In statistics, modes of variation are a continuously indexed set of vectors or functions that are centered at a mean and are used to depict the variation in a population or sample. Typically, variation patterns in the data can be decomposed in descending order of eigenvalues with the directions represented by the corresponding eigenvectors or eigenfunctions. Modes of variation provide a visualization of this decomposition and an efficient description of variation around the mean. Both in principal component analysis (PCA) and in functional principal component analysis (FPCA), modes of variation play an important role in visualizing and describing the variation in the data contributed by each eigencomponent. In real-world applications, the eigencomponents and associated modes of variation aid to interpret complex data, especially in exploratory data analysis (EDA). == Formulation == Modes of variation are a natural extension of PCA and FPCA. === Modes of variation in PCA === If a random vector X = ( X 1 , X 2 , ⋯ , X p ) T {\displaystyle \mathbf {X} =(X_{1},X_{2},\cdots ,X_{p})^{T}} has the mean vector μ p {\displaystyle {\boldsymbol {\mu }}_{p}} , and the covariance matrix Σ p × p {\displaystyle \mathbf {\Sigma } _{p\times p}} with eigenvalues λ 1 ≥ λ 2 ≥ ⋯ ≥ λ p ≥ 0 {\displaystyle \lambda _{1}\geq \lambda _{2}\geq \cdots \geq \lambda _{p}\geq 0} and corresponding orthonormal eigenvectors e 1 , e 2 , ⋯ , e p {\displaystyle \mathbf {e} _{1},\mathbf {e} _{2},\cdots ,\mathbf {e} _{p}} , by eigendecomposition of a real symmetric matrix, the covariance matrix Σ {\displaystyle \mathbf {\Sigma } } can be decomposed as Σ = Q Λ Q T , {\displaystyle \mathbf {\Sigma } =\mathbf {Q} \mathbf {\Lambda } \mathbf {Q} ^{T},} where Q {\displaystyle \mathbf {Q} } is an orthogonal matrix whose columns are the eigenvectors of Σ {\displaystyle \mathbf {\Sigma } } , and Λ {\displaystyle \mathbf {\Lambda } } is a diagonal matrix whose entries are the eigenvalues of Σ {\displaystyle \mathbf {\Sigma } } . By the Karhunen–Loève expansion for random vectors, one can express the centered random vector in the eigenbasis X − μ = ∑ k = 1 p ξ k e k , {\displaystyle \mathbf {X} -{\boldsymbol {\mu }}=\sum _{k=1}^{p}\xi _{k}\mathbf {e} _{k},} where ξ k = e k T ( X − μ ) {\displaystyle \xi _{k}=\mathbf {e} _{k}^{T}(\mathbf {X} -{\boldsymbol {\mu }})} is the principal component associated with the k {\displaystyle k} -th eigenvector e k {\displaystyle \mathbf {e} _{k}} , with the properties E ( ξ k ) = 0 , Var ( ξ k ) = λ k , {\displaystyle \operatorname {E} (\xi _{k})=0,\operatorname {Var} (\xi _{k})=\lambda _{k},} and E ( ξ k ξ l ) = 0 for l ≠ k . {\displaystyle \operatorname {E} (\xi _{k}\xi _{l})=0\ {\text{for}}\ l\neq k.} Then the k {\displaystyle k} -th mode of variation of X {\displaystyle \mathbf {X} } is the set of vectors, indexed by α {\displaystyle \alpha } , m k , α = μ ± α λ k e k , α ∈ [ − A , A ] , {\displaystyle \mathbf {m} _{k,\alpha }={\boldsymbol {\mu }}\pm \alpha {\sqrt {\lambda _{k}}}\mathbf {e} _{k},\alpha \in [-A,A],} where A {\displaystyle A} is typically selected as 2 or 3 {\displaystyle 2\ {\text{or}}\ 3} . === Modes of variation in FPCA === For a square-integrable random function X ( t ) , t ∈ T ⊂ R p {\displaystyle X(t),t\in {\mathcal {T}}\subset R^{p}} , where typically p = 1 {\displaystyle p=1} and T {\displaystyle {\mathcal {T}}} is an interval, denote the mean function by μ ( t ) = E ( X ( t ) ) {\displaystyle \mu (t)=\operatorname {E} (X(t))} , and the covariance function by G ( s , t ) = Cov ( X ( s ) , X ( t ) ) = ∑ k = 1 ∞ λ k φ k ( s ) φ k ( t ) , {\displaystyle G(s,t)=\operatorname {Cov} (X(s),X(t))=\sum _{k=1}^{\infty }\lambda _{k}\varphi _{k}(s)\varphi _{k}(t),} where λ 1 ≥ λ 2 ≥ ⋯ ≥ 0 {\displaystyle \lambda _{1}\geq \lambda _{2}\geq \cdots \geq 0} are the eigenvalues and { φ 1 , φ 2 , ⋯ } {\displaystyle \{\varphi _{1},\varphi _{2},\cdots \}} are the orthonormal eigenfunctions of the linear Hilbert–Schmidt operator G : L 2 ( T ) → L 2 ( T ) , G ( f ) = ∫ T G ( s , t ) f ( s ) d s . {\displaystyle G:L^{2}({\mathcal {T}})\rightarrow L^{2}({\mathcal {T}}),\,G(f)=\int _{\mathcal {T}}G(s,t)f(s)ds.} By the Karhunen–Loève theorem, one can express the centered function in the eigenbasis, X ( t ) − μ ( t ) = ∑ k = 1 ∞ ξ k φ k ( t ) , {\displaystyle X(t)-\mu (t)=\sum _{k=1}^{\infty }\xi _{k}\varphi _{k}(t),} where ξ k = ∫ T ( X ( t ) − μ ( t ) ) φ k ( t ) d t {\displaystyle \xi _{k}=\int _{\mathcal {T}}(X(t)-\mu (t))\varphi _{k}(t)dt} is the k {\displaystyle k} -th principal component with the properties E ( ξ k ) = 0 , Var ( ξ k ) = λ k , {\displaystyle \operatorname {E} (\xi _{k})=0,\operatorname {Var} (\xi _{k})=\lambda _{k},} and E ( ξ k ξ l ) = 0 for l ≠ k . {\displaystyle \operatorname {E} (\xi _{k}\xi _{l})=0{\text{ for }}l\neq k.} Then the k {\displaystyle k} -th mode of variation of X ( t ) {\displaystyle X(t)} is the set of functions, indexed by α {\displaystyle \alpha } , m k , α ( t ) = μ ( t ) ± α λ k φ k ( t ) , t ∈ T , α ∈ [ − A , A ] {\displaystyle m_{k,\alpha }(t)=\mu (t)\pm \alpha {\sqrt {\lambda _{k}}}\varphi _{k}(t),\ t\in {\mathcal {T}},\ \alpha \in [-A,A]} that are viewed simultaneously over the range of α {\displaystyle \alpha } , usually for A = 2 or 3 {\displaystyle A=2\ {\text{or}}\ 3} . == Estimation == The formulation above is derived from properties of the population. Estimation is needed in real-world applications. The key idea is to estimate mean and covariance. === Modes of variation in PCA === Suppose the data x 1 , x 2 , ⋯ , x n {\displaystyle \mathbf {x} _{1},\mathbf {x} _{2},\cdots ,\mathbf {x} _{n}} represent n {\displaystyle n} independent drawings from some p {\displaystyle p} -dimensional population X {\displaystyle \mathbf {X} } with mean vector μ {\displaystyle {\boldsymbol {\mu }}} and covariance matrix Σ {\displaystyle \mathbf {\Sigma } } . These data yield the sample mean vector x ¯ {\displaystyle {\overline {\mathbf {x} }}} , and the sample covariance matrix S {\displaystyle \mathbf {S} } with eigenvalue-eigenvector pairs ( λ ^ 1 , e ^ 1 ) , ( λ ^ 2 , e ^ 2 ) , ⋯ , ( λ ^ p , e ^ p ) {\displaystyle ({\hat {\lambda }}_{1},{\hat {\mathbf {e} }}_{1}),({\hat {\lambda }}_{2},{\hat {\mathbf {e} }}_{2}),\cdots ,({\hat {\lambda }}_{p},{\hat {\mathbf {e} }}_{p})} . Then the k {\displaystyle k} -th mode of variation of X {\displaystyle \mathbf {X} } can be estimated by m ^ k , α = x ¯ ± α λ ^ k e ^ k , α ∈ [ − A , A ] . {\displaystyle {\hat {\mathbf {m} }}_{k,\alpha }={\overline {\mathbf {x} }}\pm \alpha {\sqrt {{\hat {\lambda }}_{k}}}{\hat {\mathbf {e} }}_{k},\alpha \in [-A,A].} === Modes of variation in FPCA === Consider n {\displaystyle n} realizations X 1 ( t ) , X 2 ( t ) , ⋯ , X n ( t ) {\displaystyle X_{1}(t),X_{2}(t),\cdots ,X_{n}(t)} of a square-integrable random function X ( t ) , t ∈ T {\displaystyle X(t),t\in {\mathcal {T}}} with the mean function μ ( t ) = E ( X ( t ) ) {\displaystyle \mu (t)=\operatorname {E} (X(t))} and the covariance function G ( s , t ) = Cov ( X ( s ) , X ( t ) ) {\displaystyle G(s,t)=\operatorname {Cov} (X(s),X(t))} . Functional principal component analysis provides methods for the estimation of μ ( t ) {\displaystyle \mu (t)} and G ( s , t ) {\displaystyle G(s,t)} in detail, often involving point wise estimate and interpolation. Substituting estimates for the unknown quantities, the k {\displaystyle k} -th mode of variation of X ( t ) {\displaystyle X(t)} can be estimated by m ^ k , α ( t ) = μ ^ ( t ) ± α λ ^ k φ ^ k ( t ) , t ∈ T , α ∈ [ − A , A ] . {\displaystyle {\hat {m}}_{k,\alpha }(t)={\hat {\mu }}(t)\pm \alpha {\sqrt {{\hat {\lambda }}_{k}}}{\hat {\varphi }}_{k}(t),t\in {\mathcal {T}},\alpha \in [-A,A].} == Applications == Modes of variation are useful to visualize and describe the variation patterns in the data sorted by the eigenvalues. In real-world applications, modes of variation associated with eigencomponents allow to interpret complex data, such as the evolution of function traits and other infinite-dimensional data. To illustrate how modes of variation work in practice, two examples are shown in the graphs to the right, which display the first two modes of variation. The solid curve represents the sample mean function. The dashed, dot-dashed, and dotted curves correspond to modes of variation with α = ± 1 , ± 2 , {\displaystyle \alpha =\pm 1,\pm 2,} and ± 3 {\displaystyle \pm 3} , respectively. The first graph displays the first two modes of variation of female mortality data from 41 countries in 2003. The object of interest is log hazard function between ages 0 and 100 years. The first mode of variation suggests that the variation of female mortality is smaller for ages around 0 or 100, and larger for ages around 25. An appropriate and intuitive interpretation is that mortality around 25 is driven by accidental death, while around 0 or 100, mortality is related to congenital disease or natural death. Compared to female mortality
SAP StreamWork
SAP StreamWork is an enterprise collaboration tool from SAP SE released in March 2010, and discontinued in December 2015. StreamWork allowed real-time collaboration like Google Wave, but focused on business activities such as analyzing data, planning meetings, and making decisions. It incorporated technology from Box.net and Evernote to allow users to connect to online files and documents, and document-reader technology from Scribd allowed users to view documents directly within its environment. StreamWork supported the OpenSocial set of application programming interfaces (APIs), allowing it to connect to tools built by third-party developers, such as Google Docs. A version of StreamWork intended for large enterprises used a virtual appliance based on Novell's SUSE Linux Enterprise to connect it to business systems, including those from SAP.
Ave!Comics
Ave!Comics Production is a privately owned French company editing comics on smartphones, tablets and computers. It was founded in 2008 and it is a subsidiary of Aquafadas, a software development company in digital publishing owned by Kobo Inc. AveComics is a comic book store for digital comic books that can be used on computers, tablets, and smartphones.(iOS, Android) Readers can buy and read comic books, manga and graphic novels in French, English and Spanish. AveComics uses a technology created by Aquafadas for comics transformation, distribution and reading, based around its AVE format. The AveComics application was also a finalist in the BlackBerry Innovation Awards 2009, in the "Entertainment" category. == Company history == Aquafadas, a company working on creative software for Flash, HTML5, photo, and video editing, created the application MyComics to allow the reading of comics on mobile in 2006. This application was made available in 2008, to enable the reading and storing of comics on iPhone and iPod Touch. A reading system adapted to low resolution screens was also available. In October of the same year, the company launched a comics library on both devices, in partnership with the Angoulême International Comics Festival, Fnac and SNCF. This library included the official selection of the festival, and was downloaded over 150 000 times. In December 2008 "The Adventures of Lucky Luke n°3", at Lucky Comics was published on both devices. The comic made a 50 000 € turnover. In April 2009, "Les Blondes" 10th volume was the top-selling comic for 10 months on the AppStore. After, in August 2009, the AveComics application was launched on iPhone, iPod Touch and BlackBerry. The company's website was launched in September when more than 100 titles were available on smartphones and computers. == Catalogue == AveComics works with over 80 international publishers including Glénat, Marsu Productions, Delcourt, Casterman, Soleil, Ubisoft, Les Humanoïdes Associés and Mad Fabrik. Comics such as "Assassin's Creed", "Talisman", "Titeuf", and "Seoul District" are sold by the company. == Award == Grand Prix Software Venture Capital - Senate 2008.
Robomart
Robomart is an American technology company headquartered in Santa Monica, California that builds autonomous smart shops for cafes, ice cream parlors, and quick-service restaurants. The company’s white label platform gives retailers the option to expand their footprint at a significantly lower cost than traditional brick-and-mortar real-estate. Robomarts are equipped with a proprietary checkout-free system, temperature controlled compartments, sensors for autonomous operation, and external cameras for added security. The company licenses its technology and white label applications to retailers who manage their fleet of stores and deploy them to their consumers’ locations. After consumers have taken goods from the robomart, their order is automatically calculated, their card on file is charged and they are sent a receipt. The company has announced partnerships with Unilever, Mars, and Fatty Mart. == History == Robomart was founded by Ali Ahmed, Tigran Shahverdyan, and Emad Suhail Rahim. The company debuted at CES 2018 where it unveiled its concept of a self-driving store. At GITEX 2018 the company presented its first functional prototype of a fully driverless Robomart. At the 2019 Consumer Electronics Show the company demonstrated the technology behind its autonomous stores and checkout-free shopping experience. In January 2019, Robomart announced its first partnership with U.S. grocery chain Stop & Shop to test its driverless stores. In December 2020, Robomart deployed the Pharmacy Robomart in a trial in West Hollywood. In June 2021, the company launched its commercial service with a fleet of Pharmacy and Snacks Robomarts operating within West Hollywood and Central Hollywood. In August 2023, Robomart announced a $2 million seed round, putting its to-date funding at $3.4 million. == Partnerships == In September 2019, Robomart partnered with Avery Dennison to source the RFID tags used to enable its checkout-free shopping experience. In December 2020, Robomart partnered with Zeeba Vans to provide vehicles for its growing fleet. In June 2021, Robomart partnered with REEF Technology to provide inventory management and restocking services. In addition, REEF's Light Speed grocery division serves as the first merchant selling products through Robomart. == Products == The company currently offers three Robomart types. The frozen Robomart that stocks ice cream, the refrigerated Robomart that stocks perishable foods, and the ambient Robomart that stocks shelf-stable goods.
Acoustic model
An acoustic model is used in automatic speech recognition to represent the relationship between an audio signal and the phonemes or other linguistic units that make up speech. The model is learned from a set of audio recordings and their corresponding transcripts. It is created by taking audio recordings of speech, and their text transcriptions, and using software to create statistical representations of the sounds that make up each word. == Background == Modern speech recognition systems use both an acoustic model and a language model to represent the statistical properties of speech. The acoustic model models the relationship between the audio signal and the phonetic units in the language. The language model is responsible for modeling the word sequences in the language. These two models are combined to get the top-ranked word sequences corresponding to a given audio segment. Most modern speech recognition systems operate on the audio in small chunks known as frames with an approximate duration of 10ms per frame. The raw audio signal from each frame can be transformed by applying the mel-frequency cepstrum. The coefficients from this transformation are commonly known as mel-frequency cepstral coefficients (MFCCs) and are used as an input to the acoustic model along with other features. Recently, the use of convolutional neural networks has led to major improvements in acoustic modeling. == Speech audio characteristics == Audio can be encoded at different sampling rates (i.e. samples per second – the most common being: 8, 16, 32, 44.1, 48, and 96 kHz), and different bits per sample (the most common being: 8-bits, 16-bits, 24-bits or 32-bits). Speech recognition engines work best if the acoustic model they use was trained with speech audio which was recorded at the same sampling rate/bits per sample as the speech being recognized. == Telephony-based speech recognition == The limiting factor for telephony based speech recognition is the bandwidth at which speech can be transmitted. For example, a standard land-line telephone only has a bandwidth of 64 kbit/s at a sampling rate of 8 kHz and 8-bits per sample (8000 samples per second 8-bits per sample = 64000 bit/s). Therefore, for telephony based speech recognition, acoustic models should be trained with 8 kHz/8-bit speech audio files. In the case of voice over IP, the codec determines the sampling rate/bits per sample of speech transmission. Codecs with a higher sampling rate/bits per sample for speech transmission (which improve the sound quality) necessitate acoustic models trained with audio data that matches that sampling rate/bits per sample. == Desktop-based speech recognition == For speech recognition on a standard desktop PC, the limiting factor is the sound card. Most sound cards today can record at sampling rates of between 16–48 kHz of audio, with bit rates of 8- to 16-bits per sample, and playback at up to 96 kHz. As a general rule, a speech recognition engine works better with acoustic models trained with speech audio data recorded at higher sampling rates/bits per sample. But using audio with too high a sampling rate/bits per sample can slow the recognition engine down. A compromise is needed. Thus for desktop speech recognition, the current standard is acoustic models trained with speech audio data recorded at sampling rates of 16 kHz/16 bits per sample.
VideoPoet
VideoPoet is a large language model developed by Google Research in 2023 for video making. It can be asked to animate still images. The model accepts text, images, and videos as inputs, with a program to add feature for any input to any format generated content. VideoPoet was publicly announced on December 19, 2023. It uses an autoregressive language model.
List of 3D rendering software
3D rendering software products are the dedicated engines used for rendering computer-generated imagery. This is not the same as 3D modeling software, which involves the creation of 3D models, for which the software listed below can produce realistically rendered visualisations.General-purpose packages which can have their own built-in rendering capabilities are not listed here; these can be found in the list of 3D computer graphics software and list of 3D animation software. See 3D computer graphics software for more discussion about the distinctions.