least squares Method
In regression analysis, least squares, also known as ordinary least squares analysis, is a method for linear regression that determines the values of unknown quantities in a statistical model by minimizing the sum of the residuals (the difference between the predicted and observed values) squared. This method was first described by Carl Friedrich Gauss around 1794, close to the turn of the 19th century (Linear Algebra With Applications, 3rd Edition, by Otto Bretscher). Today, this method is available in most statistical software packages. The least-squares approach to regression analysis has been shown to be optimal in the sense that it satisfies the Gauss-Markov theorem


--------------------------------------------------
Note added at 3 hrs (2008-08-17 02:46:51 GMT) Post-grading
--------------------------------------------------

Média aritmética de um conjunto de observações só existe UMA.
NÃO HÀ DUAS OU MAIS MÉDIAS.
Existem ainda como medida de tendência central: a mediana, a moda, a média ponderada, a geométrica, a harmônica.
In 1829, Gauss was able to state that the least-squares approach to regression analysis is optimal in the sense that in a linear model where the errors have A MEAN (gr. nosso) of zero, are uncorrelated, and have equal variances, the best linear unbiased estimators of the coefficients is the least-squares estimators. This result is known as the Gauss-Markov theorem.



--------------------------------------------------
Note added at 3 hrs (2008-08-17 02:55:35 GMT) Post-grading
--------------------------------------------------

Least squares and regression analysis
In regression analysis, one replaces the relation (...)
by Y=ax+b+u, where the noise term ε (u) is a random variable with mean zero. Note that we are assuming that the x values are exact, and all the errors are in the y values. Again, we distinguish between linear regression, in which case the function f is linear in the parameters to be determined (e.g., f(x) = ax2 + bx + c), and nonlinear regression. As before, linear regression is much simpler than nonlinear regression. (It is tempting to think that the reason for the name linear regression is that the graph of the function f(x) = ax + b is a line. But fitting a curve like f(x) = ax2 + bx + c when estimating a, b, and c by least squares, is an instance of linear regression because the vector of least-square estimates of a, b, and c is a linear transformation of the vector whose components are f(xi) + εi.
Parameter estimates
By recognizing that the regression model is a system of linear equations we can express the model using data matrix X, target vector Y and parameter vector δ.


--------------------------------------------------
Note added at 3 hrs (2008-08-17 03:10:06 GMT) Post-grading
--------------------------------------------------

A incorretamente chamada MÉDIA DOS MÍNIMOS QUADRADOS é o parâmetro LINEAR da reta de regressão Y= a+bx; a=Y (barra sobre Y); o parâmetro b (beta) é o COEFICIENTE ANGULAR DA RETA DE REGRESSÃO,
b= somatório de Xi vezes Yi/ Somatório de Xi quadrado.
Constroem-se INTERVALOS DE CONFIANÇA para estes DOIS parâmetros.
Já li tudo D. Maria. Começa com um erro de tradução: MEANS=WAY, METHOD; MEAN=AVERAGE.
NÃO HÁ MEANS (=MÈDIAS) pq MÉDIA DE UM CONJUNTO DE OBS. SÓ TEM UMA!

--------------------------------------------------
Note added at 4 hrs (2008-08-17 03:12:05 GMT) Post-grading
--------------------------------------------------

A MÉDIA DOS ERROS É ZERO!
QUER Q. CHAME O GAUSS PARA DEMONSTRAR DE NOVO...

--------------------------------------------------
Note added at 1 day2 hrs (2008-08-18 01:34:51 GMT) Post-grading
--------------------------------------------------

FEIO é traduzir ERRADO e não aceitar correção fundamentada, em termos polidos, e oferecida de boa-vontade, o que demonstra a arrogância da ILUSTRE CONSULENTE:
"Means" é, EM BOM PORTUGUÊS, meio, método, e não "MÉDIAS".
As palavras "barracks, species, series, MEANS, works and compounds like gas-works, iron-works, etc.), ARE THE THE SAME IN SINGULAR AND PLURAL.
EM BOM PORTUGUÊS, também temos palvs. q. só se usam no SING. e, quando vão ao plural (nomes de metais e os nomes abstratos), MUDAM DE SENTIDO, como "MEAN" E "MEANS" em Ing.:
Ex.: "FERRO, OURO, COBRE; FÉ, ESPERANÇA, CARIDADE." Compare-se:
COBRE (METAL) e COBRES (DINHEIRO), FERRO (METAL) e FERROS (APARELHO, FERRAMENTAS).
Substantivos de um só número:
Em BOM PORTUGUÊS, há subst. q. só se empregam no plural, ex.:
"Alvíssaras, anais, antolhos, câs, condolências, esponsais, fezes, matinas, núpcias, primícias, víveres, copas (naipe), arredores, belas-artes, calendas, exéquias, fastos, férias, óculos, olheiras, pêsames, espadas (naipe), ouros (naipe) paus (naipe).


--------------------------------------------------
Note added at 1 day14 hrs (2008-08-18 13:25:03 GMT) Post-grading
--------------------------------------------------

Obrigado pelo rapaz,
mas isso eu já não sou mais.
Mas ainda sou capaz...
de achar no dicionário
a palavra mordaz
da qual estamos atrás...

--------------------------------------------------
Note added at 1 day14 hrs (2008-08-18 13:44:02 GMT) Post-grading
--------------------------------------------------

Em Inglês ocorre o mesmo fenômeno: palavras q. mudam de sentido no plural:
colours=1-hues/2- regimental flags;
compasses=1-inst. fr. navigation /2- inst. fr. drawing circles;
customs=1-habits /2- taxes on imported goods;
draughts=1- a game /2- current of air;
dominoes=1-a game/2-kinds of masks;
effects=1-results/2-goods, personal property;
forces=1- powers/ 2-organized bodiies of men (armed forces, police forces);
glasses=1-vessels f. drinkg. from/2-spetacles/3- binoculars;
grounds=1-land/2-reasos/3-dregs;
letters=1-signs/2-epistles/3-learning;
manners=1-ways/2-behaviour;
minutes=1-space of time/2-secretary's record of proceedings;
pains=1-sufferings/2-care;
premises=1-things assumed as true as basis for an argument/2-buildings;
quarters=1-fourth parts/2-lodgings;
returns=1-recurrences/2-statistical details;
spetacles=1-sights/2-glasses;
spirits=1-souls/2-alcoholic liquors/mental or moral attitude.