Writer invariant, also called authorial invariant or author's invariant, is a property of a text which is invariant of its author, that is, it will be similar in all texts of a given author and different in texts of different authors. It can be used to find plagiarism or discover who is real author of anonymously published text. Writer invariant is also an author's pattern of writing a letter in handwritten text recognition. While it is generally recognised that writer invariants exist, it is not agreed what properties of a text should be used. Among the first ones used was distribution of word lengths; other proposed invariants include average sentence length, average word length, noun, verb or adjective usage frequency, vocabulary richness, and frequency of function words, or specific function words. Of these, average sentence lengths can be very similar in works of different authors or vary significantly even within a single work; average word lengths likewise turn out to be very similar in works of different authors. Analysis of function words shows promise because they are used by authors unconsciously.
Neural processing unit
A neural processing unit (NPU), also known as an AI accelerator or deep learning processor, is a class of specialized hardware accelerator or computer system designed to accelerate artificial intelligence and machine learning applications, including artificial neural networks and computer vision. == Use == Their purpose is either to efficiently execute already trained AI models (inference) or to train AI models. NPUs can be more efficient in terms of speed or power consumption. NPU applications include algorithms for robotics, Internet of things, and data-intensive or sensor-driven tasks. They are often manycore or spatial designs and focus on low-precision arithmetic, novel dataflow architectures, or in-memory computing capability. As of 2024, a widely used datacenter-grade AI integrated circuit chip, the Nvidia H100 GPU, contains tens of billions of MOSFETs. === Consumer devices === AI accelerators are used in Apple silicon, Qualcomm, Samsung, Huawei, and Google Tensor smartphone processors. Vision processing units are accelerators specialized for machine vision algorithms such as CNN (convolutional neural networks) and SIFT (scale-invariant feature transform). They are used in devices that need to keep track of objects visually such as AR headsets and drones. It is more recently (circa 2017) added to processors from Apple and (circa 2022) to processors from Intel and AMD. All models of Intel Meteor Lake processors have a built-in versatile processor unit (VPU) for accelerating inference for computer vision and deep learning. On consumer devices, the NPU is intended to be small, power-efficient, but reasonably fast when used to run small models. To do this they are designed to support low-bitwidth operations using data types such as INT4, INT8, FP8, and FP16. A common metric is trillions of operations per second (TOPS). Although TOPS does not explicitly specify the kind of operations, it is typically INT8 additions and multiplications. === Datacenters === Accelerators are used in cloud computing servers: e.g., tensor processing units (TPU) for Google Cloud Platform, and Trainium and Inferentia chips for Amazon Web Services. Many vendor-specific terms exist for devices in this category, and it is an emerging technology without a dominant design. Since the late 2010s, graphics processing units designed by companies such as Nvidia and AMD often include AI-specific hardware in the form of dedicated functional units for low-precision matrix-multiplication operations. These GPUs are commonly used as AI accelerators, both for training and inference. === Scientific computation === Although NPUs are tailored for low-precision (e.g., FP16, INT8) matrix multiplication operations, they can be used to emulate higher-precision matrix multiplications in scientific computing. As modern GPUs place much focus on making the NPU part fast, using emulated FP64 (Ozaki scheme) on NPUs can potentially outperform native FP64. This has been demonstrated using FP16-emulated FP64 on NVIDIA TITAN RTX and using INT8-emulated FP64 on NVIDIA consumer GPUs and the A100 GPU. Consumer GPUs especially benefited as they have limited FP64 hardware capacity, showing a 6× speedup. Since CUDA Toolkit 13.0 Update 2, cuBLAS automatically uses INT8-emulated FP64 matrix multiplication of the equivalent precision if it is faster than native. This is in addition to the FP16-emulated FP32 feature introduced in version 12.9. == Programming == An operating system or a higher-level library may provide application programming interfaces such as TensorFlow with LiteRT Next (Android), CoreML (iOS, macOS) or DirectML (Windows). Formats such as ONNX are used to represent trained neural networks. Consumer CPU-integrated NPUs are accessible through vendor-specific APIs. AMD (Ryzen AI), Intel (OpenVINO), Apple silicon (CoreML), and Qualcomm (SNPE) each have their own APIs, which can be built upon by a higher-level library. GPUs generally use existing GPGPU pipelines such as CUDA and OpenCL adapted for lower precisions and specialized matrix-multiplication operations. Vulkan is also being used. Custom-built systems such as the Google TPU use private interfaces. There are a large number of separate underlying acceleration APIs and compilers/runtimes in use in the AI field, causing a great increase in software development effort due to the many combinations involved. As of 2025, the open standard organization Khronos Group is pursuing standardization of AI-related interfaces to reduce the amount of work needed. Khronos is working on three separate fronts: expansion of data types and intrinsic operations in OpenCL and Vulkan, inclusion of compute graphs in SPIR-V, and a NNEF/SkriptND file format for describing a neural network.
Gaussian adaptation
Gaussian adaptation (GA), also called normal or natural adaptation (NA) is an evolutionary algorithm designed for the maximization of manufacturing yield due to statistical deviation of component values of signal processing systems. In short, GA is a stochastic adaptive process where a number of samples of an n-dimensional vector x[xT = (x1, x2, ..., xn)] are taken from a multivariate Gaussian distribution, N(m, M), having mean m and moment matrix M. The samples are tested for fail or pass. The first- and second-order moments of the Gaussian restricted to the pass samples are m and M. The outcome of x as a pass sample is determined by a function s(x), 0 < s(x) < q ≤ 1, such that s(x) is the probability that x will be selected as a pass sample. The average probability of finding pass samples (yield) is P ( m ) = ∫ s ( x ) N ( x − m ) d x {\displaystyle P(m)=\int s(x)N(x-m)\,dx} Then the theorem of GA states: For any s(x) and for any value of P < q, there always exist a Gaussian p. d. f. [ probability density function ] that is adapted for maximum dispersion. The necessary conditions for a local optimum are m = m and M proportional to M. The dual problem is also solved: P is maximized while keeping the dispersion constant (Kjellström, 1991). Proofs of the theorem may be found in the papers by Kjellström, 1970, and Kjellström & Taxén, 1981. Since dispersion is defined as the exponential of entropy/disorder/average information it immediately follows that the theorem is valid also for those concepts. Altogether, this means that Gaussian adaptation may carry out a simultaneous maximisation of yield and average information (without any need for the yield or the average information to be defined as criterion functions). The theorem is valid for all regions of acceptability and all Gaussian distributions. It may be used by cyclic repetition of random variation and selection (like the natural evolution). In every cycle a sufficiently large number of Gaussian distributed points are sampled and tested for membership in the region of acceptability. The centre of gravity of the Gaussian, m, is then moved to the centre of gravity of the approved (selected) points, m. Thus, the process converges to a state of equilibrium fulfilling the theorem. A solution is always approximate because the centre of gravity is always determined for a limited number of points. It was used for the first time in 1969 as a pure optimization algorithm making the regions of acceptability smaller and smaller (in analogy to simulated annealing, Kirkpatrick 1983). Since 1970 it has been used for both ordinary optimization and yield maximization. == Natural evolution and Gaussian adaptation == It has also been compared to the natural evolution of populations of living organisms. In this case s(x) is the probability that the individual having an array x of phenotypes will survive by giving offspring to the next generation; a definition of individual fitness given by Hartl 1981. The yield, P, is replaced by the mean fitness determined as a mean over the set of individuals in a large population. Phenotypes are often Gaussian distributed in a large population and a necessary condition for the natural evolution to be able to fulfill the theorem of Gaussian adaptation, with respect to all Gaussian quantitative characters, is that it may push the centre of gravity of the Gaussian to the centre of gravity of the selected individuals. This may be accomplished by the Hardy–Weinberg law. This is possible because the theorem of Gaussian adaptation is valid for any region of acceptability independent of the structure (Kjellström, 1996). In this case the rules of genetic variation such as crossover, inversion, transposition etcetera may be seen as random number generators for the phenotypes. So, in this sense Gaussian adaptation may be seen as a genetic algorithm. == How to climb a mountain == Mean fitness may be calculated provided that the distribution of parameters and the structure of the landscape is known. The real landscape is not known, but figure below shows a fictitious profile (blue) of a landscape along a line (x) in a room spanned by such parameters. The red curve is the mean based on the red bell curve at the bottom of figure. It is obtained by letting the bell curve slide along the x-axis, calculating the mean at every location. As can be seen, small peaks and pits are smoothed out. Thus, if evolution is started at A with a relatively small variance (the red bell curve), then climbing will take place on the red curve. The process may get stuck for millions of years at B or C, as long as the hollows to the right of these points remain, and the mutation rate is too small. If the mutation rate is sufficiently high, the disorder or variance may increase and the parameter(s) may become distributed like the green bell curve. Then the climbing will take place on the green curve, which is even more smoothed out. Because the hollows to the right of B and C have now disappeared, the process may continue up to the peaks at D. But of course the landscape puts a limit on the disorder or variability. Besides — dependent on the landscape — the process may become very jerky, and if the ratio between the time spent by the process at a local peak and the time of transition to the next peak is very high, it may as well look like a punctuated equilibrium as suggested by Gould (see Ridley). == Computer simulation of Gaussian adaptation == Thus far the theory only considers mean values of continuous distributions corresponding to an infinite number of individuals. In reality however, the number of individuals is always limited, which gives rise to an uncertainty in the estimation of m and M (the moment matrix of the Gaussian). And this may also affect the efficiency of the process. Unfortunately very little is known about this, at least theoretically. The implementation of normal adaptation on a computer is a fairly simple task. The adaptation of m may be done by one sample (individual) at a time, for example m(i + 1) = (1 – a) m(i) + ax where x is a pass sample, and a < 1 a suitable constant so that the inverse of a represents the number of individuals in the population. M may in principle be updated after every step y leading to a feasible point x = m + y according to: M(i + 1) = (1 – 2b) M(i) + 2byyT, where yT is the transpose of y and b << 1 is another suitable constant. In order to guarantee a suitable increase of average information, y should be normally distributed with moment matrix μ2M, where the scalar μ > 1 is used to increase average information (information entropy, disorder, diversity) at a suitable rate. But M will never be used in the calculations. Instead we use the matrix W defined by WWT = M. Thus, we have y = Wg, where g is normally distributed with the moment matrix μU, and U is the unit matrix. W and WT may be updated by the formulas W = (1 – b)W + bygT and WT = (1 – b)WT + bgyT because multiplication gives M = (1 – 2b)M + 2byyT, where terms including b2 have been neglected. Thus, M will be indirectly adapted with good approximation. In practice it will suffice to update W only W(i + 1) = (1 – b)W(i) + bygT. This is the formula used in a simple 2-dimensional model of a brain satisfying the Hebbian rule of associative learning; see the next section (Kjellström, 1996 and 1999). The figure below illustrates the effect of increased average information in a Gaussian p.d.f. used to climb a mountain Crest (the two lines represent the contour line). Both the red and green cluster have equal mean fitness, about 65%, but the green cluster has a much higher average information making the green process much more efficient. The effect of this adaptation is not very salient in a 2-dimensional case, but in a high-dimensional case, the efficiency of the search process may be increased by many orders of magnitude. == The evolution in the brain == In the brain the evolution of DNA-messages is supposed to be replaced by an evolution of signal patterns and the phenotypic landscape is replaced by a mental landscape, the complexity of which will hardly be second to the former. The metaphor with the mental landscape is based on the assumption that certain signal patterns give rise to a better well-being or performance. For instance, the control of a group of muscles leads to a better pronunciation of a word or performance of a piece of music. In this simple model it is assumed that the brain consists of interconnected components that may add, multiply and delay signal values. A nerve cell kernel may add signal values, a synapse may multiply with a constant and An axon may delay values. This is a basis of the theory of digital filters and neural networks consisting of components that may add, multiply and delay signalvalues and also of many brain models, Levine 1991. In the figure below the brain stem is supposed to deliver Gaussian distributed signal patterns. This may be possible since certai
Detrended correspondence analysis
Detrended correspondence analysis (DCA) is a multivariate statistical technique widely used by ecologists to find the main factors or gradients in large, species-rich but usually sparse data matrices that typify ecological community data. DCA is frequently used to suppress artifacts inherent in most other multivariate analyses when applied to gradient data. == History == DCA was created in 1979 by Mark Hill of the United Kingdom's Institute for Terrestrial Ecology (now merged into Centre for Ecology and Hydrology) and implemented in FORTRAN code package called DECORANA (Detrended Correspondence Analysis), a correspondence analysis method. DCA is sometimes erroneously referred to as DECORANA; however, DCA is the underlying algorithm, while DECORANA is a tool implementing it. == Issues addressed == According to Hill and Gauch, DCA suppresses two artifacts inherent in most other multivariate analyses when applied to gradient data. An example is a time-series of plant species colonising a new habitat; early successional species are replaced by mid-successional species, then by late successional ones (see example below). When such data are analysed by a standard ordination such as a correspondence analysis: the ordination scores of the samples will exhibit the 'edge effect', i.e. the variance of the scores at the beginning and the end of a regular succession of species will be considerably smaller than that in the middle, when presented as a graph the points will be seen to follow a horseshoe shaped curve rather than a straight line ('arch effect'), even though the process under analysis is a steady and continuous change that human intuition would prefer to see as a linear trend. Outside ecology, the same artifacts occur when gradient data are analysed (e.g. soil properties along a transect running between 2 different geologies, or behavioural data over the lifespan of an individual) because the curved projection is an accurate representation of the shape of the data in multivariate space. Ter Braak and Prentice (1987, p. 121) cite a simulation study analysing two-dimensional species packing models resulting in a better performance of DCA compared to CA. == Method == DCA is an iterative algorithm that has shown itself to be a highly reliable and useful tool for data exploration and summary in community ecology (Shaw 2003). It starts by running a standard ordination (CA or reciprocal averaging) on the data, to produce the initial horse-shoe curve in which the 1st ordination axis distorts into the 2nd axis. It then divides the first axis into segments (default = 26), and rescales each segment to have mean value of zero on the 2nd axis - this effectively squashes the curve flat. It also rescales the axis so that the ends are no longer compressed relative to the middle, so that 1 DCA unit approximates to the same rate of turnover all the way through the data: the rule of thumb is that 4 DCA units mean that there has been a total turnover in the community. Ter Braak and Prentice (1987, p. 122) warn against the non-linear rescaling of the axes due to robustness issues and recommend using detrending-by-polynomials only. == Drawbacks == No significance tests are available with DCA, although there is a constrained (canonical) version called DCCA in which the axes are forced by Multiple linear regression to correlate optimally with a linear combination of other (usually environmental) variables; this allows testing of a null model by Monte-Carlo permutation analysis. == Example == The example shows an ideal data set: The species data is in rows, samples in columns. For each sample along the gradient, a new species is introduced but another species is no longer present. The result is a sparse matrix. Ones indicate the presence of a species in a sample. Except at the edges each sample contains five species. The plot of the first two axes of the correspondence analysis result on the right hand side clearly shows the disadvantages of this procedure: the edge effect, i.e. the points are clustered at the edges of the first axis, and the arch effect. == Software == An open source implementation of DCA, based on the original FORTRAN code, is available in the vegan R-package.
Training, validation, and test data sets
In machine learning, a common task is the study and construction of algorithms that can learn from and make predictions on data. Such algorithms function by making data-driven predictions or decisions, through building a mathematical model from input data. These input data used to build the model are usually divided into multiple data sets. In particular, three data sets are commonly used in different stages of the creation of the model: training, validation, and testing sets. The model is initially fit on a training data set, which is a set of examples used to fit the parameters (e.g. weights of connections between neurons in artificial neural networks) of the model. The model (e.g. a naive Bayes classifier) is trained on the training data set using a supervised learning method, for example using optimization methods such as gradient descent or stochastic gradient descent. In practice, the training data set often consists of pairs of an input vector (or scalar) and the corresponding output vector (or scalar), where the answer key is commonly denoted as the target (or label). The current model is run with the training data set and produces a result, which is then compared with the target, for each input vector in the training data set. Based on the result of the comparison and the specific learning algorithm being used, the parameters of the model are adjusted. The model fitting can include both variable selection and parameter estimation. Successively, the fitted model is used to predict the responses for the observations in a second data set called the validation data set. The validation data set provides an unbiased evaluation of a model fit on the training data set while tuning the model's hyperparameters (e.g. the number of hidden units—layers and layer widths—in a neural network). Validation data sets can be used for regularization by early stopping (stopping training when the error on the validation data set increases, as this is a sign of over-fitting to the training data set). This simple procedure is complicated in practice by the fact that the validation data set's error may fluctuate during training, producing multiple local minima. This complication has led to the creation of many ad-hoc rules for deciding when over-fitting has truly begun. Finally, the test data set is a data set used to provide an unbiased evaluation of a model fit on the training data set. When the data in the test data set has never been used (for example in cross-validation), the test data set is called a holdout data set. The term "validation set" is sometimes used instead of "test set" in some literature (e.g., if the original data set was partitioned into only two subsets, the test set might be referred to as the validation set). Deciding the sizes and strategies for data set division in training, test and validation sets is very dependent on the problem and data available. == Training data set == A training data set is a data set of examples used during the learning process and is used to fit the parameters (e.g., weights) of, for example, a classifier. For classification tasks, a supervised learning algorithm looks at the training data set to determine, or learn, the optimal combinations of variables that will generate a good predictive model. The goal is to produce a trained (fitted) model that generalizes well to new, unknown data. The fitted model is evaluated using “new” examples from the held-out data sets (validation and test data sets) to estimate the model’s accuracy in classifying new data. To reduce the risk of issues such as over-fitting, the examples in the validation and test data sets should not be used to train the model. Most approaches that search through training data for empirical relationships tend to overfit the data, meaning that they can identify and exploit apparent relationships in the training data that do not hold in general. When a training set is continuously expanded with new data, then this is incremental learning. == Validation data set == A validation data set is a data set of examples used to tune the hyperparameters (i.e. the architecture) of a model. It is sometimes also called the development set or the "dev set". An example of a hyperparameter for artificial neural networks includes the number of hidden units in each layer. It, as well as the testing set (as mentioned below), should follow the same probability distribution as the training data set. In order to avoid overfitting, when any classification parameter needs to be adjusted, it is necessary to have a validation data set in addition to the training and test data sets. For example, if the most suitable classifier for the problem is sought, the training data set is used to train the different candidate classifiers, the validation data set is used to compare their performances and decide which one to take and, finally, the test data set is used to obtain the performance characteristics such as accuracy, sensitivity, specificity, F-measure, and so on. The validation data set functions as a hybrid: it is training data used for testing, but neither as part of the low-level training nor as part of the final testing. The basic process of using a validation data set for model selection (as part of training data set, validation data set, and test data set) is: Since our goal is to find the network having the best performance on new data, the simplest approach to the comparison of different networks is to evaluate the error function using data which is independent of that used for training. Various networks are trained by minimization of an appropriate error function defined with respect to a training data set. The performance of the networks is then compared by evaluating the error function using an independent validation set, and the network having the smallest error with respect to the validation set is selected. This approach is called the hold out method. Since this procedure can itself lead to some overfitting to the validation set, the performance of the selected network should be confirmed by measuring its performance on a third independent set of data called a test set. An application of this process is in early stopping, where the candidate models are successive iterations of the same network, and training stops when the error on the validation set grows, choosing the previous model (the one with minimum error). == Test data set == A test data set is a data set that is independent of the training data set, but that follows the same probability distribution as the training data set. A test set is therefore a set of examples used only to assess the performance (i.e. generalization) of a specified classifier on unseen data. To do this, the model is used to predict classifications of examples in the test set. Those predictions are compared to the examples' true classifications to assess the model's accuracy. If a model fit to the training and validation data set also fits the test data set well, minimal overfitting has taken place (see figure below). A better fitting of the training or validation data sets as opposed to the test data set usually points to overfitting. In the scenario where a data set has a low number of samples, it is usually partitioned into a training set and a validation data set, where the model is trained on the training set and refined using the validation set to improve accuracy, but this approach will lead to overfitting. The holdout method can also be employed, where the test set is used at the end, after training on the training set. Other techniques, such as cross-validation and bootstrapping, are used on small data sets. The bootstrap method generates numerous simulated data sets of the same size by randomly sampling with replacement from the original data, allowing the random data points to serve as test sets for evaluating model performance. Cross-validation splits the data set into multiple folds, with a single sub-fold used as test data; the model is trained on the remaining folds, and all folds are cross-validated (with results averaged and models consolidated) to estimate final model performance. Note that some sources advise against using a single split, as it can lead to overfitting as well as biased model performance estimates. For this reason, data sets are split into three partitions: training, validation and test data sets. The standard machine learning practice is to train on the training set and tune hyperparameters using the validation set, where the validation process selects the model with the lowest validation loss, which is then tested on the test data set (normally held out) to assess the final model. The holdout method for the test set reduces computation by avoiding using the test set after each epoch. The test data set should never be used for validating the training model or fine-tuning hyperparameters, as it provides an accurate and honest evaluation of the model's final performance on unseen dat
Tail latency
Tail latency is a term used to describe the high-percentile response times seen in a system. This is usually measured at the 95th, 99th, or 99.9th percentile, not the average latency. In distributed systems, cloud computing, and large-scale web services, even a small number of slow requests can make the user experience and system performance much worse. Tail latency often happens because of things like resource contention, network variability, garbage collection pauses, and hardware heterogeneity. A major problem in system design is managing tail latency, because lowering average latency doesn't always make the worst-case performance better. To lessen its effects, people often use techniques like request hedging, replication, load balancing, and adaptive timeouts. In latency-sensitive applications like search engines, financial systems, and real-time services, where service-level objectives (SLOs) are often based on high-percentile latencies, it is especially important to understand and improve tail latency.
Silhouette (clustering)
Silhouette is a method of interpretation and validation of consistency within clusters of data. The technique provides a succinct graphical representation of how well each object has been classified. It was proposed by Belgian statistician Peter Rousseeuw in 1987. The silhouette value is a measure of how similar an object is to its own cluster (cohesion) compared to other clusters (separation). The silhouette value ranges from −1 to +1, where a high value indicates that the object is well matched to its own cluster and poorly matched to neighboring clusters. If most objects have a high value, then the clustering configuration is appropriate. If many points have a low or negative value, then the clustering configuration may have too many or too few clusters. A clustering with an average silhouette width of over 0.7 is considered to be "strong", a value over 0.5 "reasonable", and over 0.25 "weak". However, with an increasing dimensionality of the data, it becomes difficult to achieve such high values because of the curse of dimensionality, as the distances become more similar. The silhouette score is specialized for measuring cluster quality when the clusters are convex-shaped, and may not perform well if the data clusters have irregular shapes or are of varying sizes. The silhouette value can be calculated with any distance metric, such as Euclidean distance or Manhattan distance. == Definition == Assume the data have been clustered via any technique, such as k-medoids or k-means, into k {\displaystyle k} clusters. For data point i ∈ C i {\displaystyle i\in C_{i}} (data point i {\displaystyle i} in the cluster C i {\displaystyle C_{i}} ), calculate a ( i ) {\displaystyle a(i)} , the average distance that i {\displaystyle i} is from all other points in that cluster: a ( i ) = 1 | C i | − 1 ∑ j ∈ C i , i ≠ j d ( i , j ) {\displaystyle a(i)={\frac {1}{|C_{i}|-1}}\sum _{j\in C_{i},i\neq j}d(i,j)} where | C i | {\displaystyle |C_{i}|} is the number of points belonging to cluster C i {\displaystyle C_{i}} , and d ( i , j ) {\displaystyle d(i,j)} is the distance between data points i {\displaystyle i} and j {\displaystyle j} in the cluster C i {\displaystyle C_{i}} (we divide by | C i | − 1 {\displaystyle |C_{i}|-1} because the distance d ( i , i ) {\displaystyle d(i,i)} is not included in the sum). a ( i ) {\displaystyle a(i)} can be interpreted as a measure of how well i {\displaystyle i} is assigned to its cluster (the smaller the value, the better the assignment). We then define the mean dissimilarity of point i {\displaystyle i} to some cluster C j {\displaystyle C_{j}} as the mean of the distance from i {\displaystyle i} to all points in C j {\displaystyle C_{j}} (where C j ≠ C i {\displaystyle C_{j}\neq C_{i}} ). For each data point i ∈ C i {\displaystyle i\in C_{i}} , we now define b ( i ) {\displaystyle b(i)} as the average distance between i {\displaystyle i} and the points in the closest cluster (hence: "min") that i {\displaystyle i} does not belong to: b ( i ) = min j ≠ i 1 | C j | ∑ l ∈ C j d ( i , l ) {\displaystyle b(i)=\min _{j\neq i}{\frac {1}{|C_{j}|}}\sum _{l\in C_{j}}d(i,l)} The cluster with the smallest mean dissimilarity is said to be the "neighboring cluster" of i {\displaystyle i} because it is the next best fit cluster for point i {\displaystyle i} . We now define a silhouette (value) of one data point i {\displaystyle i} s ( i ) = b ( i ) − a ( i ) max { a ( i ) , b ( i ) } {\displaystyle s(i)={\frac {b(i)-a(i)}{\max\{a(i),b(i)\}}}} , if | C i | > 1 {\displaystyle |C_{i}|>1} and s ( i ) = 0 {\displaystyle s(i)=0} , if | C i | = 1 {\displaystyle |C_{i}|=1} , which can also be written as s ( i ) = { 1 − a ( i ) b ( i ) , if a ( i ) < b ( i ) 0 , if a ( i ) = b ( i ) b ( i ) a ( i ) − 1 , if a ( i ) > b ( i ) {\displaystyle s(i)={\begin{cases}1-{\frac {a(i)}{b(i)}},&{\mbox{ if }}a(i)