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Казанский национальный исследовательский технический университет им. А. Н. Туполева

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Цифровая обработка сигналов

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Лекции Нигматуллин / Lecture 5_SS_processes_p1.pptx

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Optimization hypothesis

Meteo-Data

			NH_meanT(C0)					rp1=4.54513,
	1.0		H0(t)					rp2=1.53165
tempearture			H1(t)					r =2.22379
	0.5

meanNH	initialfit
distributionof the	0.0
	-0.5
	andits
The	-10	0	10	20	30	40	50	60	70	80	90	100 110 120 130 140
	-10	0	10	20	30	40	50	60	70	80	90	100 110 120 130 140
					1(1881) < years <131(2011)

Fig.18. This plot demonstrates the selection of the proper hypothesis. The influence of the fitting function H1(t) is essential in comparison with

the function H0(t) (from (5)). The

values of the ratios given above on this figure and expression (A10) help to choose the proper hypothesis corresponding to case 4.

Case 4 ( two complex-conjugated roots and one real root). rp1, rp2 1.45, r 2.45. The condition of applicability:

2	Acp ycp (t,v) Asp ysp (t,v) ,
H (t,v) A0 A1t 1 (v) A2t 2 (v)	Acp ycp (t,v) Asp ysp (t,v) ,
p 1	(A11)

ycp (t, v) exp( p (v) ln t) cos p ln t , ysp (t, v) exp( p (v) ln t) sin p ln t .

Meteo-Data

Final hypothesis (4K+7 < N - number of the points in initial sequence)

H f (t) A0 A1t 1

Ack(1) yck(1) (t) Ask(1) ysk(1) (t)

k 1

A2t 2 Ack(2) yck(2) (t) Ask(2) ysk(2) (t) ,

k 1

(1)

ln t

yck

(t) exp( 1

ln t) cos 2 k

(1)

2 k

ln t

ysk

(t) exp( 1

ln t) sin

(2)

(t) exp( 2

ln t) cos

2k a 2

ln t

yck

(2)

2k a 2

ln t

ysk

(t) exp( 2

ln t) sin

All necessary parameters corresponding to the case 4.

ln a			a
	,	2		,
			r

2

exp ln 1 , 2 exp ln 2 ,2 1 a 2 , a 2,1.

S(t 3 ) a				S(t 2 ) a S(t ) a S(t) s ,
		2						1			0	0
a2	2		cos 2 ,							2
a2	2		cos 2 ,
a 2						2	cos				,
a 2							cos				,
1
a		2			2
a		2
0
s A 1 a a a											.
0	0						2 1		0

(A12)

Meteo-Data

			NH_Tmean(C0)
	1.0		Optimal fit
the NH			Final fit
the NH
	0.5
	0.0
optimal and finalfitsof m eanperaturetem
The	-0.5
The
	-10	0	10	20	30	40	50	60	70	80	90	100 110 120 130 140
							1 < years < 131
nc tio n s			Ac1k
-p e rio d ic fu m p e a rtu re			Ac1k
	0.006		As1k
			Ac2k
	0.004		As2k
	0.004

fo r lo g e a n te	0.002
es m
d H
m p litu th e N	0.000
m p litu th e N
a g	-0.002
n o f rib in	-0.002
n o f rib in
tio sc	-0.004
u e	-0.004
istrib d			0				10				20	30
D								1 < k < 30
								1 < k < 30

Fig.19. The optimal (solid black line) and the final fit that are obtained in the frame of hypotheses (A11) and (A12). The values of the fitting parameters are collected in Table 2.

Fig.20. Distributions of amplitudes of two-log periodic functions that enter to expression (A12). They describe the distribution of T(C0) in the NH. Namely, these specific amplitude- frequency responses (AFRs) describing the behavior of initial random functions depicted on Fig.15 determine the ground of the whole random process studied.

Meteo-Data

			mean T in C0(SH)
			H0(t)
Distribution of m ean tem pearture in South Hem isphere and its fit	0.6		H1(t)
			Optimal fit
	0.4		Final fit
	0.4
	0.2
	0.0
	-0.2
	-0.4
	-10	0	10	20	30	40	50	60	70	80	90	100 110 120 130 140
n ctio n s S H					1(1881) < year < 131 (2011)
n ctio n s S H
fu in			Ac1k
ic T			Ac1k
ic T			As1k
-p e rio d f m e a n	0.10		As1k
			Ac2k
			As2k
lo g n o
tw o u tio	0.05
tw o u tio
es fo r d istrib
d e	0.00
p litu to th	0.00
p litu to th
istrib u tio n o f a m c o rre sp o n d in g	-0.05
			0				10				20	30
								1 < k < 31
D								1 < k < 31

Fig.21. The optimal (solid cyan line) and the final fit (solid blue line) that describe the distribution of mean temperature in SH. This high quality fit is realized again with the help of hypotheses (A11) and (A12). The values of the fitting parameters are collected in Table 2.

Fig.22. Distributions of amplitudes of two-log periodic functions that enter to expression (A12). These distributions (serving as the specific AFRs) describe the distribution of T(C0) in SH (see previous figure).

Meteo-Data

		curve		mean T (NH)								mean T(NH)
			1.0	prediction curve								optimal prediction trend
its prediction												optimal prediction trend
	1.0	averaged
		averaged	0.5						forecast horizon
		) for the							for 27 years
		) for the
		0	0.0
		ofmeanT(C	0.0
	0.5	ofmeanT(C
(NH) and	0.5	Distribution	-0.5
			-0.5
			0		10	20	30	40		50
					1 < 3years to 1 year < 43
												horizon of
temperature	0.0											horizon of
	0.0											forecast 10 years
												forecast 10 years
												Cooling!
	-0.5
Mean	-0.5
Mean
	-10	0	10	20	30	40	50	60	70	80	90	100 110 120 130 140
						1(1881) < year < 140(2021)

p re d iction		S H an d its p red iction		T (SH)								T(SH)
			0.8	Prediction curve								T(SH)
			0.8									Prediction trend
			0.6									Prediction trend
			0.6
	0.6		0.4							horizon 27 years
	0.6									horizon 27 years

its		tu re in	0.2
its		tu re in	0.0
d	0.4	era	0.0									Forecast
d		era
n		m p	-0.2
n		m p	-0.2
a		te										horizon for
H)		M e an										horizon for
			-0.4									10 years
	0.2											10 years
(S	0.2		0		10	20		30	40	50
(S					1 < 3years to 1 year < 43
ture	0.0
e ra	0.0
e ra
te m p	-0.2											Warming!
te m p												Warming!
ea n	-0.4
M
	-10	0	10	20	30	40	50	60	70	80	90	100 110 120 130 140
						1(1881) < 141(2021)

Fig.23. Distribution of mean temperature for NH and its forecast for the nearest 10 years. In accordance with this "prediction" we should expect the tendency to the general cooling. The same tendency is observed for the compressed curve (placed in the small frame above) if we continue this curve on the nearest 27 years.

Fig.24. Distribution of mean temperature for SH and its forecast for the nearest 10 years. In contrast with the previous "prediction" we should expect the tendency to general warming. The same tendency is observed for the compressed curve (placed in the small frame above) if we continue this curve on the nearest 27 years.

Meteo-Data

Table 2(a). The table of additional and scaling parameters that describe the mean temperature data T(C0) (the 1-st part).

Number of	< 1>	ln( )	A1	A0		RelErr(%)	K/2
file

T mn (NH)	1.76572	0.32278	4.59067E-4	-0.37172	-4.34845	5.26015	31
130

T mn (NH)	1.24181	0.32278	0.00726	-0.39043	-4.34845	1.53648E-6	5
(43)

T up (NH)	1.39462	0.32278	0.02812	-0.33276	-4.34845	1.82013E-6	5
(43)

T dn (NH)	0.7179	0.32278	0.07095	-0.57355	-4.34845	3.10656E-7	5
(43)

T mn (SH)	0.73562	0.51028	0.0161	-0.48081	-5.402	3.26819	31
130

T mn (SH)	0.54818	0.51028	0.07595	-0.12483	-5.402	2.67959E-12	5
(43)

T up (SH)	0.43572	0.51028	0.09562	0.05575	-5.402	1.68926E-12	5
(43)

T dn (SH)	0.58567	0.51028	0.12695	-1.20849	05.402	2.63292E-12	5
(43)

Meteo-Data

Table 2(b). The table of additional and scaling parameters that describe the mean temperature data T(C0) (the 2-nd part)

Number of	< 2>	A2	b2	b1	b0	s0
file

T mn (NH)	7.47739	3.36388E-4	-2915.23	-13669.9	-125878	-10.0722
130
T mn (NH)	5.8845	-0.02704	-461.835	-1205.18	-6103.54	-11.2439
(43)
T up (NH)	6.34909	-0.21924	-62.4825	-112.623	-368.852	-19.0733
(43)
T dn (NH)	4.29161	-0.02515	-93.6704	-144.359	-418.556	-16.0342
(43)
T mn (SH)	0.11018	0.01649	2.04037	-2.61815	-7.20148	-19.4399
130
T mn (SH)	-0.19663	0.00769	1.66488	-1.78249	-2.94034	-6.04195
(43)
T up (SH)	-0.38072	0.01732	1.59147	-1.6495	-1.8612	2.11418
(43)
T dn (SH)	-0.13527	0.00487	1.62114	-1.66886	-3.25222	-72.5342
(43)

Comments to Tables 2. The fitting parameters collected in these tables reflect the basic parameters of the final fitting function (A12) and the parameters figuring in the scaling equation (A14) and its solution (A13). The 8-th column in Table2(a) shows the number of modes that can restore the unknown log- periodic function with accuracy given in column 7. We should note here that increasing the number of modes up to the limiting value 4K+4 N sharply increases the accuracy (see column 7 in Table 2(a)) and makes the fitting function practically exact.

Results and Discussions

Analysis of these results based on the fitting functions found from SS principle allows us to put forward the following idea.

For the fitting of different dataset having initially a random behavior we had before only one type of the fitting functions. They followed from the justified (“best fit”) model and help to reduce N initial data points to a few stationary parameters that describe the general features of the time series under analysis. In the case of success the researchers can identify these strongly-correlated relationships as laws of nature. Any other transformations realized with initial sequence (like the Fourier, numerous wavelet and other transforms) can be considered as a convenient presentation of the initial data in order to receive an additional source of information.

The SS principle gives a researcher another possibility to fit the sufficiently large initial data points and use an intermediate model. The fitting functions that follow from the SS principle occupy in intermediate position between the laws (that contain minimal number of the fitting parameters Pmin) and transformations, where the number of data points N

close to number of parameters that present of the initial data set in another presentation (frequency, number of moments and etc.).

Pmin	2(4)K 4(7)	N			Initial data
			Model	Principle	Initial data
					points
					points

Mathematical Appendix

Used for treatment of economical data

Mathematical Appendix

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