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{{merge|Linear predictor function|discuss=Talk:Linear predictor function#Proposed merge with Linear model|date=January 2014}}
In [[statistics]], the term '''linear model''' is used in different ways according to the context. The most common occurrence is in connection with regression models and the term is often taken as synonymous with [[linear regression]] model.  However, the term is also used in [[time series analysis]] with a different meaning. In each case, the designation "linear" is used to identify a subclass of models for which substantial reduction in the complexity of the related [[statistical theory]] is possible.
 
==Linear regression models==
 
For the regression case, the [[statistical model]] is as follows. Given a (random) sample <math> (Y_i, X_{i1}, \ldots, X_{ip}), \, i = 1, \ldots, n </math> the relation between the observations ''Y<sub>i</sub>'' and the [[independent variables]] ''X<sub>ij</sub>'' is formulated as
 
:<math>Y_i = \beta_0 + \beta_1 \phi_1(X_{i1}) + \cdots + \beta_p \phi_p(X_{ip}) + \varepsilon_i \qquad i = 1, \ldots, n </math>
 
where <math> \phi_1, \ldots, \phi_p </math> may be nonlinear functions. In the above, the quantities ''ε<sub>i</sub>'' are random variables representing errors in the relationship. The "linear" part of the designation relates to the appearance of the [[regression coefficient]]s, ''β<sub>j</sub>'' in a linear way in the above relationship. Alternatively, one may say that the predicted values corresponding to the above model, namely
:<math>\hat{Y}_i = \beta_0 + \beta_1 \phi_1(X_{i1}) + \cdots + \beta_p \phi_p(X_{ip}) \qquad (i = 1, \ldots, n), </math>
are linear functions of the ''β<sub>j</sub>''.
 
Given that estimation is undertaken on the basis of a [[least squares]] analysis, estimates of the unknown parameters ''β<sub>j</sub>'' are determined by minimising a sum of squares function
:<math>S = \sum_{i = 1}^n \left(Y_i - \beta_0 - \beta_1 \phi_1(X_{i1}) - \cdots - \beta_p \phi_p(X_{ip})\right)^2 .</math>
From this, it can readily be seen that the "linear" aspect of the model means the following:
:*the function to be minimised is a quadratic function of the ''β<sub>j</sub>'' for which minimisation is a relatively simple problem;
:*the derivatives of the function are linear functions of the ''β<sub>j</sub>'' making it easy to find the minimising values;
:*the  minimising values ''β<sub>j</sub>'' are linear functions of the observations ''Y<sub>i</sub>'';
:*the  minimising values ''β<sub>j</sub>'' are linear functions of the random errors ''ε<sub>i</sub>'' which makes it relatively easy to determine the statistical properties of the estimated values of ''β<sub>j</sub>''.
 
==Time series models==
 
An example of a linear time series model is an [[autoregressive moving average model]]. Here the model for values {''X<sub>t</sub>''} in a time series can be written in the form
 
:<math> X_t = c + \varepsilon_t +  \sum_{i=1}^p \varphi_i X_{t-i} + \sum_{i=1}^q \theta_i \varepsilon_{t-i}.\,</math>
 
where again the quantities ''ε<sub>t</sub>'' are random variables representing [[Innovation (signal processing)|innovations]] which are new random effects that appear at a certain time but also affect values of ''X'' at later times. In this instance the use of the term "linear model" refers to the structure of the above relationship in representing ''X<sub>t</sub>'' as a linear function of past values of the same time series and of current and past values of the innovations.<ref>Priestley, M.B. (1988) ''Non-linear and Non-stationary time series analysis'', Academic Press. ISBN 0-12-564911-8</ref> This particular aspect of the structure means that it is relative simple to derive relations for the mean and [[covariance]] properties of the time series. Note that here the "linear" part of the term "linear model" is not referring to the coefficients ''φ<sub>i</sub>'' and ''θ<sub>i</sub>'', as it would be in the case of a regression model, which looks structurally similar.
 
==Other uses in statistics==
 
There are some other instances where "nonlinear model" is used to contrast with a linearly structured model, although the term "linear model" is not usually applied. One example of this is [[nonlinear dimensionality reduction]].  
 
==See also==
* [[General linear model]]
* [[Linear system]]
* [[Statistical model]]
 
==References==
<references/>
 
{{Statistics}}
 
[[Category:Statistical models]]
[[Category:Time series models]]
[[Category:Regression analysis]]
[[Category:Statistical terminology]]
 
[[ar:نموذج الانحدار الخطي]]
[[de:Lineares Modell]]
[[fr:Modèle linéaire]]

Latest revision as of 18:15, 24 November 2014

Eating olives, walnuts, pecans, cashews, linseeds, chia seeds and peanuts are all great ways of increasing oleic acid and other omega 6 oils in your diet each day. Attendees help carry a giant phallus through the streets to the shrine. "[Bieber] gave selena a std and Bieber stole my credit card to buy him and sean p-ditty combs cocaine to use in drug free school zones. Penis massage oils also work on the principle of increasing blood circulation to the penis. This means any man's penis will fill her vagina completely, unless you happen to be one of those rare guys with an erect penile length of less than four inches.

Increasing the flow of blood to the penis is a major element in penis enlargement. A penis that is too short sometimes can lead you to failing to satisfy your partner which is very risky in any relationship. It is advised to stop usage once this occurs to promote tissue health and safety. One of which was that he despises plastic surgery making these claims that he went under the knife suspect. This is vital and it will need to be amongst the prime prerequisites when it arrives down to deciding upon male enhancement products.

This article focuses on the genetics, and the different types of androgen insensitivity syndrome. For more informative articles, visit Penile Enlargement Blog. , L-carnitine was found to be highly effective in reducing symptoms associated with neuropathy, and aiding the regeneration of nerve sensitivity and function. As far as Bieber's penis enlargement goes, this lawsuit was filed in Michigan by someone who claims they are Selena Gomez's father. As a matter of fact, you ought to note that they operate additional like weights, the far more they are employed, the higher the chances of making certain that you notice significance charge of male enhancement up to your sought after duration.

If the reason 'why you want to enlarge your penis size' is because you think it is small or your girlfriend or wife thinks so, or you already have an 8 inch monster and want to make it bigger, then you should go for the safest, most efficient and natural way possible ' penis enlargement pills. By practicing the workout routines appropriately Penile Strategies has been demonstrated to treatment several kinds of impotence, usually due to poor blood blood flow, diminish premature ejaculation, broaden the width and length of the head of the penis, and boost your standard sexual overall health and nicely becoming. Enhancement pills have a very simple function and that is to improve circulation and expand blood vessels throughout the body, including the male genitalia. Tea constipation or just constipation takes place if a person has a difficulty removing his bowels. These penis exercises work in the same way as the Size - Genetics.

Nonetheless, a fear of extenders stems from wrong facts on how they operate and understanding how they operate and how safe they are, are the very first actions to receiving rid of these fears. This would not be premature ejaculation if his partner were to climax in three minutes, so - wham bam thank you Sam. According to TMZ and Hollywood Gossip, here are the highlights:. He used a dumbbell weight to try to stretch his penis. All of these products claim that they have positive results, both in length and girth.

If you loved this information and you would such as to obtain even more information regarding does creatine make your penis bigger kindly see our web site.

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