Final Cut Express was a video editing software suite created by Apple Inc. It was the consumer version of Final Cut Pro and was designed for advanced editing of digital video as well as high-definition video, which was used by many amateur and professional videographers. Final Cut Express was considered a step above iMovie in terms of capabilities, but a step underneath Final Cut Pro and its suite of applications. As of June 21, 2011, Final Cut Express was discontinued in favor of Final Cut Pro X. == History == Final Cut Express 1.0, based on Final Cut Pro 3, was released at Macworld Conference and Expo in San Francisco in 2003. The second version, based on Final Cut Pro 4, was released at Macworld San Francisco in 2004. The third version, capable of editing high definition video, was also announced at Macworld San Francisco a year later, and was released as Final Cut Express HD in February 2005. It was based on Final Cut Pro HD (version 4.5) and included LiveType 1.2 and Soundtrack 1.2. Final Cut Express version 3.5 was released with little fanfare in May 2006 as a Universal Binary. In addition to improving real-time rendering with Dynamic RT, version 3.5 upgraded LiveType to version 2.0 and Soundtrack to version 1.5. In November 2007, Apple released Final Cut Express 4, which although it did not support real-time editing in the AVCHD format (it only allowed for transcoding AVCHD to Apple Intermediate Codec (AIC) provided that the camera was actually attached to the computer - it did not convert AVCHD files stored elsewhere and is currently for Intel processors only), imported iMovie '08 projects and included 50 new filters. It did not include Soundtrack 1.5, but it still included LiveType which enables users to create advanced text for the movies they created in Final Cut. The price was dropped from $299 for version 3.5 to $199 for version 4.0. In June 2011, Final Cut Express was officially discontinued, in favor of Final Cut Pro X. == Features == Final Cut Express' interface was identical to that of Final Cut Pro, but lacks some film-specific features, including Cinema Tools, multi-cam editing, batch capture, and a time code view. The program performed 32 undo operations, while Final Cut Pro did 99 [2]. Features the program did include were: The ability to keyframe filters Dynamic RT, which changes real-time settings on-the-fly Motion path keyframing Opacity keyframing Ripple, roll, slip, slide and blade edits Picture-in-picture and split-screen effects Up to 99 video tracks and 12 compositing modes Up to 99 audio tracks Motion project import Two-way color correction. Chroma key One feature of Final Cut Express that was not available in Final Cut Pro is the ability to import iMovie '08 projects (though transitions are not preserved). === RT Extreme === Inherited from Final Cut Pro, Final Cut Express features RT Extreme, which allows previews of some video filters and transitions without rendering. Audio that is not in the native AIFF file format needs rendering before it can be played back. RT Extreme has three modes: 'Safe', for seeing multiple video layers at a quality that more or less guarantees a smooth playback; 'Unlimited', which allows the maximum number of composited video layers to be viewed at the same time; and 'Dynamic', which alternates between these settings depending on how many simultaneous video tracks are present. Frame dropping may result from using 'Unlimited' on low-resource machines. === Boris Calligraphy === Like Final Cut Pro, Express also comes with Boris Calligraphy, a plugin for advanced titling and scrolling/crawling titles more sophisticated than the ones that can be created with the built-in title overlays. Calligraphy has a WYSIWYG interface and features wrapping, alignment, leading, kerning and tracking features, as well as allowing up to five custom outlines and five custom drop shadows to be defined for a selected portion of the title. == Soundtrack == Prior to version 4, Final Cut Express included Soundtrack 1.5, a music program similar to the consumer-level GarageBand, but designed for videographers who wish to add music to their films. Soundtrack comes with around 4,000 professionally recorded instrument loops and sound effects that can be arranged in multiple tracks beneath the video track. To use Soundtrack, users export their Final Cut Express sequence, or a marked portion thereof, as a reference file, which can include scoring markers defined in the timeline. This reference file can be imported as the video track in Soundtrack. Soundtrack is functionally and visually identical to Soundtrack Pro's multitrack editing mode, but includes fewer Logic plugins and lacks the highly regarded noise removal tool. Soundtrack was removed from Final Cut Express 4, which lowered its price and may have encouraged people to buy Logic Express.
Wetware computer
A wetware computer is an organic computer (which can also be known as an artificial organic brain or a neurocomputer) composed of organic material "wetware" such as "living" neurons. Wetware computers composed of neurons are different than conventional computers because they use biological materials, and offer the possibility of substantially more energy-efficient computing. While a wetware computer is still largely conceptual, there has been limited success with construction and prototyping, which has acted as a proof of the concept's realistic application to computing in the future. The most notable prototypes have stemmed from the research completed by biological engineer William Ditto during his time at the Georgia Institute of Technology. His work constructing a simple neurocomputer capable of basic addition from leech neurons in 1999 was a significant discovery for the concept. This research was a primary example driving interest in creating these artificially constructed, but still organic brains. == Origins and theoretical foundations == The term wetware came from cyberpunk fiction, notably through Gibson's Neuromancer, but was quickly taken up in scientific literature to explain computation by biological material. Theories of early biological computation borrowed from Alan Turing's morphogenesis model, which showed that chemical interactions could produce complex patterns without centralized control. Hopfield's associative memory networks also provided a foundation for biological information systems with fault tolerance and self-organization. == Major characteristics and processes == Biological wetware systems demonstrate dynamic reconfigurability underpinned by neuroplasticity and enable continuous learning and adaptation. Reaction-diffusion-based computing and molecular logic gates allow spatially parallel information processing unachievable in conventional systems. These systems also show fault tolerance and self-repair at the cellular and network level. The development of cerebral organoids—miniature lab-grown brains—demonstrates spontaneous learning behavior and suggests biological tissue as a viable computational substrate. == Overview == The concept of wetware is an application of specific interest to the field of computer manufacturing. Moore's law, which states that the number of transistors which can be placed on a silicon chip is doubled roughly every two years, has acted as a goal for the industry for decades, but as the size of computers continues to decrease, the ability to meet this goal has become more difficult, threatening to reach a plateau. Due to the difficulty in reducing the size of computers because of size limitations of transistors and integrated circuits, wetware provides an unconventional alternative. A wetware computer composed of neurons is an ideal concept because, unlike conventional materials which operate in binary (on/off), a neuron can shift between thousands of states, constantly altering its chemical conformation, and redirecting electrical pulses through over 200,000 channels in any of its many synaptic connections. Because of this large difference in the possible settings for any one neuron, compared to the binary limitations of conventional computers, the space limitations are far fewer. == Background == The concept of wetware is distinct and unconventional and draws slight resonance with both hardware and software from conventional computers. While hardware is understood as the physical architecture of traditional computational devices, comprising integrated circuits and supporting infrastructure, software represents the encoded architecture of storage and instructions. Wetware is a separate concept that uses the formation of organic molecules, mostly complex cellular structures (such as neurons), to create a computational device such as a computer. In wetware, the ideas of hardware and software are intertwined and interdependent. The molecular and chemical composition of the organic or biological structure would represent not only the physical structure of the wetware but also the software, being continually reprogrammed by the discrete shifts in electrical pulses and chemical concentration gradients as the molecules change their structures to communicate signals. The responsiveness of a cell, proteins, and molecules to changing conformations, both within their structures and around them, ties the idea of internal programming and external structure together in a way that is alien to the current model of conventional computer architecture. The structure of wetware represents a model where the external structure and internal programming are interdependent and unified; meaning that changes to the programming or internal communication between molecules of the device would represent a physical change in the structure. The dynamic nature of wetware borrows from the function of complex cellular structures in biological organisms. The combination of "hardware" and "software" into one dynamic, and interdependent system which uses organic molecules and complexes to create an unconventional model for computational devices is a specific example of applied biorobotics. === The cell as a model of wetware === Cells in many ways can be seen as their form of naturally occurring wetware, similar to the concept that the human brain is the preexisting model system for complex wetware. In his book Wetware: A Computer in Every Living Cell (2009) Dennis Bray explains his theory that cells, which are the most basic form of life, are just a highly complex computational structure, like a computer. To simplify one of his arguments a cell can be seen as a type of computer, using its structured architecture. In this architecture, much like a traditional computer, many smaller components operate in tandem to receive input, process the information, and compute an output. In an overly simplified, non-technical analysis, cellular function can be broken into the following components: Information and instructions for execution are stored as DNA in the cell, RNA acts as a source for distinctly encoded input, processed by ribosomes and other transcription factors to access and process the DNA and to output a protein. Bray's argument in favor of viewing cells and cellular structures as models of natural computational devices is important when considering the more applied theories of wetware to biorobotics. === Biorobotics === Wetware and biorobotics are closely related concepts, which both borrow from similar overall principles. A biorobotic structure can be defined as a system modeled from a preexisting organic complex or model such as cells (neurons) or more complex structures like organs (brain) or whole organisms. Unlike wetware, the concept of biorobotics is not always a system composed of organic molecules, but instead could be composed of conventional material which is designed and assembled in a structure similar or derived from a biological model. Biorobotics have many applications and are used to address the challenges of conventional computer architecture. Conceptually, designing a program, robot, or computational device after a preexisting biological model such as a cell, or even a whole organism, provides the engineer or programmer the benefits of incorporating into the structure the evolutionary advantages of the model. == Effects on users == Wetware technologies such as BCIs and neuromorphic chips offer new possibilities for user autonomy. For those with disabilities, such systems could restore motor or sensory functions and enhance quality of life. However, these technologies raise ethical questions: cognitive privacy, consent over biological data, and risk of exploitation. Without proper oversight, wetware technologies may also widen inequality, favoring those with access to cognitive enhancements. Open governance frameworks and ethical AI design grounded in neuro ethics will be essential. With the development of wetware devices, disparities in access could exacerbate social inequalities, benefiting those who have resources to enhance cognitive or physical abilities. It is necessary to create strong ethical frameworks, inclusive development practices, and open systems of governance to reduce risks and make sure that wetware advances are beneficial to all segments of society. == Applications and goals == === Basic neurocomputer composed of leech neurons === In 1999 William Ditto and his team of researchers at Georgia Institute of Technology and Emory University created a basic form of a wetware computer capable of simple addition by harnessing leech neurons. Leeches were used as a model organism due to the large size of their neuron, and the ease associated with their collection and manipulation. However, these results have never been published in a peer-reviewed journal, prompting questions about the validity of the claims. The computer was able to complete basic addition through electrical probes
Grid-oriented storage
Grid-oriented Storage (GOS) was a term used for data storage by a university project during the era when the term grid computing was popular. == Description == GOS was a successor of the term network-attached storage (NAS). GOS systems contained hard disks, often RAIDs (redundant arrays of independent disks), like traditional file servers. GOS was designed to deal with long-distance, cross-domain and single-image file operations, which is typical in Grid environments. GOS behaves like a file server via the file-based GOS-FS protocol to any entity on the grid. Similar to GridFTP, GOS-FS integrates a parallel stream engine and Grid Security Infrastructure (GSI). Conforming to the universal VFS (Virtual Filesystem Switch), GOS-FS can be pervasively used as an underlying platform to best utilize the increased transfer bandwidth and accelerate the NFS/CIFS-based applications. GOS can also run over SCSI, Fibre Channel or iSCSI, which does not affect the acceleration performance, offering both file level protocols and block level protocols for storage area network (SAN) from the same system. In a grid infrastructure, resources may be geographically distant from each other, produced by differing manufacturers, and have differing access control policies. This makes access to grid resources dynamic and conditional upon local constraints. Centralized management techniques for these resources are limited in their scalability both in terms of execution efficiency and fault tolerance. Provision of services across such platforms requires a distributed resource management mechanism and the peer-to-peer clustered GOS appliances allow a single storage image to continue to expand, even if a single GOS appliance reaches its capacity limitations. The cluster shares a common, aggregate presentation of the data stored on all participating GOS appliances. Each GOS appliance manages its own internal storage space. The major benefit of this aggregation is that clustered GOS storage can be accessed by users as a single mount point. GOS products fit the thin-server categorization. Compared with traditional “fat server”-based storage architectures, thin-server GOS appliances deliver numerous advantages, such as the alleviation of potential network/grid bottle-necks, CPU and OS optimized for I/O only, ease of installation, remote management and minimal maintenance, low cost and Plug and Play, etc. Examples of similar innovations include NAS, printers, fax machines, routers and switches. An Apache server has been installed in the GOS operating system, ensuring an HTTPS-based communication between the GOS server and an administrator via a Web browser. Remote management and monitoring makes it easy to set up, manage, and monitor GOS systems. == History == Frank Zhigang Wang and Na Helian proposed a funding proposal to the UK government titled “Grid-Oriented Storage (GOS): Next Generation Data Storage System Architecture for the Grid Computing Era” in 2003. The proposal was approved and granted one million pounds in 2004. The first prototype was constructed in 2005 at Centre for Grid Computing, Cambridge-Cranfield High Performance Computing Facility. The first conference presentation was at IEEE Symposium on Cluster Computing and Grid (CCGrid), 9–12 May 2005, Cardiff, UK. As one of the five best work-in-progress, it was included in the IEEE Distributed Systems Online. In 2006, the GOS architecture and its implementations was published in IEEE Transactions on Computers, titled “Grid-oriented Storage: A Single-Image, Cross-Domain, High-Bandwidth Architecture”. Starting in January 2007, demonstrations were presented at Princeton University, Cambridge University Computer Lab and others. By 2013, the Cranfield Centre still used future tense for the project. Peer-to-peer file sharings use similar techniques.
Algorithmic transparency
Algorithmic transparency is the principle that the factors that influence the decisions made by algorithms should be visible, or transparent, to the people who use, regulate, and are affected by systems that employ those algorithms. Although the phrase was coined in 2016 by Nicholas Diakopoulos and Michael Koliska about the role of algorithms in deciding the content of digital journalism services, the underlying principle dates back to the 1970s and the rise of automated systems for scoring consumer credit. The phrases "algorithmic transparency" and "algorithmic accountability" are sometimes used interchangeably – especially since they were coined by the same people – but they have subtly different meanings. Specifically, "algorithmic transparency" states that the inputs to the algorithm and the algorithm's use itself must be known, but they need not be fair. "Algorithmic accountability" implies that the organizations that use algorithms must be accountable for the decisions made by those algorithms, even though the decisions are being made by a machine, and not by a human being. Current research around algorithmic transparency interested in both societal effects of accessing remote services running algorithms, as well as mathematical and computer science approaches that can be used to achieve algorithmic transparency. In the United States, the Federal Trade Commission's Bureau of Consumer Protection studies how algorithms are used by consumers by conducting its own research on algorithmic transparency and by funding external research. In the European Union, the data protection laws that came into effect in May 2018 include a "right to explanation" of decisions made by algorithms, though it is unclear what this means. Furthermore, the European Union founded The European Center for Algorithmic Transparency (ECAT).
Informedia Digital Library
The Informedia Digital Library is an ongoing research program at Carnegie Mellon University to build search engines and information visualization technology for many types of media. The program has carried out research on spoken document retrieval, video information retrieval, video segmentation, face recognition, and cross-language information retrieval. The Lycos search engine was an early product of the Informedia Digital Library Project. The project is led by Howard Wactlar. Researchers on the project have included: Michael Mauldin, Alex Hauptmann, Michael Christel, Michael Witbrock, Raj Reddy, Takeo Kanade and Scott Stevens.
Color moments
Color moments are measures that characterise color distribution in an image in the same way that central moments uniquely describe a probability distribution. Color moments are mainly used for color indexing purposes as features in image retrieval applications in order to compare how similar two images are based on color. Usually one image is compared to a database of digital images with pre-computed features in order to find and retrieve a similar Image. Each comparison between images results in a similarity score, and the lower this score is the more identical the two images are supposed to be. == Overview == Color moments are scaling and rotation invariant. It is usually the case that only the first three color moments are used as features in image retrieval applications as most of the color distribution information is contained in the low-order moments. Since color moments encode both shape and color information they are a good feature to use under changing lighting conditions, but they cannot handle occlusion very successfully. Color moments can be computed for any color model. Three color moments are computed per channel (e.g. 9 moments if the color model is RGB and 12 moments if the color model is CMYK). Computing color moments is done in the same way as computing moments of a probability distribution. === Mean === The first color moment can be interpreted as the average color in the image, and it can be calculated by using the following formula E i = ∑ j = 1 N 1 N p i j {\displaystyle E_{i}=\textstyle \sum _{j=1}^{N}{\frac {1}{N}}p_{ij}} where N is the number of pixels in the image and p i j {\displaystyle p_{ij}} is the value of the j-th pixel of the image at the i-th color channel. === Standard Deviation === The second color moment is the standard deviation, which is obtained by taking the square root of the variance of the color distribution. σ i = ( 1 N ∑ j = 1 N ( p i j − E i ) 2 ) {\displaystyle \sigma _{i}={\sqrt {({\frac {1}{N}}\textstyle \sum _{j=1}^{N}(p_{ij}-E_{i})^{2})}}} where E i {\displaystyle E_{i}} is the mean value, or first color moment, for the i-th color channel of the image. === Skewness === The third color moment is the skewness. It measures how asymmetric the color distribution is, and thus it gives information about the shape of the color distribution. Skewness can be computed with the following formula: s i = ( 1 N ∑ j = 1 N ( p i j − E i ) 3 ) 3 σ i {\displaystyle s_{i}={\frac {\sqrt[{3}]{\left({\frac {1}{N}}\textstyle \sum _{j=1}^{N}(p_{ij}-E_{i})^{3}\right)}}{\sigma _{i}}}} === Kurtosis === Kurtosis is the fourth color moment, and, similarly to skewness, it provides information about the shape of the color distribution. More specifically, kurtosis is a measure of how extreme the tails are in comparison to the normal distribution. === Higher-order color moments === Higher-order color moments are usually not part of the color moments feature set in image retrieval tasks as they require more data in order to obtain a good estimate of their value, and also the lower-order moments generally provide enough information. == Applications == Color moments have significant applications in image retrieval. They can be used in order to compare how similar two images are. This is a relatively new approach to color indexing. The greatest advantage of using color moments comes from the fact that there is no need to store the complete color distribution. This greatly speeds up image retrieval since there are less features to compare. In addition, the first three color moments have the same units, which allows for comparison between them. === Color indexing === Color indexing is the main application of color moments. Images can be indexed, and the index will contain the computed color moments. Then, if someone has a particular image and wants to find similar images in the database, the color moments of the image of interest will also be computed. After that the following function will be used in order to compute a similarity score between the image of interest and all the images in the database: d m o m ( H , I ) = ∑ i = 1 r w i 1 | E i 1 − E i 2 | + w i 2 | σ i 1 − σ i 2 | + w i 3 | s i 1 − s i 2 | {\displaystyle d_{mom}(H,I)=\textstyle \sum _{i=1}^{r}w_{i1}|E_{i}^{1}-E_{i}^{2}|+w_{i2}|\sigma _{i}^{1}-\sigma _{i}^{2}|+w_{i3}|s_{i}^{1}-s_{i}^{2}|} where: H and I are the color distributions of the two images that are being compared i is the channel index and r is the total number of channels E i 1 {\displaystyle E_{i}^{1}} and E i 2 {\displaystyle E_{i}^{2}} are the first order moments computed for the image distributions. σ i 1 {\displaystyle \sigma _{i}^{1}} and σ i 2 {\displaystyle \sigma _{i}^{2}} are the second order moments computed for the image distributions. s_i^1 and s_i^2 are the third order moments computed for the image distributions. w i 1 {\displaystyle w_{i1}} , w i 2 {\displaystyle w_{i2}} , and w i 3 {\displaystyle w_{i3}} are weights, specified by the user, for each of the three color moments used. Finally, the images in the database will be ranked according to the computed similarity score with the image of interest, and the database images with the lowest d m o m ( H , I ) {\displaystyle d_{mom}(H,I)} value should be retrieved. "A retrieval based on d m o m ( H , I ) {\displaystyle d_{mom}(H,I)} may produce false positives because the index contains no information about the correlation between the color channels". == Example == A simple and concise example of the use of color moments for image retrieval tasks is illustrated in. Consider having several test images in a database and a "New Image". The goal is to retrieve images from the database that are similar to the "New Image". The first three color moments are used as features. There are several steps in this computation. Image preprocessing (Optional) - The image preprocessing step of the computation process is optional. For example, in this step all images could be modified to be the same size (in terms of pixels). However, since color moments are invariant to scaling, it is not necessary to make all images the same width and height. Computing the features - Use the color moments formulae in order to compute the first three moments for each of the color channels in the image. For example, if the HSV color space is used, this means that for each of the images, 9 features in total will be computed (the first three order moments for the Hue, Saturation, and Value channels). Calculating the similarity score - After computing the color moments the weights for each of the moments in the d m o m ( H , I ) {\displaystyle d_{mom}(H,I)} function should be determined by the user. The weights have to be adjusted each time in accordance with the application or condition and quality of the images. Following that the d m o m ( H , I ) {\displaystyle d_{mom}(H,I)} function is used to calculate a similarity score for the "New Image" and each of the images in the database. Ranking and image retrieval - From the previous step the d m o m ( H , I ) {\displaystyle d_{mom}(H,I)} values were obtained. Now a comparison of these values can be made in order to decide which of the images in the database are more similar to the "New Image", and thus rank the database images accordingly. The smaller the d m o m ( H , I ) {\displaystyle d_{mom}(H,I)} value is the more similar the two color distributions are supposed to be. Finally, some of the top ranked images (the ones with the smallest d m o m ( H , I ) {\displaystyle d_{mom}(H,I)} value) from the database are retrieved.
Sparse identification of non-linear dynamics
Sparse identification of nonlinear dynamics (SINDy) is a data-driven algorithm for obtaining dynamical systems from data. Given a series of snapshots of a dynamical system and its corresponding time derivatives, SINDy performs a sparsity-promoting regression (such as LASSO and sparse Bayesian inference) on a library of nonlinear candidate functions of the snapshots against the derivatives to find the governing equations. This procedure relies on the assumption that most physical systems only have a few dominant terms which dictate the dynamics, given an appropriately selected coordinate system and quality training data. It has been applied to identify the dynamics of fluids, based on proper orthogonal decomposition, as well as other complex dynamical systems, such as biological networks. == Mathematical Overview == First, consider a dynamical system of the form x ˙ = d d t x ( t ) = f ( x ( t ) ) , {\displaystyle {\dot {\textbf {x}}}={\frac {d}{dt}}{\textbf {x}}(t)={\textbf {f}}({\textbf {x}}(t)),} where x ( t ) ∈ R n {\displaystyle {\textbf {x}}(t)\in \mathbb {R} ^{n}} is a state vector (snapshot) of the system at time t {\displaystyle t} and the function f ( x ( t ) ) {\displaystyle {\textbf {f}}({\textbf {x}}(t))} defines the equations of motion and constraints of the system. The time derivative may be either prescribed or numerically approximated from the snapshots. With x {\displaystyle {\textbf {x}}} and x ˙ {\displaystyle {\dot {\textbf {x}}}} sampled at m {\displaystyle m} equidistant points in time ( t 1 , t 2 , ⋯ , t m {\displaystyle t_{1},t_{2},\cdots ,t_{m}} ), these can be arranged into matrices of the form X = [ x T ( t 1 ) x T ( t 2 ) ⋮ x T ( t m ) ] = [ x 1 ( t 1 ) x 2 ( t 1 ) ⋯ x n ( t 1 ) x 1 ( t 2 ) x 2 ( t 2 ) ⋯ x n ( t 2 ) ⋮ ⋮ ⋱ ⋮ x 1 ( t m ) x 2 ( t m ) ⋯ x n ( t m ) ] , {\displaystyle {\bf {{X}={\begin{bmatrix}\mathbf {x} ^{\mathsf {T}}(t_{1})\\\mathbf {x} ^{\mathsf {T}}(t_{2})\\\vdots \\\mathbf {x} ^{\mathsf {T}}(t_{m})\end{bmatrix}}={\begin{bmatrix}x_{1}(t_{1})&x_{2}(t_{1})&\cdots &x_{n}(t_{1})\\x_{1}(t_{2})&x_{2}(t_{2})&\cdots &x_{n}(t_{2})\\\vdots &\vdots &\ddots &\vdots \\x_{1}(t_{m})&x_{2}(t_{m})&\cdots &x_{n}(t_{m})\end{bmatrix}},}}} and similarly for X ˙ {\displaystyle {\dot {\mathbf {X} }}} . Next, a library Θ ( X ) {\displaystyle \mathbf {\Theta } (\mathbf {X} )} of nonlinear candidate functions of the columns of X {\displaystyle {\textbf {X}}} is constructed, which may be constant, polynomial, or more exotic functions (like trigonometric and rational terms, and so on): Θ ( X ) = [ | | | | | | 1 X X 2 X 3 ⋯ sin ( X ) cos ( X ) ⋯ | | | | | | ] {\displaystyle \ \ \ {\bf {{\Theta }({\bf {{X})={\begin{bmatrix}\vline &\vline &\vline &\vline &&\vline &\vline &\\1&{\bf {X}}&{\bf {{X}^{2}}}&{\bf {{X}^{3}}}&\cdots &\sin({\bf {{X})}}&\cos({\bf {{X})}}&\cdots \\\vline &\vline &\vline &\vline &&\vline &\vline &\end{bmatrix}}}}}}} The number of possible model structures from this library is combinatorially high. f ( x ( t ) ) {\displaystyle {\textbf {f}}({\textbf {x}}(t))} is then substituted by Θ ( X ) {\displaystyle {\bf {{\Theta }({\textbf {X}})}}} and a vector of coefficients Ξ = [ ξ 1 ξ 2 ⋯ ξ n ] {\displaystyle {\bf {{\Xi }=\left[{\bf {{\xi }_{1}{\bf {{\xi }_{2}\cdots {\bf {{\xi }_{n}}}}}}}\right]}}} determining the active terms in f ( x ( t ) ) {\displaystyle {\textbf {f}}({\textbf {x}}(t))} : X ˙ = Θ ( X ) Ξ {\displaystyle {\dot {\bf {X}}}={\bf {{\Theta }({\bf {{X}){\bf {\Xi }}}}}}} Because only a few terms are expected to be active at each point in time, an assumption is made that f ( x ( t ) ) {\displaystyle {\textbf {f}}({\textbf {x}}(t))} admits a sparse representation in Θ ( X ) {\displaystyle {\bf {{\Theta }({\textbf {X}})}}} . This then becomes an optimization problem in finding a sparse Ξ {\displaystyle {\bf {\Xi }}} which optimally embeds X ˙ {\displaystyle {\dot {\textbf {X}}}} . In other words, a parsimonious model is obtained by performing least squares regression on the system (4) with sparsity-promoting ( L 1 {\displaystyle L_{1}} ) regularization ξ k = arg min ξ k ′ | | X ˙ k − Θ ( X ) ξ k ′ | | 2 + λ | | ξ k ′ | | 1 , {\displaystyle {\bf {{\xi }_{k}={\underset {\bf {{\xi }'_{k}}}{\arg \min }}\left|\left|{\dot {\bf {X}}}_{k}-{\bf {{\Theta }({\bf {{X}){\bf {{\xi }'_{k}}}}}}}\right|\right|_{2}+\lambda \left|\left|{\bf {{\xi }'_{k}}}\right|\right|_{1},}}} where λ {\displaystyle \lambda } is a regularization parameter. Finally, the sparse set of ξ k {\displaystyle {\bf {{\xi }_{k}}}} can be used to reconstruct the dynamical system: x ˙ k = Θ ( x ) ξ k {\displaystyle {\dot {x}}_{k}={\bf {{\Theta }({\bf {{x}){\bf {{\xi }_{k}}}}}}}}