Apache Hama is a distributed computing framework based on bulk synchronous parallel computing techniques for massive scientific computations e.g., matrix, graph and network algorithms. Originally a sub-project of Hadoop, it became an Apache Software Foundation top level project in 2012. It was created by Edward J. Yoon, who named it (short for "Hadoop Matrix Algebra"), and Hama also means hippopotamus in Yoon's native Korean language (하마), following the trend of naming Apache projects after animals and zoology (such as Apache Pig). Hama was inspired by Google's Pregel large-scale graph computing framework described in 2010. When executing graph algorithms, Hama showed a fifty-fold performance increase relative to Hadoop. Retired in April 2020, project resources are made available as part of the Apache Attic. Yoon cited issues of installation, scalability, and a difficult programming model for its lack of adoption. == Architecture == Hama consists of three major components: BSPMaster, GroomServers and Zookeeper. === BSPMaster === BSPMaster is responsible for: Maintaining groom server status Controlling super steps in a cluster Maintaining job progress information Scheduling jobs and assigning tasks to groom servers Disseminating execution class across groom servers Controlling fault Providing users with the cluster control interface. A BSP Master and multiple grooms are started by the script. Then, the bsp master starts up with a RPC server for groom servers. Groom servers starts up with a BSPPeer instance and a RPC proxy to contact the bsp master. After started, each groom periodically sends a heartbeat message that encloses its groom server status, including maximum task capacity, unused memory, and so on. Each time the BSP master receives a heartbeat message, it brings the groom server status up-to-date. The bsp master makes use of groom servers' status in order to assign tasks to idle groom servers - and returns a heartbeat response containing assigned tasks and others actions for a groom server to do. Currently BSP master has a FIFO job scheduler and simple task assignment algorithms. === GroomServer === A groom server (shortly referred to as groom) is a process that performs BSP tasks assigned by BSPMaster. Each groom contacts the BSPMaster, and it takes assigned tasks and reports its status by means of periodical piggybacks with BSPMaster. Each groom is designed to run with HDFS or other distributed storages. Basically, a groom server and a data node should be run on one physical node. === Zookeeper === A Zookeeper is used to manage the efficient barrier synchronisation of the BSPPeers.
Multi-task learning
Multi-task learning (MTL) is a subfield of machine learning in which multiple learning tasks are solved at the same time, while exploiting commonalities and differences across tasks. This can result in improved learning efficiency and prediction accuracy for the task-specific models, when compared to training the models separately. Inherently, Multi-task learning is a multi-objective optimization problem having trade-offs between different tasks. Early versions of MTL were called "hints". In a widely cited 1997 paper, Rich Caruana gave the following characterization:Multitask Learning is an approach to inductive transfer that improves generalization by using the domain information contained in the training signals of related tasks as an inductive bias. It does this by learning tasks in parallel while using a shared representation; what is learned for each task can help other tasks be learned better. In the classification context, MTL aims to improve the performance of multiple classification tasks by learning them jointly. One example is a spam-filter, which can be treated as distinct but related classification tasks across different users. To make this more concrete, consider that different people have different distributions of features which distinguish spam emails from legitimate ones, for example an English speaker may find that all emails in Russian are spam, not so for Russian speakers. Yet there is a definite commonality in this classification task across users, for example one common feature might be text related to money transfer. Solving each user's spam classification problem jointly via MTL can let the solutions inform each other and improve performance. Further examples of settings for MTL include multiclass classification and multi-label classification. Multi-task learning works because regularization induced by requiring an algorithm to perform well on a related task can be superior to regularization that prevents overfitting by penalizing all complexity uniformly. One situation where MTL may be particularly helpful is if the tasks share significant commonalities and are generally slightly under sampled. However, as discussed below, MTL has also been shown to be beneficial for learning unrelated tasks. == Methods == The key challenge in multi-task learning, is how to combine learning signals from multiple tasks into a single model. This may strongly depend on how well different task agree with each other, or contradict each other. There are several ways to address this challenge: === Task grouping and overlap === Within the MTL paradigm, information can be shared across some or all of the tasks. Depending on the structure of task relatedness, one may want to share information selectively across the tasks. For example, tasks may be grouped or exist in a hierarchy, or be related according to some general metric. Suppose, as developed more formally below, that the parameter vector modeling each task is a linear combination of some underlying basis. Similarity in terms of this basis can indicate the relatedness of the tasks. For example, with sparsity, overlap of nonzero coefficients across tasks indicates commonality. A task grouping then corresponds to those tasks lying in a subspace generated by some subset of basis elements, where tasks in different groups may be disjoint or overlap arbitrarily in terms of their bases. Task relatedness can be imposed a priori or learned from the data. Hierarchical task relatedness can also be exploited implicitly without assuming a priori knowledge or learning relations explicitly. For example, the explicit learning of sample relevance across tasks can be done to guarantee the effectiveness of joint learning across multiple domains. === Exploiting unrelated tasks: Auxiliary learning === In auxiliary learning, one attempts learning a group of principal tasks using a group of auxiliary tasks, unrelated to the principal ones. With the right unrelated tasks, joint learning of unrelated tasks which use the same input data have been shown to be beneficial, and provide significant improvement over standard MTL. The reason is that prior knowledge about task relatedness can lead to sparser and more informative representations for each task grouping, essentially by screening out idiosyncrasies of the data distribution. It has been proposed to build on a prior multitask methodology by favoring a shared low-dimensional representation within each task grouping, and imposing a penalty on tasks from different groups which encourages the two representations to be orthogonal. Learning with auxiliary unrelated tasks poses two major challenges: Finding useful auxiliary tasks and combining losses of all tasks in a useful way. Some methods can learn these from data together with the training process, and combine tasks efficiently. === Transfer of knowledge === Related to multi-task learning is the concept of knowledge transfer. Whereas traditional multi-task learning implies that a shared representation is developed concurrently across tasks, transfer of knowledge implies a sequentially shared representation. Large scale machine learning projects such as the deep convolutional neural network GoogLeNet, an image-based object classifier, can develop robust representations which may be useful to further algorithms learning related tasks. For example, the pre-trained model can be used as a feature extractor to perform pre-processing for another learning algorithm. Or the pre-trained model can be used to initialize a model with similar architecture which is then fine-tuned to learn a different classification task. === Multiple non-stationary tasks === Traditionally Multi-task learning and transfer of knowledge are applied to stationary learning settings. Their extension to non-stationary environments is termed Group online adaptive learning (GOAL). Sharing information could be particularly useful if learners operate in continuously changing environments, because a learner could benefit from previous experience of another learner to quickly adapt to their new environment. Such group-adaptive learning has numerous applications, from predicting financial time-series, through content recommendation systems, to visual understanding for adaptive autonomous agents. === Multi-task optimization === Multi-task optimization focuses on solving optimizing the whole process. The paradigm has been inspired by the well-established concepts of transfer learning and multi-task learning in predictive analytics. The key motivation behind multi-task optimization is that if optimization tasks are related to each other in terms of their optimal solutions or the general characteristics of their function landscapes, the search progress can be transferred to substantially accelerate the search on the other. The success of the paradigm is not necessarily limited to one-way knowledge transfers from simpler to more complex tasks. In practice an attempt is to intentionally solve a more difficult task that may unintentionally solve several smaller problems. There is a direct relationship between multitask optimization and multi-objective optimization. In some cases, the simultaneous training of seemingly related tasks may hinder performance compared to single-task models. Commonly, MTL models employ task-specific modules on top of a joint feature representation obtained using a shared module. Since this joint representation must capture useful features across all tasks, MTL may hinder individual task performance if the different tasks seek conflicting representation, i.e., the gradients of different tasks point to opposing directions or differ significantly in magnitude. This phenomenon is commonly referred to as negative transfer. To mitigate this issue, various MTL optimization methods have been proposed. It has been reported that meta-knowledge transfer could help avoid negative transfer.Besides, the per-task gradients are combined into a joint update direction through various aggregation algorithms or heuristics. There are several common approaches for multi-task optimization: Bayesian optimization, evolutionary computation, and approaches based on Game theory. ==== Multi-task Bayesian optimization ==== Multi-task Bayesian optimization is a modern model-based approach that leverages the concept of knowledge transfer to speed up the automatic hyperparameter optimization process of machine learning algorithms. The method builds a multi-task Gaussian process model on the data originating from different searches progressing in tandem. The captured inter-task dependencies are thereafter utilized to better inform the subsequent sampling of candidate solutions in respective search spaces. ==== Evolutionary multi-tasking ==== Evolutionary multi-tasking has been explored as a means of exploiting the implicit parallelism of population-based search algorithms to simultaneously progress multiple distinct optimization tasks. By mapping all task
AI agent
In the context of generative artificial intelligence, AI agents (also referred to as compound AI systems or agentic AI) are a class of intelligent agents that can pursue goals, use tools, and take actions with varying degrees of autonomy. In practice, they usually operate within human-defined objectives, constraints, and available tools. == Overview == AI agents possess several key attributes, including goal-directed behavior, natural language interfaces, the capacity to use external tools, and the ability to perform multi-step tasks. Their control flow is frequently driven by large language models (LLMs). Agent systems may also include memory components, planning logic, tool interfaces, and orchestration software for coordinating agent components. AI agents do not have a standard definition. NIST describes agentic AI as an emerging area requiring standards for secure operation, interoperability, and reliable interaction with external systems. A common application of AI agents is task automation: for example, booking travel plans based on a user's prompted request. Companies such as Google, Microsoft and Amazon Web Services have offered platforms for deploying pre-built AI agents. Several protocols have been proposed for standardizing inter-agent communication, with examples including the Model Context Protocol, Gibberlink, and many others. Some of these protocols are also used for connecting agents to external applications. In December 2025, Linux Foundation announced the formation of the Agentic AI Foundation (AAIF), with the goal of ensuring agentic AI evolves transparently and collaboratively. == History == AI agents have been traced back to research from the 1990s, with Harvard professor Milind Tambe noting that the definition of an AI agent was not clear at the time. Researcher Andrew Ng has been credited with spreading the term "agentic" to a wider audience in 2024. == Training and testing == Researchers have attempted to build world models and reinforcement learning environments to train or evaluate AI agents. For example, video games such as Minecraft and No Man's Sky as well as replicas of company websites, have also been used for training such agents. == Autonomous capabilities == The Financial Times compared the autonomy of AI agents to the SAE classification of self-driving cars, likening most applications to level 2 or level 3, with some achieving level 4 in highly specialized circumstances, and level 5 being theoretical. == Cognitive architecture == The following are some internal design options for reasoning within an agent: Retrieval-augmented generation ReAct (Reason + Act) pattern is an iterative process in which an AI agent alternates between reasoning and taking actions, receives observations from the environment or external tools, and integrates these observations into subsequent reasoning steps. Reflexion, which uses an LLM to create feedback on the agent's plan of action and stores that feedback in a memory cache. A tool/agent registry, for organizing software functions or other agents that the agent can use. One-shot model querying, which queries the model once to create the plan of action. === Reference architecture === Ken Huang proposed an AI agent reference architecture, which consists of seven interconnected layers, with each layer building on the functionality of the layers beneath it: Layer 1: Foundation models - provide the core AI engines to power agent capabilities. Layer 2: Data operations - manage the complex data infrastructure required for AI agent operations, including Vector database, data loaders, RAG. Layer 3: Agent frameworks - sophisticated software and tools that simplify the development and management of the AI agents. Layer 4: Deployment and infrastructure - provide the robust technical foundation for running AI agents. Layer 5: Evaluation and observability - focus on assessing the safety and performance of AI agents. Layer 6: Security and compliance - a crucial protective framework ensuring AI agents operate safely, securely, and conform to regulatory boundaries. At this layer security and compliance features embedded into all the AI agent stack layers are integrated together. Layer 7: Agent ecosystem - represents the AI agents' interface with real-world applications and users. == Orchestration patterns == To execute complex tasks, autonomous agents are often integrated with other agents or specialized tools. These configurations, known as orchestration patterns or workflows, include the following: Prompt chaining: A sequence where the output of one step serves as the input for the next. Routing: The classification of an input to direct it to a specialized downstream task or tool. Parallelization: The simultaneous execution of multiple tasks. Sequential processing: A fixed, linear progression of tasks through a predefined pipeline. Planner-critic: An iterative pattern where one agent generates a proposal and another evaluates it to provide feedback for refinement. == Multimodal AI agents == In addition to large language models (LLMs), vision-language models (VLMs) and multimodal foundation models can be used as the basis for agents. In September 2024, Allen Institute for AI released an open-source vision-language model. Nvidia released a framework for developers to use VLMs, LLMs and retrieval-augmented generation for building AI agents that can analyze images and videos, including video search and video summarization. Microsoft released a multimodal agent model – trained on images, video, software user interface interactions, and robotics data – that the company claimed can manipulate software and robots. == Applications == As of April 2025, per the Associated Press, there are few real-world applications of AI agents. As of June 2025, per Fortune, many companies are primarily experimenting with AI agents. The Information divided AI agents into seven archetypes: business-task agents, for acting within enterprise software; conversational agents, which act as chatbots for customer support; research agents, for querying and analyzing information (such as OpenAI Deep Research); analytics agents, for analyzing data to create reports; software developer or coding agents (such as Cursor); domain-specific agents, which include specific subject matter knowledge; and web browser agents (such as OpenAI Operator). By mid-2025, AI agents have been used in video game development, gambling (including sports betting), cryptocurrency wallets (including cryptocurrency trading and meme coins) and social media. In August 2025, New York Magazine described software development as the most definitive use case of AI agents. Likewise, by October 2025, noting a decline in expectations, The Information noted AI coding agents and customer support as the primary use cases by businesses. In November 2025, The Wall Street Journal reported that few companies that deployed AI agents have received a return on investment. === Applications in government === Several government bodies in the United States and United Kingdom have deployed or announced the deployment of agents, at the local and national level. The city of Kyle, Texas deployed an AI agent from Salesforce in March 2025 for 311 customer service. In November 2025, the Internal Revenue Service stated that it would use Agentforce, AI agents from Salesforce, for the Office of Chief Counsel, Taxpayer Advocate Services and the Office of Appeals. That same month, Staffordshire Police announced that they would trial Agentforce agents for handling non-emergency 101 calls in the United Kingdom starting in 2026. In December 2025, the Department of Neighborhoods in Detroit, Michigan, in partnership with a local business, deployed a pilot project in two Detroit districts for an AI agent to be used for customer service calls. In February 2025, Thomas Shedd, the director of the Technology Transformation Services, proposed using AI coding agents across the United States federal government. A recruiter for the Department of Government Efficiency proposed in April 2025 to use AI agents to automate the work of about 70,000 United States federal government employees, as part of a startup with funding from OpenAI and a partnership agreement with Palantir. This proposal was criticized by experts for its impracticality, if not impossibility, and the lack of corresponding widespread adoption by businesses. In December 2025, the Food and Drug Administration announced that it would offer "agentic AI capabilities" to its staff for "meeting management, pre-market reviews, review validation, post-market surveillance, inspections and compliance and administrative functions." That same month, the United States Department of Defense launched GenAI.mil, an internal platform for American military personnel to use generative AI-based applications based on Google Gemini, including "intelligent agentic workflows". Defense Secretary Pete Hegseth listed applications such as "[conducting] deep r
Cognitive robotics
Cognitive robotics or cognitive technology is a subfield of robotics concerned with endowing a robot with intelligent behavior by providing it with a processing architecture that will allow it to learn and reason about how to behave in response to complex goals in a complex world. Cognitive robotics may be considered the engineering branch of embodied cognitive science and embodied embedded cognition, consisting of robotic process automation, artificial intelligence, machine learning, deep learning, optical character recognition, image processing, process mining, analytics, software development and system integration. == Core issues == While traditional cognitive modeling approaches have assumed symbolic coding schemes as a means for depicting the world, translating the world into these kinds of symbolic representations has proven to be problematic if not untenable. Perception and action and the notion of symbolic representation are therefore core issues to be addressed in cognitive robotics. == Starting point == Cognitive robotics views human or animal cognition as a starting point for the development of robotic information processing, as opposed to more traditional artificial intelligence techniques. Target robotic cognitive capabilities include perception processing, attention allocation, anticipation, planning, complex motor coordination, reasoning about other agents and perhaps even about their own mental states. Robotic cognition embodies the behavior of intelligent agents in the physical world (or a virtual world, in the case of simulated cognitive robotics). Ultimately, the robot must be able to act in the real world. == Learning techniques == === Motor Babble === A preliminary robot learning technique called motor babbling involves correlating pseudo-random complex motor movements by the robot with resulting visual and/or auditory feedback such that the robot may begin to expect a pattern of sensory feedback given a pattern of motor output. Desired sensory feedback may then be used to inform a motor control signal. This is thought to be analogous to how a baby learns to reach for objects or learns to produce speech sounds. For simpler robot systems, where, for instance, inverse kinematics may feasibly be used to transform anticipated feedback (desired motor result) into motor output, this step may be skipped. === Imitation === Once a robot can coordinate its motors to produce a desired result, the technique of learning by imitation may be used. The robot monitors the performance of another agent and then the robot tries to imitate that agent. It is often a challenge to transform imitation information from a complex scene into a desired motor result for the robot. Note that imitation is a high-level form of cognitive behavior and imitation is not necessarily required in a basic model of embodied animal cognition. === Knowledge acquisition === A more complex learning approach is "autonomous knowledge acquisition": the robot is left to explore the environment on its own. A system of goals and beliefs is typically assumed. A somewhat more directed mode of exploration can be achieved by "curiosity" algorithms, such as Intelligent Adaptive Curiosity or Category-Based Intrinsic Motivation. These algorithms generally involve breaking sensory input into a finite number of categories and assigning some sort of prediction system (such as an artificial neural network) to each. The prediction system keeps track of the error in its predictions over time. Reduction in prediction error is considered learning. The robot then preferentially explores categories in which it is learning (or reducing prediction error) the fastest. == Other architectures == Some researchers in cognitive robotics have tried using architectures such as (ACT-R and Soar (cognitive architecture)) as a basis of their cognitive robotics programs. These highly modular symbol-processing architectures have been used to simulate operator performance and human performance when modeling simplistic and symbolized laboratory data. The idea is to extend these architectures to handle real-world sensory input as that input continuously unfolds through time. What is needed is a way to somehow translate the world into a set of symbols and their relationships. == Questions == Some of the fundamental questions to be answered in cognitive robotics are: How much human programming should or can be involved to support the learning processes? How can one quantify progress? Some of the adopted ways are reward and punishment. But what kind of reward and what kind of punishment? In humans, when teaching a child, for example, the reward would be candy or some encouragement, and the punishment can take many forms. But what is an effective way with robots?
Situated
In artificial intelligence and cognitive science, the term situated refers to an agent which is embedded in an environment. The term situated is commonly used to refer to robots, but some researchers argue that software agents can also be situated if: they exist in a dynamic (rapidly changing) environment, which they can manipulate or change through their actions, and which they can sense or perceive. Examples might include web-based agents, which can alter data or trigger processes (such as purchases) over the internet, or virtual-reality bots which inhabit and change virtual worlds, such as Second Life. Being situated is generally considered to be part of being embodied, but it is useful to consider each perspective individually. The situated perspective emphasizes that intelligent behaviour derives from the environment and the agent's interactions with it. The nature of these interactions are defined by an agent's embodiment.
Image texture
An image texture is the small-scale structure perceived on an image, based on the spatial arrangement of color or intensities. It can be quantified by a set of metrics calculated in image processing. Image texture metrics give us information about the whole image or selected regions. Image textures can be artificially created or found in natural scenes captured in an image. Image textures are one way that can be used to help in segmentation or classification of images. For more accurate segmentation the most useful features are spatial frequency and an average grey level. To analyze an image texture in computer graphics, there are two ways to approach the issue: structured approach and statistical approach. == Structured approach == A structured approach sees an image texture as a set of primitive texels in some regular or repeated pattern. This works well when analyzing artificial textures. To obtain a structured description a characterization of the spatial relationship of the texels is gathered by using Voronoi tessellation of the texels. == Statistical approach == A statistical approach sees an image texture as a quantitative measure of the arrangement of intensities in a region. In general this approach is easier to compute and is more widely used, since natural textures are made of patterns of irregular subelements. === Edge detection === The use of edge detection is to determine the number of edge pixels in a specified region, helps determine a characteristic of texture complexity. After edges have been found the direction of the edges can also be applied as a characteristic of texture and can be useful in determining patterns in the texture. These directions can be represented as an average or in a histogram. Consider a region with N pixels. the gradient-based edge detector is applied to this region by producing two outputs for each pixel p: the gradient magnitude Mag(p) and the gradient direction Dir(p). The edgeness per unit area can be defined by F e d g e n e s s = | { p | M a g ( p ) > T } | N {\displaystyle F_{edgeness}={\frac {|\{p|Mag(p)>T\}|}{N}}} for some threshold T. To include orientation with edgeness histograms for both gradient magnitude and gradient direction can be used. Hmag(R) denotes the normalized histogram of gradient magnitudes of region R, and Hdir(R) denotes the normalized histogram of gradient orientations of region R. Both are normalized according to the size NR Then F m a g , d i r = ( H m a g ( R ) , H d i r ( R ) ) {\displaystyle F_{mag,dir}=(H_{mag}(R),H_{dir}(R))} is a quantitative texture description of region R. === Co-occurrence matrices === The co-occurrence matrix captures numerical features of a texture using spatial relations of similar gray tones. Numerical features computed from the co-occurrence matrix can be used to represent, compare, and classify textures. The following are a subset of standard features derivable from a normalized co-occurrence matrix: A n g u l a r 2 n d M o m e n t = ∑ i ∑ j p [ i , j ] 2 C o n t r a s t = ∑ i = 1 N g ∑ j = 1 N g n 2 p [ i , j ] , where | i − j | = n C o r r e l a t i o n = ∑ i = 1 N g ∑ j = 1 N g ( i j ) p [ i , j ] − μ x μ y σ x σ y E n t r o p y = − ∑ i ∑ j p [ i , j ] l n ( p [ i , j ] ) {\displaystyle {\begin{aligned}Angular{\text{ }}2nd{\text{ }}Moment&=\sum _{i}\sum _{j}p[i,j]^{2}\\Contrast&=\sum _{i=1}^{Ng}\sum _{j=1}^{Ng}n^{2}p[i,j]{\text{, where }}|i-j|=n\\Correlation&={\frac {\sum _{i=1}^{Ng}\sum _{j=1}^{Ng}(ij)p[i,j]-\mu _{x}\mu _{y}}{\sigma _{x}\sigma _{y}}}\\Entropy&=-\sum _{i}\sum _{j}p[i,j]ln(p[i,j])\\\end{aligned}}} where p [ i , j ] {\displaystyle p[i,j]} is the [ i , j ] {\displaystyle [i,j]} th entry in a gray-tone spatial dependence matrix, and Ng is the number of distinct gray-levels in the quantized image. One negative aspect of the co-occurrence matrix is that the extracted features do not necessarily correspond to visual perception. It is used in dentistry for the objective evaluation of lesions [DOI: 10.1155/2020/8831161], treatment efficacy [DOI: 10.3390/ma13163614; DOI: 10.11607/jomi.5686; DOI: 10.3390/ma13173854; DOI: 10.3390/ma13132935] and bone reconstruction during healing [DOI: 10.5114/aoms.2013.33557; DOI: 10.1259/dmfr/22185098; EID: 2-s2.0-81455161223; DOI: 10.3390/ma13163649]. === Laws texture energy measures === Another approach is to use local masks to detect various types of texture features. Laws originally used four vectors representing texture features to create sixteen 2D masks from the outer products of the pairs of vectors. The four vectors and relevant features were as follows: L5 = [ +1 +4 6 +4 +1 ] (Level) E5 = [ -1 -2 0 +2 +1 ] (Edge) S5 = [ -1 0 2 0 -1 ] (Spot) R5 = [ +1 -4 6 -4 +1 ] (Ripple) To these 4, a fifth is sometimes added: W5 = [ -1 +2 0 -2 +1 ] (Wave) From Laws' 4 vectors, 16 5x5 "energy maps" are then filtered down to 9 in order to remove certain symmetric pairs. For instance, L5E5 measures vertical edge content and E5L5 measures horizontal edge content. The average of these two measures is the "edginess" of the content. The resulting 9 maps used by Laws are as follows: L5E5/E5L5 L5R5/R5L5 E5S5/S5E5 S5S5 R5R5 L5S5/S5L5 E5E5 E5R5/R5E5 S5R5/R5S5 Running each of these nine maps over an image to create a new image of the value of the origin ([2,2]) results in 9 "energy maps," or conceptually an image with each pixel associated with a vector of 9 texture attributes. === Autocorrelation and power spectrum === The autocorrelation function of an image can be used to detect repetitive patterns of textures. == Texture segmentation == The use of image texture can be used as a description for regions into segments. There are two main types of segmentation based on image texture, region based and boundary based. Though image texture is not a perfect measure for segmentation it is used along with other measures, such as color, that helps solve segmenting in image. === Region based === Attempts to group or cluster pixels based on texture properties. === Boundary based === Attempts to group or cluster pixels based on edges between pixels that come from different texture properties.
Kernel density estimation
In statistics, kernel density estimation (KDE) is the application of kernel smoothing for probability density estimation, i.e., a non-parametric method to estimate the probability density function of a random variable based on kernels as weights. KDE answers a fundamental data smoothing problem where inferences about the population are made based on a finite data sample. In some fields such as signal processing and econometrics it is also termed the Parzen–Rosenblatt window method, after Emanuel Parzen and Murray Rosenblatt, who are usually credited with independently creating it in its current form. One of the famous applications of kernel density estimation is in estimating the class-conditional marginal densities of data when using a naive Bayes classifier, which can improve its prediction accuracy. == Definition == Let x = ( x 1 , x 2 , x 3 , . . . ) {\displaystyle \mathbf {x} =\left(x_{1},x_{2},x_{3},...\right)} be independent and identically distributed samples drawn from some univariate distribution with an unknown density f at any given point x. We are interested in estimating the shape of this function f. Its kernel density estimator is f ^ h ( x ) = 1 n ∑ i = 1 n K h ( x − x i ) = 1 n h ∑ i = 1 n K ( x − x i h ) , {\displaystyle {\hat {f}}_{h}(x)={\frac {1}{n}}\sum _{i=1}^{n}K_{h}(x-x_{i})={\frac {1}{nh}}\sum _{i=1}^{n}K{\left({\frac {x-x_{i}}{h}}\right)},} where K is the kernel — a non-negative function — and h > 0 is a smoothing parameter called the bandwidth or simply width. A kernel with subscript h is called the scaled kernel and defined as Kh(x) = 1/h K(x/h). Intuitively one wants to choose h as small as the data will allow; however, there is always a trade-off between the bias of the estimator and its variance. The choice of bandwidth is discussed in more detail below. A range of kernel functions are commonly used: uniform, triangular, biweight, triweight, Epanechnikov (parabolic), normal, and others. The Epanechnikov kernel is optimal in a mean square error sense, though the loss of efficiency is small for the kernels listed previously. Due to its convenient mathematical properties, the normal kernel is often used, which means K(x) = ϕ(x), where ϕ is the standard normal density function. The kernel density estimator then becomes f ^ h ( x ) = 1 n ∑ i = 1 n 1 h 2 π exp ( − ( x − x i ) 2 2 h 2 ) , {\displaystyle {\hat {f}}_{h}(x)={\frac {1}{n}}\sum _{i=1}^{n}{\frac {1}{h{\sqrt {2\pi }}}}\exp \left({\frac {-(x-x_{i})^{2}}{2h^{2}}}\right),} where h {\displaystyle h} is the standard deviation of the sample x {\displaystyle \mathbf {x} } . The construction of a kernel density estimate finds interpretations in fields outside of density estimation. For example, in thermodynamics, this is equivalent to the amount of heat generated when heat kernels (the fundamental solution to the heat equation) are placed at each data point locations xi. Similar methods are used to construct discrete Laplace operators on point clouds for manifold learning (e.g. diffusion map). == Example == Kernel density estimates are closely related to histograms, but can be endowed with properties such as smoothness or continuity by using a suitable kernel. The diagram below based on these 6 data points illustrates this relationship: For the histogram, first, the horizontal axis is divided into sub-intervals or bins which cover the range of the data: In this case, six bins each of width 2. Whenever a data point falls inside this interval, a box of height 1/12 is placed there. If more than one data point falls inside the same bin, the boxes are stacked on top of each other. For the kernel density estimate, normal kernels with a standard deviation of 1.5 (indicated by the red dashed lines) are placed on each of the data points xi. The kernels are summed to make the kernel density estimate (solid blue curve). The smoothness of the kernel density estimate (compared to the discreteness of the histogram) illustrates how kernel density estimates converge faster to the true underlying density for continuous random variables. == Bandwidth selection == The bandwidth of the kernel is a free parameter which exhibits a strong influence on the resulting estimate. To illustrate its effect, we take a simulated random sample from the standard normal distribution (plotted at the blue spikes in the rug plot on the horizontal axis). The grey curve is the true density (a normal density with mean 0 and variance 1). In comparison, the red curve is undersmoothed since it contains too many spurious data artifacts arising from using a bandwidth h = 0.05, which is too small. The green curve is oversmoothed since using the bandwidth h = 2 obscures much of the underlying structure. The black curve with a bandwidth of h = 0.337 is considered to be optimally smoothed since its density estimate is close to the true density. An extreme situation is encountered in the limit h → 0 {\displaystyle h\to 0} (no smoothing), where the estimate is a sum of n delta functions centered at the coordinates of analyzed samples. In the other extreme limit h → ∞ {\displaystyle h\to \infty } the estimate retains the shape of the used kernel, centered on the mean of the samples (completely smooth). The most common optimality criterion used to select this parameter is the expected L2 risk function, also termed the mean integrated squared error: MISE ( h ) = E [ ∫ ( f ^ h ( x ) − f ( x ) ) 2 d x ] {\displaystyle \operatorname {MISE} (h)=\operatorname {E} \!\left[\int \!{\left({\hat {f}}\!_{h}(x)-f(x)\right)}^{2}dx\right]} Under weak assumptions on f and K, (f is the, generally unknown, real density function), MISE ( h ) = AMISE ( h ) + o ( ( n h ) − 1 + h 4 ) {\displaystyle \operatorname {MISE} (h)=\operatorname {AMISE} (h)+{\mathcal {o}}{\left((nh)^{-1}+h^{4}\right)}} where o is the little o notation, and n the sample size (as above). The AMISE is the asymptotic MISE, i. e. the two leading terms, AMISE ( h ) = R ( K ) n h + 1 4 m 2 ( K ) 2 h 4 R ( f ″ ) {\displaystyle \operatorname {AMISE} (h)={\frac {R(K)}{nh}}+{\frac {1}{4}}m_{2}(K)^{2}h^{4}R(f'')} where R ( g ) = ∫ g ( x ) 2 d x {\textstyle R(g)=\int g(x)^{2}\,dx} for a function g, m 2 ( K ) = ∫ x 2 K ( x ) d x {\textstyle m_{2}(K)=\int x^{2}K(x)\,dx} and f ″ {\displaystyle f''} is the second derivative of f {\displaystyle f} and K {\displaystyle K} is the kernel. The minimum of this AMISE is the solution to this differential equation ∂ ∂ h AMISE ( h ) = − R ( K ) n h 2 + m 2 ( K ) 2 h 3 R ( f ″ ) = 0 {\displaystyle {\frac {\partial }{\partial h}}\operatorname {AMISE} (h)=-{\frac {R(K)}{nh^{2}}}+m_{2}(K)^{2}h^{3}R(f'')=0} or h AMISE = R ( K ) 1 / 5 m 2 ( K ) 2 / 5 R ( f ″ ) 1 / 5 n − 1 / 5 = C n − 1 / 5 {\displaystyle h_{\operatorname {AMISE} }={\frac {R(K)^{1/5}}{m_{2}(K)^{2/5}R(f'')^{1/5}}}n^{-1/5}=Cn^{-1/5}} Neither the AMISE nor the hAMISE formulas can be used directly since they involve the unknown density function f {\displaystyle f} or its second derivative f ″ {\displaystyle f''} . To overcome that difficulty, a variety of automatic, data-based methods have been developed to select the bandwidth. Several review studies have been undertaken to compare their efficacies, with the general consensus that the plug-in selectors and cross validation selectors are the most useful over a wide range of data sets. Substituting any bandwidth h which has the same asymptotic order n−1/5 as hAMISE into the AMISE gives that AMISE(h) = O(n−4/5), where O is the big O notation. It can be shown that, under weak assumptions, there cannot exist a non-parametric estimator that converges at a faster rate than the kernel estimator. Note that the n−4/5 rate is slower than the typical n−1 convergence rate of parametric methods. If the bandwidth is not held fixed, but is varied depending upon the location of either the estimate (balloon estimator) or the samples (pointwise estimator), this produces a particularly powerful method termed adaptive or variable bandwidth kernel density estimation. Bandwidth selection for kernel density estimation of heavy-tailed distributions is relatively difficult. === A rule-of-thumb bandwidth estimator === If Gaussian basis functions are used to approximate univariate data, and the underlying density being estimated is Gaussian, the optimal choice for h (that is, the bandwidth that minimises the mean integrated squared error) is: h = ( 4 σ ^ 5 3 n ) 1 / 5 ≈ 1.06 σ ^ n − 1 / 5 , {\displaystyle h={\left({\frac {4{\hat {\sigma }}^{5}}{3n}}\right)}^{1/5}\approx 1.06\,{\hat {\sigma }}\,n^{-1/5},} An h {\displaystyle h} value is considered more robust when it improves the fit for long-tailed and skewed distributions or for bimodal mixture distributions. This is often done empirically by replacing the standard deviation σ ^ {\displaystyle {\hat {\sigma }}} by the parameter A {\displaystyle A} below: A = min ( σ ^ , I Q R 1.34 ) {\displaystyle A=\min \left({\hat {\sigma }},{\frac {\mathrm {IQR} }{1.34}}\right)} where IQR is the