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Deconvolution
In mathematics, deconvolution is the inverse of convolution. Both operations are used in signal processing and image processing. For example, it may be possible to recover the original signal after a filter (convolution) by using a deconvolution method with a certain degree of accuracy. Due to the measurement error of the recorded signal or image, it can be demonstrated that the worse the signal-to-noise ratio (SNR), the worse the reversing of a filter will be; hence, inverting a filter is not always a good solution as the error amplifies. Deconvolution offers a solution to this problem. The foundations for deconvolution and time-series analysis were largely laid by Norbert Wiener of the Massachusetts Institute of Technology in his book Extrapolation, Interpolation, and Smoothing of Stationary Time Series (1949). The book was based on work Wiener had done during World War II but that had been classified at the time. Some of the early attempts to apply these theories were in the fields of weather forecasting and economics. == Description == In general, the objective of deconvolution is to find the solution f of a convolution equation of the form: f ∗ g = h {\displaystyle fg=h\,} Usually, h is some recorded signal, and f is some signal that we wish to recover, but has been convolved with a filter or distortion function g, before we recorded it. Usually, h is a distorted version of f and the shape of f can't be easily recognized by the eye or simpler time-domain operations. The function g represents the impulse response of an instrument or a driving force that was applied to a physical system. If we know g, or at least know the form of g, then we can perform deterministic deconvolution. However, if we do not know g in advance, then we need to estimate it. This can be done using methods of statistical estimation or building the physical principles of the underlying system, such as the electrical circuit equations or diffusion equations. There are several deconvolution techniques, depending on the choice of the measurement error and deconvolution parameters: === Raw deconvolution === When the measurement error is very low (ideal case), deconvolution collapses into a filter reversing. This kind of deconvolution can be performed in the Laplace domain. By computing the Fourier transform of the recorded signal h and the system response function g, you get H and G, with G as the transfer function. Using the convolution theorem, F = H / G {\displaystyle F=H/G\,} where F is the estimated Fourier transform of f. Finally, the inverse Fourier transform of the function F is taken to find the estimated deconvolved signal f. Note that G is at the denominator and could amplify elements of the error model if present. === Deconvolution with noise === In physical measurements, the situation is usually closer to ( f ∗ g ) + ε = h {\displaystyle (fg)+\varepsilon =h\,} In this case ε is noise that has entered our recorded signal. If a noisy signal or image is assumed to be noiseless, the statistical estimate of g will be incorrect. In turn, the estimate of ƒ will also be incorrect. The lower the signal-to-noise ratio, the worse the estimate of the deconvolved signal will be. That is the reason why inverse filtering the signal (as in the "raw deconvolution" above) is usually not a good solution. However, if at least some knowledge exists of the type of noise in the data (for example, white noise), the estimate of ƒ can be improved through techniques such as Wiener deconvolution. == Applications == === Seismology === The concept of deconvolution had an early application in reflection seismology. In 1950, Enders Robinson was a graduate student at MIT. He worked with others at MIT, such as Norbert Wiener, Norman Levinson, and economist Paul Samuelson, to develop the "convolutional model" of a reflection seismogram. This model assumes that the recorded seismogram s(t) is the convolution of an Earth-reflectivity function e(t) and a seismic wavelet w(t) from a point source, where t represents recording time. Thus, our convolution equation is s ( t ) = ( e ∗ w ) ( t ) . {\displaystyle s(t)=(ew)(t).\,} The seismologist is interested in e, which contains information about the Earth's structure. By the convolution theorem, this equation may be Fourier transformed to S ( ω ) = E ( ω ) W ( ω ) {\displaystyle S(\omega )=E(\omega )W(\omega )\,} in the frequency domain, where ω {\displaystyle \omega } is the frequency variable. By assuming that the reflectivity is white, we can assume that the power spectrum of the reflectivity is constant, and that the power spectrum of the seismogram is the spectrum of the wavelet multiplied by that constant. Thus, | S ( ω ) | ≈ k | W ( ω ) | . {\displaystyle |S(\omega )|\approx k|W(\omega )|.\,} If we assume that the wavelet is minimum phase, we can recover it by calculating the minimum phase equivalent of the power spectrum we just found. The reflectivity may be recovered by designing and applying a Wiener filter that shapes the estimated wavelet to a Dirac delta function (i.e., a spike). The result may be seen as a series of scaled, shifted delta functions (although this is not mathematically rigorous): e ( t ) = ∑ i = 1 N r i δ ( t − τ i ) , {\displaystyle e(t)=\sum _{i=1}^{N}r_{i}\delta (t-\tau _{i}),} where N is the number of reflection events, r i {\displaystyle r_{i}} are the reflection coefficients, t − τ i {\displaystyle t-\tau _{i}} are the reflection times of each event, and δ {\displaystyle \delta } is the Dirac delta function. In practice, since we are dealing with noisy, finite bandwidth, finite length, discretely sampled datasets, the above procedure only yields an approximation of the filter required to deconvolve the data. However, by formulating the problem as the solution of a Toeplitz matrix and using Levinson recursion, we can relatively quickly estimate a filter with the smallest mean squared error possible. We can also do deconvolution directly in the frequency domain and get similar results. The technique is closely related to linear prediction. === Optics and other imaging === In optics and imaging, the term "deconvolution" is specifically used to refer to the process of reversing the optical distortion that takes place in an optical microscope, electron microscope, telescope, or other imaging instrument, thus creating clearer images. It is usually done in the digital domain by a software algorithm, as part of a suite of microscope image processing techniques. Deconvolution is also practical to sharpen images that suffer from fast motion or jiggles during capturing. Early Hubble Space Telescope images were distorted by a flawed mirror and were sharpened by deconvolution. The usual method is to assume that the optical path through the instrument is optically perfect, convolved with a point spread function (PSF), that is, a mathematical function that describes the distortion in terms of the pathway a theoretical point source of light (or other waves) takes through the instrument. Usually, such a point source contributes a small area of fuzziness to the final image. If this function can be determined, it is then a matter of computing its inverse or complementary function, and convolving the acquired image with that. The result is the original, undistorted image. In practice, finding the true PSF is impossible, and usually an approximation of it is used, theoretically calculated or based on some experimental estimation by using known probes. Real optics may also have different PSFs at different focal and spatial locations, and the PSF may be non-linear. The accuracy of the approximation of the PSF will dictate the final result. Different algorithms can be employed to give better results, at the price of being more computationally intensive. Since the original convolution discards data, some algorithms use additional data acquired at nearby focal points to make up some of the lost information. Regularization in iterative algorithms (as in expectation-maximization algorithms) can be applied to avoid unrealistic solutions. When the PSF is unknown, it may be possible to deduce it by systematically trying different possible PSFs and assessing whether the image has improved. This procedure is called blind deconvolution. Blind deconvolution is a well-established image restoration technique in astronomy, where the point nature of the objects photographed exposes the PSF thus making it more feasible. It is also used in fluorescence microscopy for image restoration, and in fluorescence spectral imaging for spectral separation of multiple unknown fluorophores. The most common iterative algorithm for the purpose is the Richardson–Lucy deconvolution algorithm; the Wiener deconvolution (and approximations) are the most common non-iterative algorithms. For some specific imaging systems such as laser pulsed terahertz systems, PSF can be modeled mathematically. As a result, as shown in the figure, deconvolution of the modeled PS
Digital scrapbooking
Digital scrapbooking is the term for the creation of a new 2D artwork by re-combining various graphic elements. It is a form of scrapbooking that is done using a personal computer, digital or scanned photos and computer graphics software. It is a relatively new form of the traditional print scrapbooking. Recent advances in technology now enable the craft to be pursued on tablets and smart devices utilising imaging apps as well as hobby specific apps, some of which have been created specifically by brands for use with their own image products. Digital scrapbooking kits are available to purchase and download at many websites that specialize in the craft. Kits contain graphics and word-art and are usually themed and color-coordinated. They usually consist of a mix of background images and "cut out" [extracted] images containing alpha channels. Once a kit has been downloaded to the computer or device, it can then be used over and over again to make new scrapbook pages (scrapbook layouts) within the software program that one chooses to use, often in combination with the users's own family photographs, scanned keepsakes and other unique personal elements scanned on a flatbed scanner. Scanning is usually done at 300dpi, to make the resulting images suitable for print. == Licensing and Copyright == Kits are sometimes licensed differently from other forms of traditional royalty-free stock images that may be purchased per-item or in sets at online stock photography sites. Some kit packs will be wholly royalty-free, but some kit makers may restrict usage to non-commercial work only. Some may specifically forbid the use of their work in projects for commercial gain, for example greetings cards and gift tags that may be made with their kits. Licensing often varies from kit to kit, even from the same maker. Some kits include derivative works of public domain material. In contrast to stock, creators of digital scrapbooking kits often require a credit or byline to indicate that their image elements have been used in a new creation. == Uses == Some artistic individuals combine digital scrapbooking with traditional scrapbooking to create what's known as hybrid scrapbooking projects. Hybrid scrapbooking involves creating layouts on the computer using digital supplies that will then be printed and combined with traditional supplies such as buttons, ribbons and other elements. Conversely, a hybrid scrapbook project may also be created using traditional paper supplies and augmented with digital elements that have been printed and cut out specifically for use on the project. Journaling may be done within the software programs to accompany images and to create digital storybooks, or scrapbooks, which are then published in photo books via various popular print-on-demand services, printed and added to traditional scrapbooks, burned to CDs or posted on the Web. Digital Scrapbooking may also be done online by uploading photos to a specialist scrapbooking website and utilising their custom built platforms and decorative image elements to complete the projects for print to finished products, for example photo books and holiday greeting cards. == Market Size == The traditional scrapbooking market appeared to decline somewhat in the USA since 2010, probably due to the 2008 financial crisis, and the digital scrapbooking market (being potentially a much cheaper form of scrapbooking) may have increased accordingly. Both markets currently appear to have recovered lost ground and expanded since the beginning of the COVID-19 pandemic as many people sought to productively fill their time during lockdowns, quarantines and self-isolation / stay at home directions. == Digital scrapbooking software == The main software programs that are typically used are Adobe Photoshop, Adobe Photoshop Elements, paint.net (freeware), Filter Forge, Corel Paintshop Pro, and GIMP. Additionally Adobe offer the Photoshop iOS product using the same code base as the desktop version to drive the app version. == Digital scrapbooking supplies == Digital scrapbooking supplies are downloaded from the Internet and then stored on a computer or external hardrive, DVD or CD media, SD cards, or in the cloud, to be used as needed. Both paid and free digital scrapbooking supplies available from numerous designers on their blogs or in e-commerce stores either as solo designers or as part of a wide cohort of designers working cooperatively in large full service e-commerce websites. Usually designed at 300ppi image resolution, digital scrapbooking product offerings and supplies often include: Full coordinated kits containing digital background “papers”, decorative alphabets, and diverse embellishments generally containing a mixture of .JPG and .PNG files; "Quick pages", flattened files containing a completed page layout with transparent photo windows in .PNG file format; Digital templates, fully layered layouts i.e. pages that have had the composition pre-designed ready for use in an imaging program or app, fully customizable for color schemes, kit choices, photographs and other embellishments, generally supplied in either .PSD or .TIF file format; Hybrid “quick pages”, i.e. layouts that are both fully designed and fully layered for customization, generally supplied in either .PSD or .TIF file format; Adobe Photoshop actions, brushes, custom shapes, paths and styles, saved in their respective native Photoshop file formats; and Corel PaintShop Pro equivalent tools.
Microformat
Microformats (μF) are predefined HTML markup (like HTML classes) created to serve as descriptive and consistent metadata about elements, designating them as representing a certain type of data (such as contact information, geographic coordinates, events, products, recipes, etc.). They allow software to process the information reliably by having set classes refer to a specific type of data rather than being arbitrary. Microformats emerged around 2005 and were predominantly designed for use by search engines, web syndication and aggregators such as RSS. Google confirmed in 2020 that it still parses microformats for use in content indexing. Microformats are referenced in several W3C social web specifications, including IndieAuth and Webmention. Although the content of web pages has been capable of some "automated processing" since the inception of the web, such processing is difficult because the markup elements used to display information on the web do not describe what the information means. Microformats can bridge this gap by attaching semantics, and thereby obviating other, more complicated, methods of automated processing, such as natural language processing or screen scraping. The use, adoption and processing of microformats enables data items to be indexed, searched for, saved or cross-referenced, so that information can be reused or combined. As of 2013, microformats allow the encoding and extraction of event details, contact information, social relationships and similar information. Microformats2, abbreviated as mf2, is the updated version of microformats. Mf2 provides an easier way of interpreting HTML structured syntax and vocabularies than the earlier ways that made use of RDFa and microdata. == Background == Microformats emerged around 2005 as part of a grassroots movement to make recognizable data items (such as events, contact details or geographical locations) capable of automated processing by software, as well as directly readable by end-users. Link-based microformats emerged first. These include vote links that express opinions of the linked page, which search engines can tally into instant polls. CommerceNet, a nonprofit organization that promotes e-commerce on the Internet, has helped sponsor and promote the technology and support the microformats community in various ways. CommerceNet also helped co-found the Microformats.org community site. Neither CommerceNet nor Microformats.org operates as a standards body. The microformats community functions through an open wiki, a mailing list, and an Internet relay chat (IRC) channel. Most of the existing microformats originated at the Microformats.org wiki and the associated mailing list by a process of gathering examples of web-publishing behaviour, then codifying it. Some other microformats (such as rel=nofollow and unAPI) have been proposed, or developed, elsewhere. == Technical overview == XHTML and HTML standards allow for the embedding and encoding of semantics within the attributes of markup elements. Microformats take advantage of these standards by indicating the presence of metadata using the following attributes: class Classname rel relationship, description of the target address in an anchor-element (...) rev reverse relationship, description of the referenced document (in one case, otherwise deprecated in microformats) For example, in the text "The birds roosted at 52.48, -1.89" is a pair of numbers which may be understood, from their context, to be a set of geographic coordinates. With wrapping in spans (or other HTML elements) with specific class names (in this case geo, latitude and longitude, all part of the geo microformat specification): Software agents can recognize exactly what each value represents and can then perform a variety of tasks such as indexing, locating it on a map and exporting it to a GPS device. === Examples === In this example, the contact information is presented as follows: With hCard microformat markup, that becomes: Here, the formatted name (fn), organisation (org), telephone number (tel) and web address (url) have been identified using specific class names and the whole thing is wrapped in class="vcard", which indicates that the other classes form an hCard (short for "HTML vCard") and are not merely coincidentally named. Other, optional, hCard classes also exist. Software, such as browser plug-ins, can now extract the information, and transfer it to other applications, such as an address book. == Specific microformats == Several microformats have been developed to enable semantic markup of particular types of information. However, only hCard and hCalendar have been ratified, the others remaining as drafts: hAtom (superseded by h-entry and h-feed) – for marking up Atom feeds from within standard HTML hCalendar – for events hCard – for contact information; includes: adr – for postal addresses geo – for geographical coordinates (latitude, longitude) hMedia – for audio/video content hAudio – for audio content hNews – for news content hProduct – for products hRecipe – for recipes and foodstuffs. hReview – for reviews rel-directory – for distributed directory creation and inclusion rel-enclosure – for multimedia attachments to web pages rel-license – specification of copyright license rel-nofollow, an attempt to discourage third-party content spam (e.g. spam in blogs) rel-tag – for decentralized tagging (Folksonomy) XHTML Friends Network (XFN) – for social relationships XOXO – for lists and outlines == Uses == Using microformats within HTML code provides additional formatting and semantic data that applications can use. For example, applications such as web crawlers can collect data about online resources, or desktop applications such as e-mail clients or scheduling software can compile details. The use of microformats can also facilitate "mash ups" such as exporting all of the geographical locations on a web page into (for example) Google Maps to visualize them spatially. Several browser extensions, such as Operator for Firefox and Oomph for Internet Explorer, provide the ability to detect microformats within an HTML document. When hCard or hCalendar are involved, such browser extensions allow microformats to be exported into formats compatible with contact management and calendar utilities, such as Microsoft Outlook. When dealing with geographical coordinates, they allow the location to be sent to applications such as Google Maps. Yahoo! Query Language can be used to extract microformats from web pages. On 12 May 2009 Google announced that they would be parsing the hCard, hReview and hProduct microformats, and using them to populate search result pages. They subsequently extended this in 2010 to use hCalendar for events and hRecipe for cookery recipes. Similarly, microformats are also processed by Bing and Yahoo!. As of late 2010, these are the world's top three search engines. Microsoft said in 2006 that they needed to incorporate microformats into upcoming projects, as did other software companies. Alex Faaborg summarizes the arguments for putting the responsibility for microformat user interfaces in the web browser rather than making more complicated HTML: Only the web browser knows what applications are accessible to the user and what the user's preferences are It lowers the barrier to entry for web site developers if they only need to do the markup and not handle "appearance" or "action" issues Retains backwards compatibility with web browsers that do not support microformats The web browser presents a single point of entry from the web to the user's computer, which simplifies security issues == Evaluation == Various commentators have offered review and discussion on the design principles and practical aspects of microformats. Microformats have been compared to other approaches that seek to serve the same or similar purpose. As of 2007, there had been some criticism of one, or all, microformats. The spread and use of microformats was being advocated as of 2007. Opera Software CTO and CSS creator Håkon Wium Lie said in 2005 "We will also see a bunch of microformats being developed, and that’s how the semantic web will be built, I believe." However, in August 2008 Toby Inkster, author of the "Swignition" (formerly "Cognition") microformat parsing service, pointed out that no new microformat specifications had been published since 2005. === Design principles === Computer scientist and entrepreneur, Rohit Khare stated that reduce, reuse, and recycle is "shorthand for several design principles" that motivated the development and practices behind microformats. These aspects can be summarized as follows: Reduce: favor the simplest solutions and focus attention on specific problems; Reuse: work from experience and favor examples of current practice; Recycle: encourage modularity and the ability to embed, valid XHTML can be reused in blog posts, RSS feeds, and anywhere else you can access the web. === Accessibi
Mike Little
Mike Little (born 12 May 1962) is an English web developer and writer. He is the co-founder of the free and open source web publishing software WordPress. == Biography == Mike Little was born in Manchester, England in 1962 to a Nigerian father, who was a mathematics lecturer and musician, and an English mother who worked as a primary school teacher. Little was placed into foster care when he was four months of age, and was later adopted by the same family. He grew up on a council estate in Brinnington, Stockport, and was educated at Stockport School. In 2003, Little and Matt Mullenweg started working on a project in which they built on b2/cafelog and later named it WordPress, releasing the first version on 27 May 2003. Little states that, despite not being invited to join his co-founder's for-profit business Automattic, he and Mullenweg remain on good terms. He clarified: "I don’t want it to sound like he cheated me out of something or ripped me off in some way. He didn’t." In June 2013, Little was awarded the SAScon's "Outstanding Contribution to Digital" award for his part in co-founding and developing WordPress. Little has been described as "modest" and living in "virtual anonymity". He has one daughter. He identifies as a follower of Stoicism and a humanist, and in 2021, he became a patron of charity Humanists UK.
Model-based clustering
In statistics, cluster analysis is the algorithmic grouping of objects into homogeneous groups based on numerical measurements. Model-based clustering based on a statistical model for the data, usually a mixture model. This has several advantages, including a principled statistical basis for clustering, and ways to choose the number of clusters, to choose the best clustering model, to assess the uncertainty of the clustering, and to identify outliers that do not belong to any group. == Model-based clustering == Suppose that for each of n {\displaystyle n} observations we have data on d {\displaystyle d} variables, denoted by y i = ( y i , 1 , … , y i , d ) {\displaystyle y_{i}=(y_{i,1},\ldots ,y_{i,d})} for observation i {\displaystyle i} . Then model-based clustering expresses the probability density function of y i {\displaystyle y_{i}} as a finite mixture, or weighted average of G {\displaystyle G} component probability density functions: p ( y i ) = ∑ g = 1 G τ g f g ( y i ∣ θ g ) , {\displaystyle p(y_{i})=\sum _{g=1}^{G}\tau _{g}f_{g}(y_{i}\mid \theta _{g}),} where f g {\displaystyle f_{g}} is a probability density function with parameter θ g {\displaystyle \theta _{g}} , τ g {\displaystyle \tau _{g}} is the corresponding mixture probability where ∑ g = 1 G τ g = 1 {\displaystyle \sum _{g=1}^{G}\tau _{g}=1} . Then in its simplest form, model-based clustering views each component of the mixture model as a cluster, estimates the model parameters, and assigns each observation to cluster corresponding to its most likely mixture component. === Gaussian mixture model === The most common model for continuous data is that f g {\displaystyle f_{g}} is a multivariate normal distribution with mean vector μ g {\displaystyle \mu _{g}} and covariance matrix Σ g {\displaystyle \Sigma _{g}} , so that θ g = ( μ g , Σ g ) {\displaystyle \theta _{g}=(\mu _{g},\Sigma _{g})} . This defines a Gaussian mixture model. The parameters of the model, τ g {\displaystyle \tau _{g}} and θ g {\displaystyle \theta _{g}} for g = 1 , … , G {\displaystyle g=1,\ldots ,G} , are typically estimated by maximum likelihood estimation using the expectation-maximization algorithm (EM); see also EM algorithm and GMM model. Bayesian inference is also often used for inference about finite mixture models. The Bayesian approach also allows for the case where the number of components, G {\displaystyle G} , is infinite, using a Dirichlet process prior, yielding a Dirichlet process mixture model for clustering. === Choosing the number of clusters === An advantage of model-based clustering is that it provides statistically principled ways to choose the number of clusters. Each different choice of the number of groups G {\displaystyle G} corresponds to a different mixture model. Then standard statistical model selection criteria such as the Bayesian information criterion (BIC) can be used to choose G {\displaystyle G} . The integrated completed likelihood (ICL) is a different criterion designed to choose the number of clusters rather than the number of mixture components in the model; these will often be different if highly non-Gaussian clusters are present. === Parsimonious Gaussian mixture model === For data with high dimension, d {\displaystyle d} , using a full covariance matrix for each mixture component requires estimation of many parameters, which can result in a loss of precision, generalizabity and interpretability. Thus it is common to use more parsimonious component covariance matrices exploiting their geometric interpretation. Gaussian clusters are ellipsoidal, with their volume, shape and orientation determined by the covariance matrix. Consider the eigendecomposition of a matrix Σ g = λ g D g A g D g T , {\displaystyle \Sigma _{g}=\lambda _{g}D_{g}A_{g}D_{g}^{T},} where D g {\displaystyle D_{g}} is the matrix of eigenvectors of Σ g {\displaystyle \Sigma _{g}} , A g = diag { A 1 , g , … , A d , g } {\displaystyle A_{g}={\mbox{diag}}\{A_{1,g},\ldots ,A_{d,g}\}} is a diagonal matrix whose elements are proportional to the eigenvalues of Σ g {\displaystyle \Sigma _{g}} in descending order, and λ g {\displaystyle \lambda _{g}} is the associated constant of proportionality. Then λ g {\displaystyle \lambda _{g}} controls the volume of the ellipsoid, A g {\displaystyle A_{g}} its shape, and D g {\displaystyle D_{g}} its orientation. Each of the volume, shape and orientation of the clusters can be constrained to be equal (E) or allowed to vary (V); the orientation can also be spherical, with identical eigenvalues (I). This yields 14 possible clustering models, shown in this table: It can be seen that many of these models are more parsimonious, with far fewer parameters than the unconstrained model that has 90 parameters when G = 4 {\displaystyle G=4} and d = 9 {\displaystyle d=9} . Several of these models correspond to well-known heuristic clustering methods. For example, k-means clustering is equivalent to estimation of the EII clustering model using the classification EM algorithm. The Bayesian information criterion (BIC) can be used to choose the best clustering model as well as the number of clusters. It can also be used as the basis for a method to choose the variables in the clustering model, eliminating variables that are not useful for clustering. Different Gaussian model-based clustering methods have been developed with an eye to handling high-dimensional data. These include the pgmm method, which is based on the mixture of factor analyzers model, and the HDclassif method, based on the idea of subspace clustering. The mixture-of-experts framework extends model-based clustering to include covariates. == Example == We illustrate the method with a dateset consisting of three measurements (glucose, insulin, sspg) on 145 subjects for the purpose of diagnosing diabetes and the type of diabetes present. The subjects were clinically classified into three groups: normal, chemical diabetes and overt diabetes, but we use this information only for evaluating clustering methods, not for classifying subjects. The BIC plot shows the BIC values for each combination of the number of clusters, G {\displaystyle G} , and the clustering model from the Table. Each curve corresponds to a different clustering model. The BIC favors 3 groups, which corresponds to the clinical assessment. It also favors the unconstrained covariance model, VVV. This fits the data well, because the normal patients have low values of both sspg and insulin, while the distributions of the chemical and overt diabetes groups are elongated, but in different directions. Thus the volumes, shapes and orientations of the three groups are clearly different, and so the unconstrained model is appropriate, as selected by the model-based clustering method. The classification plot shows the classification of the subjects by model-based clustering. The classification was quite accurate, with a 12% error rate as defined by the clinical classification. Other well-known clustering methods performed worse with higher error rates, such as single-linkage clustering with 46%, average link clustering with 30%, complete-linkage clustering also with 30%, and k-means clustering with 28%. == Outliers in clustering == An outlier in clustering is a data point that does not belong to any of the clusters. One way of modeling outliers in model-based clustering is to include an additional mixture component that is very dispersed, with for example a uniform distribution. Another approach is to replace the multivariate normal densities by t {\displaystyle t} -distributions, with the idea that the long tails of the t {\displaystyle t} -distribution would ensure robustness to outliers. However, this is not breakdown-robust. A third approach is the "tclust" or data trimming approach which excludes observations identified as outliers when estimating the model parameters. == Non-Gaussian clusters and merging == Sometimes one or more clusters deviate strongly from the Gaussian assumption. If a Gaussian mixture is fitted to such data, a strongly non-Gaussian cluster will often be represented by several mixture components rather than a single one. In that case, cluster merging can be used to find a better clustering. A different approach is to use mixtures of complex component densities to represent non-Gaussian clusters. == Non-continuous data == === Categorical data === Clustering multivariate categorical data is most often done using the latent class model. This assumes that the data arise from a finite mixture model, where within each cluster the variables are independent. === Mixed data === These arise when variables are of different types, such as continuous, categorical or ordinal data. A latent class model for mixed data assumes local independence between the variable. The location model relaxes the local independence assumption. The clustMD approach assumes that the observed variables are manifestations of underlying continuous Gaussian latent
Influence-for-hire
Influence-for-hire or collective influence, refers to the economy that has emerged around buying and selling influence on social media platforms. == Overview == Companies that engage in the influence-for-hire industry range from content farms to high-end public relations agencies. Traditionally influence operations have largely been confined to public sector actors like intelligence agencies, in the influence-for-hire industry the groups conduction the operations are private with commerce being their primary consideration. However many of the clients in the influence-for-hire industry are countries or countries acting through proxies. They are often located in countries with less expensive digital labor. == History == In May 2021, Facebook took a Ukrainian influence-for-hire network offline. Facebook attributed the network to organizations and consultants linked to Ukrainian politicians including Andriy Derkach. During the COVID-19 pandemic state sponsored misinformation was spread through influence-for-hire networks. In August 2021, a report published by the Australian Strategic Policy Institute implicated the Chinese government and the ruling Chinese Communist Party in campaigns of online manipulation conducted against Australia and Taiwan using influence-for-hire.