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Processo gaussiano ガウス過程 Proces gaussowski Gauß-Prozess Gaussian process Proceso de Gauss Гауссівський процес 高斯过程 Processus gaussien Гауссовский процесс Processo gaussiano
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In probability theory and statistics, a Gaussian process is a stochastic process (a collection of random variables indexed by time or space), such that every finite collection of those random variables has a multivariate normal distribution, i.e. every finite linear combination of them is normally distributed. The distribution of a Gaussian process is the joint distribution of all those (infinitely many) random variables, and as such, it is a distribution over functions with a continuous domain, e.g. time or space. Гауссівський процес в теорії випадкових процесів — це процес, чиї гауссовські. Ein Gaußprozess (nach Carl Friedrich Gauß) ist in der Wahrscheinlichkeitstheorie ein stochastischer Prozess, bei dem jede endliche Teilmenge von Zufallsvariablen mehrdimensional normalverteilt (gaußverteilt) ist. Ein Gaußprozess repräsentiert im Allgemeinen eine Funktion, deren Funktionswerte aufgrund eines Mangels an Information nur mit bestimmten Unsicherheiten und Wahrscheinlichkeiten modelliert werden können. Konstruiert wird er aus geeigneten Funktionen der Erwartungswerte, Varianzen und Kovarianzen und beschreibt damit die Funktionswerte als ein Kontinuum aus korrelierten Zufallsvariablen in Form einer unendlichdimensionalen Normalverteilung. Ein Gaußprozess ist somit eine Wahrscheinlichkeitsverteilung von Funktionen. Eine Stichprobe daraus ergibt eine zufällige Funktion mit bestimmt Un proceso de Gauss es un proceso estocástico que muestra en el tiempo de manera tal que no afecte la finitud de una combinación lineal que se tenga (o más generalmente cualquier funcional lineal de la función de muestra ), combinación lineal que se distribuirá normalmente. 在概率论和统计学中，高斯过程（英語：Gaussian process）是观测值出现在一个连续域（例如时间或空间）的随机过程。在高斯过程中，连续输入空间中每个点都是与一个正态分布的随机变量相关联。此外，这些随机变量的每个有限集合都有一个多元正态分布，换句话说他们的任意有限线性组合是一个正态分布。高斯过程的分布是所有那些（無限多个）随机变量的联合分布，正因如此，它是连续域（例如时间或空间）上函数的分布。 高斯過程被認為是一種機器學習算法，是以方式，利用點與點之間同質性的度量作為，以從輸入的訓練數據預測未知點的值。其預測結果不僅包含該點的值，而同時包含不確定性的資料－它的一維高斯分佈（即該點的邊際分佈）。 對於某些核函數，可以使用矩陣代數（見條目）來計算預測值。若核函數有代數參數，則通常使用軟體以擬合高斯過程的模型。 由於高斯過程是基於高斯分佈（正態分佈）的概念，故其以卡爾·弗里德里希·高斯為名。可以把高斯過程看成多元正態分佈的無限維廣義延伸。 高斯過程常用於統計建模中，而使用高斯過程的模型可以得到高斯過程的屬性。举例来说，如果把一隨機過程用高斯過程建模，我们可以显示求出各種導出量的分布，这些导出量可以是例如隨機過程在一定範圍次數內的平均值，及使用小範圍採樣次數及採樣值進行平均值預測的誤差。 Proces gaussowski – proces stochastyczny którego rozkłady skończenie wymiarowe są gaussowskie. Najbardziej znanymi przykładami procesów gaussowskich są proces Wienera i most Browna. In teoria delle probabilità un processo gaussiano è un processo stocastico f(x) tale che prendendo un qualsiasi numero finito di variabili aleatorie, dalla collezione che forma il processo aleatorio stesso, esse hanno una distribuzione di probabilità congiunta gaussiana. Un processo gaussiano è specificato interamente dalla sua media (x) e dalla covarianza (x,x'), e viene indicato nel modo seguente: En théorie des probabilités et en statistiques, un processus gaussien est un processus stochastique (une collection de variables aléatoires avec un index temporel ou spatial) de telle sorte que chaque collection finie de ces variables aléatoires suit une loi normale multidimensionnelle ; c'est-à-dire que chaque combinaison linéaire est normalement distribuée. La distribution d'un processus gaussien est la loi jointe de toutes ces variables aléatoires. Ses réalisations sont donc des fonctions avec un domaine continu. ガウス過程（ガウス-かてい、英: Gaussian process）は連続時間確率過程の一種である。この概念はカール・フリードリッヒ・ガウスの名にちなんでいるが、それは単に正規分布がガウス分布とも呼ばれるためであり、しかも正規分布はガウスが最初に研究したというわけでもない。いくつかの文献（たとえば下記のSimonの著書）では、確率変数 Xt の期待値が 0 であることを仮定する場合もある。 Em teoria da probabilidade e estatística, um processo gaussiano é um modelo estatístico em que as observações ocorrem em um domínio contínuo, por exemplo, tempo ou espaço. Em um processo gaussiano, cada ponto em algum espaço de entrada contínua está associada com uma variável aleatória com distribuição normal. Além disso, cada conjunto finito dessas variáveis ​​aleatórias tem uma distribuição normal multivariada. A distribuição de um processo gaussiano é a distribuição conjunta de todas as infinitas variáveis aleatórias, e, como tal, é uma distribuição de funções com um domínio contínuo. В теории вероятностей и статистике гауссовский процесс — это стохастический процесс (совокупность случайных величин, индексированных некоторым параметром, чаще всего временем или координатами), такой что любой конечный набор этих случайных величин имеет многомерное нормальное распределение, то есть любая конечная линейная комбинация из них нормально распределена. Распределение гауссовского процесса – это совместное распределение всех его случайных величин и, в силу чего, является распределением функций с непрерывной областью определения.
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Em teoria da probabilidade e estatística, um processo gaussiano é um modelo estatístico em que as observações ocorrem em um domínio contínuo, por exemplo, tempo ou espaço. Em um processo gaussiano, cada ponto em algum espaço de entrada contínua está associada com uma variável aleatória com distribuição normal. Além disso, cada conjunto finito dessas variáveis ​​aleatórias tem uma distribuição normal multivariada. A distribuição de um processo gaussiano é a distribuição conjunta de todas as infinitas variáveis aleatórias, e, como tal, é uma distribuição de funções com um domínio contínuo. Visto como um algoritmo de aprendizado de máquina, um processo gaussiano utiliza "aprendizagem preguiçosa" e uma medida da similaridade entre os pontos (a função kernel) para prever o valor de um ponto invisível a partir de dados de treinamento. A previsão não é apenas uma estimativa para esse ponto, mas tem também a informação da incerteza. É uma distribuição gaussiana unidimensional (que é uma distribuição marginal nesse ponto). Em algumas funções kernel, a álgebra de matrizes pode ser usada para calcular as previsões, como descrito no artigo krigagem. Quando um kernel parametrizado é utilizado, o software de otimização é tipicamente usado para ajustar um modelo de processo gaussiano. O conceito de processos gaussianos tem o nome de Carl Friedrich Gauss, porque se baseia na noção da distribuição gaussiana (distribuição normal). Processos gaussianos podem ser visto como uma generalização infinito-dimensional de distribuições normais multivariadas. Processos gaussianos são úteis na modelagem estatística, beneficiando-se de propriedades herdadas do normal. Por exemplo, se um processo aleatório é modelado como um processo gaussiano, as distribuições de várias derivadas de grandezas podem ser obtidas de forma explícita. Tais grandezas incluem o valor médio do processo em um dado intervalo de tempo e o erro na estimativa da média usando valores de amostra de um curto intervalo de tempo. 在概率论和统计学中，高斯过程（英語：Gaussian process）是观测值出现在一个连续域（例如时间或空间）的随机过程。在高斯过程中，连续输入空间中每个点都是与一个正态分布的随机变量相关联。此外，这些随机变量的每个有限集合都有一个多元正态分布，换句话说他们的任意有限线性组合是一个正态分布。高斯过程的分布是所有那些（無限多个）随机变量的联合分布，正因如此，它是连续域（例如时间或空间）上函数的分布。 高斯過程被認為是一種機器學習算法，是以方式，利用點與點之間同質性的度量作為，以從輸入的訓練數據預測未知點的值。其預測結果不僅包含該點的值，而同時包含不確定性的資料－它的一維高斯分佈（即該點的邊際分佈）。 對於某些核函數，可以使用矩陣代數（見條目）來計算預測值。若核函數有代數參數，則通常使用軟體以擬合高斯過程的模型。 由於高斯過程是基於高斯分佈（正態分佈）的概念，故其以卡爾·弗里德里希·高斯為名。可以把高斯過程看成多元正態分佈的無限維廣義延伸。 高斯過程常用於統計建模中，而使用高斯過程的模型可以得到高斯過程的屬性。举例来说，如果把一隨機過程用高斯過程建模，我们可以显示求出各種導出量的分布，这些导出量可以是例如隨機過程在一定範圍次數內的平均值，及使用小範圍採樣次數及採樣值進行平均值預測的誤差。 Гауссівський процес в теорії випадкових процесів — це процес, чиї гауссовські. En théorie des probabilités et en statistiques, un processus gaussien est un processus stochastique (une collection de variables aléatoires avec un index temporel ou spatial) de telle sorte que chaque collection finie de ces variables aléatoires suit une loi normale multidimensionnelle ; c'est-à-dire que chaque combinaison linéaire est normalement distribuée. La distribution d'un processus gaussien est la loi jointe de toutes ces variables aléatoires. Ses réalisations sont donc des fonctions avec un domaine continu. Un processus stochastique X sur un ensemble fini de sites S est dit gaussien si, pour toute partie finie A⊂S et toute suite réelle (a) sur A, ∑s∈A as X(s) est une variable gaussienne. Posant mA et ΣA la moyenne et la covariance de X sur A, si ΣA est inversible, alors XA = (Xs,s∈A) admet pour densité (ou vraisemblance) par rapport à la mesure de Lebesgue sur ℝcard(A) : В теории вероятностей и статистике гауссовский процесс — это стохастический процесс (совокупность случайных величин, индексированных некоторым параметром, чаще всего временем или координатами), такой что любой конечный набор этих случайных величин имеет многомерное нормальное распределение, то есть любая конечная линейная комбинация из них нормально распределена. Распределение гауссовского процесса – это совместное распределение всех его случайных величин и, в силу чего, является распределением функций с непрерывной областью определения. Если рассматривать гауссовский процесс как способ решения задач машинного обучения, то используется ленивое обучение и мера подобия между точками (функция ядра) для получения прогноза значения невидимой точки из обучающей выборки. В понятие прогноза, помимо самой оценки точки, входит информация о неопределенности — одномерное гауссовское распределение. Для вычисления прогнозов некоторых функций ядра используют метод матричной алгебры, кригинг. Гауссовский процесс назван так в честь Карла Фридриха Гаусса, поскольку в его основе лежит понятие гауссовского распределения (нормального распределения). Гауссовский процесс может рассматриваться как бесконечномерное обобщение многомерных нормальных распределений. Эти процессы применяются в статистическом моделировании; в частности используются свойства нормальности. Например, если случайный процесс моделируется как гауссовский, то распределения различных производных величин, такие как среднее значение процесса в течение определенного промежутка времени и погрешность его оценки с использованием выборки значений, могут быть получены явно. In probability theory and statistics, a Gaussian process is a stochastic process (a collection of random variables indexed by time or space), such that every finite collection of those random variables has a multivariate normal distribution, i.e. every finite linear combination of them is normally distributed. The distribution of a Gaussian process is the joint distribution of all those (infinitely many) random variables, and as such, it is a distribution over functions with a continuous domain, e.g. time or space. The concept of Gaussian processes is named after Carl Friedrich Gauss because it is based on the notion of the Gaussian distribution (normal distribution). Gaussian processes can be seen as an infinite-dimensional generalization of multivariate normal distributions. Gaussian processes are useful in statistical modelling, benefiting from properties inherited from the normal distribution. For example, if a random process is modelled as a Gaussian process, the distributions of various derived quantities can be obtained explicitly. Such quantities include the average value of the process over a range of times and the error in estimating the average using sample values at a small set of times. While exact models often scale poorly as the amount of data increases, multiple approximation methods have been developed which often retain good accuracy while drastically reducing computation time. Un proceso de Gauss es un proceso estocástico que muestra en el tiempo de manera tal que no afecte la finitud de una combinación lineal que se tenga (o más generalmente cualquier funcional lineal de la función de muestra ), combinación lineal que se distribuirá normalmente. Ein Gaußprozess (nach Carl Friedrich Gauß) ist in der Wahrscheinlichkeitstheorie ein stochastischer Prozess, bei dem jede endliche Teilmenge von Zufallsvariablen mehrdimensional normalverteilt (gaußverteilt) ist. Ein Gaußprozess repräsentiert im Allgemeinen eine Funktion, deren Funktionswerte aufgrund eines Mangels an Information nur mit bestimmten Unsicherheiten und Wahrscheinlichkeiten modelliert werden können. Konstruiert wird er aus geeigneten Funktionen der Erwartungswerte, Varianzen und Kovarianzen und beschreibt damit die Funktionswerte als ein Kontinuum aus korrelierten Zufallsvariablen in Form einer unendlichdimensionalen Normalverteilung. Ein Gaußprozess ist somit eine Wahrscheinlichkeitsverteilung von Funktionen. Eine Stichprobe daraus ergibt eine zufällige Funktion mit bestimmten bevorzugten Eigenschaften. ガウス過程（ガウス-かてい、英: Gaussian process）は連続時間確率過程の一種である。この概念はカール・フリードリッヒ・ガウスの名にちなんでいるが、それは単に正規分布がガウス分布とも呼ばれるためであり、しかも正規分布はガウスが最初に研究したというわけでもない。いくつかの文献（たとえば下記のSimonの著書）では、確率変数 Xt の期待値が 0 であることを仮定する場合もある。 Proces gaussowski – proces stochastyczny którego rozkłady skończenie wymiarowe są gaussowskie. Najbardziej znanymi przykładami procesów gaussowskich są proces Wienera i most Browna. In teoria delle probabilità un processo gaussiano è un processo stocastico f(x) tale che prendendo un qualsiasi numero finito di variabili aleatorie, dalla collezione che forma il processo aleatorio stesso, esse hanno una distribuzione di probabilità congiunta gaussiana. Un processo gaussiano è specificato interamente dalla sua media (x) e dalla covarianza (x,x'), e viene indicato nel modo seguente: Talvolta si assume che la media sia pari a zero e spesso si sceglie come insieme indice quello temporale cosicché il processo gaussiano risulti definito sul tempo. Accade di frequente nell'ambito delle telecomunicazioni, dove vari segnali vengono interpretati come processi gaussiani (ad esempio il rumore gaussiano).
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