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Regularization is most commonly done using a squared regularizing function , which is equivalent to placing a zero-mean Gaussian prior distribution on the coefficients, but other regularizers are also possible.

Whether or not regularization is used, it is usually not possible to find a closed-form solution; instead, an iterative numerical method must be used, such as iteratively reweighted least squares IRLS or, more commonly these days, a quasi-Newton method such as the L-BFGS method.

An equivalent formula uses the inverse of the logit function, which is the logistic function , i. The formula can also be written as a probability distribution specifically, using a probability mass function :.

The above model has an equivalent formulation as a latent-variable model. This formulation is common in the theory of discrete choice models and makes it easier to extend to certain more complicated models with multiple, correlated choices, as well as to compare logistic regression to the closely related probit model.

Then Y i can be viewed as an indicator for whether this latent variable is positive:. The choice of modeling the error variable specifically with a standard logistic distribution, rather than a general logistic distribution with the location and scale set to arbitrary values, seems restrictive, but in fact, it is not.

It must be kept in mind that we can choose the regression coefficients ourselves, and very often can use them to offset changes in the parameters of the error variable's distribution.

Similarly, an arbitrary scale parameter s is equivalent to setting the scale parameter to 1 and then dividing all regression coefficients by s.

Note that this predicts that the irrelevancy of the scale parameter may not carry over into more complex models where more than two choices are available.

It turns out that this formulation is exactly equivalent to the preceding one, phrased in terms of the generalized linear model and without any latent variables.

This can be shown as follows, using the fact that the cumulative distribution function CDF of the standard logistic distribution is the logistic function , which is the inverse of the logit function , i.

This formulation—which is standard in discrete choice models—makes clear the relationship between logistic regression the "logit model" and the probit model , which uses an error variable distributed according to a standard normal distribution instead of a standard logistic distribution.

Both the logistic and normal distributions are symmetric with a basic unimodal, "bell curve" shape. The only difference is that the logistic distribution has somewhat heavier tails , which means that it is less sensitive to outlying data and hence somewhat more robust to model mis-specifications or erroneous data.

This model has a separate latent variable and a separate set of regression coefficients for each possible outcome of the dependent variable.

The reason for this separation is that it makes it easy to extend logistic regression to multi-outcome categorical variables, as in the multinomial logit model.

In such a model, it is natural to model each possible outcome using a different set of regression coefficients.

It is also possible to motivate each of the separate latent variables as the theoretical utility associated with making the associated choice, and thus motivate logistic regression in terms of utility theory.

In terms of utility theory, a rational actor always chooses the choice with the greatest associated utility. This is the approach taken by economists when formulating discrete choice models, because it both provides a theoretically strong foundation and facilitates intuitions about the model, which in turn makes it easy to consider various sorts of extensions.

See the example below. The choice of the type-1 extreme value distribution seems fairly arbitrary, but it makes the mathematics work out, and it may be possible to justify its use through rational choice theory.

It turns out that this model is equivalent to the previous model, although this seems non-obvious, since there are now two sets of regression coefficients and error variables, and the error variables have a different distribution.

In fact, this model reduces directly to the previous one with the following substitutions:. An intuition for this comes from the fact that, since we choose based on the maximum of two values, only their difference matters, not the exact values — and this effectively removes one degree of freedom.

Another critical fact is that the difference of two type-1 extreme-value-distributed variables is a logistic distribution, i.

As an example, consider a province-level election where the choice is between a right-of-center party, a left-of-center party, and a secessionist party e.

We would then use three latent variables, one for each choice. Then, in accordance with utility theory , we can then interpret the latent variables as expressing the utility that results from making each of the choices.

We can also interpret the regression coefficients as indicating the strength that the associated factor i. A voter might expect that the right-of-center party would lower taxes, especially on rich people.

This would give low-income people no benefit, i. On the other hand, the left-of-center party might be expected to raise taxes and offset it with increased welfare and other assistance for the lower and middle classes.

This would cause significant positive benefit to low-income people, perhaps a weak benefit to middle-income people, and significant negative benefit to high-income people.

Finally, the secessionist party would take no direct actions on the economy, but simply secede. Yet another formulation combines the two-way latent variable formulation above with the original formulation higher up without latent variables, and in the process provides a link to one of the standard formulations of the multinomial logit.

Here, instead of writing the logit of the probabilities p i as a linear predictor, we separate the linear predictor into two, one for each of the two outcomes:.

This term, as it turns out, serves as the normalizing factor ensuring that the result is a distribution.

This can be seen by exponentiating both sides:. In this form it is clear that the purpose of Z is to ensure that the resulting distribution over Y i is in fact a probability distribution , i.

This means that Z is simply the sum of all un-normalized probabilities, and by dividing each probability by Z , the probabilities become " normalized ".

That is:. This shows clearly how to generalize this formulation to more than two outcomes, as in multinomial logit.

Note that this general formulation is exactly the softmax function as in. In fact, it can be seen that adding any constant vector to both of them will produce the same probabilities:.

As a result, we can simplify matters, and restore identifiability, by picking an arbitrary value for one of the two vectors. Note that most treatments of the multinomial logit model start out either by extending the "log-linear" formulation presented here or the two-way latent variable formulation presented above, since both clearly show the way that the model could be extended to multi-way outcomes.

In general, the presentation with latent variables is more common in econometrics and political science , where discrete choice models and utility theory reign, while the "log-linear" formulation here is more common in computer science , e.

This functional form is commonly called a single-layer perceptron or single-layer artificial neural network. A single-layer neural network computes a continuous output instead of a step function.

With this choice, the single-layer neural network is identical to the logistic regression model. This function has a continuous derivative, which allows it to be used in backpropagation.

This function is also preferred because its derivative is easily calculated:. A closely related model assumes that each i is associated not with a single Bernoulli trial but with n i independent identically distributed trials, where the observation Y i is the number of successes observed the sum of the individual Bernoulli-distributed random variables , and hence follows a binomial distribution :.

An example of this distribution is the fraction of seeds p i that germinate after n i are planted.

In terms of expected values , this model is expressed as follows:. In a Bayesian statistics context, prior distributions are normally placed on the regression coefficients, usually in the form of Gaussian distributions.

There is no conjugate prior of the likelihood function in logistic regression. When Bayesian inference was performed analytically, this made the posterior distribution difficult to calculate except in very low dimensions.

However, when the sample size or the number of parameters is large, full Bayesian simulation can be slow, and people often use approximate methods such as variational Bayesian methods and expectation propagation.

A detailed history of the logistic regression is given in Cramer The logistic function was independently developed in chemistry as a model of autocatalysis Wilhelm Ostwald , This naturally gives rise to the logistic equation for the same reason as population growth: the reaction is self-reinforcing but constrained.

They were initially unaware of Verhulst's work and presumably learned about it from L. Gustave du Pasquier , but they gave him little credit and did not adopt his terminology.

The probit model influenced the subsequent development of the logit model and these models competed with each other. By , the logit model achieved parity with the probit model in use in statistics journals and thereafter surpassed it.

This relative popularity was due to the adoption of the logit outside of bioassay, rather than displacing the probit within bioassay, and its informal use in practice; the logit's popularity is credited to the logit model's computational simplicity, mathematical properties, and generality, allowing its use in varied fields.

Various refinements occurred during that time, notably by David Cox , as in Cox The multinomial logit model was introduced independently in Cox and Thiel , which greatly increased the scope of application and the popularity of the logit model.

Most statistical software can do binary logistic regression. Notably, Microsoft Excel 's statistics extension package does not include it.

From Wikipedia, the free encyclopedia. Statistical model for a binary dependent variable. It is not to be confused with Logit function.

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October Main article: One in ten rule. Mathematics portal. Trauma Score and the Injury Severity Score". The Journal of Trauma.

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Definition from Wiktionary, the free dictionary. See also: Appendix:Variations of "tam". English Wikipedia has an article on: tam as a cap.

English Wikipedia has an article on: picul. Sextus tam iratus erat ut fratrem interficere vellet Sextus was so angry that he wished to kill his brother.

Spinoza, Ethica Liber V: Sed omnia praeclara tam difficilia, quam rara sunt. But all things excellent are as rare as they are difficult.

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