Method of Forecasting of Random Sequences
based on the
Prony Decomposition: Analysis of
Finance/Economic Data
Dr. Prof. Raoul R. Nigmatullin1)
Radioelectronic and Informative –Measurements Technics Department,
Kazan National Research Technical University (KNITU-KAI)
Kazan, K. Marx str., 420011, Russian Federation.
Dr. Prof. Jose Tenreiro Machado2)
ISEP-Institute of Engineering, Polytechnic of Porto, Dept. of Electrical
Engineering
Rua Dr. Antonio Bernardino de Almeida, 431
4200-072 Porto, Portugal
Abstract. In this presentation we suggest an original method for forecasting data based on the Prony
decomposition. Under forecasting procedure we understand a possible prolongation of the fitting
function which describes of the known optimal trend in terms of the Prony decomposition. We give
some arguments justifying the selection of the Prony function as the fitting function. This fitting function
should describe the segment of a random curve with high accuracy that is supposed to be known and can
be continued out of the fitting interval. This type of technical prediction is possible if we simply continue
tendencies collected in the past for the future temporal interval. Naturally, the boundaries of the
sequential future interval are limited by the influence of the future events that can change the
tendencies "stretched out" from the known past. This determination in general coincides with definition
of a forecasting procedure given in the book [1]. We establish the relationships between modes of the
quasi-periodic Prony decomposition with the well-known K-(Kondratiev) waves (or Elliot-waves) that
were defined in economy for forecasting of long random time-series with trends. See, for example, the
site: http://www.onlinetradingconcepts.com/TechnicalAnalysis/. Our method ads new features to the
conventional procedure and corrects the general conception of K(E)-waves for short temporal cycles. It
allows to find additional criteria for identification of the K-waves (especially in short sequences) in the
given random sequence analyzed. We suggest at least two independent methods of forecasting: (a)
from the segment that starts from the nearest past and (b) from the curve that is remained as the selfsimilar curve to the initial one. In both cases we show that the results of prolongation are similar to each
other that increase the reliability of the suggested forecasting procedure. As an example we consider the
data associated with 4 basic fund markets (Dow Jones, Japan (Nikkei), Nasdaq (100) and Switzerland
Swiss market) and current prices of the 4 precious metals (Ag, Au, Pd and Pt).
All trading days data cover the period (1.01.2013-31.03.2015)-(108 working weeks (27 months)) and
were taken from the site: http://export.rbc.ru/expdocs/free.index.0.shtml. We do hope that this new
method will give to traders and economists additional possibilities for analysis and forecasting of random
time-series associated with economic data having different lengths.
2
1. The formulation of the problem
The problem that can be considered in this presentation can be formulated as follows. Is it
possible to formulate a general concept (with its proper justification) in order to fit finance
data in the case of absence of the "best fit" (specific model) in wide range of scales?
If this general concept can be found then it will help to realize the fit of finance/economic
data and realize the most important problem that exist in modern economy – technical
forecasting, when the fitting function can be extended out from the given (temporal) interval.
We should specify the term "forecasting" that will be used in this paper. Under forecasting
procedure we understand a possible extension of the known fitting function which describes
of the optimal trend in terms of the Prony decomposition. Below we give some justified
arguments in favor of selection of the Prony function as the fitting function for the fitting of
financial data.
So, it is necessary to find a new (or generalize the existing ones) concept that can be effective
in realization of the problem formulated above. In this paper we want to combine/unify two
concepts that could be effective in solution of the forecasting problem that is important for
the modern economy.
The first problem is related with cyclic waves that are associated with the names of Russian
economist N.D. Kondratiev-[11] (http://en.wikipedia.org/wiki/Kondratiev_wave) and
American economist R.N. Elliott waves (http://en.wikipedia.org/wiki/Elliott_wave_principle).
They put forward the conceptions of scaling waves that are appeared in complex economic
systems as stock markets.
3
Here we show typical
pictures associated
with EWP analysis.
The basic critical remark can be expressed as – "the EWP is too vague to be useful, since it
cannot consistently identify when a wave begins or ends, and that Elliott wave forecasts
are prone to subjective revision“
The second principle that we want to combine with the first principle is related to
justification of the quasi-periodic (QP) processes that can be justified and they are widely
observed in many random processes that take place in Mother-Nature. We remind here the
basic publications , where the definition of these processes are given and confirmed on
available data. Because of importance of these processes it is necessary to repeat some key
points that will be helpful in understanding and construction of the suitable algorithm
described below.
4
Pr  t  T   Pr(t )

t
t 


Pr(t )  A0    Ack cos  2k   Ask sin 2k  
T
T 


k 1 

Fourier
F (t  T )  aF (t )  b
Quasi-Periodic processes ,
their definitions and
solutions
L 1
a  1: F (t )  exp  t  Pr(t )  c0 ,  
ln(a)
b
, c0 
,
T
1 a
t
a  1: F (t )  Pr(t )  b .
T
L 1
t
b

(
A
)
a

1:
F
(
t
)

exp
ln

Pr
(
t
)

c
,
c

,


F (t  LT )  as F (t  sT )  b


s
r
0
0

 r
L 1
T


s 0
r 1
s 0
1 a

L

s 0
L 1
L 1
P()   L   as  s  0.
s 0
s
t
t
b

( B)  as  1: F (t )   exp  ln   r   Prr (t )  c1 , c1 
.
L 1
T
T

s 0
r 1
L   s  as
L
s 0
5
In this sense the Fourier solution (decomposition) can be interpreted as Markovian process!
Pr (t )  A 
(T )
r
(r )
0
 (r )
t
t 


(r )
Ac
cos
2

k

As
sin
2

k

k




 k
T
T  


k 1 
K 1
t

F (t )  exp  ln 
T

Pr
(a)
(t ) 
K 1

k 1
 (a)
(a)
(a)
 Pr (t ), Pr (t  T )   Pr (t ),


t

Ac
cos
2
k

1




 k
T


Prr (t )  A 
 (r )
t

Ac
cos
2
k



 k
T

k 1 
K 1
t 


(r )

As
sin
2
k

k


 .
T 


t

F (t )  exp  ln  c   Prc (t ),
T

Prc (t ) 

t
t 


Ac
cos
2
k



As
sin
2
k







k



 .
 k
T
T




k 0 
K 1
The root is negative
t 



As
sin
2
k

1



k


 .
T  


t  g 1 r

F (t )  exp  ln   g      t  Prr (t ) ,
T  r 0

(r )
0
The root is real
The root is g-fold
degenerated
The couple of roots is
positive and complexconjugated
6
One can mark the following useful property of solutions that will be used in the next
section. If we realize the scaling transformation ( is an arbitrary scaling parameter)
t  t /  t'


T T / T '
This property signifies that the Prony decomposition is applicable for a wide range of scales.
In particular, scaling relationship allows to find the range of the interval [min ,max] for values
of the r = ln(r) and facilitate the fitting procedure associated with the usage of the Prony
functions . Finishing this section one can suppose that the single E-wave can be identified
with a couple of non-periodic functions (r = I,2,…,L, k=1,2,…,K)

t
Eck   r ,
 Tr

  exp   r   t / Tr   cos   2k  r    t / Tr   ,


t 
Esk   r ,   exp   r   t / Tr   sin   2k  r    t / Tr   ,
 Tr 
r  0, 1, Im   r  .
New presentation of the Elliott’s (K-) waves.
7
This fitting function is approximate and follows from the fact that the set of parameters as in
coincides with constants. In reality, these constants should be replaced by the temporal
variables as(t) but we do not know how to find the analytical solutions of for this case.
2. Description of the data treatment procedure
Before to apply this general function (14) it is necessary to outline a general algorithm that
can be useful for forecasting of many noisy/filtered data including the finance data, which
play key role in economy. For clarity we divide the original algorithm on some steps and use
for illustration the real data. As an example we consider the data associated with 4 basic
fund markets (Dow Jones, Japan (Nikkei), Nasdaq (100) and Switzerland - Swiss market)
and dynamics of the current prices for 4 precious metals (Ag, Au, Pd and Pt).
All day data cover the period (1.01.2013-31.03.2015)-(108 working weeks (27 months)) and
were taken from the site: http://export.rbc.ru/expdocs/free.index.0.shtml. For simplicity, we
suppose that in each year we have fixed 5 working days and the total set of data expressed
in trading days includes 4(wks)x27(mnth)x5(ds)=540 days (data points). So, we ignore real
fluctuations related with different days in a month and deviations related with different
trading days (4-6) related to local holidays in different countries. The tests that were realized
on available data shows that new approach allows to realize three types of forecasting if the
initial data have 4 types of "prices" expressed in terms of the normalized fund market index:
maximal price, open price, closed price and minimal price. As the test function we take the
simplest one that follows from (14)
8
K
E (t ; , T , K )  A0  B0  exp( opt t / Topt )    Ack yck (t )  Ask ysk (t ) ,
k 1

t
yck (t )  exp( opt t / Topt ) cos  2k

Topt



t
,
ys
(
t
)

exp(

t
/
T
)
sin
2

k


k
opt
opt
Topt


E (t  T )  a  E (t )  b,

 .

The simplest selection of the
Prony function.
b
, a  1,   ln( a),
1 a
T ( s)  Tmin  s Tmax  Tmin  , s  [0,1],
t 
E (t )  exp    t / T  Pr(t )  b   , a  1.
( s )   min  s   max   min  ,
T 
E (t )  exp    t / T  Pr(t ) 
1
mean( price)   open( price)  close( price) 
2
 min  min  ln(aup ),ln  adn   ,  max  max  ln(aup ),ln  adn   ,
Tmin  0.5  T , Tmx  2  T , T  length  (t  interval ).
Yup(t )  aupYmn(t )  bup , Ydn(t )  adnYmn(t )  bdn .
 stdev  ymn(t )  E (t , , T , K )  
Re lErr (, T , K )  
 100%
mean
ymn
(
t
)


The basic expressions for the
final fitting
function
K (v)  Kmin  v   Kmax  Kmin  , v [0,1].
9
We should stress also that the value of the optimal period Topt is not known but it is located
in the interval [0.5T, 2T], where T is the known length of the temporal interval of the random
sequence considered. Now everything is ready for realization of the fitting procedure.
But besides the actual fit of the finance data in the given temporal interval we should know
the length of the interval taken in the past for its continuation to the future. The problem of
the finding of the length of this interval is not trivial and needs a special research. Here we
want to suggest the following solution, based on analysis of cumulative (integrated) curve.
This curve is calculated as
1
t j  t j 1    y j 1  y j  ,

2
1 N
y j  y j  mean( y ), mean( y )   y j .
N j 1
Jy j  Jy j 1 
It has two peculiarities that are important for forecasting purpose (a) it eliminates the highfrequency fluctuations (noise) and (b) it reflects the tendencies of long-range fluctuations for
their increasing/decreasing. The cumulative curve correctly reflects the long-range
increasing/decreasing tendencies and detection of the tendency is important for forecasting
purpose. The figures 1(a,b) explain how to choose the segment of the curve on the initial
segment in order to continue it into the future interval.
10
DJ_index_total
Index_cut
108 weeks
the range of
forecasting
in the past
52 working
weeks
(Dow Jones fund index)/10
3
18
16
56 weeks
14
Fig.1(a). This plot shows the evolution of the
Dow Jones index (decreased for simplicity in
1000 times) covering the period of 108
working weeks. Each working week contains 5
working days. The end of the red line
(containing 108-52 = 56 weeks) determines
the starting forecasting point from the event
taking place in the past. In order to find it we
recommend analyzing the cumulative curve.
12
0
40
80
120
time(working weeks)
Jtot
Jcut
Cumulative curves (total and cut)
0
-20
-40
Fig.1(b). Here we show the cumulative curve
that contains the information about longrange fluctuations. The tendency of these
fluctuations on the previous figure is
unnoticeable but in the cumulative curve the
high-frequency fluctuations are suppressed
and the remained of the long-range
fluctuations clearly express their tendencies to
increasing/decreasing. The knowledge of
these tendencies is important for forecasting.
-60
0
40
80
time in working weeks
120
11
max(Indx)=Yup, min(Indx)=ydn
Yup(t)
Ydn(t)
Ymn(t)
16
Fig.2(a). This figure shows the strong correlations
between maximal/minimal values of the DJ index
with respect to its mean value. Because of the
strong correlations one can approximate the
functions Yup(t) and Ydn(t) by straight lines and
find the confidence interval for the value of .
14
12
12
14
16
Mean(Indx)=Ymn
Rup
Rupsm
Rdn
Rdnsm
The ratios and their smoothed curves
1.02
1.01
aup=1.0076
1.00
adn=0.9907
0.99
0
20
40
time (working weeks)
60
Fig.2(b). It is instructive to demonstrate these
curves that show the basic approximation that was
accepted in description of Elliott's waves by Prony
functions. Here we show the ratios
Rup(t)=Yup(t)/Ymn(t), Rdn(t)=Ydn(t)/Ymn(t) of
maximal and minimal DJ indexes with respect to
time. The solid red and cyan curves show their
smoothed values. In fact, we make a supposition
that these curves have narrow range of deviations
and can be replaced approximately by horizontal
lines. The desired mean values are given on the
right-hand side.
12
Ycut
for the complete curve (140 wks)
total curve (108 wks)
forcast for the cut curve
Forecast for the cut and complete curve
21
140 weeks
18
108 working weeks
56 working weeks
15
Fig.3. Here we compare two types of forecasting.
Forecasting from the cut off curve (starting from
the "past" point (corresponding to 56 working
weeks) and continued to the future (cyan line).
The complete curve does not have the past
segment (solid blue line) and directly was
continued to the future for the same period (140
weeks) –red line. As one can notice from this
figure they are very close to each other.
12
0
60
120
t
21
Yscl-forecast
Yscl
Ymn
Yscaled and its forecast
140 weeks
18
108 weeks
Fig.4. This figure demonstrates the coincidence
of the forecasting curves realized for the scaled
curve (we decrease the scale in three times) and
for the total curve Ymn(t). As before the horizon
of forecasting covers 32 working weeks.
15
12
0
70
time (working weeks)
140
13
Ymean_(JN-index)/1000
Ycut (84 weeks)
108 weeks
Ymean(108 weeks)
Ycut(84 weeks)
20
local minimum
15
84 weeks
Fig.5(a). The selection of the "past"
segment for Japan (Nikkei) fund market.
The past segment should contain
approximately 84 weeks. So, for moving for
the future we should keep the segment
containing 108-84=24 weeks.
10
0
50
100
time (working weeks)
Selected
peculiarity
Cum_curve_total
Cum_curve_cut
Total cumulative curve and its part (blue)
containing the increasing tendency
0
108 weeks
-30
small plateau
84 weeks
-60
0
50
time (working weeks)
Fig.5(b). This plot explains the selection of
the "past" segment on the previous figure.
We deliberately kept a small plateau that
corresponds to the local minimum near the
84th week shown on Fig.5(a). On the next
figures we will see how this small plateau is
reflected on the accuracy of the
forecasting.
100
14
Ack
Ask
2
2 1/2
Amdk=(Ack +Ask )
Ack,Ask.Amdk
0.8
0.0
Fig.6. Typical distribution of the amplitudes Ack
and Ask (k=1,2,…,31) (K=31) corresponding to
the decomposition of the random signal to the
Prony spectrum. Other fitting parameters are
collected in Table 2. Other distributions
corresponding to other fund markets have a
similar character and are not given.
-0.8
0
10
20
30
Number of modes 0 < k < 31(K=31)
Ycut_forecast
Ycut
Ymean_forecast
Ymean
108 wks
Ymean, Yfcst
20
140 weeks
84 weeks
15
10
0
70
time (working weeks)
140
Fig.7. Here we reproduce all necessary plots
that explain the accuracy of forecasting taken
from the cut off curve (past segment)
corresponding to 84 weeks. The forecasting line
coincides with cyan curve. The horizon of
forecasting for the total curve is shown by red
solid line. Again, we notice that these curves are
close to each other. The local minimum near the
84th week is not described by the local
forecasting (cyan) line because we deliberately
include a small plateau (clearly seen on Fig. 5b)
in the forecasting horizon. As it has been
mentioned in the text the second type of
forecasting based on the scaled curve is not
effective.
15
The cut off segment and the total
segment (Nasdaq 100)
Forecast horizon for the cut curve
The cut curve
The total curve
Frecast horizon for the total curve
140 weeks
5
108 weeks
78 weeks
4
Fig.8. Here we show the horizons of forecasting
for the Nasdaq (100) fund index normalized for
the value 1000. The past segment defined from
the cumulative curve is moved for 30 weeks into
the past. If we compare the forecasting curves
realized for the cut and total curves then one can
notice that are very close to each other.
3
0
70
140
time (working weeks)
Forecast curve obtained from 78 weeks
The segment obtained by the shifting on 30 weeks into the past
Forecast curve obtained from the total curve
The fit of the total curve
10
140 weeks
78 weeks
Hcut
9
108 weeks
8
7
0
70
140
Fig.9. Here we demonstrate the result for the
Switzerland-Swiss fund market. We observe
strong deviations between two types of forecast
horizons in comparison with the previous cases.
We explain these deviations by strong local
fluctuations that were happened during 20
weeks in the past. These strong deviations
distort the forecasting curve that is obtained for
the total curve. The forecast curve obtained for
the cut off curve which is started from the 78th
week does not "notice" these strong deviations.
These deviations can be explained from analysis
of the cumulative curve shown below.
time (working weeks)
16
The total cumulative curve
The cut off cumulative curve
0
103
Cumulative curves
plateau-3
90
-10
plateau-2
80
Fig.10. This plot explains the deviations that are
appeared between two forecast curves. The
deep peaks are reflected by three plateaus
appeared at 80, 90 and 103 week. So, the
horizon for the forecast curve for this curve is
short and for reliable prediction it should start
from 103 week.
plateau-1
-20
0
40
80
120
time (working weeks)
The cut off segment_100 weeks
The total segment_ 108 weeks
The initial silover prices covering the period
108 weeks. The cut off curve covers 100 weeks
40
100 weeks
32
108 weeks
Fig.11(a). Here we show the finance data
associated with dynamics of the silver prices. As
before we use the same temporal interval – 108
weeks. From analysis of the cumulative curve it
flows that we should go back to the "past" only
for 8 weeks.
24
16
0
40
80
time (working weeks)
17
The cut off segment
The total segment
Cumulative curves for the total silver
prices and for the segment covering 100 weeks
80
108 weeks
0
Fig.11(b). This plot demonstrates the
selection of the cut off segment which is
necessary for forecasting purposes. As one
can see from this curve the selection of the
interval covering 8 weeks back is acceptable.
-80
100 weeks
0
40
80
time (working weeks)
Fig.11(c). This curve explains the idea of
creation the desired "tube" for the given
mean price. As it is explained in the text it is
necessary to create initially the cut off curve
and then decrease the scale and reduced to
three incident points. In our case we chose
the minimal value of scale equaled 3.
Reduced curve obtained after reduction
to 3 incident points
max(price)
mean(price)
min(price)
35
mean number of weeks 100/3 = 33.3
28
21
0
50
time (mean number of weeks)
100
18
max(price) and min(price) with respect to
mean(price)
35
Mup(t)=max(price)
Mdn(t)=min(price)
Lup_fit_max(price)
Ldn_fit_min(price)
Lup
Ldn
28
Lup=1.0641*Mn-1.1227
Ldn=0.95005*Mn+0.7962
21
21
28
Fig.11(d). These plot shows that the
curves Mup(t)=max(price) and
Mdn(t)=min(price) being plotted to
respect the Mn(t)=mean(price) are
strongly-correlated and can be
approximated by the segments of the
straight lines. The calculated slopes help
to find the limits for the parameter opt.
35
The ratios of prices and their approximation
Mn=mean(price)
1.12
Rup(t)=Mup(t)/Mn(t)
Smoothed Rup(t)
Rdn(t)=Mdn(t)/Mn(t)
Smoothed Rdn(t)
1.04
mean(Rup)=1.0165
mean(Rdn)=0.9838
0.96
0
50
weeks
100
Fig.11(e). This plot demonstrates the
basic approximation related to Prony
decompositions. If we form the ratios of
the prices Rup(t)=Mup(t)/Mn(t) and
Rdn(t)=Mdn(t)/Mn(t) we note that are
changed with time. But their variations
are small and we replace these functions
by their mean values. They are shown on
the right-hand side. These values contain
an additional information for calculation
of the opt value.
19
The dynamics of the silver prices
and their forecast
Scaled curve (166 data points)
Forecast curve (750 data points)
Total curve (540 data points)
Forecast curve (750 data curve)
Ag
100
35
The cut curve forecast
108
28
150 week
The total curve
forecast
21
0
40
80
120
Scaled curve (166 data points)
Forecast curve (750 data points)
Total curve (540 data points)
Forecast curve (750 data points)
Forecast curves for
cut and total curves
Au
The cut curve forecast
2400
100
108
150
The total curve forecast
1800
1200
0
40
Fig.12. This plot shows two types of forecasting
for silver (Ag) prices obtained from the scaled
and the shifted curve (cyan line) and forecast
obtained from the total curve (red line). We
should stress here that the scaled curve contain
only 166 data points (each data point
correspond to one trading day) but the total
curve contains 540 data points. In spite of this
difference the forecasting segments are close to
each other and reflect the same tendencies.
80
time (working weeks)
120
Fig.13. This plot shows two types of forecasting
for gold (Au) prices obtained from the scaled
and the shifted curve (cyan line) and forecast
obtained from the total curve (red line). We
should stress here that the scaled curve contain
only 166 data points (each data point
correspond to one trading day) but the total
curve contains 540 data points. In spite of this
difference the forecasting segments are close to
each other and reflect the same tendencies.
160
20
Scaled curve (166 data points)
Forecast curve (750 data points)
Total curve (540 data points)
Forecast curve (750 data points)
Pd
108
150 weeks
The fit of the prices for Palladium
and their fit
1600
100
(the extreme value missing)
1200
150
800
0
40
80
120
160
time (working weeks)
Dynamics of the prices for platinum
and their forecast
Scaled curve (166 data points)
Forecast curve (750 data points)
Total curve (540 data points)
Forecast curve (750 data points)
Pt
150
2400
100
108
150
2000
1600
0
40
80
time (working weeks)
120
160
Fig.14. The fit and subsequent forecast of
current prices for palladium (Pd). In
order to stress the sensitivity to extreme
points we chose this instructive example.
If we ignore the extreme point (where
the price achieves its local maximum)
then forecasting price becomes different
if we extend the cut curve (cyan line) and
compare it with the forecast curve
obtained from the total curve (red line).
So, any local extremum should be taken
into account.
Fig.15. This plot illustrates the dynamics
of the prices for platinum (Pt) and its
forecast for the future 50 weeks. We
notice again that these forecast curves
reflect the same tendencies and actually
determine the sealing and floor of these
prices that are important for traders. The
low level of the credibility in comparison
with forecast of fund indexes is explained
in the text.
21
Table 1. The optimal number of modes K that provides acceptable forecasting. Here we accept the following units.
Each trading week includes 5 working days. In each month we have 5x4 =20 working days and so in a year
we have 240 working days. The real data are taken from the site covering the period from 1.01.2013 – 1.04.2015
that gives 540 working days. We do not take into account possible fluctuations in working days that
can be important in real trading sessions. For us it is important to demonstrate a new principle associated with fitting
of data based on the Prony decomposition.
The type of
the fund
market
Type of the
curve
The length of
the curve
T (in working
weeks)
The length of
the past
segment
Sh
The horizon of
forecasting
H
Number of
optimal
modes
K
DJones
Ycut(t)
Ytot(t)
56+52=108
108
52
0
140-56=84
140-108=32
38
37
Japan_Nikkei
Ycut(t)
84+24=108
24
140-84=56
38
Ytot(t)
108
0
140-108=32
35
Ycut(t)
78+30=108
30
140-78=62
15
Ytot(t)
108
0
140-108=32
42
Ycut(t)
78+30=108
30
140-78=62
18
Ytot(t)
108
0
140-108=32
38
Nasdaq-100
Switzeland
22
Precious
metals
Type of
curve
The length of
curve (in
working weeks)
The length of
the past
segment
The horizon of
forecasting
H
Number of
optimal
modes
K
100/3=33.3
(166 data
points)
108
8
150-33=117
6
0
150-108=42
18
8
150-33=117
5
Ytot(t)
100/3=33.3
(166 data
points)
108
0
150-108=42
17
Palladium
Yscl(t)
Ytot(t)
98/3=33
108
10
0
150-33=117
150-108=42
7
23
Platinum
Yscl(t)
Ytot(t)
98/3=33
108
10
0
150-33=117
150-108=42
5
23
Silver (USD for Yscl(t)
troy ounce)
Ytot(t)
Gold (in USD
for troy
ounce)
Yscl(t)
the
23
Table 2. The additional fitting parameters that enter to the simple fitting function. They are given for the
4 fund indexes.
Type of the file
Topt
opt
A0
B0
Range (Amd)
RelError(%)
DJ_Ycut
56.762
-0.00359
700.506
-686.618
0.24328
0.10935
DY_Ymean
109.08
-0.00359
1129.27
-1115.24
0.19242
0.09841
JN_Ycut
82.516
-0.00381
1372.22
-1360.67
1.05102
0.63628
JN_Ymean
109.282
-0.00365
1783.34
-1771.69
0.8251
0.70455
Nq(100)_Ycut
82.516
-0.00381
1372.22
-1360.67
1.05102
0.63628
Nq(100)_Ymean
109.282
-0.00365
1783.34
-1771.69
0.8251
0.70455
Swd-S_Ycut
78.982
-0.00458
307.927
-300.54
0.08393
0.52402
Swd-S_Ymean
109.282
-0.00458
280.941
-273.314
0.10824
0.42647
Table 3. The additional fitting parameters that enter to the fitting function (15). They are given for the 4
prices associated with precious metals (Ag, Au, Pd, Pt).
Type of the file
for the given
metal
Topt
opt
A0
B0
Range (Amd)
RelError(%)
Ag_Yscaled
103.584
-0.01043
936.584
-917.009
4.61607
2.95656
Ag_Ytotal
109.282
-0.0127
335.859
-313.022
4.18467
2.01744
Au_Yscaled
103.584
-0.016
62637.6
-61578.2
322.878
2.87613
Au_Ytotal
106.036
-0.02395
30471.3
-29242.8
295.973
2.05649
Pd_Yscaled
101.712
-0.0083
40149.3
-39430.9
107.218
1.99709
Pd_Ytotal
109.282
-0.01078
91755.1
-91303
187.01
1.79653
Pt_Yscaled
107.58
-0.00507
131587
-130249
178.894
2.2666
Pt_Ytotal
109.282
-0.01789
54173.3
-52952.1
239.811
1.50004
24
Russian oil and gas/100 – data from RBC – period January-1/01/15- September03/09/15
Time- working
weeks,
Right figure –
cumulative index
August - 2015
End of
September
2015
25
Russian industry/1000
26
USD/RuR (period 0.1/01/2015-0.3/0.9/2015)
USD vs RuR during 8 months (32 weeks) . Each
week contains 5 working days
Cumulative index. USD has a tendency to fall down.
Then in autumn we should observe the further
weakening of the RuR.
27
The fit with the help of the Prony’s
spectrum.
End of this year (USD is becoming stronger)
February of the next year
USD will became weaker in the end of the
winter of 2016.
28
The basic conclusions:
1. The proper selection of the segment in the past with the help of cumulative
curve is important.
2. The natural data helping to form a “tube” of prices is important. The artificial
creation of the desired tube aggravate the forecasting horizon.
3. The presentation of the E(K)-waves in the form of Prony decomposition is
important for technical trading.
4. The suggested method is rather flexible and allows in further modifications.
29
30