Note that if the canonical link function is used, then they are the same. 3 bayesian methods edit In general, the posterior distribution cannot be found in closed form and so must be approximated, usually using Laplace approximations or some type of Markov chain Monte carlo method such as Gibbs sampling. Examples edit general linear models edit a possible point of confusion has to do with the distinction between generalized linear models and the general linear model, two broad statistical models. The general linear model may be viewed as a special case of the generalized linear model with identity link and responses normally distributed. As most exact results of interest are obtained only for the general linear model, the general linear model has undergone a somewhat longer historical development. Results for the generalized linear model with non-identity link are asymptotic (tending to work well with large samples). Linear regression edit a simple, very important example of a generalized linear model (also an example of a general linear model) is linear regression.
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The resulting model is known as logistic regression (or multinomial logistic regression in the case that K-way rather than binary values are being predicted). For the bernoulli and binomial distributions, the parameter is a single probability, indicating the likelihood of occurrence of a single event. The bernoulli still satisfies the basic condition of the generalized linear model in that, writing even though a single outcome will always be either 0 or 1, the expected value will nonetheless be a real-valued probability,. The probability of occurrence of a "yes" (or 1) outcome. Similarly, in a binomial distribution, the expected value is Np,. The expected proportion of "yes" outcomes will be the probability to be predicted. For categorical and multinomial distributions, the parameter to be predicted is a k -vector of probabilities, with the further restriction that all probabilities must add up. Each probability indicates the likelihood of occurrence of one of the k possible values. For the multinomial distribution, and for the vector form of the categorical distribution, the expected values of the elements of the vector can be related to the predicted probabilities similarly to the binomial and Bernoulli distributions. Fitting edit maximum likelihood edit The maximum likelihood estimates can be found writes using an iteratively reweighted least squares algorithm or a newtonRaphson method with updates of the form: boldsymbol beta (t1)boldsymbol beta (t)mathcal J-1(boldsymbol beta (t)u(boldsymbol beta (t where J(β(t)displaystyle mathcal J(boldsymbol beta (t).
Binomial integer: 0,1 Ndisplaystyle 0,1,ldots, n count of of "yes" occurrences out of n yes/no occurrences Categorical integer: 0,K)displaystyle 0,K) outcome of single k-way occurrence k-vector of integer: 0,1displaystyle 0,1, where exactly one element in the vector has the value 1 Multinomial k -vector. K ) out of N total k -way occurrences In the cases of the exponential and gamma distributions, the domain of the canonical link function is not the same as the permitted range of the mean. In particular, the linear predictor may be positive, which would give an impossible negative mean. When reviews maximizing the likelihood, precautions must be taken to avoid this. An alternative is to use a noncanonical link function. Note also that in the case of the bernoulli, binomial, categorical and multinomial distributions, the support of the distributions is not the same type of data as the parameter being predicted. In all of these cases, the predicted parameter is one or more probabilities,. Real numbers in the range 0,1displaystyle 0,1.
Μxβdisplaystyle mu mathbf X boldsymbol writings beta! Exponential real: (0 displaystyle (0,infty ) Exponential-response data, scale parameters Negative inverse xβμ1displaystyle mathbf X boldsymbol beta -mu -1! Μ(Xβ)1displaystyle mu -(mathbf X boldsymbol beta )-1! Gamma Inverse gaussian real: (0 displaystyle (0,infty ) Inverse squared xβμ2displaystyle mathbf X boldsymbol beta mu -2! Μ(Xβ)1/2displaystyle mu (mathbf X boldsymbol beta )-1/2! Poisson integer: 0,1,2,displaystyle 0,1,2,ldots count of occurrences in fixed amount of time/space log Xβln(μ)displaystyle mathbf X boldsymbol beta ln(mu! Μexp(Xβ)displaystyle mu exp(mathbf statement X boldsymbol beta! Bernoulli integer: 0,1displaystyle 0,1 outcome of single yes/no occurrence logit Xβln(μ1μ)displaystyle mathbf X boldsymbol beta ln left(frac mu 1-mu right! mu frac exp(mathbf X boldsymbol beta )1exp(mathbf X boldsymbol beta )frac 11exp(-mathbf X boldsymbol beta!
There is always a well-defined canonical link function which is derived from the exponential of the response's density function. However, in some cases it makes sense to try to match the domain of the link function to the range of the distribution function's mean, or use a non-canonical link function for algorithmic purposes, for example bayesian probit regression. When using a distribution function with a canonical parameter θdisplaystyle theta, the canonical link function is the function that expresses θdisplaystyle theta in terms of μdisplaystyle mu,. Θb(μ)displaystyle theta b(mu ). For the most common distributions, the mean μdisplaystyle mu is one of the parameters in the standard form of the distribution's density function, and then b(μ)displaystyle b(mu ) is the function as defined above that maps the density function into its canonical form. When using the canonical link function, b(μ)θXβdisplaystyle b(mu )theta mathbf X boldsymbol beta, which allows xtydisplaystyle mathbf X rm Tmathbf y to be a sufficient statistic for βdisplaystyle boldsymbol beta. Following is a table of several exponential-family distributions in common use and the data they are typically used for, along with the canonical link functions and their inverses (sometimes referred to as the mean function, as done here). Common distributions with typical uses and canonical link functions Distribution Support of distribution Typical uses Link name link function, xβg(μ)displaystyle mathbf X boldsymbol beta g(mu! Mean function Normal real: displaystyle (-infty, infty ) Linear-response data Identity xβμdisplaystyle mathbf X boldsymbol beta mu!
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If, in addition, T(y)displaystyle mathbf T (y) is the report identity and τdisplaystyle tau is known, then θdisplaystyle boldsymbol theta is called the canonical parameter (or natural parameter ) and is related to the mean through μE(Y)A(θ).displaystyle boldsymbol mu operatorname e (mathbf Y )nabla A(boldsymbol theta. For scalar Ydisplaystyle y and θdisplaystyle theta, this reduces to μe(Y)A(θ).displaystyle mu operatorname e (Y)A theta ).! Under this scenario, the variance of the distribution can be shown to be 2 Var(Y)2A(θ)d(τ).displaystyle operatorname var (mathbf Y )nabla 2A(boldsymbol theta )d(tau ).! For scalar Ydisplaystyle y and θdisplaystyle theta, this reduces to var(Y)A(θ)d(τ).displaystyle operatorname var (Y)A theta )d(tau ).! Linear predictor edit The linear predictor is the quantity which incorporates the information about the independent variables into the model.
The symbol η ( Greek " eta denotes a linear predictor. It is related to the expected value of the data (thus, "predictor through the link function. Η is expressed as linear combinations (thus, "linear of unknown parameters. The coefficients of the linear combination are represented as the matrix of independent variables. Η can thus be expressed as ηxβ. Displaystyle eta mathbf X boldsymbol beta., link function edit The link function provides the relationship between the linear predictor and the mean of the distribution function. There are many commonly used link functions, and their choice is informed by several considerations.
The unknown parameters, β, are typically estimated with maximum likelihood, maximum quasi-likelihood, or bayesian techniques. Model components edit The glm consists of three elements:. A probability distribution from the exponential family. A linear predictor η. A link function g such that E( Y ) μ g 1( η ).
Probability distribution edit The overdispersed exponential family of distributions is a generalization of the exponential family and exponential dispersion model of distributions and includes those probability distributions, parameterized by θdisplaystyle boldsymbol theta and τdisplaystyle tau, whose density functions f (or probability mass function, for the. The dispersion parameter, τdisplaystyle tau, typically is known and is usually related to the variance of the distribution. The functions h(y,τ)displaystyle h(mathbf y, tau ), b(θ)displaystyle mathbf b (boldsymbol theta ), t(y)displaystyle mathbf T (y), a(θ)displaystyle A(boldsymbol theta ), and d(τ)displaystyle d(tau ) are known. Many common distributions are in this family, including the normal, exponential, gamma, poisson, bernoulli, and (for fixed number of trials) binomial, multinomial, and negative binomial. For scalar Ydisplaystyle y and θdisplaystyle theta, this reduces to f_Y(ymid theta, tau )h(y,tau )exp left(frac b(theta )T(y)-A(theta )d(tau )right).! Θdisplaystyle boldsymbol theta is related to the mean of the distribution. If b(θ)displaystyle mathbf b (boldsymbol theta ) is the identity function, then the distribution is said to be in canonical form (or natural form ). Note that any distribution can be converted to canonical form by rewriting θdisplaystyle boldsymbol theta as θdisplaystyle boldsymbol theta ' and then applying the transformation θb(θ)displaystyle boldsymbol theta mathbf b (boldsymbol theta. It is always possible to convert A(θ)displaystyle A(boldsymbol theta ) in terms of the new parametrization, even if b(θ)displaystyle mathbf b (boldsymbol theta is not a one-to-one function; see comments in the page on the exponential family.
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Rather, it is the odds that are doubling: from 2:1 odds, to 4:1 odds, to 8:1 odds, etc. Such a model is a log-odds or logistic model. Generalized linear models cover all these situations by allowing for response variables that have arbitrary distributions (rather than simply normal distributions and for an arbitrary function of the response variable (the link function ) to vary linearly with the predicted values (rather than assuming that. For example, the case above of predicted number of beach attendees would typically be modeled with a poisson distribution and a log link, while the case of predicted probability of beach attendance would typically be modeled with a bernoulli distribution (or binomial distribution, depending. Overview edit In a generalized linear model (glm each outcome y of the dependent variables is assumed to be generated from a particular distribution in the exponential family, a large range of probability distributions that includes the normal, binomial, biography poisson and gamma distributions, among others. The mean, μ, of the distribution depends on the independent variables, x, through: E(Y)μg1(Xβ)displaystyle operatorname e (mathbf Y )boldsymbol mu g-1(mathbf X boldsymbol beta ) where E( Y ) is the expected value of Y ; x β is the linear predictor, a linear combination. In this framework, the variance is typically a function, v, of the mean: operatorname var (mathbf Y )operatorname v (boldsymbol mu )operatorname v (g-1(mathbf X boldsymbol beta ). It is convenient if V follows from the exponential family summary distribution, but it may simply be that the variance is a function of the predicted value.
to have the impossible attendance. Logically, a more realistic model would instead predict a constant rate of increased beach attendance (e.g. An increase in 10 degrees leads to a doubling in beach attendance, and a drop in 10 degrees leads to a halving in attendance). Such a model is termed an exponential-response model (or log-linear model, since the logarithm of the response is predicted to vary linearly). Similarly, a model that predicts a probability of making a yes/no choice (a bernoulli variable ) is even less suitable as a linear-response model, since probabilities are bounded on both ends (they must be between 0 and 1). Imagine, for example, a model that predicts the likelihood of a given person going to the beach as a function of temperature. A reasonable model might predict, for example, that a change in 10 degrees makes a person two times more or less likely to go to the beach. But what does "twice as likely" mean in terms of a probability? It cannot literally mean to double the probability value (e.g. 50 becomes 100, 75 becomes 150, etc.).
Contents, intuition edit, ordinary linear regression predicts the expected value of a given unknown quantity (the response variable, a random variable ) as a linear combination of a set of observed values ( predictors ). This implies that a constant change in a predictor leads to a constant change in the response variable (i.e. A linear-response model ). This is appropriate when the response variable has a normal distribution (intuitively, when a response variable can vary essentially indefinitely in either direction with no fixed "zero value or more generally for any quantity that only varies by a relatively small amount,. However, these assumptions are inappropriate for some types of response variables. For example, in cases where the response variable is expected to be always positive and varying over a wide range, constant input changes lead to geometrically varying, rather than constantly varying, output changes. As an example, a prediction model might predict that 10 degree temperature decrease would lead to 1,000 fewer people visiting the beach is unlikely to generalize well over both small beaches (e.g. Those where the expected attendance was 50 at a particular temperature) and large beaches (e.g. Those where the expected attendance was 10,000 at a low temperature).
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Not to be confused with essays general linear model or generalized least squares. In statistics, the generalized linear model gLM ) is a flexible generalization of ordinary linear regression that allows for response variables that have error distribution models other than a normal distribution. The glm generalizes linear regression by allowing the linear model to be related to the response variable via a link function and by allowing the magnitude of the variance of each measurement to be a function of its predicted value. Generalized linear models were formulated by, john Nelder and. Robert Wedderburn as a way of unifying various other statistical models, including linear regression, logistic regression and, poisson regression. 1, they proposed an iteratively reweighted least squares method for maximum likelihood estimation of the model parameters. Maximum-likelihood estimation remains popular and is the default method on many statistical computing packages. Bayesian approaches and least squares fits to variance stabilized responses, have been developed.