Нигматуллин Р.Р. (лекция 2)

Basic question:

In what cases one can obtain the basic linear relationship for the function that initially contains a set of nonlinear fitting parameters?

Example: R(x; A) = B xμ exp(-a1 x- a2x2/2).

	ln R = ln B +μln(x) −a x −a				x2	,
					2
			1	2
	x dR = μR −a xR −a x2 R
	dx	1	2
	R = BX4 (x), B =		(R(x) X	4 (x))			,
			(X4 (x) X4 (x))

X4 (x) = xμ exp(−a1x −a2 x2 / 2).

N

(A B) = ∑Aj Bj

j=1

3

Y (x) = ∑C X (x),

k k

k =1

Y (x) = x R(x) − ∫x R(u)du − ... ,

x0

X1 (x) = ∫x R(u)du − ... , C1 = μ,

x0

X2 (x) = ∫x uR(u)du − ... , C2 = −a1,

x0

X3 (x) = ∫x u2 R(u)du − ... , C3 = −a2.

x0

Here the value x0 – corresponds to the initial point of the discrete data considered. As before we should eliminate all possible constants in expression (9)

obeying the expression (9) to the condition . From expressions (10) one can find one can find

the unknown fitting constants μ, a1,2.

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So, one can conclude that many functions containing initially the unknown set of the fitting parameters obey to differential equations where the initial set of parameters forms a linear combination. In this case the ECs method reduces the

problem of the nonlinear fitting to the well known LLSM. It opens a quite new possibility of the calculation of the fitting parameters of many statistical distributions by means of simple and well-developed method.

2. The usage of the orthogonal variables

As it is known that in practical applications of the LLSM the determinant might have the value close to zero. In these uncomfortable cases the values of the fitting parameters Ck found from (5) contains large errors and sometimes cannot be calculated. For these cases one suggests the transformation of initial variables Xk(x) to another set of variables that orthogonal each other. It makes the functions statistically independent to each other and helps to avoid the zeros in the corresponding determinants, which are appeared in calculation of the fitting coefficients Cp. We present some BLR in the form

Y (xj ) = ∑s Ck Xk (xj )

k =1

The relationship reminds the decomposition of a wave function Y(xj) over the finite set of eigenfunctions

{X(xj )}k=1,2,…,s. Using the process of orthogonalization one can choose the set of orthogonal functions {Ψ(xj )}k=1,2,…,s and present the initial function Y(xj) in the form of linear combination of

{Ψ(xj )}k. This transformation is realized with the help of the following formulae.

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Y (xj ) = C1Ψ1 (xj ) +...+CsΨs (xj )

Ψk (xj ) = Xk (xj ) −∑k −1 ((Ψp Xk )Ψp (xj ), Ψ1 (xj ) = X1 (xj ),

p=1 Ψp Ψp

(Ψp Xk ) = ∑N Ψp (xj )Xk (xj ), j=1

Ψ1 (xj ) = X1 (xj ),

Ψ3 (xj ) = X3 (xj )

Ψ2 (xj ) = X2 (xj ) − (Ψ1 X2 ) Ψ1 (xj ),

(Ψ1 Ψ1 )

	−	(Ψ1 X3 )	Ψ	(x	) −	(Ψ2 X3 )	Ψ	(x	),...
		(Ψ1 Ψ1 )				(Ψ2 Ψ2 )
			1	j			2	j

These expressions realize the orthogonal transformations. Important that (det(T) = 1).

(Ψp , Ψq ) = (Ψp , Ψp )δpq ,

Ck =

(Y, Ψk )

= Ck + ∑ss

(Ψk , XX p )

Cp , Cs = Cs ,

(Ψk , Ψk )

p=k +1 (Ψk , Ψk )

1, (Ψ1, X2 ),.......

(Ψ1, X s )

(Ψ1, Ψ1 )

(Ψ1, Ψ1 )

(Ψ

, X

3

)

(Ψ

,

X

s

)

2

2

(Ψ2 , Ψ2 )

0, 1, (Ψ2 , Ψ2 ),

T = ...........................................

(Ψ

s−1

, X

s

)

0..........

0,

1,

(Ψs−1, Ψs−1 )

0,.............., 0..........

1

3. Is it possible to select the most suitable hypothesis from two alternative ones? The basic idea. Each verified function has own differential equation and if the value of the initial error is rather small then there is one-to-one correspondence between the function and its differential equation which should be satisfied by this partial solution (Picard theorem). So, a "strange" function ystr(x) being presented in the eigen-coordinates belonging to a "native" function ynat(x) should receive an additional dependence of the constants Ck (k=1,2,…,s) against the variable x.

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In other words, the set of straight lines

s

Y (x) = Ck Xk (x), Y (x) =Y (x) − ∑ Cp X p (x)

p ( p≠k )

identifying the "native" function (Ck = const (k=1,2,…,s)) is distorted when the "strange" function (which does not satisfy the set of equalities (27)) is verified.

Example:

y1 (x) =

A1

, y2 (x) =

A2

Y2

(2)

(2)

(2)

X

(2)

θ

ν

(1+b1x)

1+b2 x

(x) = C1

X1 (x)

+C2

2 (x)

Y2 (x) = x y2 (x) − ∫x

Y (x) = C(1) X

(1) (x) +C(1)

X (1)

(x)

y2 (u)dx − <... >,

x0

1

1

1

2

2

Y1 (x) = y1 (x)− <... >,

x

(2)

∫

(2)

= −ν,

X

(1)

= x y (x)− <... >, C(1) = −b ,

X1

(x) =

y2 (u)du − <... >, C1

x0

1

1

1

1

x

ν

x

(y2 (u))2 du− <

X2(2) (x) = ∫

... >, C2(2) =

.

X

(1)

= ∫ y1 (u)du− <

(1)

= b1 (1−θ).

2

... >, C2

x0

A2

x0

(y (x) X

(x))

(Yb (x) Xb (x))

Yb (x) = b2 Xb (x), b2

=

,

A

(Xb (x) Xb (x))

y (x) = A X

(x), A

=

1

,

A

(X A (x) X A (x))

1

1

1

X

(x) = x

ν

, Y (x) =

A

−1.

−θ

b

2

X

(x) = (1+b x) .

b

y2 (x)

A

1

14

(Y(x) X2 (x))

(X1(x) X2 (x))

Y(x) −

X2

(x)

=C1 X1(x) −

X2

(x)

,

(X2 (x) X2 (x))

(X2 (x) X2 (x))

Y(x) −

(Y(x) X1(x))

X

(x) =C

X

(x)−

(X1(x) X2 (x)) X

(x) .

(X1(x) X1(x))

1

2

2

(X1(x) X1(x))

1

Presentation the curves in the form of straight lines.

Here we show two different functions marked by red triangle points (y1(x)) and green stars (y2(x)) respectively. The values of the parameters are the following: (b1=0.5, A1=10, θ=0.7; b2=0.37, A2=10, ν = 0.84). The value of the error does not exceed 2% .

Final criterion:

	stdev ( y j				− yft j )
										100%,
			mean		yft
RelErr(%) =
	1	N	(Aj −		A )		2	1/2
Stdev(A) =										,
	N ∑j=1

	1	N
mean(A) = A =		∑Aj
	N	j=1

15

0.2	errn

	0.1
errn	0.0
-0.4975 -0.1
	0	20	40
		x
	distortions are
	deviated
	0		5

Normal presentation (left). The differences are unnoticeable. Presentation in the form of straight lines (differences are noticeible).

Useful recommendations:

Recommendation 1. Try to work with "clean" data containing large number of measured points. If the values of the external error are rather high (as it is presented on Fig2a, for example, then it is useful to use the POLS for smoothing of initial data.

Recommendation 2. If it is possible, for each selected hypothesis it is necessary to calculate its BLR that was preliminary verified on mimic data.

Recommendation 3. If the calculation of the BLR is not possible the smaller values of the relative error (Eqn.(11)) can serve as a quantitative criterion in selection of the proper hypothesis.

Recommendation 4. The specific behavior of the constants C1,2 and their distortions calculated from expressions (31) will give additional information for selection of true hypothesis.

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Further generalizations and some recommendations to the usage of the ECs method.

At first we should explain why this method received the definition as the "eigen-

coordinates" (ECs) method. The origin of this definition is the following. As one can see above besides the fitting procedure corresponding to the global fitting minimum

(the seed values of the fitting parameters are absent) it has a possibility to select

the most suitable hypothesis. Imagine that we want to verify numerically the eigenvalues for the stationary Shrodinger quantum equation

H Ψk = Ek Ψk , H Ψk ≡Yk , Yk = Ek Ψk

If the set of Yk does not coincide with eigen-functions of the chosen Hamiltonian

then this set of segments is distorted and the eigen-values Ek start to depend on some current variable. In the definition of this method we want to stress this important property.

This method is based on the presentation of some analytical function F(x, A) initially containing a set of non-linear fitting parameters to a new set of the fitting

parameters С(A), which are becoming linear with respect to the chosen function y

i

≡ F({x(j)}, A). Such presentation becomes possible if the chosen function satisfies to linear/nonlinear differential equation with a new set of parameters С(A) forming a linear combination with respect to independent variable x, dependent variable y and the corresponding derivatives.

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In other words, the applicability of the ECs method is based on the following structure of the corresponding linear/nonlinear differential equation

d 2 y1 +C1 dy1 +C2 y1 = 0, C1 = λ1 +λ2 ,C2 = −λ1λ2 .

dx2 dx

y (x) = A exp(λ x) + A exp(λ x						dy2
1	1	1	2	2	x		= C y	2	+C		y	2	ln y		2	+C ln(x) y			,
y2	(x) = Axν exp(−γxμ )					dx	1			2						3		2
y (x) = A x−ν1		+ A x−ν2			C1 = ν−μln A, C2							= μ, C3 = −μν.
3	1	2															d
					D2 y		+C Dy			+C		y		= 0, D ≡ x				,
					3		1	3			2	3

dx

C1 = ν1 +ν2 , C2 = −ν1ν2.

The general structure of the ordinary differential equation

Y (x, y, y′,...) = C1 X1 (x, y, y′,...) +...+Cs Xs (x, y, y′,...)

Y (x, y, y′, y′′,...), Xk (x, y, y′, y′′,...) F(x, A)

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The usage of a priori information. Let us suppose that a priori some constant Ck is known. It means that it is located in some interval: ak ≤ Ck ≤ bk. How to take into account this information and makes the calculation in the frame of the LLSM more

definite and statistically stable? In this case we introduce a new variable t located in the interval

Ck (t) = ak + bkt, t [0,1], bk =bk −ak

The limiting values of ak, bk are supposed to be known. Then basic equations minimizing the value of dispersion accepts the form

εj (t) =

−ak Xk

s

≡ ε0, j −tε1, j

,

j =1, 2..., N

Yj

−∑Cp X p

−t bk Xk , j

p≠k

1

N

s

2

1

N

2

σ =

∑ Yj

−ak

Xk , j − ∑

(Cp X p )−t bk Xk , j ≡

∑(ε0, j −tε1, j )

N

N

j=1

p( p≠k )

j=1

Cp (t) = ap + bpt, t [0,1],

s

−(ak +t bk )(Xk Xm )

∑ Cp (X p Xm ) =(Y Xm )

bp = bp −ap , p =1, 2..., s, p ≠ k.

p( p≠k )

From the solution of linear system of equations it follows that the supposition leads to the conclusion that any constant Cp starts to depend on variable t,

forming a "fork" located between the unknown constants a and b . We remember p bp We remember

that all set of constants {C} are linear independent.

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εj		s			s	bp
	= Yj	−ak Xk , j − ∑ ap	X p, j	−t	bk Xk , j + ∑		X p, j
		p, p≠k			p, p≠k
	≡ ε0, j	−tε1, j .

The variable t is the independent variable and so the unknown limits entering in (53) are calculated at t = 0 and t = 1. So, for these limiting cases we have the relationships:

Yj −ak Xk , j = ∑s	X
p, p≠
s
bk Xk , j = − ∑	X
p, p≠

p, j

p, j

,

1

N

N

2

σ(t) =

N

∑ε0, j −t∑ε1, j ≡ ε02 −2t ε0 ε1 +t2 ε12 .

.

j=1

j=1

t

min

= ε0 ε1 , 0 <t

min

<1, σ

min

= ε2

− ε

0

ε t

min

ε2

0

1

1

Cp (tmin ) = ap +

bptmin ,

bp

=bp −ap ,

p =1, 2..., s, p ≠ k.

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