Batch normalization

In artificial neural networks, batch normalization (also known as batch norm) is a normalization technique used to make training faster and more stable by adjusting the inputs to each layer—re-centering them around zero and re-scaling them to a standard size. It was introduced by Sergey Ioffe and Christian Szegedy in 2015. Experts still debate why batch normalization works so well. It was initially thought to tackle internal covariate shift, a problem where parameter initialization and changes in the distribution of the inputs of each layer affect the learning rate of the network. However, newer research suggests it doesn’t fix this shift but instead smooths the objective function—a mathematical guide the network follows to improve—enhancing performance. In very deep networks, batch normalization can initially cause a severe gradient explosion—where updates to the network grow uncontrollably large—but this is managed with shortcuts called skip connections in residual networks. Another theory is that batch normalization adjusts data by handling its size and path separately, speeding up training. == Internal covariate shift == Each layer in a neural network has inputs that follow a specific distribution, which shifts during training due to two main factors: the random starting values of the network’s settings (parameter initialization) and the natural variation in the input data. This shifting pattern affecting the inputs to the network’s inner layers is called internal covariate shift. While a strict definition isn’t fully agreed upon, experiments show that it involves changes in the means and variances of these inputs during training. Batch normalization was first developed to address internal covariate shift. During training, as the parameters of preceding layers adjust, the distribution of inputs to the current layer changes accordingly, such that the current layer needs to constantly readjust to new distributions. This issue is particularly severe in deep networks, because small changes in shallower hidden layers will be amplified as they propagate within the network, resulting in significant shift in deeper hidden layers. Batch normalization was proposed to reduced these unwanted shifts to speed up training and produce more reliable models. Beyond possibly tackling internal covariate shift, batch normalization offers several additional advantages. It allows the network to use a higher learning rate—a setting that controls how quickly the network learns—without causing problems like vanishing or exploding gradients, where updates become too small or too large. It also appears to have a regularizing effect, improving the network’s ability to generalize to new data, reducing the need for dropout, a technique used to prevent overfitting (when a model learns the training data too well and fails on new data). Additionally, networks using batch normalization are less sensitive to the choice of starting settings or learning rates, making them more robust and adaptable. == Procedures == === Transformation === In a neural network, batch normalization is achieved through a normalization step that fixes the means and variances of each layer's inputs. Ideally, the normalization would be conducted over the entire training set, but to use this step jointly with stochastic optimization methods, it is impractical to use the global information. Thus, normalization is restrained to each mini-batch in the training process. Let us use B to denote a mini-batch of size m of the entire training set. The empirical mean and variance of B could thus be denoted as μ B = 1 m ∑ i = 1 m x i {\displaystyle \mu _{B}={\frac {1}{m}}\sum _{i=1}^{m}x_{i}} and σ B 2 = 1 m ∑ i = 1 m ( x i − μ B ) 2 {\displaystyle \sigma _{B}^{2}={\frac {1}{m}}\sum _{i=1}^{m}(x_{i}-\mu _{B})^{2}} . For a layer of the network with d-dimensional input, x = ( x ( 1 ) , . . . , x ( d ) ) {\displaystyle x=(x^{(1)},...,x^{(d)})} , each dimension of its input is then normalized (i.e. re-centered and re-scaled) separately, x ^ i ( k ) = x i ( k ) − μ B ( k ) ( σ B ( k ) ) 2 + ϵ {\displaystyle {\hat {x}}_{i}^{(k)}={\frac {x_{i}^{(k)}-\mu _{B}^{(k)}}{\sqrt {\left(\sigma _{B}^{(k)}\right)^{2}+\epsilon }}}} , where k ∈ [ 1 , d ] {\displaystyle k\in [1,d]} and i ∈ [ 1 , m ] {\displaystyle i\in [1,m]} ; μ B ( k ) {\displaystyle \mu _{B}^{(k)}} and σ B ( k ) {\displaystyle \sigma _{B}^{(k)}} are the per-dimension mean and standard deviation, respectively. ϵ {\displaystyle \epsilon } is added in the denominator for numerical stability and is an arbitrarily small positive constant. The resulting normalized activation x ^ ( k ) {\displaystyle {\hat {x}}^{(k)}} have zero mean and unit variance, if ϵ {\displaystyle \epsilon } is not taken into account. To restore the representation power of the network, a transformation step then follows as y i ( k ) = γ ( k ) x ^ i ( k ) + β ( k ) {\displaystyle y_{i}^{(k)}=\gamma ^{(k)}{\hat {x}}_{i}^{(k)}+\beta ^{(k)}} , where the parameters γ ( k ) {\displaystyle \gamma ^{(k)}} and β ( k ) {\displaystyle \beta ^{(k)}} are subsequently learned in the optimization process. Formally, the operation that implements batch normalization is a transform B N γ ( k ) , β ( k ) : x 1... m ( k ) → y 1... m ( k ) {\displaystyle BN_{\gamma ^{(k)},\beta ^{(k)}}:x_{1...m}^{(k)}\rightarrow y_{1...m}^{(k)}} called the Batch Normalizing transform. The output of the BN transform y ( k ) = B N γ ( k ) , β ( k ) ( x ( k ) ) {\displaystyle y^{(k)}=BN_{\gamma ^{(k)},\beta ^{(k)}}(x^{(k)})} is then passed to other network layers, while the normalized output x ^ i ( k ) {\displaystyle {\hat {x}}_{i}^{(k)}} remains internal to the current layer. === Backpropagation === The described BN transform is a differentiable operation, and the gradient of the loss l {\displaystyle l} with respect to the different parameters can be computed directly with the chain rule. Specifically, ∂ l ∂ y i ( k ) {\displaystyle {\frac {\partial l}{\partial y_{i}^{(k)}}}} depends on the choice of activation function, and the gradient against other parameters could be expressed as a function of ∂ l ∂ y i ( k ) {\displaystyle {\frac {\partial l}{\partial y_{i}^{(k)}}}} : ∂ l ∂ x ^ i ( k ) = ∂ l ∂ y i ( k ) γ ( k ) {\displaystyle {\frac {\partial l}{\partial {\hat {x}}_{i}^{(k)}}}={\frac {\partial l}{\partial y_{i}^{(k)}}}\gamma ^{(k)}} , ∂ l ∂ γ ( k ) = ∑ i = 1 m ∂ l ∂ y i ( k ) x ^ i ( k ) {\displaystyle {\frac {\partial l}{\partial \gamma ^{(k)}}}=\sum _{i=1}^{m}{\frac {\partial l}{\partial y_{i}^{(k)}}}{\hat {x}}_{i}^{(k)}} , ∂ l ∂ β ( k ) = ∑ i = 1 m ∂ l ∂ y i ( k ) {\displaystyle {\frac {\partial l}{\partial \beta ^{(k)}}}=\sum _{i=1}^{m}{\frac {\partial l}{\partial y_{i}^{(k)}}}} , ∂ l ∂ σ B ( k ) 2 = ∑ i = 1 m ∂ l ∂ y i ( k ) ( x i ( k ) − μ B ( k ) ) ( − γ ( k ) 2 ( σ B ( k ) 2 + ϵ ) − 3 / 2 ) {\displaystyle {\frac {\partial l}{\partial \sigma _{B}^{(k)^{2}}}}=\sum _{i=1}^{m}{\frac {\partial l}{\partial y_{i}^{(k)}}}(x_{i}^{(k)}-\mu _{B}^{(k)})\left(-{\frac {\gamma ^{(k)}}{2}}(\sigma _{B}^{(k)^{2}}+\epsilon )^{-3/2}\right)} , ∂ l ∂ μ B ( k ) = ∑ i = 1 m ∂ l ∂ y i ( k ) − γ ( k ) σ B ( k ) 2 + ϵ + ∂ l ∂ σ B ( k ) 2 1 m ∑ i = 1 m ( − 2 ) ⋅ ( x i ( k ) − μ B ( k ) ) {\displaystyle {\frac {\partial l}{\partial \mu _{B}^{(k)}}}=\sum _{i=1}^{m}{\frac {\partial l}{\partial y_{i}^{(k)}}}{\frac {-\gamma ^{(k)}}{\sqrt {\sigma _{B}^{(k)^{2}}+\epsilon }}}+{\frac {\partial l}{\partial \sigma _{B}^{(k)^{2}}}}{\frac {1}{m}}\sum _{i=1}^{m}(-2)\cdot (x_{i}^{(k)}-\mu _{B}^{(k)})} , and ∂ l ∂ x i ( k ) = ∂ l ∂ x ^ i ( k ) 1 σ B ( k ) 2 + ϵ + ∂ l ∂ σ B ( k ) 2 2 ( x i ( k ) − μ B ( k ) ) m + ∂ l ∂ μ B ( k ) 1 m {\displaystyle {\frac {\partial l}{\partial x_{i}^{(k)}}}={\frac {\partial l}{\partial {\hat {x}}_{i}^{(k)}}}{\frac {1}{\sqrt {\sigma _{B}^{(k)^{2}}+\epsilon }}}+{\frac {\partial l}{\partial \sigma _{B}^{(k)^{2}}}}{\frac {2(x_{i}^{(k)}-\mu _{B}^{(k)})}{m}}+{\frac {\partial l}{\partial \mu _{B}^{(k)}}}{\frac {1}{m}}} . === Inference === During the training stage, the normalization steps depend on the mini-batches to ensure efficient and reliable training. However, in the inference stage, this dependence is not useful any more. Instead, the normalization step in this stage is computed with the population statistics such that the output could depend on the input in a deterministic manner. The population mean, E [ x ( k ) ] {\displaystyle E[x^{(k)}]} , and variance, Var ⁡ [ x ( k ) ] {\displaystyle \operatorname {Var} [x^{(k)}]} , are computed as: E [ x ( k ) ] = E B [ μ B ( k ) ] {\displaystyle E[x^{(k)}]=E_{B}[\mu _{B}^{(k)}]} , and Var ⁡ [ x ( k ) ] = m m − 1 E B [ ( σ B ( k ) ) 2 ] {\displaystyle \operatorname {Var} [x^{(k)}]={\frac {m}{m-1}}E_{B}[\left(\sigma _{B}^{(k)}\right)^{2}]} . The population statistics thus is a complete representation of the mini-batches. The BN transform in the inference step thus becomes y ( k ) = B N γ ( k ) , β ( k ) inf ( x ( k ) ) = γ ( k ) x ( k ) − E [ x ( k ) ] Var ⁡ [ x ( k ) ] + ϵ + β

A Logical Calculus of the Ideas Immanent in Nervous Activity

"A Logical Calculus of the Ideas Immanent in Nervous Activity" is a 1943 paper written by Warren Sturgis McCulloch and Walter Pitts, published in the journal The Bulletin of Mathematical Biophysics. The paper proposed a mathematical model of the nervous system as a network of simple logical elements, later known as artificial neurons, or McCulloch–Pitts neurons. These neurons receive inputs, perform a weighted sum, and fire an output signal based on a threshold function. By connecting these units in various configurations, McCulloch and Pitts demonstrated that their model could perform all logical functions. It is a seminal work in cognitive science, computational neuroscience, computer science, and artificial intelligence. It was a foundational result in automata theory. John von Neumann cited it as a significant result. == Mathematics == The artificial neuron used in the original paper is slightly different from the modern version. They considered neural networks that operate in discrete steps of time t = 0 , 1 , … {\displaystyle t=0,1,\dots } . The neural network contains a number of neurons. Let the state of a neuron i {\displaystyle i} at time t {\displaystyle t} be N i ( t ) {\displaystyle N_{i}(t)} . The state of a neuron can either be 0 or 1, standing for "not firing" and "firing". Each neuron also has a firing threshold θ {\displaystyle \theta } , such that it fires if the total input exceeds the threshold. Each neuron can connect to any other neuron (including itself) with positive synapses (excitatory) or negative synapses (inhibitory). That is, each neuron can connect to another neuron with a weight w {\displaystyle w} taking an integer value. A peripheral afferent is a neuron with no incoming synapses. We can regard each neural network as a directed graph, with the nodes being the neurons, and the directed edges being the synapses. A neural network has a circle or a circuit if there exists a directed circle in the graph. Let w i j ( t ) {\displaystyle w_{ij}(t)} be the connection weight from neuron j {\displaystyle j} to neuron i {\displaystyle i} at time t {\displaystyle t} , then its next state is N i ( t + 1 ) = H ( ∑ j = 1 n w i j ( t ) N j ( t ) − θ i ( t ) ) , {\displaystyle N_{i}(t+1)=H\left(\sum _{j=1}^{n}w_{ij}(t)N_{j}(t)-\theta _{i}(t)\right),} where H {\displaystyle H} is the Heaviside step function (outputting 1 if the input is greater than or equal to 0, and 0 otherwise). === Symbolic logic === The paper used, as a logical language for describing neural networks, "Language II" from The Logical Syntax of Language by Rudolf Carnap with some notations taken from Principia Mathematica by Alfred North Whitehead and Bertrand Russell. Language II covers substantial parts of classical mathematics, including real analysis and portions of set theory. To describe a neural network with peripheral afferents N 1 , N 2 , … , N p {\displaystyle N_{1},N_{2},\dots ,N_{p}} and non-peripheral afferents N p + 1 , N p + 2 , … , N n {\displaystyle N_{p+1},N_{p+2},\dots ,N_{n}} they considered logical predicate of form P r ( N 1 , N 2 , … , N p , t ) {\displaystyle Pr(N_{1},N_{2},\dots ,N_{p},t)} where P r {\displaystyle Pr} is a first-order logic predicate function (a function that outputs a boolean), N 1 , … , N p {\displaystyle N_{1},\dots ,N_{p}} are predicates that take t {\displaystyle t} as an argument, and t {\displaystyle t} is the only free variable in the predicate. Intuitively speaking, N 1 , … , N p {\displaystyle N_{1},\dots ,N_{p}} specifies the binary input patterns going into the neural network over all time, and P r ( N 1 , N 2 , … , N n , t ) {\displaystyle Pr(N_{1},N_{2},\dots ,N_{n},t)} is a function that takes some binary input patterns, and constructs an output binary pattern P r ( N 1 , N 2 , … , N n , 0 ) , P r ( N 1 , N 2 , … , N n , 1 ) , … {\displaystyle Pr(N_{1},N_{2},\dots ,N_{n},0),Pr(N_{1},N_{2},\dots ,N_{n},1),\dots } . A logical sentence P r ( N 1 , N 2 , … , N n , t ) {\displaystyle Pr(N_{1},N_{2},\dots ,N_{n},t)} is realized by a neural network iff there exists a time-delay T ≥ 0 {\displaystyle T\geq 0} , a neuron i {\displaystyle i} in the network, and an initial state for the non-peripheral neurons N p + 1 ( 0 ) , … , N n ( 0 ) {\displaystyle N_{p+1}(0),\dots ,N_{n}(0)} , such that for any time t {\displaystyle t} , the truth-value of the logical sentence is equal to the state of the neuron i {\displaystyle i} at time t + T {\displaystyle t+T} . That is, ∀ t = 0 , 1 , 2 , … , P r ( N 1 , N 2 , … , N p , t ) = N i ( t + T ) {\displaystyle \forall t=0,1,2,\dots ,\quad Pr(N_{1},N_{2},\dots ,N_{p},t)=N_{i}(t+T)} === Equivalence === In the paper, they considered some alternative definitions of artificial neural networks, and have shown them to be equivalent, that is, neural networks under one definition realizes precisely the same logical sentences as neural networks under another definition. They considered three forms of inhibition: relative inhibition, absolute inhibition, and extinction. The definition above is relative inhibition. By "absolute inhibition" they meant that if any negative synapse fires, then the neuron will not fire. By "extinction" they meant that if at time t {\displaystyle t} , any inhibitory synapse fires on a neuron i {\displaystyle i} , then θ i ( t + j ) = θ i ( 0 ) + b j {\displaystyle \theta _{i}(t+j)=\theta _{i}(0)+b_{j}} for j = 1 , 2 , 3 , … {\displaystyle j=1,2,3,\dots } , until the next time an inhibitory synapse fires on i {\displaystyle i} . It is required that b j = 0 {\displaystyle b_{j}=0} for all large j {\displaystyle j} . Theorem 4 and 5 state that these are equivalent. They considered three forms of excitation: spatial summation, temporal summation, and facilitation. The definition above is spatial summation (which they pictured as having multiple synapses placed close together, so that the effect of their firing sums up). By "temporal summation" they meant that the total incoming signal is ∑ τ = 0 T ∑ j = 1 n w i j ( t ) N j ( t − τ ) {\displaystyle \sum _{\tau =0}^{T}\sum _{j=1}^{n}w_{ij}(t)N_{j}(t-\tau )} for some T ≥ 1 {\displaystyle T\geq 1} . By "facilitation" they meant the same as extinction, except that b j ≤ 0 {\displaystyle b_{j}\leq 0} . Theorem 6 states that these are equivalent. They considered neural networks that do not change, and those that change by Hebbian learning. That is, they assume that at t = 0 {\displaystyle t=0} , some excitatory synaptic connections are not active. If at any t {\displaystyle t} , both N i ( t ) = 1 , N j ( t ) = 1 {\displaystyle N_{i}(t)=1,N_{j}(t)=1} , then any latent excitatory synapse between i , j {\displaystyle i,j} becomes active. Theorem 7 states that these are equivalent. === Logical expressivity === They considered "temporal propositional expressions" (TPE), which are propositional formulas with one free variable t {\displaystyle t} . For example, N 1 ( t ) ∨ N 2 ( t ) ∧ ¬ N 3 ( t ) {\displaystyle N_{1}(t)\vee N_{2}(t)\wedge \neg N_{3}(t)} is such an expression. Theorem 1 and 2 together showed that neural nets without circles are equivalent to TPE. For neural nets with loops, they noted that "realizable P r {\displaystyle Pr} may involve reference to past events of an indefinite degree of remoteness". These then encodes for sentences like "There was some x such that x was a ψ" or ( ∃ x ) ( ψ x ) {\displaystyle (\exists x)(\psi x)} . Theorems 8 to 10 showed that neural nets with loops can encode all first-order logic with equality and conversely, any looped neural networks is equivalent to a sentence in first-order logic with equality, thus showing that they are equivalent in logical expressiveness. As a remark, they noted that a neural network, if furnished with a tape, scanners, and write-heads, is equivalent to a Turing machine, and conversely, every Turing machine is equivalent to some such neural network. Thus, these neural networks are equivalent to Turing computability and Church's lambda-definability. == Context == === Previous work === The paper built upon several previous strands of work. In the symbolic logic side, it built on the previous work by Carnap, Whitehead, and Russell. This was contributed by Walter Pitts, who had a strong proficiency with symbolic logic. Pitts provided mathematical and logical rigor to McCulloch’s vague ideas on psychons (atoms of psychological events) and circular causality. In the neuroscience side, it built on previous work by the mathematical biology research group centered around Nicolas Rashevsky, of which McCulloch was a member. The paper was published in the Bulletin of Mathematical Biophysics, which was founded by Rashevsky in 1939. During the late 1930s, Rashevsky's research group was producing papers that had difficulty publishing in other journals at the time, so Rashevsky decided to found a new journal exclusively devoted to mathematical biophysics. Also in the Rashevsky's group was Alston Scott Householder, who in 1941 published an abstract model

Wilkinson's Grammar of Graphics

The Grammar of Graphics (GoG) is a grammar-based system for representing graphics to provide grammatical constraints on the composition of data and information visualizations. A graphical grammar differs from a graphics pipeline as it focuses on semantic components such as scales and guides, statistical functions, coordinate systems, marks and aesthetic attributes. For example, a bar chart can be converted into a pie chart by specifying a polar coordinate system without any other change in graphical specification. The grammar of graphics concept was launched by Leland Wilkinson in 2001 (Wilkinson et al., 2001; Wilkinson, 2005) and graphical grammars have since been written in a variety of languages with various parameterisations and extensions. The major implementations of graphical grammars are nViZn created by a team at SPSS/IBM, followed by Polaris focusing on multidimensional relational databases which is commercialised as Tableau, a revised Layered Grammar of Graphics by Hadley Wickham in Ggplot2, and Vega-Lite which is a visualisation grammar with added interactivity. The grammar of graphics continues to evolve with alternate parameterisations, extensions, or new specifications. == Wilkinson's Grammar of Graphics == === Theory === Wilkinson conceived the seven elements of a graphics to be Variables: mapping of objects to values represented in a graphic Algebra: operations to combine variables and specify dimensions of graphs Geometry: creation of geometric graphs from variables Aesthetics: sensory attributes Statistics: functions to change the appearance and representation of graphs Scales: represent variables on measured dimensions Coordinates: mapping to coordinate systems With these, Wilkinson hypothesised that These seven constructs are orthogonal and virtually all known statistical charts can be generated relatively parsimoniously This computational system is not a taxonomy of charts and rather it describes the meaning of what we do when we construct statistical graphics. === Implementations === Wilkinson wrote SYSTAT, a statistical software package, in the early 1980s. This program was noted for its comprehensive graphics, including the first software implementation of the heatmap display now widely used among biologists. After his company grew to 50 employees, he sold it to SPSS in 1995. At SPSS, he assembled a team of graphics programmers who developed the nViZn platform that produces the visualizations in SPSS, Clementine, and other analytics products. While at Stanford, Tableau founders Hanrahan and Stolte, as well as Diane Tang, created the predecessor to Tableau, named Polaris. Polaris was a data visualization software tool, built with the support of a United States Department of Energy defense program, the Accelerated Strategic Computing Initiative (ASCI). The main differences between Wilkinson's system and Polaris are the use of SQL relational algebra for database services and using shelves instead of cross and nest operators. == Wickham's Layered Grammar of Graphics == === Theory === Hadley Wickham conceived an alternate parameterisation of the syntax Wilkinson had derived, creating a layered grammar of graphics which he implemented as ggplot2 for R (programming language) users. This added a hierarchy of defaults based around the idea of building up a graphic from multiple layers. Wickham conceived these elements to be: Defaults: consists of data and mapping Data: dataset Mapping: aesthetic mappings Layer: consists of data, mapping, geom, stat, and position Data: dataset, or inherit from defaults Mapping: aesthetic mappings, or inherit from defaults Geom: geometric object Stat: statistical transformation Position: position adjustment Scale: mapping of data to aesthetic attributes Coord: mapping of data to the plane of the plot Facet: split up the data === Reception === Wilkinson is generally positive on Wickham's parameterisation and implementation of ggplot2, praising its elegance and expressivity whilst claiming that his original Grammar of Graphics is capable of representing a wider range of statistical graphics. === Implementations === ggplot2 is the first implementation of a layered grammar of graphics in R and implementations in other programming languages have ensued. These include direct ports plotnine for Python, gramm for MATLAB, Lets-Plot for Kotlin and gadfly for Julia. Projects inspired by elements of Wickham's grammar include Vega-Lite which specifies plots in JSON and uses a JavaScript engine. Implementations for Python include Vega-Altair (built on top of Vega-Lite). == Vega-Lite: A Grammar of Interactive Graphics == === Theory === Vega-Lite combines ideas from Wilkinson's Grammar of Graphics and Wickham's Layered Grammar of Graphics with a composition algebra for layered and multi-view displays with a grammar of interaction. The Vega-Lite specification is instantiated in JSON and rendered by the lower-level Vega. The graphical grammar implemented by Vega-Lite is composed of the following: Unit: consists of data, transforms, mark-type and encoding Data: relational table consisting of records (rows) and named attributes (columns) Transforms: data transformations Mark-type: geometric object for visual encoding Encodings: mapping of data attributes to visual marks properties where each encoding consists of: Channel: e.g. colour, shape, size, or text Field: data attribute Data-type: e.g. nominal, ordinal, quantitative, or temporal Value: use a literal instead of a data-type Functions: e.g. binning, aggregation, and sorting Scale: maps from data domain to visual range Guide: axis or legend for visualising scale Composite Views: compose views from multiple unit specifications with operators: Layer: charts plotted on top of each other Hconcat/Vconcat: place views side-by-side Facet: subset data to produce a trellis plot Repeat: multiple plots similar to facet but with full data replication in each cell Interaction: selections identify the set of points a user is interested in manipulating, with components: Selection: get the minimal number of backing points Name: reference Type: how many backing values are stored Predicate: determine the set of selected points e.g. single, list, interval Domain|Range: store data domain or visual range Event: e.g. mouseover, mousedown, mouseup, Init: initialise with specific backing points Transforms: e.g. project, toggle, translate, zoom, and nearest Resolve: resolve selections to union or intersect ==== Implementations ==== Whilst Vega-Lite is the sole implementation of this graphics grammar specification with compilation to Vega, other implementations do create JSON files which can be interpreted by Vega-Lite. == Related projects == Ggplot2 is an R package for plotting Tableau Software (originally known as Polaris) is a commercial software built using the Grammar of Graphics nViZn built by Wilkinson. SYSTAT (statistics package) built by Wilkinson ggpy, ggplot for Python, but has not been updated since 20 November 2016 plotnine started as an effort to improve the scalability of ggplot for Python and is largely compatible with ggplot2 syntax. Plotly - Interactive, online ggplot2 graphs gramm, a plotting class for MATLAB inspired by ggplot2 gadfly, a system for plotting and visualization written in Julia, based largely on ggplot2 Chart::GGPlot - ggplot2 port in Perl, but has not been updated since 16 March 2023 The Lets-Plot for Python library includes a native backend and a Python API, which was mostly based on the ggplot2 package. Lets-Plot Kotlin API is an open-source plotting library for statistical data implemented using the Kotlin programming language, and is built on the principles of layered graphics first described in the Leland Wilkinson's work The Grammar of Graphics. ggplotnim, plotting library using the Nim programming language inspired by ggplot2. Vega and Vega-Lite are plotting libraries that use JSON to specify plots. Vega-Altair, a Python library built on top of Vega-Lite chart-parts - React-friendly Grammar of Graphics, but has not been updated since 10 Dec 2021 g2 - a JavaScript library

Screenpal

ScreenPal (formerly known as Screencast-O-Matic) is cross-platform screen capture and screen recording software originally developed in 2006. == History == The company was founded by AJ Gregory in 2006 as Screencast-O-Matic. The software includes features for screen recording, screenshot capture, video editing, image editing, and a video and image hosting service. It is available for Windows and Mac operating systems, and has mobile apps for iOS and Android. The company launched a video editor in 2015. It began offering free video and image hosting in 2019, with premium hosting options for subscribers. In 2023, it was rebranded as ScreenPal.

Non-native speech database

A non-native speech database is a speech database of non-native pronunciations of English. Such databases are used in the development of: multilingual automatic speech recognition systems, text to speech systems, pronunciation trainers, and second language learning systems. == List == The actual table with information about the different databases is shown in Table 2. === Legend === In the table of non-native databases some abbreviations for language names are used. They are listed in Table 1. Table 2 gives the following information about each corpus: The name of the corpus, the institution where the corpus can be obtained, or at least further information should be available, the language which was actually spoken by the speakers, the number of speakers, the native language of the speakers, the total amount of non-native utterances the corpus contains, the duration in hours of the non-native part, the date of the first public reference to this corpus, some free text highlighting special aspects of this database and a reference to another publication. The reference in the last field is in most cases to the paper which is especially devoted to describe this corpus by the original collectors. In some cases it was not possible to identify such a paper. In these cases a paper is referenced which is using this corpus is. Some entries are left blank and others are marked with unknown. The difference here is that blank entries refer to attributes where the value is just not known. Unknown entries, however, indicate that no information about this attribute is available in the database itself. As an example, in the Jupiter weather database no information about the origin of the speakers is given. Therefore this data would be less useful for verifying accent detection or similar issues. Where possible, the name is a standard name of the corpus, for some of the smaller corpora, however, there was no established name and hence an identifier had to be created. In such cases, a combination of the institution and the collector of the database is used. In the case where the databases contain native and non-native speech, only attributes of the non-native part of the corpus are listed. Most of the corpora are collections of read speech. If the corpus instead consists either partly or completely of spontaneous utterances, this is mentioned in the Specials column.

Phrase structure grammar

The term phrase structure grammar was originally introduced by Noam Chomsky as the term for grammar studied previously by Emil Post and Axel Thue (Post canonical systems). Some authors, however, reserve the term for more restricted grammars in the Chomsky hierarchy: context-sensitive grammars or context-free grammars. In a broader sense, phrase structure grammars are also known as constituency grammars. The defining character of phrase structure grammars is thus their adherence to the constituency relation, as opposed to the dependency relation of dependency grammars. == History == In 1956, Chomsky wrote, "A phrase-structure grammar is defined by a finite vocabulary (alphabet) Vp, and a finite set Σ of initial strings in Vp, and a finite set F of rules of the form: X → Y, where X and Y are strings in Vp." == Constituency relation == In linguistics, phrase structure grammars are all those grammars that are based on the constituency relation, as opposed to the dependency relation associated with dependency grammars; hence, phrase structure grammars are also known as constituency grammars. Any of several related theories for the parsing of natural language qualify as constituency grammars, and most of them have been developed from Chomsky's work, including Government and binding theory Generalized phrase structure grammar Head-driven phrase structure grammar Lexical functional grammar The minimalist program Nanosyntax Further grammar frameworks and formalisms also qualify as constituency-based, although they may not think of themselves as having spawned from Chomsky's work, e.g. Arc pair grammar, and Categorial grammar.

Computer appliance

A computer appliance is a computer system with a combination of hardware, software, or firmware that is specifically designed to provide a particular computing resource. Such devices became known as appliances because of the similarity in role or management to a home appliance, which are generally closed and sealed, and are not serviceable by the user or owner. The hardware and software are delivered as an integrated product and may even be pre-configured before delivery to a customer, to provide a turn-key solution for a particular application. Unlike general purpose computers, appliances are generally not designed to allow the customers to change the software and the underlying operating system, or to flexibly reconfigure the hardware. Another form of appliance is the virtual appliance, which has similar functionality to a dedicated hardware appliance, but is distributed as a software virtual machine image for a hypervisor-equipped device. == Overview == Traditionally, software applications run on top of a general-purpose operating system, which uses the hardware resources of the computer (primarily memory, disk storage, processing power, and networking bandwidth) to meet the computing needs of the user. The main issue with the traditional model is related to complexity. It is complex to integrate the operating system and applications with a hardware platform, and complex to support it afterwards. By tightly constraining the variations of the hardware and software, the appliance becomes easily deployable, and can be used without nearly as wide (or deep) IT knowledge. Additionally, when problems and errors appear, the supporting staff very rarely needs to explore them deeply to understand the matter thoroughly. The staff needs merely training on the appliance management software to be able to resolve most of problems. In all forms of the computer appliance model, customers benefit from easy operations. The appliance has exactly one combination of hardware and operating system and application software, which has been pre-installed at the factory. This prevents customers from needing to perform complex integration work, and dramatically simplifies troubleshooting. In fact, this "turnkey operation" characteristic is the driving benefit that customers seek when purchasing appliances. To be considered an appliance, the (hardware) device needs to be integrated with software, and both are supplied as a package. This distinguishes appliances from "home grown" solutions, or solutions requiring complex implementations by integrators or value-added resellers (VARs). The appliance approach helps to decouple the various systems and applications, for example in the data center. Once a resource is decoupled, in theory it can be also centralized to become shared among many systems, centrally managed and optimized, all without requiring changes to any other system. == Tradeoffs of the computer appliance approach == The major disadvantage of deploying a computer appliance is that since they are designed to supply a specific resource, they most often include a customized operating system running over specialized hardware, neither of which are likely to be compatible with the other systems previously deployed. Customers lose flexibility. One may believe that a proprietary embedded operating system, or operating system within an application, can make the appliance much more secure from common cyber attacks. However, the opposite is true. Security by obscurity is a poor security decision, and appliances are often plagued by security issues as evidenced by the proliferation of IoT devices. == Types of appliances == The variety of computer appliances reflects the wide range of computing resources they provide to applications. Some examples: Storage appliances provide large amounts of storage, often available to many machines on the network. See Network-attached storage and Storage area network. Network appliances are general purpose routers which may also provide firewall protection, Transport Layer Security (TLS), messaging, access to specialized networking protocols (like the ebXML Message Service) and bandwidth multiplexing for the multiple systems they front-end. Backup and disaster recovery appliances computer appliances that are integrated backup software and backup targets, sometimes with hypervisors to support local DR of protected servers. They are often a gateway to a full DRaaS solution. Firewall and Security appliances Dedicated network appliances that are designed to protect computer networks from unwanted traffic. IIoT and MES Gateway appliances Computer appliances that are designed to translate data bidirectionally between control systems and enterprise systems. Proprietary, embedded, firmware applications running on the appliance use point-to-point connections to translate data between field devices in their native automation protocols and MES systems through their APIs, ODBC, or RESTful interfaces. Anti-spam appliances for e-mail spam Software appliances A single application server appliance, with just enough operating system (JeOS) for it to run. Virtual machine appliances consist of a "hypervisor style" embedded operating system running on appliance hardware. The hypervisor layer is matched to the hardware of the appliance, and cannot be varied by the customer, but the customer may load other operating systems and applications onto the appliance in the form of virtual machines. == Consumer appliances == Aside from its deployment within data centers, many computer appliances are directly used by the general public. These include: Digital video recorder Residential gateway Network-attached storage (NAS) Video game console Consumer uses stress the need for an appliance to have easy installation, configuration, and operation, with little or no technical knowledge being necessary. == Appliances in industrial automation == The world of industrial automation has been rich in appliances. These appliances have been hardened to withstand temperature and vibration extremes. These appliances are also highly configurable, enabling customization to meet a wide variety of applications. The key benefits of an appliance in automation are: Reduced downtime - a failed appliance is typically replaced with a COTS replacement and its task is quickly and easily reloaded from a backup. Highly scalable - appliances are typically targeted solutions for an area of a plant or process. As the requirements change, scalability is achieved through the installation of another appliance. Automation concepts are easily replicated throughout the enterprise by standardizing on appliances to perform the needed tasks, as opposed to the development of custom automation schemes for each task. Low TCO (total cost of ownership) - appliances are developed, tested and supported by automation product vendors and undergo a much broader level of quality testing than custom designed automation solutions. The use of appliances in automation reduce the level of testing needed in each individual application. Reduced design time - appliances perform specific functions and although they are highly configurable, they are typically self documenting. This enables appliance based solutions to be transferred from engineer to engineer with minimal need for training and documentation. Types of automation appliances: PLC (programmable logic controller) - Programmable logic controllers are appliances that are typically used for discrete control and offer a wide range of Input and Output options. They are configured through standardized programming languages such as IEC-1131. PID (proportional–integral–derivative controller) - PID controllers are appliances that monitor a process variable and, based on an error term, effect change on a control output (manipulated variable) to drive the process variable to a setpoint. PAC (programmable automation controller) - Programmable automation controllers are appliances that embody properties of both PLCs and PID controllers enabling the integration of both analog and discrete control. Universal gateway - A universal gateway appliance has the ability to communicate with a variety of devices through their respective communication protocols, and will affect data transactions between them. This in increasingly important as manufacturing strives to improve agility, quality, production rates, production costs and reduce downtime through enhanced M2M (machine to machine) communications. EATMs (Enterprise Appliance Transaction Modules) - Enterprise appliance transaction modules are appliances that affect data transactions from plant floor automation systems to enterprise business systems. They communicate to plant floor equipment through various vendor automation protocols, and communicate to business systems through database communication protocols such as JMS (Java Message Service) and SQL (Structured Query Language). == Internal structure == There are several