Deltahedron: /* References */ expand bibliodata

2012-09-21T21:03:46Z

References: expand bibliodata

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			In [[statistics]] and [[econometrics]], the '''multinomial probit model''' is a generalization of the [[probit model]] used when there are several possible categories that the [[dependent variable]] can fall into. As such, it is an alternative to the [[multinomial logit]] model as one method of [[multiclass classification]]. It is not to be confused with the ''multivariate'' probit model, which is used to model correlated binary outcomes for more than one dependent variable.

			==General specification==
			It is assumed that we have a series of observations ''Y''<sub>''i''</sub>, for ''i'' = 1...''n'', of the outcomes of multi-way choices from a [[categorical distribution]] of size ''m'' (there are ''m'' possible choices). Along with each observation ''Y''<sub>''i''</sub> is a set of ''k'' observed values ''x''<sub>''1,i''</sub>, ..., ''x''<sub>''k,i''</sub> of explanatory variables (also known as [[independent variable]]s, predictor variables, features, etc.). Some examples:
			*The observed outcomes might be "has disease A, has disease B, has disease C, has none of the diseases" for a set of rare diseases with similar symptoms, and the explanatory variables might be characteristics of the patients thought to be pertinent (sex, race, age, [[blood pressure]], [[body-mass index]], presence or absence of various symptoms, etc.).
			*The observed outcomes are the votes of people for a given party or candidate in a multi-way election, and the explanatory variables are the demographic characteristics of each person (e.g. sex, race, age, income, etc.).

			The multinomial probit model is a [[statistical model]] that can be used to predict the likely outcome of an unobserved multi-way trial given the associated explanatory variables. In the process, the model attempts to explain the relative effect of differing explanatory variables on the different outcomes.

			Formally, the outcomes ''Y''<sub>''i''</sub> are described as being [[categorical distribution\|categorically-distributed]] data, where each outcome value ''h'' for observation ''i'' occurs with an unobserved probability ''p''<sub>''i,h''</sub> that is specific to the observation ''i'' at hand because it is determined by the values of the explanatory variables associated with that observation. That is:

			:<math>Y_i\|x_{1,i},\ldots,x_{k,i} \ \sim \operatorname{Categorical}(p_i,\ldots,p_m),\text{ for }i = 1, \dots , n</math>

			or equivalently

			:<math>\Pr[Y_i=h\|x_{1,i},\ldots,x_{k,i}] = p_{i,h},\text{ for }i = 1, \dots , n,</math>

			for each of ''m'' possible values of ''h''.

			===Latent variable model===

			Multinomial probit is often written in terms of a [[latent variable model]]:

			: <math>
			\begin{align}
			Y_i^{1\ast} &= \boldsymbol\beta_0 \cdot \mathbf{X}_i + \varepsilon_1 \, \\
			Y_i^{2\ast} &= \boldsymbol\beta_1 \cdot \mathbf{X}_i + \varepsilon_2 \, \\
			\ldots & \ldots \\
			Y_i^{m\ast} &= \boldsymbol\beta_m \cdot \mathbf{X}_i + \varepsilon_m \, \\
			\end{align}
			</math>

			where

			: <math>\boldsymbol\varepsilon \sim \mathcal{N}(0,\boldsymbol\Sigma)</math>

			Then

			: <math> Y_i = \begin{cases}
			1 & \text{if }Y_i^{1\ast} > Y_i^{2\ast},\ldots,Y_i^{m\ast} \\
			2 & \text{if }Y_i^{2\ast} > Y_i^{1\ast},Y_i^{3\ast},\ldots,Y_i^{m\ast} \\
			\ldots & \ldots \\
			m &\text{otherwise.} \end{cases} </math>

			That is,
			:<math> Y_i = \arg\max_{h=1}^m Y_i^{h\ast}</math>

			Note that this model allows for arbitrary correlation between the [[error variable]]s, so that it doesn't necessarily respect [[independence of irrelevant alternatives]].

			When <math>\scriptstyle\boldsymbol\Sigma</math> is the identity matrix (such that there is no correlation or [[heteroscedasticity]]), the model is called '''independent probit'''.

			[[Category:Regression analysis]]

en>Fred Gandt: /* Pseudo algebraically closed fields */ Linking Rational point

2012-01-25T01:26:16Z

Pseudo algebraically closed fields: Linking Rational point

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Deltahedron: /* References */ expand bibliodata

en>Fred Gandt: /* Pseudo algebraically closed fields */ Linking Rational point