;.\end{aligned}}} This formulation is also known as the clique partitioning problem. It can be shown that all four problems that are formulated above are equivalent. This means that a clustering that is optimal with respect to any of the four objectives is optimal for all of the four objectives. == Algorithms == If the graph admits a clustering with zero disagreements, then deleting all negative edges and computing the connected components of the remaining graph yields an optimal clustering. A necessary and sufficient condition for the existence of such a clustering was given by Davis: no cycle in the graph may contain exactly one negative edge. Bansal et al. discuss the NP-completeness proof and also present both a constant factor approximation algorithm and polynomial-time approximation scheme to find the clusters in this setting. Ailon et al. propose a randomized 3-approximation algorithm for the same problem. CC-Pivot(G=(V,E+,E−)) Pick random pivot i ∈ V Set C = { i } {\displaystyle C=\{i\}} , V'=Ø For all j ∈ V, j ≠ i; If (i,j) ∈ E+ then Add j to C Else (If (i,j) ∈ E−) Add j to V' Let G' be the subgraph induced by V' Return clustering C,CC-Pivot(G') The authors show that the above algorithm is a 3-approximation algorithm for correlation clustering. The best polynomial-time approximation algorithm known at the moment for this problem achieves a ~2.06 approximation by rounding a linear program, as shown by Chawla, Makarychev, Schramm, and Yaroslavtsev. Karpinski and Schudy proved existence of a polynomial time approximation scheme (PTAS) for that problem on complete graphs and fixed number of clusters. == Optimal number of clusters == In 2011, it was shown by Bagon and Galun that the optimization of the correlation clustering functional is closely related to well known discrete optimization methods. In their work they proposed a probabilistic analysis of the underlying implicit model that allows the correlation clustering functional to estimate the
Persian Speech Corpus
The Persian Speech Corpus is a Modern Persian speech corpus for speech synthesis. The corpus contains phonetic and orthographic transcriptions of about 2.5 hours of Persian speech aligned with recorded speech on the phoneme level, including annotations of word boundaries. Previous spoken corpora of Persian include FARSDAT, which consists of read aloud speech from newspaper texts from 100 Persian speakers and the Telephone FARsi Spoken language DATabase (TFARSDAT) which comprises seven hours of read and spontaneous speech produced by 60 native speakers of Persian from ten regions of Iran. The Persian Speech Corpus was built using the same methodologies laid out in the doctoral project on Modern Standard Arabic of Nawar Halabi at the University of Southampton. The work was funded by MicroLinkPC, who own an exclusive license to commercialise the corpus, though the corpus is available for non-commercial use through the corpus' website. It is distributed under the Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License. The corpus was built for speech synthesis purposes, but has been used for building HMM based voices in Persian. It can also be used to automatically align other speech corpora with their phonetic transcript and could be used as part of a larger corpus for training speech recognition systems. == Contents == The corpus is downloadable from its website, and contains the following: 396 .wav files containing spoken utterances 396 .lab files containing text utterances 396 .TextGrid files containing the phoneme labels with time stamps of the boundaries where these occur in the .wav files. phonetic-transcript.txt which has the form "[wav_filename]" "[Phoneme Sequence]" in every line orthographic-transcript.txt which has the form "[wav_filename]" "[Orthographic Transcript]" in every line
Andrej Mrvar
Andrej Mrvar is a Slovenian computer scientist and a professor at the University of Ljubljana's Faculty of Social Sciences. He is known for his work in network analysis, graph drawing, decision making, virtual reality, timing and data processing of sports competitions. == Education and career == He is well known for his work on Pajek, a free software for analysis and visualization of large networks. Mrvar began work on Pajek in 1996 with Vladimir Batagelj. His book Exploratory Social Network Analysis with Pajek, coauthored with Wouter de Nooy and Vladimir Batagelj, is his most cited work. It was published by Cambridge University Press in three editions (first 2005, second 2011, and third 2018). The book was translated into Japanese (2009) and Chinese (first edition 2012, second 2014). With Anuška Ferligoj, he was a founding co-editor-in-chief of the Metodološki zvezki - Advances in Methodology and Statistics journal. == Awards and honors == Vidmar Award (Faculty of Electrical and Computer Engineering, University of Ljubljana): 1988, 1990 First prizes for contributions (with Vladimir Batagelj) to Graph Drawing Contests in years: 1995, 1996, 1997, 1998, 1999, 2000 and 2005 / Graph Drawing Hall of Fame. Award of University of Ljubljana for contributions in education and research (Svečana listina Univerze v Ljubljani za pomembne dosežke na področju vzgojnoizobraževalnega in znanstvenoraziskovalega dela): 2001 The INSNA's William D. Richards Software award for work on Pajek (with Vladimir Batagelj): 2013 Award of Faculty of Social Sciences, University of Ljubljana for scientific excellence (Priznanje za znanstveno odličnost): 2013 == Selected publications == Wouter de Nooy, Andrej Mrvar, Vladimir Batagelj, Mark Granovetter (Series Editor), Exploratory Social Network Analysis with Pajek (Structural Analysis in the Social Sciences), Cambridge University Press (First Edition: 2005, Second Edition: 2011, Third Edition: 2018 ). Japanese Translation (2010). Chinese Translation (First Edition: 2012, Second Edition: 2014) Andrej Mrvar and Vladimir Batagelj, Analysis and visualization of large networks with program package Pajek. Complex Adaptive Systems Modeling, 4:6. SpringerOpen, 2016 Vladimir Batagelj and Andrej Mrvar, Some Analyses of Erdős Collaboration Graph, Social Networks, 22, 173–186, 2000 Vladimir Batagelj and Andrej Mrvar, A Subquadratic Triad Census Algorithm for Large Sparse Networks with Small Maximum Degree. Social Networks, 23, 237–243, 2001 Patrick Doreian and Andrej Mrvar, A Partitioning Approach to Structural Balance, Social Networks, 18, 149–168, 1996 Patrick Doreian and Andrej Mrvar, Partitioning Signed Social Networks, Social Networks, 31, 1–11, 2009 Andrej Mrvar and Patrick Doreian, Partitioning Signed Two-Mode Networks, Journal of Mathematical Sociology, 33, 196–221, 2009 Patrick Doreian and Andrej Mrvar, The international reach of the Koch brothers network. In: Antonyuk, A. and Basov, N. (Eds.): Networks in the Global World V. NetGloW 2020. Lecture Notes in Networks and Systems, 181, 225–235. Springer, 2021 Patrick Doreian and Andrej Mrvar, Delineating Changes in the Fundamental Structure of Signed Networks, Frontiers in Physics, 294, 1–11, 2021 Patrick Doreian and Andrej Mrvar, Hubs and Authorities in the Koch Brothers Network. Social Networks, Social Networks, 64, 148–157, 2021 Patrick Doreian and Andrej Mrvar, Public issues, policy proposals, social movements, and the interests of the Koch Brothers network of allies, Quality and Quantity, 56, 305–322, 2022 Douglas R. White, Vladimir Batagelj, Andrej Mrvar, Analyzing Large Kinship and Marriage Networks with Pgraph and Pajek. Social Science Computer Review, 17, 245–274, 1999 Ion Georgiou, Ronald Concer, Andrej Mrvar, A Systemic Approach to Sociometric Group Research: Advancing The Work of Leslie Day Zeleny, 1939–1947, Social Networks, 63, 174–200, 2020
Vero (app)
Vero (stylized as VERO) is a social media platform and mobile app company. Vero markets itself as a social network free from advertisements, data mining and algorithms. == History == The app was founded by French-Lebanese billionaire Ayman Hariri who is the son of former Lebanese prime minister Rafic Hariri. The name is taken from the Italian word for true. The app launched officially in 2015 as an alternative to Facebook and their popular photo-blogging app Instagram. Within weeks of its release the app surged in popularity although users expressed mixed reports with some feeling confused about how the app worked. Cosplayers were early to adopt the app as their photo-sharing platform of choice, favouring the app's pinch and zoom magnification feature over Instagram's zoom feature. Other creative communities soon followed, and the app became popular with niche groups of makeup artists, tattoo artists, and skateboarders. In March 2018, Vero's popularity surged, partly helped by an exodus from Facebook and Instagram following the Cambridge Analytica data scandal. In the wake of the scandal, Vero devised an advertising campaign aimed at defected Facebook and Instagram users, hoping the app's policies and privacy settings would assuage concerns over sharing personal information on the internet. Within the space of one week, the app went from being a small service, akin to Ello or Peach, to being the most downloaded app in eighteen countries. In December 2020, Vero released its most significant update to date, Vero 2.0 which introduced new features including voice and video calls, game and app posts and bookmarks, and refinements to the UI. In October 2021, Vero introduced their Desktop app (beta) with multiple post options and a re-sizable multi-column feed. == Concept and funding == Vero's content feed resembles Instagram's although users can share a wider variety of content and the app has a chronological content feed whereas Facebook and Instagram's feeds are algorithm based. Vero's business plan is also distinct from similar social media apps. Whereas its competitors such as Facebook or Instagram make money from in-app advertising revenue and the sale of user data, Vero's business plan was to invite the first one million users to use the app for free then charge any subsequent users a subscription fee. The app was entirely funded by its founder and generated additional revenues by charging affiliate fees when someone buys a product they find on Vero. == Awards == Vero was recognized at the 2021 Webbys, being named as an Honoree in the Best Visual Design - Aesthetic Category. == Controversies == === Privacy === Vero has faced some criticism over the wording of their manifesto, in particular, the statement "Vero only collects the data we believe is necessary to provide users with a great experience and to ensure the security of their accounts." Because this policy does not explicitly state that the app will not sell data on to third parties some users fear that the need to monetise the app through data might prove too tempting. Users have also complained about not being able to delete their accounts. While this was never the case, the option was hidden deep in the app's settings. === Russian involvement === Although Vero remains transparent about the app's Russian development team, they have been caught up in concerns about Russian interference on social media platforms. The app's founder Ayman Hariri was quick to dismiss the remarks as xenophobic and defend the nationality of his employees, stating in an interview with Time Magazine; "At the end of the day, where people are from is really not how anybody should judge anyone". === Criticism of the app's founder === Until 2013, Vero's founder Ayman Harari was deputy CEO and chairman of Saudi Oger, the Saudi Arabian construction company which collapsed in 2017, mired by controversies over the welfare and treatment of their employees. However, Hariri is quick to point out that he divested from the firm in 2014 and the worker's rights violations occurred after he had left the company.
Sliced inverse regression
Sliced inverse regression (SIR) is a tool for dimensionality reduction in the field of multivariate statistics. In statistics, regression analysis is a method of studying the relationship between a response variable y and its input variable x _ {\displaystyle {\underline {x}}} , which is a p-dimensional vector. There are several approaches in the category of regression. For example, parametric methods include multiple linear regression, and non-parametric methods include local smoothing. As the number of observations needed to use local smoothing methods scales exponentially with high-dimensional data (as p grows), reducing the number of dimensions can make the operation computable. Dimensionality reduction aims to achieve this by showing only the most important dimension of the data. SIR uses the inverse regression curve, E ( x _ | y ) {\displaystyle E({\underline {x}}\,|\,y)} , to perform a weighted principal component analysis. == Model == Given a response variable Y {\displaystyle \,Y} and a (random) vector X ∈ R p {\displaystyle X\in \mathbb {R} ^{p}} of explanatory variables, SIR is based on the model Y = f ( β 1 ⊤ X , … , β k ⊤ X , ε ) ( 1 ) {\displaystyle Y=f(\beta _{1}^{\top }X,\ldots ,\beta _{k}^{\top }X,\varepsilon )\quad \quad \quad \quad \quad (1)} where β 1 , … , β k {\displaystyle \beta _{1},\ldots ,\beta _{k}} are unknown projection vectors, k {\displaystyle \,k} is an unknown number smaller than p {\displaystyle \,p} , f {\displaystyle \;f} is an unknown function on R k + 1 {\displaystyle \mathbb {R} ^{k+1}} as it only depends on k {\displaystyle \,k} arguments, and ε {\displaystyle \varepsilon } is a random variable representing error with E [ ε | X ] = 0 {\displaystyle E[\varepsilon |X]=0} and a finite variance of σ 2 {\displaystyle \sigma ^{2}} . The model describes an ideal solution, where Y {\displaystyle \,Y} depends on X ∈ R p {\displaystyle X\in \mathbb {R} ^{p}} only through a k {\displaystyle \,k} dimensional subspace; i.e., one can reduce the dimension of the explanatory variables from p {\displaystyle \,p} to a smaller number k {\displaystyle \,k} without losing any information. An equivalent version of ( 1 ) {\displaystyle \,(1)} is: the conditional distribution of Y {\displaystyle \,Y} given X {\displaystyle \,X} depends on X {\displaystyle \,X} only through the k {\displaystyle \,k} dimensional random vector ( β 1 ⊤ X , … , β k ⊤ X ) {\displaystyle (\beta _{1}^{\top }X,\ldots ,\beta _{k}^{\top }X)} . It is assumed that this reduced vector is as informative as the original X {\displaystyle \,X} in explaining Y {\displaystyle \,Y} . The unknown β i ′ s {\displaystyle \,\beta _{i}'s} are called the effective dimension reducing directions (EDR-directions). The space that is spanned by these vectors is denoted by the effective dimension reducing space (EDR-space). == Relevant linear algebra background == Given a _ 1 , … , a _ r ∈ R n {\displaystyle {\underline {a}}_{1},\ldots ,{\underline {a}}_{r}\in \mathbb {R} ^{n}} , then V := L ( a _ 1 , … , a _ r ) {\displaystyle V:=L({\underline {a}}_{1},\ldots ,{\underline {a}}_{r})} , the set of all linear combinations of these vectors is called a linear subspace and is therefore a vector space. The equation says that vectors a _ 1 , … , a _ r {\displaystyle {\underline {a}}_{1},\ldots ,{\underline {a}}_{r}} span V {\displaystyle \,V} , but the vectors that span space V {\displaystyle \,V} are not unique. The dimension of V ( ∈ R n ) {\displaystyle \,V(\in \mathbb {R} ^{n})} is equal to the maximum number of linearly independent vectors in V {\displaystyle \,V} . A set of n {\displaystyle \,n} linear independent vectors of R n {\displaystyle \mathbb {R} ^{n}} makes up a basis of R n {\displaystyle \mathbb {R} ^{n}} . The dimension of a vector space is unique, but the basis itself is not. Several bases can span the same space. Dependent vectors can still span a space, but the linear combinations of the latter are only suitable to a set of vectors lying on a straight line. == Inverse regression == Computing the inverse regression curve (IR) means instead of looking for E [ Y | X = x ] {\displaystyle \,E[Y|X=x]} , which is a curve in R p {\displaystyle \mathbb {R} ^{p}} it is actually E [ X | Y = y ] {\displaystyle \,E[X|Y=y]} , which is also a curve in R p {\displaystyle \mathbb {R} ^{p}} , but consisting of p {\displaystyle \,p} one-dimensional regressions. The center of the inverse regression curve is located at E [ E [ X | Y ] ] = E [ X ] {\displaystyle \,E[E[X|Y]]=E[X]} . Therefore, the centered inverse regression curve is E [ X | Y = y ] − E [ X ] {\displaystyle \,E[X|Y=y]-E[X]} which is a p {\displaystyle \,p} dimensional curve in R p {\displaystyle \mathbb {R} ^{p}} . == Inverse regression versus dimension reduction == The centered inverse regression curve lies on a k {\displaystyle \,k} -dimensional subspace spanned by Σ x x β i ′ s {\displaystyle \,\Sigma _{xx}\beta _{i}\,'s} . This is a connection between the model and inverse regression. Given this condition and ( 1 ) {\displaystyle \,(1)} , the centered inverse regression curve E [ X | Y = y ] − E [ X ] {\displaystyle \,E[X|Y=y]-E[X]} is contained in the linear subspace spanned by Σ x x β k ( k = 1 , … , K ) {\displaystyle \,\Sigma _{xx}\beta _{k}(k=1,\ldots ,K)} , where Σ x x = C o v ( X ) {\displaystyle \,\Sigma _{xx}=Cov(X)} . == Estimation of the EDR-directions == After having had a look at all the theoretical properties, the aim now is to estimate the EDR-directions. For that purpose, weighted principal component analyses are needed. If the sample means m ^ h ′ s {\displaystyle \,{\hat {m}}_{h}\,'s} , X {\displaystyle \,X} would have been standardized to Z = Σ x x − 1 / 2 { X − E ( X ) } {\displaystyle \,Z=\Sigma _{xx}^{-1/2}\{X-E(X)\}} . Corresponding to the theorem above, the IR-curve m 1 ( y ) = E [ Z | Y = y ] {\displaystyle \,m_{1}(y)=E[Z|Y=y]} lies in the space spanned by ( η 1 , … , η k ) {\displaystyle \,(\eta _{1},\ldots ,\eta _{k})} , where η i = Σ x x 1 / 2 β i {\displaystyle \,\eta _{i}=\Sigma _{xx}^{1/2}\beta _{i}} . As a consequence, the covariance matrix c o v [ E [ Z | Y ] ] {\displaystyle \,cov[E[Z|Y]]} is degenerate in any direction orthogonal to the η i ′ s {\displaystyle \,\eta _{i}\,'s} . Therefore, the eigenvectors η k ( k = 1 , … , K ) {\displaystyle \,\eta _{k}(k=1,\ldots ,K)} associated with the largest K {\displaystyle \,K} eigenvalues are the standardized EDR-directions. == Algorithm == === SIR algorithm === The algorithm from Li, K-C. (1991) to estimate the EDR-directions via SIR is as follows. 1. Let Σ x x {\displaystyle \,\Sigma _{xx}} be the covariance matrix of X {\displaystyle \,X} . Standardize X {\displaystyle \,X} to Z = Σ x x − 1 / 2 { X − E ( X ) } {\displaystyle \,Z=\Sigma _{xx}^{-1/2}\{X-E(X)\}} ( 1 ) {\displaystyle \,(1)} can also be rewritten as Y = f ( η 1 ⊤ Z , … , η k ⊤ Z , ε ) {\displaystyle Y=f(\eta _{1}^{\top }Z,\ldots ,\eta _{k}^{\top }Z,\varepsilon )} where η k = β k Σ x x 1 / 2 ∀ k {\displaystyle \,\eta _{k}=\beta _{k}\Sigma _{xx}^{1/2}\quad \forall \;k} .) 2. Divide the range of y i {\displaystyle \,y_{i}} into S {\displaystyle \,S} non-overlapping slices H s ( s = 1 , … , S ) . n s {\displaystyle \,H_{s}(s=1,\ldots ,S).\;n_{s}} is the number of observations within each slice and I H s {\displaystyle \,I_{H_{s}}} is the indicator function for the slice: n s = ∑ i = 1 n I H s ( y i ) {\displaystyle n_{s}=\sum _{i=1}^{n}I_{H_{s}}(y_{i})} 3. Compute the mean of z i {\displaystyle \,z_{i}} over all slices, which is a crude estimate m ^ 1 {\displaystyle \,{\hat {m}}_{1}} of the inverse regression curve m 1 {\displaystyle \,m_{1}} : z ¯ s = n s − 1 ∑ i = 1 n z i I H s ( y i ) {\displaystyle \,{\bar {z}}_{s}=n_{s}^{-1}\sum _{i=1}^{n}z_{i}I_{H_{s}}(y_{i})} 4. Calculate the estimate for C o v { m 1 ( y ) } {\displaystyle \,Cov\{m_{1}(y)\}} : V ^ = n − 1 ∑ i = 1 S n s z ¯ s z ¯ s ⊤ {\displaystyle \,{\hat {V}}=n^{-1}\sum _{i=1}^{S}n_{s}{\bar {z}}_{s}{\bar {z}}_{s}^{\top }} 5. Identify the eigenvalues λ ^ i {\displaystyle \,{\hat {\lambda }}_{i}} and the eigenvectors η ^ i {\displaystyle \,{\hat {\eta }}_{i}} of V ^ {\displaystyle \,{\hat {V}}} , which are the standardized EDR-directions. 6. Transform the standardized EDR-directions back to the original scale. The estimates for the EDR-directions are given by: β ^ i = Σ ^ x x − 1 / 2 η ^ i {\displaystyle \,{\hat {\beta }}_{i}={\hat {\Sigma }}_{xx}^{-1/2}{\hat {\eta }}_{i}} (which are not necessarily orthogonal.)