Hierarchical Risk Parity (HRP) is an advanced investment portfolio optimization framework developed in 2016 by Marcos López de Prado at Guggenheim Partners and Cornell University. HRP is a probabilistic graph-based alternative to the prevailing mean-variance optimization (MVO) framework developed by Harry Markowitz in 1952, and for which he received the Nobel Prize in economic sciences. HRP algorithms apply discrete mathematics and machine learning techniques to create diversified and robust investment portfolios that outperform MVO methods out-of-sample. HRP aims to address the limitations of traditional portfolio construction methods, particularly when dealing with highly correlated assets. Following its publication, HRP has been implemented in numerous open-source libraries, and received multiple extensions. == Key features == HRP portfolios have been proposed as a robust alternative to traditional quadratic optimization methods, including the Critical Line Algorithm (CLA) of Markowitz. HRP addresses three central issues commonly associated with quadratic optimizers: numerical instability, excessive concentration in a small number of assets, and poor out-of-sample performance. HRP leverages techniques from graph theory and machine learning to construct diversified portfolios using only the information embedded in the covariance matrix. Unlike quadratic programming methods, HRP does not require the covariance matrix to be invertible. Consequently, HRP remains applicable even in cases where the covariance matrix is ill-conditioned or singular—conditions under which standard optimizers fail. Monte Carlo simulations indicate that HRP achieves lower out-of-sample variance than CLA, despite the fact that minimizing variance is the explicit optimization objective of CLA. Furthermore, HRP portfolios exhibit lower realized risk compared to those generated by traditional risk parity methodologies. Empirical backtests have demonstrated that HRP would have historically outperformed conventional portfolio construction techniques. Algorithms within the HRP framework are characterized by the following features: Machine Learning Approach: HRP employs hierarchical clustering, a machine learning technique, to group similar assets based on their correlations. This allows the algorithm to identify the underlying hierarchical structure of the portfolio, and avoid that errors spread through the entire network. Risk-Based Allocation: The algorithm allocates capital based on risk, ensuring that assets only compete with similar assets for representation in the portfolio. This approach leads to better diversification across different risk sources, while avoiding the instability associated with noisy returns estimates. Covariance Matrix Handling: Unlike traditional methods like Mean-Variance Optimization, HRP does not require inverting the covariance matrix. This makes it more stable and applicable to portfolios with a large number of assets, particularly when the covariance matrix's condition number is high. == The problem: Markowitz's Curse == Portfolio construction is perhaps the most recurrent financial problem. On a daily basis, investment managers must build portfolios that incorporate their views and forecasts on risks and returns. Despite the theoretical elegance of Markowitz's mean-variance framework, its practical implementation is hindered by several limitations that undermine the reliability of solutions derived from the Critical Line Algorithm (CLA). A principal concern is the high sensitivity of optimal portfolios to small perturbations in expected returns: even minor forecasting errors can result in significantly different allocations (Michaud, 1998). Given the inherent difficulty of producing accurate return forecasts, numerous researchers have advocated for approaches that forgo expected returns entirely and instead rely solely on the covariance structure of asset returns. This has given rise to risk-based allocation methods, among which risk parity is a widely cited example (Jurczenko, 2015). While eliminating return forecasts mitigates some instability, it does not eliminate it. Quadratic programming techniques employed in portfolio optimization require the inversion of a positive-definite covariance matrix, meaning all eigenvalues must be strictly positive. When the matrix is numerically ill-conditioned—that is, when the ratio of its largest to smallest eigenvalue (its condition number) is large—matrix inversion becomes unreliable and prone to significant numerical errors (Bailey and López de Prado, 2012). The condition number of a covariance, correlation, or any symmetric (and thus diagonalizable) matrix is defined as the absolute value of the ratio between its largest and smallest eigenvalues in modulus. The figure on the right presents the sorted eigenvalues of several correlation matrices; the condition number is represented by the ratio of the first to last eigenvalues in each sequence. A diagonal correlation matrix, which is equal to its own inverse, exhibits the minimum possible condition number. As the number of correlated (or multicollinear) assets in a portfolio increases, the condition number rises. At high levels, this leads to severe numerical instability, whereby slight modifications in any matrix entry may result in drastically different inverses. This phenomenon, often referred to as Markowitz’s curse, encapsulates the paradox wherein increased correlation among assets heightens the theoretical need for diversification, yet simultaneously increases the likelihood of unstable optimization outcomes. Consequently, the potential benefits of diversification are frequently overshadowed by estimation errors. These problems are exacerbated as the dimensionality of the covariance matrix increases. The estimation of each covariance term consumes degrees of freedom, and in general, a minimum of 1 2 N ( N + 1 ) {\displaystyle {\frac {1}{2}}N(N+1)} independent and identically distributed (IID) observations is required to estimate a non-singular covariance matrix of dimension N {\displaystyle N} . For example, constructing an invertible covariance matrix of dimension 50 necessitates at least five years of daily IID observations. However, empirical evidence suggests that the correlation structure of financial assets is highly unstable over such extended periods. These difficulties are highlighted by the observation that even naïve allocation strategies—such as equally weighted portfolios—have frequently outperformed both mean-variance and risk-based optimizations in out-of-sample tests (De Miguel et al., 2009). == The solution: Hierarchical Risk Parity == The HRP algorithm addresses Markowitz's curse in three steps: Hierarchical Clustering: Assets are grouped into clusters based on their correlations, forming a hierarchical tree structure. Quasi-Diagonalization: The correlation matrix is reordered based on the clustering results, revealing a block diagonal structure. Recursive Bisection: Weights are assigned to assets through a top-down approach, splitting the portfolio into smaller sub-portfolios and allocating capital based on inverse variance. === Step 1: Hierarchical clustering === Given a T × N {\displaystyle T\times N} matrix of asset returns X {\displaystyle X} , where each column represents a time series of returns for one of N {\displaystyle N} assets over T {\displaystyle T} time periods, a hierarchical clustering process can be used to construct a tree-based representation of asset relationships. First, we compute the N × N {\displaystyle N\times N} correlation matrix ρ = ρ i , j i , j = 1 . . . N {\displaystyle \rho ={\rho _{i,j}}\;{i,j=1\;...\;N}} , where ρ i , j = c o r r ( X i , X j ) {\displaystyle \rho _{i,j}=\mathrm {corr} (X_{i},X_{j})} . From this, a pairwise distance matrix D = d i , j {\displaystyle D={d_{i,j}}} is defined using the transformation: d i , j = 1 2 ( 1 − ρ i , j ) {\displaystyle d_{i,j}={\sqrt {{\frac {1}{2}}(1-\rho _{i,j})}}} This distance function defines a proper metric space, satisfying non-negativity, identity of indiscernibles, symmetry, and the triangle inequality. Next, a secondary distance matrix D ~ = d ~ i , j {\displaystyle {\tilde {D}}={{\tilde {d}}_{i,j}}} is computed, where each entry measures the Euclidean distance between the distance profiles of two assets: d ~ i , j = ∑ n = 1 N ( d n , i − d n , j ) 2 {\displaystyle {\tilde {d}}_{i,j}={\sqrt {\sum _{n=1}^{N}(d_{n,i}-d_{n,j})^{2}}}} While d i , j {\displaystyle d_{i,j}} reflects correlation-based proximity between two assets, d ~ i , j {\displaystyle {\tilde {d}}_{i,j}} quantifies dissimilarity across the entire system, as it depends on all pairwise distances. Hierarchical clustering proceeds by identifying the pair ( i , j ) {\displaystyle (i,j)} with the smallest value of d ~ i , j {\displaystyle {\tilde {d}}_{i,j}} (for i ≠ j {\displaystyle i\neq j} ), and forming a new cluster u [ 1 ] = ( i , j ) {\displaystyle u[1]=(i,j)} .
Proximal gradient methods for learning
Proximal gradient (forward backward splitting) methods for learning is an area of research in optimization and statistical learning theory which studies algorithms for a general class of convex regularization problems where the regularization penalty may not be differentiable. One such example is ℓ 1 {\displaystyle \ell _{1}} regularization (also known as Lasso) of the form min w ∈ R d 1 n ∑ i = 1 n ( y i − ⟨ w , x i ⟩ ) 2 + λ ‖ w ‖ 1 , where x i ∈ R d and y i ∈ R . {\displaystyle \min _{w\in \mathbb {R} ^{d}}{\frac {1}{n}}\sum _{i=1}^{n}(y_{i}-\langle w,x_{i}\rangle )^{2}+\lambda \|w\|_{1},\quad {\text{ where }}x_{i}\in \mathbb {R} ^{d}{\text{ and }}y_{i}\in \mathbb {R} .} Proximal gradient methods offer a general framework for solving regularization problems from statistical learning theory with penalties that are tailored to a specific problem application. Such customized penalties can help to induce certain structure in problem solutions, such as sparsity (in the case of lasso) or group structure (in the case of group lasso). == Relevant background == Proximal gradient methods are applicable in a wide variety of scenarios for solving convex optimization problems of the form min x ∈ H F ( x ) + R ( x ) , {\displaystyle \min _{x\in {\mathcal {H}}}F(x)+R(x),} where F {\displaystyle F} is convex and differentiable with Lipschitz continuous gradient, R {\displaystyle R} is a convex, lower semicontinuous function which is possibly nondifferentiable, and H {\displaystyle {\mathcal {H}}} is some set, typically a Hilbert space. The usual criterion of x {\displaystyle x} minimizes F ( x ) + R ( x ) {\displaystyle F(x)+R(x)} if and only if ∇ ( F + R ) ( x ) = 0 {\displaystyle \nabla (F+R)(x)=0} in the convex, differentiable setting is now replaced by 0 ∈ ∂ ( F + R ) ( x ) , {\displaystyle 0\in \partial (F+R)(x),} where ∂ φ {\displaystyle \partial \varphi } denotes the subdifferential of a real-valued, convex function φ {\displaystyle \varphi } . Given a convex function φ : H → R {\displaystyle \varphi :{\mathcal {H}}\to \mathbb {R} } an important operator to consider is its proximal operator prox φ : H → H {\displaystyle \operatorname {prox} _{\varphi }:{\mathcal {H}}\to {\mathcal {H}}} defined by prox φ ( u ) = arg min x ∈ H φ ( x ) + 1 2 ‖ u − x ‖ 2 2 , {\displaystyle \operatorname {prox} _{\varphi }(u)=\operatorname {arg} \min _{x\in {\mathcal {H}}}\varphi (x)+{\frac {1}{2}}\|u-x\|_{2}^{2},} which is well-defined because of the strict convexity of the ℓ 2 {\displaystyle \ell _{2}} norm. The proximal operator can be seen as a generalization of a projection. We see that the proximity operator is important because x ∗ {\displaystyle x^{}} is a minimizer to the problem min x ∈ H F ( x ) + R ( x ) {\displaystyle \min _{x\in {\mathcal {H}}}F(x)+R(x)} if and only if x ∗ = prox γ R ( x ∗ − γ ∇ F ( x ∗ ) ) , {\displaystyle x^{}=\operatorname {prox} _{\gamma R}\left(x^{}-\gamma \nabla F(x^{})\right),} where γ > 0 {\displaystyle \gamma >0} is any positive real number. === Moreau decomposition === One important technique related to proximal gradient methods is the Moreau decomposition, which decomposes the identity operator as the sum of two proximity operators. Namely, let φ : X → R {\displaystyle \varphi :{\mathcal {X}}\to \mathbb {R} } be a lower semicontinuous, convex function on a vector space X {\displaystyle {\mathcal {X}}} . We define its Fenchel conjugate φ ∗ : X → R {\displaystyle \varphi ^{}:{\mathcal {X}}\to \mathbb {R} } to be the function φ ∗ ( u ) := sup x ∈ X ⟨ x , u ⟩ − φ ( x ) . {\displaystyle \varphi ^{}(u):=\sup _{x\in {\mathcal {X}}}\langle x,u\rangle -\varphi (x).} The general form of Moreau's decomposition states that for any x ∈ X {\displaystyle x\in {\mathcal {X}}} and any γ > 0 {\displaystyle \gamma >0} that x = prox γ φ ( x ) + γ prox φ ∗ / γ ( x / γ ) , {\displaystyle x=\operatorname {prox} _{\gamma \varphi }(x)+\gamma \operatorname {prox} _{\varphi ^{}/\gamma }(x/\gamma ),} which for γ = 1 {\displaystyle \gamma =1} implies that x = prox φ ( x ) + prox φ ∗ ( x ) {\displaystyle x=\operatorname {prox} _{\varphi }(x)+\operatorname {prox} _{\varphi ^{}}(x)} . The Moreau decomposition can be seen to be a generalization of the usual orthogonal decomposition of a vector space, analogous with the fact that proximity operators are generalizations of projections. In certain situations it may be easier to compute the proximity operator for the conjugate φ ∗ {\displaystyle \varphi ^{}} instead of the function φ {\displaystyle \varphi } , and therefore the Moreau decomposition can be applied. This is the case for group lasso. == Lasso regularization == Consider the regularized empirical risk minimization problem with square loss and with the ℓ 1 {\displaystyle \ell _{1}} norm as the regularization penalty: min w ∈ R d 1 n ∑ i = 1 n ( y i − ⟨ w , x i ⟩ ) 2 + λ ‖ w ‖ 1 , {\displaystyle \min _{w\in \mathbb {R} ^{d}}{\frac {1}{n}}\sum _{i=1}^{n}(y_{i}-\langle w,x_{i}\rangle )^{2}+\lambda \|w\|_{1},} where x i ∈ R d and y i ∈ R . {\displaystyle x_{i}\in \mathbb {R} ^{d}{\text{ and }}y_{i}\in \mathbb {R} .} The ℓ 1 {\displaystyle \ell _{1}} regularization problem is sometimes referred to as lasso (least absolute shrinkage and selection operator). Such ℓ 1 {\displaystyle \ell _{1}} regularization problems are interesting because they induce sparse solutions, that is, solutions w {\displaystyle w} to the minimization problem have relatively few nonzero components. Lasso can be seen to be a convex relaxation of the non-convex problem min w ∈ R d 1 n ∑ i = 1 n ( y i − ⟨ w , x i ⟩ ) 2 + λ ‖ w ‖ 0 , {\displaystyle \min _{w\in \mathbb {R} ^{d}}{\frac {1}{n}}\sum _{i=1}^{n}(y_{i}-\langle w,x_{i}\rangle )^{2}+\lambda \|w\|_{0},} where ‖ w ‖ 0 {\displaystyle \|w\|_{0}} denotes the ℓ 0 {\displaystyle \ell _{0}} "norm", which is the number of nonzero entries of the vector w {\displaystyle w} . Sparse solutions are of particular interest in learning theory for interpretability of results: a sparse solution can identify a small number of important factors. === Solving for L1 proximity operator === For simplicity we restrict our attention to the problem where λ = 1 {\displaystyle \lambda =1} . To solve the problem min w ∈ R d 1 n ∑ i = 1 n ( y i − ⟨ w , x i ⟩ ) 2 + ‖ w ‖ 1 , {\displaystyle \min _{w\in \mathbb {R} ^{d}}{\frac {1}{n}}\sum _{i=1}^{n}(y_{i}-\langle w,x_{i}\rangle )^{2}+\|w\|_{1},} we consider our objective function in two parts: a convex, differentiable term F ( w ) = 1 n ∑ i = 1 n ( y i − ⟨ w , x i ⟩ ) 2 {\displaystyle F(w)={\frac {1}{n}}\sum _{i=1}^{n}(y_{i}-\langle w,x_{i}\rangle )^{2}} and a convex function R ( w ) = ‖ w ‖ 1 {\displaystyle R(w)=\|w\|_{1}} . Note that R {\displaystyle R} is not strictly convex. Let us compute the proximity operator for R ( w ) {\displaystyle R(w)} . First we find an alternative characterization of the proximity operator prox R ( x ) {\displaystyle \operatorname {prox} _{R}(x)} as follows: u = prox R ( x ) ⟺ 0 ∈ ∂ ( R ( u ) + 1 2 ‖ u − x ‖ 2 2 ) ⟺ 0 ∈ ∂ R ( u ) + u − x ⟺ x − u ∈ ∂ R ( u ) . {\displaystyle {\begin{aligned}u=\operatorname {prox} _{R}(x)\iff &0\in \partial \left(R(u)+{\frac {1}{2}}\|u-x\|_{2}^{2}\right)\\\iff &0\in \partial R(u)+u-x\\\iff &x-u\in \partial R(u).\end{aligned}}} For R ( w ) = ‖ w ‖ 1 {\displaystyle R(w)=\|w\|_{1}} it is easy to compute ∂ R ( w ) {\displaystyle \partial R(w)} : the i {\displaystyle i} th entry of ∂ R ( w ) {\displaystyle \partial R(w)} is precisely ∂ | w i | = { 1 , w i > 0 − 1 , w i < 0 [ − 1 , 1 ] , w i = 0. {\displaystyle \partial |w_{i}|={\begin{cases}1,&w_{i}>0\\-1,&w_{i}<0\\\left[-1,1\right],&w_{i}=0.\end{cases}}} Using the recharacterization of the proximity operator given above, for the choice of R ( w ) = ‖ w ‖ 1 {\displaystyle R(w)=\|w\|_{1}} and γ > 0 {\displaystyle \gamma >0} we have that prox γ R ( x ) {\displaystyle \operatorname {prox} _{\gamma R}(x)} is defined entrywise by ( prox γ R ( x ) ) i = { x i − γ , x i > γ 0 , | x i | ≤ γ x i + γ , x i < − γ , {\displaystyle \left(\operatorname {prox} _{\gamma R}(x)\right)_{i}={\begin{cases}x_{i}-\gamma ,&x_{i}>\gamma \\0,&|x_{i}|\leq \gamma \\x_{i}+\gamma ,&x_{i}<-\gamma ,\end{cases}}} which is known as the soft thresholding operator S γ ( x ) = prox γ ‖ ⋅ ‖ 1 ( x ) {\displaystyle S_{\gamma }(x)=\operatorname {prox} _{\gamma \|\cdot \|_{1}}(x)} . === Fixed point iterative schemes === To finally solve the lasso problem we consider the fixed point equation shown earlier: x ∗ = prox γ R ( x ∗ − γ ∇ F ( x ∗ ) ) . {\displaystyle x^{}=\operatorname {prox} _{\gamma R}\left(x^{}-\gamma \nabla F(x^{})\right).} Given that we have computed the form of the proximity operator explicitly, then we can define a standard fixed point iteration procedure. Namely, fix some initial w 0 ∈ R d {\displaystyle w^{0}\in \mathbb {R} ^{d}} , and for k = 1 , 2 , … {\displaystyle k=1,2,\ldots } define w k + 1 = S γ ( w k − γ ∇ F ( w k ) ) . {\displaystyle w^{k+1}=S_{\gamma }\left(w^{k}-\gamma \nabla F\l
Sample exclusion dimension
In computational learning theory, sample exclusion dimensions arise in the study of exact concept learning with queries. In algorithmic learning theory, a concept over a domain X is a Boolean function over X. Here we only consider finite domains. A partial approximation S of a concept c is a Boolean function over Y ⊆ X {\displaystyle Y\subseteq X} such that c is an extension to S. Let C be a class of concepts and c be a concept (not necessarily in C). Then a specifying set for c w.r.t. C, denoted by S is a partial approximation S of c such that C contains at most one extension to S. If we have observed a specifying set for some concept w.r.t. C, then we have enough information to verify a concept in C with at most one more mind change. The exclusion dimension, denoted by XD(C), of a concept class is the maximum of the size of the minimum specifying set of c' with respect to C, where c' is a concept not in C.
PVLV
The primary value learned value (PVLV) model is a possible explanation for the reward-predictive firing properties of dopamine (DA) neurons. It simulates behavioral and neural data on Pavlovian conditioning and the midbrain dopaminergic neurons that fire in proportion to unexpected rewards. It is an alternative to the temporal-differences (TD) algorithm. It is used as part of Leabra.
Relationship square
In statistics, the relationship square is a graphical representation for use in the factorial analysis of a table individuals x variables. This representation completes classical representations provided by principal component analysis (PCA) or multiple correspondence analysis (MCA), namely those of individuals, of quantitative variables (correlation circle) and of the categories of qualitative variables (at the centroid of the individuals who possess them). It is especially important in factor analysis of mixed data (FAMD) and in multiple factor analysis (MFA). == Definition of relationship square in the MCA frame == The first interest of the relationship square is to represent the variables themselves, not their categories, which is all the more valuable as there are many variables. For this, we calculate for each qualitative variable j {\displaystyle j} and each factor F s {\displaystyle F_{s}} ( F s {\displaystyle F_{s}} , rank s {\displaystyle s} factor, is the vector of coordinates of the individuals along the axis of rank s {\displaystyle s} ; in PCA, F s {\displaystyle F_{s}} is called principal component of rank s {\displaystyle s} ), the square of the correlation ratio between the F s {\displaystyle F_{s}} and the variable j {\displaystyle j} , usually denoted : η 2 ( j , F s ) {\displaystyle \eta ^{2}(j,F_{s})} Thus, to each factorial plane, we can associate a representation of qualitative variables themselves. Their coordinates being between 0 and 1, the variables appear in the square having as vertices the points (0,0), ( 0,1), (1,0) and (1,1). == Example in MCA == Six individuals ( i 1 , … , i 6 ) {\displaystyle i_{1},\ldots ,i_{6})} are described by three variables ( q 1 , q 2 , q 3 ) {\displaystyle (q_{1},q_{2},q_{3})} having respectively 3, 2 and 3 categories. Example : the individual i 1 {\displaystyle i_{1}} possesses the category a {\displaystyle a} of q 1 {\displaystyle q_{1}} , d {\displaystyle d} of q 2 {\displaystyle q_{2}} and f {\displaystyle f} of q 3 {\displaystyle q_{3}} . Applied to these data, the MCA function included in the R Package FactoMineR provides to the classical graph in Figure 1. The relationship square (Figure 2) makes easier the reading of the classic factorial plane. It indicates that: The first factor is related to the three variables but especially q 3 {\displaystyle q_{3}} (which have a very high coordinate along the first axis) and then q 2 {\displaystyle q_{2}} . The second factor is related only to q 1 {\displaystyle q_{1}} and q 3 {\displaystyle q_{3}} (and not to q 2 {\displaystyle q_{2}} which has a coordinate along axis 2 equal to 0) and that in a strong and equal manner. All this is visible on the classic graphic but not so clearly. The role of the relationship square is first to assist in reading a conventional graphic. This is precious when the variables are numerous and possess numerous coordinates. == Extensions == This representation may be supplemented with those of quantitative variables, the coordinates of the latter being the square of correlation coefficients (and not of correlation ratios). Thus, the second advantage of the relationship square lies in the ability to represent simultaneously quantitative and qualitative variables. The relationship square can be constructed from any factorial analysis of a table individuals x variables. In particular, it is (or should be) used systematically: in multiple correspondences analysis (MCA); in principal components analysis (PCA) when there are many supplementary variables; in factor analysis of mixed data (FAMD). An extension of this graphic to groups of variables (how to represent a group of variables by a single point ?) is used in Multiple Factor Analysis (MFA) == History == The idea of representing the qualitative variables themselves by a point (and not the categories) is due to Brigitte Escofier. The graphic as it is used now has been introduced by Brigitte Escofier and Jérôme Pagès in the framework of multiple factor analysis == Conclusion == In MCA, the relationship square provides a synthetic view of the connections between mixed variables, all the more valuable as there are many variables having many categories. This representation iscan be useful in any factorial analysis when there are numerous mixed variables, active and/or supplementary.
Knowledge assessment methodology
The knowledge assessment methodology (KAM) is "an interactive benchmarking tool created by the World Bank's Knowledge for Development Program to help countries identify the challenges and opportunities they face in making the transition to the knowledge-based economy." KAM does so by providing information on knowledge economy indicators for 146 countries. Its products include the Knowledge Economy Index and the Knowledge Index.
Ordination (statistics)
Ordination or gradient analysis, in multivariate analysis, is a method complementary to data clustering, and used mainly in exploratory data analysis (rather than in hypothesis testing). In contrast to cluster analysis, ordination orders quantities in a (usually lower-dimensional) latent space. In the ordination space, quantities that are near each other share attributes (i.e., are similar to some degree), and dissimilar objects are farther from each other. Such relationships between the objects, on each of several axes or latent variables, are then characterized numerically and/or graphically in a biplot. The first ordination method, principal components analysis, was suggested by Karl Pearson in 1901. == Methods == Ordination methods can broadly be categorized in eigenvector-, algorithm-, or model-based methods. Many classical ordination techniques, including principal components analysis, correspondence analysis (CA) and its derivatives (detrended correspondence analysis, canonical correspondence analysis, and redundancy analysis, belong to the first group). The second group includes some distance-based methods such as non-metric multidimensional scaling, and machine learning methods such as T-distributed stochastic neighbor embedding and nonlinear dimensionality reduction. The third group includes model-based ordination methods, which can be considered as multivariate extensions of Generalized Linear Models. Model-based ordination methods are more flexible in their application than classical ordination methods, so that it is for example possible to include random-effects. Unlike in the aforementioned two groups, there is no (implicit or explicit) distance measure in the ordination. Instead, a distribution needs to be specified for the responses as is typical for statistical models. These and other assumptions, such as the assumed mean-variance relationship, can be validated with the use of residual diagnostics, unlike in other ordination methods. == Applications == Ordination can be used on the analysis of any set of multivariate objects. It is frequently used in several environmental or ecological sciences, particularly plant community ecology. It is also used in genetics and systems biology for microarray data analysis and in psychometrics.