ImageMixer is a brand name of video editing software that edits digital video and still image in camcorders and authors to VCD and DVD. It is a second-party Japanese product, distributed by Pixela Corporation, a Japanese manufacturer of PC peripheral hardware and multimedia software. == Bundling == ImageMixer is widely used for several camcorder brands, such as JVC, Hitachi and Canon. Also, Sony has chosen to package ImageMixer with its DVD and HDD Handycam. == ImageMixer series == ImageMixer has other series of software for digital camera, such as ImageMixer Label Maker and ImageMixer DVD dubbing. ImageMixer also has movie editing solution for Macintosh. == Windows Vista version of ImageMixer == A Windows Vista version of ImageMixer has been developed (ImageMixer3).
Software construction
Software construction is the process of creating working software via coding and integration. The process includes unit and integration testing although does not include higher level testing such as system testing. Construction is an aspect of the software development lifecycle and is integrated in the various software development process models with varying focus on construction as an activity separate from other activities. In the waterfall model, a software development effort consists of sequential phases including requirements analysis, design, and planning which are prerequisites for starting construction. In an iterative model such as scrum, evolutionary prototyping, or extreme programming, construction as an activity that occurs concurrently or overlapping other activities. Construction planning may include defining the order in which components are created and integrated, the software quality management processes, and the allocation of tasks to teams and developers. To facilitate project management, numerous construction aspects can be measured; these include the amount of code developed, modified, reused, and destroyed, code complexity, code inspection statistics, faults-fixed and faults-found rates, and effort expended. These measurements can be useful for aspects such as ensuring quality and improving the process. == Activities == Construction includes many activities. === Coding === The following are a few of the key aspects of the coding activity: Naming Choice of name for each identifier. One study showed that the effort required to debug a program is minimized when variable names are between 10 and 16 characters. Logic Organization into statements and routines Highly cohesive routines proved to be less error prone than routines with lower cohesion. A study of 450 routines found that 50 percent of the highly cohesive routines were fault free compared to only 18 percent of routines with low cohesion. Another study of a different 450 routines found that routines with the highest coupling-to-cohesion ratios had 7 times as many errors as those with the lowest coupling-to-cohesion ratios and were 20 times as costly to fix. Although studies showed inconclusive results regarding the correlation between routine sizes and the rate of errors in them, but one study found that routines with fewer than 143 lines of code were 2.4 times less expensive to fix than larger routines. Another study showed that the code needed to be changed least when routines averaged 100 to 150 lines of code. Another study found that structural complexity and amount of data in a routine were correlated with errors regardless of its size. Interfaces between routines are some of the most error-prone areas of a program. One study showed that 39 percent of all errors were errors in communication between routines. Unused parameters are correlated with an increased error rate. In one study, only 17 to 29 percent of routines with more than one unreferenced variable had no errors, compared to 46 percent in routines with no unused variables. The number of parameters of a routine should be 7 at maximum as research has found that people generally cannot keep track of more than about seven chunks of information at once. One experiment showed that designs which access arrays sequentially, rather than randomly, result in fewer variables and fewer variable references. One experiment found that loops-with-exit are more comprehensible than other kinds of loops. Regarding the level of nesting in loops and conditionals, studies have shown that programmers have difficulty comprehending more than three levels of nesting. Control flow complexity has been shown to correlate with low reliability and frequent errors. Modularity Structuring and refactoring the code into classes, packages and other structures. When considering containment, the maximum number of data members in a class shouldn't exceed 7±2. Research has shown that this number is the number of discrete items a person can remember while performing other tasks. When considering inheritance, the number of levels in the inheritance tree should be limited. Deep inheritance trees have been found to be significantly associated with increased fault rates. When considering the number of routines in a class, it should be kept as small as possible. A study on C++ programs has found an association between the number of routines and the number of faults. A study by NASA showed that the putting the code into well-factored classes can double the code reusability compared to the code developed using functional design. Error handling Encoding logic to handle both planned and unplanned errors and exceptions. Resource management Managing computational resource use via exclusion mechanisms and discipline in accessing serially reusable resources, including threads or database locks. Security Prevention of code-level security breaches such as buffer overrun and array index overflow. Optimization Optimization while avoiding premature optimization. Documentation Both embedded in the code as comments and as external documents. === Integration === Integration is about combining separately constructed parts. Concerns include planning the sequence in which components will be integrated, creating scaffolding to support interim versions of the software, determining the degree of testing and quality work performed on components before they are integrated, and determining points in the project at which interim versions are tested. === Testing === Testing can reduce the time between when faulty logic is inserted in the code and when it is detected. In some cases, testing is performed after code has been written, but in test-first programming, test cases are created before code is written. Construction includes at least two forms of testing, often performed by the developer who wrote the code: unit testing and integration testing. === Reuse === Software reuse entails more than creating and using libraries. It requires formalizing the practice of reuse by integrating reuse processes and activities into the software life cycle. The tasks related to reuse in software construction during coding and testing may include: selection of the reusable code, evaluation of code or test re-usability, reporting reuse metrics. === Quality assurance === Techniques for ensuring quality as software is constructed include: Testing One study found that the average defect detection rates of Unit testing and integration testing are 30% and 35% respectively. Software inspection With respect to software inspection, one study found that the average defect detection rate of formal code inspections is 60%. Regarding the cost of finding defects, a study found that code reading detected 80% more faults per hour than testing. Another study shown that it costs six times more to detect design defects by using testing than by using inspections. A study by IBM showed that only 3.5 hours were needed to find a defect through code inspections versus 15–25 hours through testing. Microsoft has found that it takes 3 hours to find and fix a defect by using code inspections and 12 hours to find and fix a defect by using testing. In a 700 thousand lines program, it was reported that code reviews were several times as cost-effective as testing. Studies found that inspections result in 20% - 30% fewer defects per 1000 lines of code than less formal review practices and that they increase productivity by about 20%. Formal inspections will usually take 10% - 15% of the project budget and will reduce overall project cost. Researchers found that having more than 2 - 3 reviewers on a formal inspection doesn't increase the number of defects found, although the results seem to vary depending on the kind of material being inspected. Technical review With respect to technical review, one study found that the average defect detection rates of informal code reviews and desk checking are 25% and 40% respectively. Walkthroughs were found to have a defect detection rate of 20% - 40%, but were found also to be expensive especially when project pressures increase. Code reading was found by NASA to detect 3.3 defects per hour of effort versus 1.8 defects per hour for testing. It also finds 20% - 60% more errors over the life of the project than different kinds of testing. A study of 13 reviews about review meetings, found that 90% of the defects were found in preparation for the review meeting while only around 10% were found during the meeting. Static analysis With respect to Static analysis (IEEE1028), studies have shown that a combination of these techniques needs to be used to achieve a high defect detection rate. Other studies showed that different people tend to find different defects. One study found that the extreme programming practices of pair programming, desk checking, unit testing, integration testing, and regression testing can achieve a 90% defect detection rate. An experiment involving exper
Automated Pain Recognition
Automated Pain Recognition (APR) is a method for objectively measuring pain and at the same time represents an interdisciplinary research area that comprises elements of medicine, psychology, psychobiology, and computer science. The focus is on computer-aided objective recognition of pain, implemented on the basis of machine learning. Automated pain recognition allows for the valid, reliable detection and monitoring of pain in people who are unable to communicate verbally. The underlying machine learning processes are trained and validated in advance by means of unimodal or multimodal body signals. Signals used to detect pain may include facial expressions or gestures and may also be of a (psycho-)physiological or paralinguistic nature. To date, the focus has been on identifying pain intensity, but visionary efforts are also being made to recognize the quality, site, and temporal course of pain. However, the clinical implementation of this approach is a controversial topic in the field of pain research. Critics of automated pain recognition argue that pain diagnosis can only be performed subjectively by humans. == Background == Pain diagnosis under conditions where verbal reporting is restricted - such as in verbally and/or cognitively impaired people or in patients who are sedated or mechanically ventilated - is based on behavioral observations by trained professionals. However, all known observation procedures (e.g., Zurich Observation Pain Assessment (ZOPA)); Pain Assessment in Advanced Dementia Scale (PAINAD) require a great deal of specialist expertise. These procedures can be made more difficult by perception- and interpretation-related misjudgments on the part of the observer. With regard to the differences in design, methodology, evaluation sample, and conceptualization of the phenomenon of pain, it is difficult to compare the quality criteria of the various tools. Even if trained personnel could theoretically record pain intensity several times a day using observation instruments, it would not be possible to measure it every minute or second. In this respect, the goal of automated pain recognition is to use valid, robust pain response patterns that can be recorded multimodally for a temporally dynamic, high-resolution, automated pain intensity recognition system. == Procedure == For automated pain recognition, pain-relevant parameters are usually recorded using non-invasive sensor technology, which captures data on the (physical) responses of the person in pain. This can be achieved with camera technology that captures facial expressions, gestures, or posture, while audio sensors record paralinguistic features. (Psycho-)physiological information such as muscle tone and heart rate can be collected via biopotential sensors (electrodes). Pain recognition requires the extraction of meaningful characteristics or patterns from the data collected. This is achieved using machine learning techniques that are able to provide an assessment of the pain after training (learning), e.g., "no pain," "mild pain," or "severe pain." == Parameters == Although the phenomenon of pain comprises different components (sensory discriminative, affective (emotional), cognitive, vegetative, and (psycho-)motor), automated pain recognition currently relies on the measurable parameters of pain responses. These can be divided roughly into the two main categories of "physiological responses" and "behavioral responses". === Physiological responses === In humans, pain almost always initiates autonomic nervous processes that are reflected measurably in various physiological signals. ==== Physiological signals ==== Measurements can include electrodermal activity (EDA, also skin conductance), electromyography (EMG), electrocardiogram (ECG), blood volume pulse (BVP), electroencephalogram (EEG), respiration, and body temperature, which are regulatory mechanisms of the sympathetic and parasympathetic systems. Physiological signals are mainly recorded using special non-invasive surface electrodes (for EDA, EMG, ECG, and EEG), a blood volume pulse sensor (BVP), a respiratory belt (respiration), and a thermal sensor (body temperature). Endocrinological and immunological parameters can also be recorded, but this requires measures that are somewhat invasive (e.g., blood sampling). === Behavioral responses === Behavioral responses to pain fulfil two functions: protection of the body (e.g., through protective reflexes) and external communication of the pain (e.g., as a cry for help). The responses are particularly evident in facial expressions, gestures, and paralinguistic features. ==== Facial expressions ==== Behavioral signals captured comprise facial expression patterns (expressive behavior), which are measured with the aid of video signals. Facial expression recognition is based on the everyday clinical observation that pain often manifests itself in the patient's facial expressions but that this is not necessarily always the case, since facial expressions can be inhibited through self-control. Despite the possibility that facial expressions may be influenced consciously, facial expression behavior represents an essential source of information for pain diagnosis and is thus also a source of information for automatic pain recognition. One advantage of video-based facial expression recognition is the contact-free measurement of the face, provided that it can be captured on video, which is not possible in every position (e.g., lying face down) or may be limited by bandages covering the face. Facial expression analysis relies on rapid, spontaneous, and temporary changes in neuromuscular activity that lead to visually detectable changes in the face. ==== Gestures ==== Gestures are also captured predominantly using non-contact camera technology. Motor pain responses vary and are strongly dependent on the type and cause of the pain. They range from abrupt protective reflexes (e.g., spontaneous retraction of extremities or doubling up) to agitation (pathological restlessness) and avoidance behavior (hesitant, cautious movements). ==== Paralinguistic features of language ==== Among other things, pain leads to nonverbal linguistic behavior that manifests itself in sounds such as sighing, gasping, moaning, whining, etc. Paralinguistic features are usually recorded using highly sensitive microphones. == Algorithms == After the recording, pre-processing (e.g., filtering), and extraction of relevant features, an optional information fusion can be performed. During this process, modalities from different signal sources are merged to generate new or more precise knowledge. The pain is classified using machine learning processes. The method chosen has a significant influence on the recognition rate and depends greatly on the quality and granularity of the underlying data. Similar to the field of affective computing, the following classifiers are currently being used: Support Vector Machine (SVM): The goal of an SVM is to find a clearly defined optimal hyperplane with the greatest minimal distance to two (or more) classes to be separated. The hyperplane acts as a decision function for classifying an unknown pattern. Random Forest (RF): RF is based on the composition of random, uncorrelated decision trees. An unknown pattern is judged individually by each tree and assigned to a class. The final classification of the patterns by the RF is then based on a majority decision. k-Nearest Neighbors (k-NN): The k-NN algorithm classifies an unknown object using the class label that most commonly classifies the k neighbors closest to it. Its neighbors are determined using a selected similarity measure (e.g., Euclidean distance, Jaccard coefficient, etc.). Artificial neural networks (ANNs): ANNs are inspired by biological neural networks and model their organizational principles and processes in a very simplified manner. Class patterns are learned by adjusting the weights of the individual neuronal connections. == Databases == In order to classify pain in a valid manner, it is necessary to create representative, reliable, and valid pain databases that are available to the machine learner for training. An ideal database would be sufficiently large and would consist of natural (not experimental), high-quality pain responses. However, natural responses are difficult to record and can only be obtained to a limited extent; in most cases they are characterized by suboptimal quality. The databases currently available therefore contain experimental or quasi-experimental pain responses, and each database is based on a different pain model. The following list shows a selection of the most relevant pain databases (last updated: April 2020): UNBC-McMaster Shoulder Pain BioVid Heat Pain EmoPain SenseEmotion X-ITE Pain
Physical neural network
A physical neural network is a type of artificial neural network in which an electrically adjustable material is used to emulate the function of a neural synapse or a higher-order (dendritic) neuron model. "Physical" neural network is used to emphasize the reliance on physical hardware used to emulate neurons as opposed to software-based approaches. More generally the term is applicable to other artificial neural networks in which a memristor or other electrically adjustable resistance material is used to emulate a neural synapse. == Types of physical neural networks == === ADALINE === In the 1960s Bernard Widrow and Ted Hoff developed ADALINE (Adaptive Linear Neuron) which used electrochemical cells called memistors (memory resistors) to emulate synapses of an artificial neuron. The memistors were implemented as 3-terminal devices operating based on the reversible electroplating of copper such that the resistance between two of the terminals is controlled by the integral of the current applied via the third terminal. The ADALINE circuitry was briefly commercialized by the Memistor Corporation in the 1960s enabling some applications in pattern recognition. However, since the memistors were not fabricated using integrated circuit fabrication techniques the technology was not scalable and was eventually abandoned as solid-state electronics became mature. === Analog VLSI === In 1989 Carver Mead published his book Analog VLSI and Neural Systems, which spun off perhaps the most common variant of analog neural networks. The physical realization is implemented in analog VLSI. This is often implemented as field effect transistors in low inversion. Such devices can be modelled as translinear circuits. This is a technique described by Barrie Gilbert in several papers around mid 1970th, and in particular his Translinear Circuits from 1981. With this method circuits can be analyzed as a set of well-defined functions in steady-state, and such circuits assembled into complex networks. === Physical Neural Network === Alex Nugent describes a physical neural network as one or more nonlinear neuron-like nodes used to sum signals and nanoconnections formed from nanoparticles, nanowires, or nanotubes which determine the signal strength input to the nodes. Alignment or self-assembly of the nanoconnections is determined by the history of the applied electric field performing a function analogous to neural synapses. Numerous applications for such physical neural networks are possible. For example, a temporal summation device can be composed of one or more nanoconnections having an input and an output thereof, wherein an input signal provided to the input causes one or more of the nanoconnection to experience an increase in connection strength thereof over time. Another example of a physical neural network is taught by U.S. Patent No. 7,039,619 entitled "Utilized nanotechnology apparatus using a neural network, a solution and a connection gap," which issued to Alex Nugent by the U.S. Patent & Trademark Office on May 2, 2006. A further application of physical neural network is shown in U.S. Patent No. 7,412,428 entitled "Application of hebbian and anti-hebbian learning to nanotechnology-based physical neural networks," which issued on August 12, 2008. Nugent and Molter have shown that universal computing and general-purpose machine learning are possible from operations available through simple memristive circuits operating the AHaH plasticity rule. More recently, it has been argued that also complex networks of purely memristive circuits can serve as neural networks. === Phase change neural network === In 2002, Stanford Ovshinsky described an analog neural computing medium in which phase-change material has the ability to cumulatively respond to multiple input signals. An electrical alteration of the resistance of the phase change material is used to control the weighting of the input signals. === Memristive neural network === Greg Snider of HP Labs describes a system of cortical computing with memristive nanodevices. The memristors (memory resistors) are implemented by thin film materials in which the resistance is electrically tuned via the transport of ions or oxygen vacancies within the film. DARPA's SyNAPSE project has funded IBM Research and HP Labs, in collaboration with the Boston University Department of Cognitive and Neural Systems (CNS), to develop neuromorphic architectures which may be based on memristive systems. === Protonic artificial synapses === In 2022, researchers reported the development of nanoscale brain-inspired artificial synapses, using the ion proton (H+), for 'analog deep learning'.
Information Harvesting
Information Harvesting (IH) was an early data mining product from the 1990s. It was invented by Ralphe Wiggins and produced by the Ryan Corp, later Information Harvesting Inc., of Cambridge, Massachusetts. Wiggins had a background in genetic algorithms and fuzzy logic. IH sought to infer rules from sets of data. It did this first by classifying various input variables into one of a number of bins, thereby putting some structure on the continuous variables in the input. IH then proceeds to generate rules, trading off generalization against memorization, that will infer the value of the prediction variable, possibly creating many levels of rules in the process. It included strategies for checking if overfitting took place and, if so, correcting for it. Because of its strategies for correcting for overfitting by considering more data, and refining the rules based on that data, IH might also be considered to be a form of machine learning. The advantage of IH, as compared with other data mining products of its time and even later, was that it provided a mechanism for finding multiple rules that would classify the data and determining, according to set criteria, the best rules to use.
Symbol level
In knowledge-based systems, agents choose actions based on the principle of rationality to move closer to a desired goal. The agent is able to make decisions based on knowledge it has about the world (see knowledge level). But for the agent to actually change its state, it must use whatever means it has available. This level of description for the agent's behavior is the symbol level. The term was coined by Allen Newell in 1982. For example, in a computer program, the knowledge level consists of the information contained in its data structures that it uses to perform certain actions. The symbol level consists of the program's algorithms, the data structures themselves, and so on.
Diffusion map
Diffusion maps is a dimensionality reduction or feature extraction algorithm introduced by Coifman and Lafon which computes a family of embeddings of a data set into Euclidean space (often low-dimensional) whose coordinates can be computed from the eigenvectors and eigenvalues of a diffusion operator on the data. The Euclidean distance between points in the embedded space is equal to the "diffusion distance" between probability distributions centered at those points. Different from linear dimensionality reduction methods such as principal component analysis (PCA), diffusion maps are part of the family of nonlinear dimensionality reduction methods which focus on discovering the underlying manifold that the data has been sampled from. By integrating local similarities at different scales, diffusion maps give a global description of the data-set. Compared with other methods, the diffusion map algorithm is robust to noise perturbation and computationally inexpensive. == Definition of diffusion maps == Following and , diffusion maps can be defined in four steps. === Connectivity === Diffusion maps exploit the relationship between heat diffusion and random walk Markov chain. The basic observation is that if we take a random walk on the data, walking to a nearby data-point is more likely than walking to another that is far away. Let ( X , A , μ ) {\displaystyle (X,{\mathcal {A}},\mu )} be a measure space, where X {\displaystyle X} is the data set and μ {\displaystyle \mu } represents the distribution of the points on X {\displaystyle X} . Based on this, the connectivity k {\displaystyle k} between two data points, x {\displaystyle x} and y {\displaystyle y} , can be defined as the probability of walking from x {\displaystyle x} to y {\displaystyle y} in one step of the random walk. Usually, this probability is specified in terms of a kernel function of the two points: k : X × X → R {\displaystyle k:X\times X\rightarrow \mathbb {R} } . For example, the popular Gaussian kernel: k ( x , y ) = exp ( − | | x − y | | 2 ϵ ) {\displaystyle k(x,y)=\exp \left(-{\frac {||x-y||^{2}}{\epsilon }}\right)} More generally, the kernel function has the following properties k ( x , y ) = k ( y , x ) {\displaystyle k(x,y)=k(y,x)} ( k {\displaystyle k} is symmetric) k ( x , y ) ≥ 0 ∀ x , y {\displaystyle k(x,y)\geq 0\,\,\forall x,y} ( k {\displaystyle k} is positivity preserving). The kernel constitutes the prior definition of the local geometry of the data-set. Since a given kernel will capture a specific feature of the data set, its choice should be guided by the application that one has in mind. This is a major difference with methods such as principal component analysis, where correlations between all data points are taken into account at once. Given ( X , k ) {\displaystyle (X,k)} , we can then construct a reversible discrete-time Markov chain on X {\displaystyle X} (a process known as the normalized graph Laplacian construction): d ( x ) = ∫ X k ( x , y ) d μ ( y ) {\displaystyle d(x)=\int _{X}k(x,y)d\mu (y)} and define: p ( x , y ) = k ( x , y ) d ( x ) {\displaystyle p(x,y)={\frac {k(x,y)}{d(x)}}} Although the new normalized kernel does not inherit the symmetric property, it does inherit the positivity-preserving property and gains a conservation property: ∫ X p ( x , y ) d μ ( y ) = 1 {\displaystyle \int _{X}p(x,y)d\mu (y)=1} === Diffusion process === From p ( x , y ) {\displaystyle p(x,y)} we can construct a transition matrix of a Markov chain ( M {\displaystyle M} ) on X {\displaystyle X} . In other words, p ( x , y ) {\displaystyle p(x,y)} represents the one-step transition probability from x {\displaystyle x} to y {\displaystyle y} , and M t {\displaystyle M^{t}} gives the t-step transition matrix. We define the diffusion matrix L {\displaystyle L} (it is also a version of graph Laplacian matrix) L i , j = k ( x i , x j ) {\displaystyle L_{i,j}=k(x_{i},x_{j})\,} We then define the new kernel L i , j ( α ) = k ( α ) ( x i , x j ) = L i , j ( d ( x i ) d ( x j ) ) α {\displaystyle L_{i,j}^{(\alpha )}=k^{(\alpha )}(x_{i},x_{j})={\frac {L_{i,j}}{(d(x_{i})d(x_{j}))^{\alpha }}}\,} or equivalently, L ( α ) = D − α L D − α {\displaystyle L^{(\alpha )}=D^{-\alpha }LD^{-\alpha }\,} where D is a diagonal matrix and D i , i = ∑ j L i , j . {\displaystyle D_{i,i}=\sum _{j}L_{i,j}.} We apply the graph Laplacian normalization to this new kernel: M = ( D ( α ) ) − 1 L ( α ) , {\displaystyle M=({D}^{(\alpha )})^{-1}L^{(\alpha )},\,} where D ( α ) {\displaystyle D^{(\alpha )}} is a diagonal matrix and D i , i ( α ) = ∑ j L i , j ( α ) . {\displaystyle {D}_{i,i}^{(\alpha )}=\sum _{j}L_{i,j}^{(\alpha )}.} p ( x j , t | x i ) = M i , j t {\displaystyle p(x_{j},t|x_{i})=M_{i,j}^{t}\,} One of the main ideas of the diffusion framework is that running the chain forward in time (taking larger and larger powers of M {\displaystyle M} ) reveals the geometric structure of X {\displaystyle X} at larger and larger scales (the diffusion process). Specifically, the notion of a cluster in the data set is quantified as a region in which the probability of escaping this region is low (within a certain time t). Therefore, t not only serves as a time parameter, but it also has the dual role of scale parameter. The eigendecomposition of the matrix M t {\displaystyle M^{t}} yields M i , j t = ∑ l λ l t ψ l ( x i ) ϕ l ( x j ) {\displaystyle M_{i,j}^{t}=\sum _{l}\lambda _{l}^{t}\psi _{l}(x_{i})\phi _{l}(x_{j})\,} where { λ l } {\displaystyle \{\lambda _{l}\}} is the sequence of eigenvalues of M {\displaystyle M} and { ψ l } {\displaystyle \{\psi _{l}\}} and { ϕ l } {\displaystyle \{\phi _{l}\}} are the biorthogonal left and right eigenvectors respectively. Due to the spectrum decay of the eigenvalues, only a few terms are necessary to achieve a given relative accuracy in this sum. ==== Parameter α and the diffusion operator ==== The reason to introduce the normalization step involving α {\displaystyle \alpha } is to tune the influence of the data point density on the infinitesimal transition of the diffusion. In some applications, the sampling of the data is generally not related to the geometry of the manifold we are interested in describing. In this case, we can set α = 1 {\displaystyle \alpha =1} and the diffusion operator approximates the Laplace–Beltrami operator. We then recover the Riemannian geometry of the data set regardless of the distribution of the points. To describe the long-term behavior of the point distribution of a system of stochastic differential equations, we can use α = 0.5 {\displaystyle \alpha =0.5} and the resulting Markov chain approximates the Fokker–Planck diffusion. With α = 0 {\displaystyle \alpha =0} , it reduces to the classical graph Laplacian normalization. === Diffusion distance === The diffusion distance at time t {\displaystyle t} between two points can be measured as the similarity of two points in the observation space with the connectivity between them. It is given by D t ( x i , x j ) 2 = ∑ y ( p ( y , t | x i ) − p ( y , t | x j ) ) 2 ϕ 0 ( y ) {\displaystyle D_{t}(x_{i},x_{j})^{2}=\sum _{y}{\frac {(p(y,t|x_{i})-p(y,t|x_{j}))^{2}}{\phi _{0}(y)}}} where ϕ 0 ( y ) {\displaystyle \phi _{0}(y)} is the stationary distribution of the Markov chain, given by the first left eigenvector of M {\displaystyle M} . Explicitly: ϕ 0 ( y ) = d ( y ) ∑ z ∈ X d ( z ) {\displaystyle \phi _{0}(y)={\frac {d(y)}{\sum _{z\in X}d(z)}}} Intuitively, D t ( x i , x j ) {\displaystyle D_{t}(x_{i},x_{j})} is small if there is a large number of short paths connecting x i {\displaystyle x_{i}} and x j {\displaystyle x_{j}} . There are several interesting features associated with the diffusion distance, based on our previous discussion that t {\displaystyle t} also serves as a scale parameter: Points are closer at a given scale (as specified by D t ( x i , x j ) {\displaystyle D_{t}(x_{i},x_{j})} ) if they are highly connected in the graph, therefore emphasizing the concept of a cluster. This distance is robust to noise, since the distance between two points depends on all possible paths of length t {\displaystyle t} between the points. From a machine learning point of view, the distance takes into account all evidences linking x i {\displaystyle x_{i}} to x j {\displaystyle x_{j}} , allowing us to conclude that this distance is appropriate for the design of inference algorithms based on the majority of preponderance. === Diffusion process and low-dimensional embedding === The diffusion distance can be calculated using the eigenvectors by D t ( x i , x j ) 2 = ∑ l λ l 2 t ( ψ l ( x i ) − ψ l ( x j ) ) 2 {\displaystyle D_{t}(x_{i},x_{j})^{2}=\sum _{l}\lambda _{l}^{2t}(\psi _{l}(x_{i})-\psi _{l}(x_{j}))^{2}\,} So the eigenvectors can be used as a new set of coordinates for the data. The diffusion map is defined as: Ψ t ( x ) = ( λ 1 t ψ 1 ( x ) , λ 2 t ψ 2 ( x ) , … , λ k t ψ k ( x ) ) {\displaystyle \Psi _{t}(x)=(\lambda _{1}^{t}\psi _{1}(x),\lambda _{2}^{t}\psi _{2}(x),\ld