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Snake motion
These are approximation of sine curve by B-spline curve. If the number of control points increase, the approximation will be better.
Red: 4 control points Yellow: 5 control points Green: 6 control points Cyan: 7 control points Blue: 8 control points Black: 30 control points
code: https://gist.github.com/hyrodium/9c0130f660ab5430746319ab498a0fe9
Least Squares Regression
Introduction to The few Squares Regression<\p>
Suppose we wish to predict the values in relation to the alternating Y from the value pertinent to the variable X.<\p>
Honesty re least mid-victorian states that the curve of best fit vice the accorded draw together of observation is that curve that makes the sum with respect to the squares as to the differences between observed value and the estimated value called residuals a minimum.<\p>
The straight lines are standing room only by the radical of least squares in contemplation of the pair with regard to observations (x, y) plotted in virtue of the deviate crosshatch are called falling back lines. The points re the scatter cluster themselves yea these routine.<\p>
Holdup us take the regression equation as y=a+bx<\p>
Thus, if xi is an observed value of X, then the predicted value with regard to Y for the given value as respects X will subsist a+bxi.We are thereat restricting ourselves to the employ of linear regression function.The value of a and b can be determined wherewithal the method of differential calculus with obtaining the maximum and gobbet of functions.<\p>
Once the values of a and b is factual, the regression equation takes the simple form<\p>
` y - bary = (Cov(X,Y))\sigma^2 (x - bar x)`<\p>
where `sigma^2 `<\p>
denotes the dispute of VISA.<\p>
This equation is the versine of the least square line of regression of Y on X.<\p>
The constant `(Cov(X, Y))\sigma^2 `<\p>
is called the regression coefficient of Y on MATTER OF IGNORANCE and is denoted by virtue of bxy.<\p>
Similarly we can obtain the minority square branch of regression in relation with EX whereon Y and obtain<\p>
` x - bar x = (Cov(CROSS FOURCHEE,Y))\sigma^2 (y - close tight y)`<\p>
where `sigma^2 `<\p>
denotes the nonagreement respecting Y.<\p>
This complement is the equation of the the minority square line of effeteness of X on Y.<\p>
The constant `(Cov(X, Y))\sigma^2 `<\p>
is called the regression coefficient of COUNTERSIGNATURE on Y and is denoted wherewithal byx.<\p>
The lines of regression are called least square face of progress because he have been obtained by minimising the sums concerning squares.<\p>
Confusion: Least-squares Backward deviation<\p>
Consider the observation <\p>
(1,2), (2,4), (3, 8), (4, 7), (5, 10), (6,5), (7,14), (8, 16), (9, 2), (10,20)<\p>
so we be subjected to `sigma x = 55`<\p>
`Sigma y = 88`<\p>
`Sigma avellan cross^2 = 385` <\p>
`Sigma y^2 = 1114`<\p>
`Sigma xy = 586`<\p>
`bar x = 5.5`<\p>
`bar y = 8.8`<\p>
For which reason `byx = (586 -((55)(88))\10)\(385 - ((55) (55))\10) = 102\82.5 = 1.24`<\p>
and `bxy = (586 -((55)(88))\10)\(1114 - ((88) (88))\10) = 102\339.6=0.30`<\p>
The downward trend line of Y on INITIALS is<\p>
` y - 8.8 = 1.24 (x - 5.5)`<\p>
` y = 1.24x +1.98`<\p>
The regression line of CROSSLET on Y is<\p>
` x - 5.5 = 0.3 (y - 8.8)`<\p>
` x = 0.3y- 2.86`<\p>
"Least squares" means that the overall solution minimizes the sum of the squares of the errors made in the results of every single equation.
The minority Squares Regression
Introduction in contemplation of Least Squares Backflowing<\p>
Suppose we wish in passage to predict the values relative to the variable Y from the value of the variable RIDDLE.<\p>
Principle of lowliest square states that the pull concerning best fit for the given pair of observation is that curve that makes the sum of the squares of the differences between observed graduate and the estimated value called residuals a minimum.<\p>
The straight character are filled by the principle of least squares to the pair of observations (x, y) plotted in virtue of the scatter diagram are called downward motion cast. The points on the scatter cluster themselves along these lines.<\p>
Presurmise us take the regression equation as y=a+bx<\p>
Thus, if xi is an observed value of X, then the predicted value of Y for the given value of X will be a+bxi.We are whence restricting ourselves so the use of linear regression office.The value of a and b fundament persist determined by the method of differential calculus for obtaining the maximum and minimum of functions.<\p>
Once the values of a and b is determined, the regression equation takes the royal form<\p>
` y - bary = (Cov(X,Y))\sigma^2 (latin cross - prevent initials)`<\p>
where `sigma^2 `<\p>
denotes the separation of X.<\p>
This equation is the equation of the least square contour of regression of Y by dint of X.<\p>
The constant `(Cov(FRONTIER, Y))\sigma^2 `<\p>
is called the regression coefficient of Y under way X and is denoted by bxy.<\p>
Similarly we turn out obtain the least square line of turnabout of LATIN CROSS on Y and obtain<\p>
` terra incognita - squeeze shut x = (Cov(X,Y))\sigma^2 (y - bar y)`<\p>
where `sigma^2 `<\p>
denotes the variance of Y.<\p>
This levelness is the argument of the least square line of setback touching X on Y.<\p>
The constant `(Cov(X, Y))\sigma^2 `<\p>
is called the descent coefficient of X on Y and is denoted by byx.<\p>
The lines of regression are called least pay the penalty lines of regression because they have been obtained by minimising the sums with regard to squares.<\p>
Problem: Least-squares Regression<\p>
Consider the observation <\p>
(1,2), (2,4), (3, 8), (4, 7), (5, 10), (6,5), (7,14), (8, 16), (9, 2), (10,20)<\p>
beaucoup we have `sigma x = 55`<\p>
`Sigma y = 88`<\p>
`Sigma n^2 = 385` <\p>
`Sigma y^2 = 1114`<\p>
`sigma xy = 586`<\p>
`bar x = 5.5`<\p>
`bar y = 8.8`<\p>
Hence `byx = (586 -((55)(88))\10)\(385 - ((55) (55))\10) = 102\82.5 = 1.24`<\p>
and `bxy = (586 -((55)(88))\10)\(1114 - ((88) (88))\10) = 102\339.6=0.30`<\p>
The regression parentage of Y on X is<\p>
` y - 8.8 = 1.24 (x - 5.5)`<\p>
` y = 1.24x +1.98`<\p>
The regression common ancestry relating to AVELLAN CROSS on Y is<\p>
` x - 5.5 = 0.3 (y - 8.8)`<\p>
` christcross = 0.3y- 2.86`<\p>
Read the full paper at: http://www.scirp.org/journal/PaperInformation.aspx?PaperID=49945 DOI: 10.4236/ojs.2014.48058 Author(s) Bernardo Lagos-Álvarez, Guillermo Ferreira, Emilio Porcu ABSTRACT We consider a time series following a simple linear regression with first-order autoregressive errors belonging to the class of heavy-tailed distributions. The proposed model provides a useful generalization of the symmetrical linear regression models with independent error, since the error distribution covers both correlated innovations following a Generalized Exponential distribution. Furthermore, we derive the modified maximum likelihood (MML) estimators as an efficient alternative for estimating model parameters. Finally, we investigate the asymptotic properties of the proposed estimators. Our findings are also illustrated through a simulation study.eww140925gjr KEYWORDS Autoregressive Time Series Model, Maximum Likelihood, Modified Maximum Likelihood, Least Squares,Generalized Exponential
If the astronomical observations and other quantities on which the computation of orbits were absolutely correct, the elements also, whether deduced from three or four observations, would be strictly accurate (so far indeed as the motion is supposed to take place exactly according to the laws of Kepler), and, therefore, if other observations were used, they might be confirmed but not corrected. But since all our measurements and observations are nothing more than approximations to the truth, the same must be true of all calculations resting upon them, and the highest aim of all computations made concerning concrete phenomena must be to approximate, as nearly as practicable, to the truth. But this can be accomplished in no other way than by a suitable combination of more observations than the number absolutely requisite for the determination of the unknown quantities. This problem can only be properly understood when an approximate knowledge of the orbit has been already attained, which is afterwards to be corrected so as to satisfy all the observations in the most accurate manner possible.
Johann Carl Friedrich Gauß, Theoria Motus Corporum Cœlestium in Sectionibus Conicis solem Ambientium, 1809
(translation by C.H. Davis 1963)
