Random forests or random decision forests is an ensemble learning method for classification, regression and other tasks that works by creating a multitude of decision trees during training. For classification tasks, the output of the random forest is the class selected by most trees. For regression tasks, the output is the average of the predictions of the trees. Random forests correct for decision trees' habit of overfitting to their training set. The first algorithm for random decision forests was created in 1995 by Tin Kam Ho using the random subspace method, which, in Ho's formulation, is a way to implement the "stochastic discrimination" approach to classification proposed by Eugene Kleinberg. An extension of the algorithm was developed by Leo Breiman and Adele Cutler, who registered "Random Forests" as a trademark in 2006 (as of 2019, owned by Minitab, Inc.). The extension combines Breiman's "bagging" idea and random selection of features, introduced first by Ho and later independently by Amit and Geman in order to construct a collection of decision trees with controlled variance. == History == The general method of random decision forests was first proposed by Salzberg and Heath in 1993, with a method that used a randomized decision tree algorithm to create multiple trees and then combine them using majority voting. This idea was developed further by Ho in 1995. Ho established that forests of trees splitting with oblique hyperplanes can gain accuracy as they grow without suffering from overtraining, as long as the forests are randomly restricted to be sensitive to only selected feature dimensions. A subsequent work along the same lines concluded that other splitting methods behave similarly, as long as they are randomly forced to be insensitive to some feature dimensions. This observation that a more complex classifier (a larger forest) gets more accurate nearly monotonically is in sharp contrast to the common belief that the complexity of a classifier can only grow to a certain level of accuracy before being hurt by overfitting. The explanation of the forest method's resistance to overtraining can be found in Kleinberg's theory of stochastic discrimination. The early development of Breiman's notion of random forests was influenced by the work of Amit and Geman who introduced the idea of searching over a random subset of the available decisions when splitting a node, in the context of growing a single tree. The idea of random subspace selection from Ho was also influential in the design of random forests. This method grows a forest of trees, and introduces variation among the trees by projecting the training data into a randomly chosen subspace before fitting each tree or each node. Finally, the idea of randomized node optimization, where the decision at each node is selected by a randomized procedure, rather than a deterministic optimization was first introduced by Thomas G. Dietterich. The proper introduction of random forests was made in a paper by Leo Breiman, that has become one of the world's most cited papers. This paper describes a method of building a forest of uncorrelated trees using a CART like procedure, combined with randomized node optimization and bagging. In addition, this paper combines several ingredients, some previously known and some novel, which form the basis of the modern practice of random forests, in particular: Using out-of-bag error as an estimate of the generalization error. Measuring variable importance through permutation. The report also offers the first theoretical result for random forests in the form of a bound on the generalization error which depends on the strength of the trees in the forest and their correlation. == Algorithm == === Preliminaries: decision tree learning === Decision trees are a popular method for various machine learning tasks. Tree learning is almost "an off-the-shelf procedure for data mining", say Hastie et al., "because it is invariant under scaling and various other transformations of feature values, is robust to inclusion of irrelevant features, and produces inspectable models. However, they are seldom accurate". In particular, trees that are grown very deep tend to learn highly irregular patterns: they overfit their training sets, i.e. have low bias, but very high variance. Random forests are a way of averaging multiple deep decision trees, trained on different parts of the same training set, with the goal of reducing the variance. This comes at the expense of a small increase in the bias and some loss of interpretability, but generally greatly boosts the performance in the final model. === Bagging === The training algorithm for random forests applies the general technique of bootstrap aggregating, or bagging, to tree learners. Given a training set X = x1, ..., xn with responses Y = y1, ..., yn, bagging repeatedly (B times) selects a random sample with replacement of the training set and fits trees to these samples: After training, predictions for unseen samples x' can be made by averaging the predictions from all the individual regression trees on x': f ^ = 1 B ∑ b = 1 B f b ( x ′ ) {\displaystyle {\hat {f}}={\frac {1}{B}}\sum _{b=1}^{B}f_{b}(x')} or by taking the plurality vote in the case of classification trees. This bootstrapping procedure leads to better model performance because it decreases the variance of the model, without increasing the bias. This means that while the predictions of a single tree are highly sensitive to noise in its training set, the average of many trees is not, as long as the trees are not correlated. Simply training many trees on a single training set would give strongly correlated trees (or even the same tree many times, if the training algorithm is deterministic); bootstrap sampling is a way of de-correlating the trees by showing them different training sets. Additionally, an estimate of the uncertainty of the prediction can be made as the standard deviation of the predictions from all the individual regression trees on x′: σ = ∑ b = 1 B ( f b ( x ′ ) − f ^ ) 2 B − 1 . {\displaystyle \sigma ={\sqrt {\frac {\sum _{b=1}^{B}(f_{b}(x')-{\hat {f}})^{2}}{B-1}}}.} The number B of samples (equivalently, of trees) is a free parameter. Typically, a few hundred to several thousand trees are used, depending on the size and nature of the training set. B can be optimized using cross-validation, or by observing the out-of-bag error: the mean prediction error on each training sample xi, using only the trees that did not have xi in their bootstrap sample. The training and test error tend to level off after some number of trees have been fit. === From bagging to random forests === The above procedure describes the original bagging algorithm for trees. Random forests also include another type of bagging scheme: they use a modified tree learning algorithm that selects, at each candidate split in the learning process, a random subset of the features. This process is sometimes called "feature bagging". The reason for doing this is the correlation of the trees in an ordinary bootstrap sample: if one or a few features are very strong predictors for the response variable (target output), these features will be selected in many of the B trees, causing them to become correlated. An analysis of how bagging and random subspace projection contribute to accuracy gains under different conditions is given by Ho. Typically, for a classification problem with p {\displaystyle p} features, p {\displaystyle {\sqrt {p}}} (rounded down) features are used in each split. For regression problems the inventors recommend p / 3 {\displaystyle p/3} (rounded down) with a minimum node size of 5 as the default. In practice, the best values for these parameters should be tuned on a case-to-case basis for every problem. === ExtraTrees === Adding one further step of randomization yields extremely randomized trees, or ExtraTrees. As with ordinary random forests, they are an ensemble of individual trees, but there are two main differences: (1) each tree is trained using the whole learning sample (rather than a bootstrap sample), and (2) the top-down splitting is randomized: for each feature under consideration, a number of random cut-points are selected, instead of computing the locally optimal cut-point (based on, e.g., information gain or the Gini impurity). The values are chosen from a uniform distribution within the feature's empirical range (in the tree's training set). Then, of all the randomly chosen splits, the split that yields the highest score is chosen to split the node. Similar to ordinary random forests, the number of randomly selected features to be considered at each node can be specified. Default values for this parameter are p {\displaystyle {\sqrt {p}}} for classification and p {\displaystyle p} for regression, where p {\displaystyle p} is the number of features in the model. === Random forests for high-dimensional data === The basic random forest procedure may
Uncertain data
In computer science, uncertain data is data that contains noise that makes it deviate from the correct, intended or original values. In the age of big data, uncertainty or data veracity is one of the defining characteristics of data. Data is constantly growing in volume, variety, velocity and uncertainty (1/veracity). Uncertain data is found in abundance today on the web, in sensor networks, within enterprises both in their structured and unstructured sources. For example, there may be uncertainty regarding the address of a customer in an enterprise dataset, or the temperature readings captured by a sensor due to aging of the sensor. In 2012 IBM called out managing uncertain data at scale in its global technology outlook report that presents a comprehensive analysis looking three to ten years into the future seeking to identify significant, disruptive technologies that will change the world. In order to make confident business decisions based on real-world data, analyses must necessarily account for many different kinds of uncertainty present in very large amounts of data. Analyses based on uncertain data will have an effect on the quality of subsequent decisions, so the degree and types of inaccuracies in this uncertain data cannot be ignored. Uncertain data is found in the area of sensor networks; text where noisy text is found in abundance on social media, web and within enterprises where the structured and unstructured data may be old, outdated, or plain incorrect; in modeling where the mathematical model may only be an approximation of the actual process. When representing such data in a database, an appropriate uncertain database model needs to be selected. == Example data model for uncertain data == One way to represent uncertain data is through probability distributions. Let us take the example of a relational database. There are three main ways to do represent uncertainty as probability distributions in such a database model. In attribute uncertainty, each uncertain attribute in a tuple is subject to its own independent probability distribution. For example, if readings are taken of temperature and wind speed, each would be described by its own probability distribution, as knowing the reading for one measurement would not provide any information about the other. In correlated uncertainty, multiple attributes may be described by a joint probability distribution. For example, if readings are taken of the position of an object, and the x- and y-coordinates stored, the probability of different values may depend on the distance from the recorded coordinates. As distance depends on both coordinates, it may be appropriate to use a joint distribution for these coordinates, as they are not independent. In tuple uncertainty, all the attributes of a tuple are subject to a joint probability distribution. This covers the case of correlated uncertainty, but also includes the case where there is a probability of a tuple not belonging in the relevant relation, which is indicated by all the probabilities not summing to one. For example, assume we have the following tuple from a probabilistic database: Then, the tuple has 10% chance of not existing in the database.
Dynamic Graphics Project
The Dynamic Graphics Project (commonly referred to as DGP) is an interdisciplinary research laboratory at the University of Toronto devoted to projects involving computer graphics, computer vision, human computer interaction, and visualization. The lab began as the computer graphics research group of Department of Computer Science Professor Leslie Mezei in 1967. Mezei invited Bill Buxton, a pioneer of human–computer interaction (HCI) to join. In 1972, Ronald Baecker, another HCI pioneer joined, establishing DGP as the first Canadian university group focused on computer graphics and human-computer interaction. According to csrankings.org, the DGP is the top research institution in the world for the combined subfields of computer graphics, HCI, and visualization. Since then, DGP has hosted many well known faculty and students in computer graphics, computer vision and HCI (e.g., Alain Fournier, Bill Reeves, Jos Stam, Demetri Terzopoulos, Marilyn Tremaine). DGP also occasionally hosts artists in residence (e.g., Oscar-winner Chris Landreth). Many past and current researchers at Autodesk (and before that Alias Wavefront) graduated after working at DGP. DGP is located in the St. George campus of University of Toronto in the Bahen Centre for Information Technology. DGP researchers regularly publish at ACM SIGGRAPH, ACM SIGCHI and ICCV. DGP hosts the Toronto User Experience (TUX) Speaker Series and the Sanders Series Lectures. == Notable alumni == Bill Buxton (MS 1978) James McCrae (PhD 2013) Dimitris Metaxas (PhD 1992) Bill Reeves (MS 1976, Ph.D. 1980) Jos Stam (MS 1991, Ph.D. 1995)
Kruti
Kruti is a multilingual AI agent and chatbot developed by the Indian company Ola Krutrim. It is designed to perform real-world tasks for users, such as booking taxis and ordering food, by integrating directly with various online services. It is notable for its ability to understand and respond in multiple Indian languages. Developed by a team founded by Bhavish Aggarwal, Kruti functions as an "agentic" AI, meaning it can reason, plan, and execute multi-step tasks to fulfill a user's request. The backend technology combines several open-source large language models with Ola's proprietary Krutrim V2 model. The system was developed to work primarily on smartphones, addressing the Indian market's specific needs, including language diversity and potential bandwidth constraints. Kruti was officially released in June 2025, replacing an earlier chatbot from the company that was also named Krutrim. Initially supporting 13 languages, the company plans to expand its capabilities to 22 Indian languages. == Background == Kruti is an improved version of Ola's Krutrim chatbot, which was first launched in 2023 and was intended to be replaced by Kruti. It was officially released on 12 June 2025 as an upgrade to passive chatbots, with support for text and voice in 13 Indian languages. As an agentic AI, it can execute tasks with customization and reasoning, providing adaptive answers based on user preferences and past interactions. Kruti is optimized for smartphone usage and designed to accommodate bandwidth constraints and usage patterns in India. To ensure scalability and cost-effective performance, it combines various open-source large language models with Ola's own Krutrim V2, which has 12 billion parameters. Its speech recognition is built to identify regional Indian languages, dialects, and accents. Due to its integration with numerous apps and services, Kruti is context-aware and can proactively complete tasks. Initially connected only with Ola ecosystem services, Krutrim intends to expand and incorporate various Indian services into Kruti, with the goal of adding services from Blinkit, Swiggy, and Uber with respective voice command support. On 20 June 2025, Krutrim acquired the AI platform BharatSah‘AI’yak to increase its involvement in government, education, and agriculture projects. This acquisition will allow Kruti to assist in broadening the scope of BharatSah'AI'yak's work on India-centric, vernacular retrieval-augmented generation AI bots. == Development == Kruti is designed to perform tasks with minimal user input, accepting documents, images, and text, without requiring users to switch between applications. Its agentic framework breaks queries into sub-tasks executed by multiple agents working sequentially or concurrently, with reported accuracy exceeding 90%. Kruti connects to company databases and APIs via the Model Context Protocol and presents responses as summaries, tables, or narratives adapted to user behaviour. The system supports payments via credit/debit cards and UPI. The underlying stack, which includes foundation models and AI training and inference systems, is intended to support adaptation across sectors such as healthcare, education, and finance. Ola Cabs and the Open Network for Digital Commerce have begun integrating Kruti into their platforms pending broader reliability testing.
CMU Pronouncing Dictionary
The CMU Pronouncing Dictionary (also known as CMUdict) is an open-source pronouncing dictionary originally created by the Speech Group at Carnegie Mellon University (CMU) for use in speech recognition research. CMUdict provides a mapping orthographic/phonetic for English words in their North American pronunciations. It is commonly used to generate representations for speech recognition (ASR), e.g. the CMU Sphinx system, and speech synthesis (TTS), e.g. the Festival system. CMUdict can be used as a training corpus for building statistical grapheme-to-phoneme (g2p) models that will generate pronunciations for words not yet included in the dictionary. The most recent release is 0.7b; it contains over 134,000 entries. An interactive lookup version is available. == Database format == The database is distributed as a plain text file with one entry to a line in the format "WORD
Scale space implementation
In the areas of computer vision, image analysis and signal processing, the notion of scale-space representation is used for processing measurement data at multiple scales, and specifically enhance or suppress image features over different ranges of scale (see the article on scale space). A special type of scale-space representation is provided by the Gaussian scale space, where the image data in N dimensions is subjected to smoothing by Gaussian convolution. Most of the theory for Gaussian scale space deals with continuous images, whereas one when implementing this theory will have to face the fact that most measurement data are discrete. Hence, the theoretical problem arises concerning how to discretize the continuous theory while either preserving or well approximating the desirable theoretical properties that lead to the choice of the Gaussian kernel (see the article on scale-space axioms). This article describes basic approaches for this that have been developed in the literature, see also for an in-depth treatment regarding the topic of approximating the Gaussian smoothing operation and the Gaussian derivative computations in scale-space theory, and for a complementary treatment regarding hybrid discretization methods. == Statement of the problem == The Gaussian scale-space representation of an N-dimensional continuous signal, f C ( x 1 , ⋯ , x N , t ) , {\displaystyle f_{C}\left(x_{1},\cdots ,x_{N},t\right),} is obtained by convolving fC with an N-dimensional Gaussian kernel: g N ( x 1 , ⋯ , x N , t ) . {\displaystyle g_{N}\left(x_{1},\cdots ,x_{N},t\right).} In other words: L ( x 1 , ⋯ , x N , t ) = ∫ u 1 = − ∞ ∞ ⋯ ∫ u N = − ∞ ∞ f C ( x 1 − u 1 , ⋯ , x N − u N , t ) ⋅ g N ( u 1 , ⋯ , u N , t ) d u 1 ⋯ d u N . {\displaystyle L\left(x_{1},\cdots ,x_{N},t\right)=\int _{u_{1}=-\infty }^{\infty }\cdots \int _{u_{N}=-\infty }^{\infty }f_{C}\left(x_{1}-u_{1},\cdots ,x_{N}-u_{N},t\right)\cdot g_{N}\left(u_{1},\cdots ,u_{N},t\right)\,du_{1}\cdots du_{N}.} However, for implementation, this definition is impractical, since it is continuous. When applying the scale space concept to a discrete signal fD, different approaches can be taken. This article is a brief summary of some of the most frequently used methods. == Separability == Using the separability property of the Gaussian kernel g N ( x 1 , … , x N , t ) = G ( x 1 , t ) ⋯ G ( x N , t ) {\displaystyle g_{N}\left(x_{1},\dots ,x_{N},t\right)=G\left(x_{1},t\right)\cdots G\left(x_{N},t\right)} the N-dimensional convolution operation can be decomposed into a set of separable smoothing steps with a one-dimensional Gaussian kernel G along each dimension L ( x 1 , ⋯ , x N , t ) = ∫ u 1 = − ∞ ∞ ⋯ ∫ u N = − ∞ ∞ f C ( x 1 − u 1 , ⋯ , x N − u N , t ) G ( u 1 , t ) d u 1 ⋯ G ( u N , t ) d u N , {\displaystyle L(x_{1},\cdots ,x_{N},t)=\int _{u_{1}=-\infty }^{\infty }\cdots \int _{u_{N}=-\infty }^{\infty }f_{C}(x_{1}-u_{1},\cdots ,x_{N}-u_{N},t)G(u_{1},t)\,du_{1}\cdots G(u_{N},t)\,du_{N},} where G ( x , t ) = 1 2 π t e − x 2 2 t {\displaystyle G(x,t)={\frac {1}{\sqrt {2\pi t}}}e^{-{\frac {x^{2}}{2t}}}} and the standard deviation of the Gaussian σ is related to the scale parameter t according to t = σ2. Separability will be assumed in all that follows, even when the kernel is not exactly Gaussian, since separation of the dimensions is the most practical way to implement multidimensional smoothing, especially at larger scales. Therefore, the rest of the article focuses on the one-dimensional case. == The sampled Gaussian kernel == When implementing the one-dimensional smoothing step in practice, the presumably simplest approach is to convolve the discrete signal fD with a sampled Gaussian kernel: L ( x , t ) = ∑ n = − ∞ ∞ f ( x − n ) G ( n , t ) {\displaystyle L(x,t)=\sum _{n=-\infty }^{\infty }f(x-n)\,G(n,t)} where G ( n , t ) = 1 2 π t e − n 2 2 t {\displaystyle G(n,t)={\frac {1}{\sqrt {2\pi t}}}e^{-{\frac {n^{2}}{2t}}}} (with t = σ2) which in turn is truncated at the ends to give a filter with finite impulse response L ( x , t ) = ∑ n = − M M f ( x − n ) G ( n , t ) {\displaystyle L(x,t)=\sum _{n=-M}^{M}f(x-n)\,G(n,t)} for M chosen sufficiently large (see error function) such that 2 ∫ M ∞ G ( u , t ) d u = 2 ∫ M t ∞ G ( v , 1 ) d v < ε . {\displaystyle 2\int _{M}^{\infty }G(u,t)\,du=2\int _{\frac {M}{\sqrt {t}}}^{\infty }G(v,1)\,dv<\varepsilon .} A common choice is to set M to a constant C times the standard deviation of the Gaussian kernel M = C σ + 1 = C t + 1 {\displaystyle M=C\sigma +1=C{\sqrt {t}}+1} where C is often chosen somewhere between 3 and 6. Using the sampled Gaussian kernel can, however, lead to implementation problems, in particular when computing higher-order derivatives at finer scales by applying sampled derivatives of Gaussian kernels. When accuracy and robustness are primary design criteria, alternative implementation approaches should therefore be considered. For small values of ε (10−6 to 10−8) the errors introduced by truncating the Gaussian are usually negligible. For larger values of ε, however, there are many better alternatives to a rectangular window function. For example, for a given number of points, a Hamming window, Blackman window, or Kaiser window will do less damage to the spectral and other properties of the Gaussian than a simple truncation will. Notwithstanding this, since the Gaussian kernel decreases rapidly at the tails, the main recommendation is still to use a sufficiently small value of ε such that the truncation effects are no longer important. == The discrete Gaussian kernel == A more refined approach is to convolve the original signal with the discrete Gaussian kernel T(n, t) L ( x , t ) = ∑ n = − ∞ ∞ f ( x − n ) T ( n , t ) {\displaystyle L(x,t)=\sum _{n=-\infty }^{\infty }f(x-n)\,T(n,t)} where T ( n , t ) = e − t I n ( t ) {\displaystyle T(n,t)=e^{-t}I_{n}(t)} and I n ( t ) {\displaystyle I_{n}(t)} denotes the modified Bessel functions of integer order, n. This is the discrete counterpart of the continuous Gaussian in that it is the solution to the discrete diffusion equation (discrete space, continuous time), just as the continuous Gaussian is the solution to the continuous diffusion equation. This filter can be truncated in the spatial domain as for the sampled Gaussian L ( x , t ) = ∑ n = − M M f ( x − n ) T ( n , t ) {\displaystyle L(x,t)=\sum _{n=-M}^{M}f(x-n)\,T(n,t)} or can be implemented in the Fourier domain using a closed-form expression for its discrete-time Fourier transform: T ^ ( θ , t ) = ∑ n = − ∞ ∞ T ( n , t ) e − i θ n = e t ( cos θ − 1 ) . {\displaystyle {\widehat {T}}(\theta ,t)=\sum _{n=-\infty }^{\infty }T(n,t)\,e^{-i\theta n}=e^{t(\cos \theta -1)}.} With this frequency-domain approach, the scale-space properties transfer exactly to the discrete domain, or with excellent approximation using periodic extension and a suitably long discrete Fourier transform to approximate the discrete-time Fourier transform of the signal being smoothed. Moreover, higher-order derivative approximations can be computed in a straightforward manner (and preserving scale-space properties) by applying small support central difference operators to the discrete scale space representation. As with the sampled Gaussian, a plain truncation of the infinite impulse response will in most cases be a sufficient approximation for small values of ε, while for larger values of ε it is better to use either a decomposition of the discrete Gaussian into a cascade of generalized binomial filters or alternatively to construct a finite approximate kernel by multiplying by a window function. If ε has been chosen too large such that effects of the truncation error begin to appear (for example as spurious extrema or spurious responses to higher-order derivative operators), then the options are to decrease the value of ε such that a larger finite kernel is used, with cutoff where the support is very small, or to use a tapered window. == Recursive filters == Since computational efficiency is often important, low-order recursive filters are often used for scale-space smoothing. For example, Young and van Vliet use a third-order recursive filter with one real pole and a pair of complex poles, applied forward and backward to make a sixth-order symmetric approximation to the Gaussian with low computational complexity for any smoothing scale. By relaxing a few of the axioms, Lindeberg concluded that good smoothing filters would be "normalized Pólya frequency sequences", a family of discrete kernels that includes all filters with real poles at 0 < Z < 1 and/or Z > 1, as well as with real zeros at Z < 0. For symmetry, which leads to approximate directional homogeneity, these filters must be further restricted to pairs of poles and zeros that lead to zero-phase filters. To match the transfer function curvature at zero frequency of the discrete Gaussian, which ensures an approximate semi-group property of additive t, two poles at Z = 1 + 2 t − ( 1 + 2 t ) 2 − 1 {\displaystyle
Grammatik
Grammatik was the first grammar-checking program for home computers. Aspen Software of Albuquerque, NM, released the earliest version of this diction and style checker for personal computers. It was first released no later than 1981, and was inspired by the Writer's Workbench. Grammatik was first available for the TRS-80, and soon had versions for CP/M and the IBM PC. Reference Software International of San Francisco, California, acquired Grammatik in 1985. Development of Grammatik continued, and it became an actual grammar checker that could detect writing errors beyond simple style checking. Subsequent versions were released for MS-DOS, Windows, Macintosh, and Unix. Grammatik was ultimately acquired by WordPerfect Corporation and is integrated into the WordPerfect word processor.