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Confidence Band
A confidence band is used in statistical analysis to represent the uncertainty in an estimate of a curve or function based on limited or noisy data. Similarly, a prediction band is used to represent the uncertainty about the value of a new data-point on the curve, but subject to noise. Confidence and prediction bands are often used as part of the graphical presentation of results of a regression analysis. Confidence bands are closely related to confidence intervals, which represent the uncertainty in an estimate of a single numerical value. "As confidence intervals, by construction, only refer to a single point, they are narrower (at this point) than a confidence band which is supposed to hold simultaneously at many points." Pointwise and simultaneous confidence bands Suppose our aim is to estimate a function ''f''(''x''). For example, ''f''(''x'') might be the proportion of people of a particular age ''x'' who support a given candidate in an election. If ''x'' is measured at t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Statistics
Statistics (from German language, German: ''wikt:Statistik#German, Statistik'', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, industrial, or social problem, it is conventional to begin with a statistical population or a statistical model to be studied. Populations can be diverse groups of people or objects such as "all people living in a country" or "every atom composing a crystal". Statistics deals with every aspect of data, including the planning of data collection in terms of the design of statistical survey, surveys and experimental design, experiments.Dodge, Y. (2006) ''The Oxford Dictionary of Statistical Terms'', Oxford University Press. When census data cannot be collected, statisticians collect data by developing specific experiment designs and survey sample (statistics), samples. Representative sampling as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Empirical Distribution Function
In statistics, an empirical distribution function (commonly also called an empirical Cumulative Distribution Function, eCDF) is the distribution function associated with the empirical measure of a sample. This cumulative distribution function is a step function that jumps up by at each of the data points. Its value at any specified value of the measured variable is the fraction of observations of the measured variable that are less than or equal to the specified value. The empirical distribution function is an estimate of the cumulative distribution function that generated the points in the sample. It converges with probability 1 to that underlying distribution, according to the Glivenko–Cantelli theorem. A number of results exist to quantify the rate of convergence of the empirical distribution function to the underlying cumulative distribution function. Definition Let be independent, identically distributed real random variables with the common cumulative distribut ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Characteristic Function (probability Theory)
In probability theory and statistics, the characteristic function of any real-valued random variable completely defines its probability distribution. If a random variable admits a probability density function, then the characteristic function is the Fourier transform of the probability density function. Thus it provides an alternative route to analytical results compared with working directly with probability density functions or cumulative distribution functions. There are particularly simple results for the characteristic functions of distributions defined by the weighted sums of random variables. In addition to univariate distributions, characteristic functions can be defined for vector- or matrix-valued random variables, and can also be extended to more generic cases. The characteristic function always exists when treated as a function of a real-valued argument, unlike the moment-generating function. There are relations between the behavior of the characteristic function of a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Survival Function
The survival function is a function that gives the probability that a patient, device, or other object of interest will survive past a certain time. The survival function is also known as the survivor function or reliability function. The term ''reliability function'' is common in engineering while the term ''survival function'' is used in a broader range of applications, including human mortality. The survival function is the complementary cumulative distribution function of the lifetime. Sometimes complementary cumulative distribution functions are called survival functions in general. Definition Let the lifetime ''T'' be a continuous random variable with cumulative distribution function ''F''(''t'') on the interval [0,∞). Its ''survival function'' or ''reliability function'' is: :S(t) = P(\) = \int_t^ f(u)\,du = 1-F(t). Examples of survival functions The graphs below show examples of hypothetical survival functions. The x-axis is time. The y-axis is the proportion o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Scatterplot Smoothing
In statistics, several scatterplot smoothing methods are available to fit a function through the points of a scatterplot to best represent the relationship between the variables. Scatterplots may be smoothed by fitting a line to the data points in a diagram. This line attempts to display the non-random component of the association between the variables in a 2D scatter plot. Smoothing attempts to separate the non-random behaviour in the data from the random fluctuations, removing or reducing these fluctuations, and allows prediction of the response based value of the explanatory variable.Dodge, Y. (2006) ''The Oxford Dictionary of Statistical Terms'', OUP. (entry for "smoothing") Smoothing is normally accomplished by using any one of the techniques mentioned below. * A straight line (simple linear regression) * A quadratic or a polynomial curve * Local regression * Smoothing splines The smoothing curve is chosen so as to provide the best fit in some sense, often defined as the fi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantile Function
In probability and statistics, the quantile function, associated with a probability distribution of a random variable, specifies the value of the random variable such that the probability of the variable being less than or equal to that value equals the given probability. Intuitively, the quantile function associates with a range at and below a probability input the likelihood that a random variable is realized in that range for some probability distribution. It is also called the percentile function, percent-point function or inverse cumulative distribution function. Definition Strictly monotonic distribution function With reference to a continuous and strictly monotonic cumulative distribution function F_X\colon \mathbb \to ,1/math> of a random variable ''X'', the quantile function Q\colon , 1\to \mathbb returns a threshold value ''x'' below which random draws from the given c.d.f. would fall ''100*p'' percent of the time. In terms of the distribution function ''F'', the qua ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spectral Density
The power spectrum S_(f) of a time series x(t) describes the distribution of power into frequency components composing that signal. According to Fourier analysis, any physical signal can be decomposed into a number of discrete frequencies, or a spectrum of frequencies over a continuous range. The statistical average of a certain signal or sort of signal (including noise) as analyzed in terms of its frequency content, is called its spectrum. When the energy of the signal is concentrated around a finite time interval, especially if its total energy is finite, one may compute the energy spectral density. More commonly used is the power spectral density (or simply power spectrum), which applies to signals existing over ''all'' time, or over a time period large enough (especially in relation to the duration of a measurement) that it could as well have been over an infinite time interval. The power spectral density (PSD) then refers to the spectral energy distribution that would be ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probability Density Function
In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) can be interpreted as providing a ''relative likelihood'' that the value of the random variable would be close to that sample. Probability density is the probability per unit length, in other words, while the ''absolute likelihood'' for a continuous random variable to take on any particular value is 0 (since there is an infinite set of possible values to begin with), the value of the PDF at two different samples can be used to infer, in any particular draw of the random variable, how much more likely it is that the random variable would be close to one sample compared to the other sample. In a more precise sense, the PDF is used to specify the probability of the random variable falling ''within a particular range of values'', as opposed to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Regression Confidence Band
Regression or regressions may refer to: Science * Marine regression, coastal advance due to falling sea level, the opposite of marine transgression * Regression (medicine), a characteristic of diseases to express lighter symptoms or less extent (mainly for tumors), without disappearing totally * Regression (psychology), a defensive reaction to some unaccepted impulses * Nodal regression, the movement of the nodes of an object in orbit, in the opposite direction to the motion of the object Statistics * Regression analysis, a statistical technique for estimating the relationships among variables. There are several types of regression: ** Linear regression ** Simple linear regression ** Logistic regression ** Nonlinear regression ** Nonparametric regression ** Robust regression ** Stepwise regression * Regression toward the mean, a common statistical phenomenon Computing * Software regression, the appearance of a bug which was absent in a previous revision ** Regression testing, a s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Regression Analysis
In statistical modeling, regression analysis is a set of statistical processes for estimating the relationships between a dependent variable (often called the 'outcome' or 'response' variable, or a 'label' in machine learning parlance) and one or more independent variables (often called 'predictors', 'covariates', 'explanatory variables' or 'features'). The most common form of regression analysis is linear regression, in which one finds the line (or a more complex linear combination) that most closely fits the data according to a specific mathematical criterion. For example, the method of ordinary least squares computes the unique line (or hyperplane) that minimizes the sum of squared differences between the true data and that line (or hyperplane). For specific mathematical reasons (see linear regression), this allows the researcher to estimate the conditional expectation (or population average value) of the dependent variable when the independent variables take on a given ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Family-wise Error Rate
In statistics, family-wise error rate (FWER) is the probability of making one or more false discoveries, or type I errors when performing multiple hypotheses tests. Familywise and Experimentwise Error Rates Tukey (1953) developed the concept of a familywise error rate as the probability of making a Type I error among a specified group, or "family," of tests. Based on Tukey (1953), Ryan (1959) proposed the related concept of an ''experimentwise error rate'', which is the probability of making a Type I error in a given experiment. Hence, an experimentwise error rate is a familywise error rate for all of the tests that are conducted within an experiment. As Ryan (1959, Footnote 3) explained, an experiment may contain two or more families of multiple comparisons, each of which relates to a particular statistical inference and each of which has its own separate familywise error rate. Hence, familywise error rates are usually based on theoretically informative collections of multiple c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
