Nonlinear and chaotic analysis of a financial complex system?

Appl. Math. Mech. -Engl. Ed. 31(10), 1305- 1316 (2010)Applied MathematicsDOI 10.1007/s10483-010- 1363-7and MechanicsCShanghai University and Springer- Verlag(English Edition)Berlin Heidelberg 2010Nonlinear and chaotic analysis of a financial complex system*Yong- xin LIN (林勇新)，Yu-shu CHEN (陈予恕)，Qing-jie CAO (曹庆杰)(School of Astronautics, Harbin Institute of Technology,Harbin 150001, P. R. China)(Contributed by Yu-shu CHEN)Abstract In this paper, determination of the characteristics of futures market in Chinais presented by the method of the phase-randomized surrogate data. There is a significantdiference in the obtained critical values when this method is used for random timeseriesnoise in the chaotic timeseries. The phase space of chaotic timeseries is decomposed intorange space and null noise space. The original chaotic timeseries in range space is re-structured. The method of strong disturbance based on the improved general constrainedrandomized method is further adopted to re-deternination. With the calculated results,an analysis on the trend of futures market of commodity is made in this paper. Theresults indicate that China's futures market of commodity is a complicated nonlinearsystem with obvious nonlinear chaotic characteristic.Key wordsnonlinear chaotic timeseries, random timeseries, phase-randomized,singular value decomposition, general constrained randomizationChinese Library Classification 0175, O2412000 Mathematics Subject Classification 65D25, 65D301 IntroductionAs we all know, the dynamical system of the real world is usually nonlinear. In fact, aquantitative change is never able to generate a qualitative change in a linear system. In otherwords, phase transitions of physics, cell mutation of biology, and the diminishing returns in eco-nomics and so on, all of which will be lost in the linear world. Linear model once undoubtedlyplayed a very positive role, such as purely autoregressive (AR) model, purely moving- average(MA) model, autoregressive moving average (ARMA) model, autoregressive integrated moving-average (ARIMA) model1l. But there are nonlinear factors existed in the most of the motion.Before the 1960s of the last century, the nonlinear research was restricted because of the phaseof development of the scientific theories. In the economic field, we could begin to see thatthe autoregressive conditional heteroscedastic (ARCH) model and the threshold autoregressive(TAR) model, which are typical representatives of the nonlinear timeseries model, appeared one* Received Apr. 4, 2010 / Revised Aug. 13, 2010Project supported by the National Natural Science Foundation of China (No. 10632040)Corresponding author Yong-xin LIN, Ph. D, E-mail: linyongxin7406@126.com中国煤化工MHCNM HG1306Yong- xin LIN, Yu shu CHEN, and Qing-jie CAOafter anotherl2- 4 until the end of 1970s and beginning of 1980s in the 20th century. In partic-ular, since 1990s, the non-parametric and semi-parametric technology, which was developed byTjvstheim et al., had given the researching state of the entire time series a totally new look. Anadvantage of this method is that there are less requirements on the transcendental informationof the model structure and it provides a useful perceptual knowledge for further parameterfitting5 8]. The recent research results of ARCH/GARCH models and TAR models also reflectthe integration of the parameter and non-parametric method in analysis of time seriesl9]. Theway of adopting nonlinear dynamics on studying time- series has developed for almost thirtyyears, including the researches of a chaotic dynamical systems reflecting low degree of freedomand patterns chaotic dynamical system reflecting high degree of freedomIf we test that a timeseries is random sequence, it should be studied by the statisticalmethods. If the sequence is deterministic chaotic, the nonlinear model may better portray thebehaviour of dynamic system. Even if the chaotic timeseries is generated by a deterministicsystem, it inevitably contains noise. It is a problem that we must be resolved that how onearth is the proportion of noise in the sequence. To sum up, we naturally raise this kind ofquestion: how to judge the time series which is deterministic or random, linear or nonlinear.The paper firstly makes a decision on the chaotic characteristic of seven varieties of futuresmarket of commodity of China by the method of the phase randomized surrogate datal12- 14.Then it adopts the singular value decomposition technology based on the non-linear dynamicstechnology to reduce the noise of timeseries in the Chinese futures market of commoditylI1iIt further adopts the method of strong disturbance to re-determine on the basis of the improvedgeneral constrained randomized method18. This paper applies the disturbance method tomake a nonlinear chaotic test on this kind of complex system- the Chinese futures market ofrespectively through a weak disturbance, noise reducing andstrong disturbance on the system. This method is used to analyse the nonlinear chaotic behaviorof a complex system and make a relevant conclusion which can give references to investors.2 Theory and method2.1 Construction of surrogate data sets and setting up the critical valueAccording to 12] which gives a different distributed phase-randomized method, we haveacquired time series x1, x2, D3， .. xN, the sampled time of time series are respectively 0,△T, 2△T, . (N-1)△T,△T is the regular interval of sampled time. The discrete Fouriertransform is applied and transform operator is FX(f)= F{x(t)}= 2 (n)e2rifnOt = A(f)eib(f),(1)where A(f) is the amplitude and φ(f) is the phase angle; f = -N△f/2,，, -Of, 0,△f,N△f/2, where△f = (1/N)AT. Then, we make φ(f) to rotate randomly the phase angle ψ(f).Here, we use the random numbers obeying rectangular distribution or Gaussian distributionwhich are generated randomly by computer in the range [0, 2π] to ψ(f), respectively. We canget父= A(f)+(0()+() = x()eiv6().(2)Do an inverse Fourier transform to (2):(f)= F-1{8(f)}= F-l{xX(f)el()}.(3Then, we can get the surrogate data ^1, 22, T3,-., TN, which are called the surrogate data oforiginal data and their imaginary part must be zero.中国煤化工MHCNM HGNonlinear and chaotic analysis of a financial complex system1307The method determining the property of data is given below. According to [14], we recon-struct the phrase space of time series, suppose0i = xi,Di+r,Li+2r, ...i+(m-1)r,小,(4where的∈Rm. In order to reduce the auto correlation character between the data in the reallife, c(r, N, W, m) should use the followving formula:N N-nc(r,N, W,m)=. 2 H[r-lv:+n-ol,(5)(N- W)(N-W+1)N=W i=1where H is the Heaviside functionH(x)=[1， x≥0,(6)l0， x< 0,and W should satisfyw≥τ(示)°(7where T is the interval of time series, m is the embedding dimension. Using the correlationdimension of time seriesd= lim limdln c(r, N, W, m)/dr(8dln(r)]/drwhere d≤m≤2d+ 1. Based on the formula In c(r, N, m, W)/ lnr, seeking the correlationdimension to the original data and surrogate data respectively, denoted by Sori and Ssur, definecriticial value Z: .Z = | - (sori>|/osur(9where <> is the statistical average; σsur is the mean square deviation of surrogate data. Thethreshold value Z = 1.96. If 8<1.96, the really measured data with 95% confidence level isthe time series mainly on random factors; when Z≥1.96 the time series mainly on nonlinear2.2 Noise reduction by singular value decomposition technologyIt is easily affected by the noise when the phase randomization method makes a non- linearchaotic detection on a complex system. So we should reduce the noise of time series. Afterreducing the noise of time series, the sequence is tested again to further prove the non-linearchaotic characteristics. The traditional filtering theory assumes that the useful data and noisedata of the observation time series have a different spectrum, i. e., the useful data and noise dataare frequency spectrum separating, and thus an ideal filter can be designed to carry out noisefiltering. Clearly, such a filter system does not take into account other features, such as the lawof operating evolution, the signal and noise statistical characteristics. Therefore there are moredefects in this method. This kind of method is not accurate to reduce noise of the observedtime series. So it needs to apply the singular value decomposition based on the nonlineardynamics to handle the noise of observed time series of the actually complex system. This kindof method studies characteristics of evolving track of the attractor to reduce the noise throughreconstructing the nonlinear dynamic systeml15- 171. The signal after reducing noise has strongrobustness but the phase-randomized method is only suitable to the sensitive system which hasa low anti-jamming ability. We suggest adopting the method of strong disturbance on the basisof the general constrained randomized method18] to make a nonlinear chaotic characteristicdicision on the signal after reducing the noise.中国煤化工MHCNM HG1308Yong-xin LIN, Yu shu CHEN, and Qing-jie CAO2.3 General constrained randomization and critical valueSupposing {xin}n=1 is the surrogate time series of {xn}n=1, we consider the requirementthat the autocorrelation function C(r)= (inTn-t) of the surrogate time series is the same asthe autocorrelation function C(T) = (xnxn-> of the original time series {xn}n=1; so we canmake the constraint asF({in})=C(r)-C(+)，τ= 1,2，.. ,Tmax,(10)also the cost function can be written asE({n})= max{|C(r)-C(r)|:τ= 1,2，,Tmax},or({]))= (E |C(r)- C()9)*.(12)time order in the original time series. This can guarantee the random process of which surrogatetime series and original time series are selected from the same (unknown) density distribution.We can complete this step through reordering of the original time series. In order to seek theminimum solution of the cost function to various different orderings of {xn}n=1, we can usethe simulated annealing. By exchanging the order of {xn}n_ 1 the configuration is updatedand annealing plan will determine whether to accept or reject. To the medium and longertime series and given constraints, surrogate time series generated by the general constrainedrandomization can be very accurate. But it is at the cost of a very large of computing time.This paper improves on the original basis, so it can more easily make a decision.Let({1))= (Z |(x)- ()*,(13)().))= (二()- (1),")*.(14)an interval when q equals to 1. The cost function (13) is the summation of the autocorrelationfunction difference of the original data and surrogate data. The cost function (14) is thesummation of the autocorrelation function difference of two sets of random numbers obtainedafter disrupting the original data. We can determine the non-linear chaotic characteristic bycomparing the different intervals of the extreme value of two cost functions. If the range isdifferent, it is chaotic timeseries. Otherwise, it is random timeseries. The determination of thisnonlinear chaotic characteristic is of essential theoretical and practical application value to thefuture modeling.3 Calculating results and analysisWe choose respectively Shanghai copper, Shanghai aluminum, Natural rubber, Dalian trans-genic soybean, Dalian soybean meal, Zhengzhou hard wheat, Zhengzhou strong wheat, sevenvarieties of Chinese futures market of commodity as the sequence data of the daily closing price中国煤化工MYHCNM HGNonlinear and chaotic analysis of a financial complex system1309in January delivery period from the very beginning of starting to the end of 2008. The data isobtained from the CSMAR database. We test the random characteristics or non-linear chaoticcharacteristics of precious metals, rubber, grains, the three types of commodity futures.Firstly, according to formula (1)- (3), we use the phase randomization method generates thecorresponding surrogate data respectively on the time series of all varieties.Secondly, according to [14], we make phase space reconstruction respectively on the originaltime series and surrogate time series. According to (4)- (8), we obtain the correlation dimensionand make a correlation dimension Fig. 1. According to (9), we obtain the critical value Z andlist in Table 1.Table 1 Critical valueDalianZhengzhouShanghai ShanghaiNaturaltransgenicsoybeantronghard wheataluminumcoppermealwheat(Sori)0.783 61.160 81.234 41.071 11.356 91.044 41.1210(sur>1.69321.749 81 .849 81.944 21.868 51.501 11.727 7Osur0.206 30.11 020.152 30.280 10.125 60.092 20.063 14.409 35.258 34.041 33.116 84.953 49.920 5Figure 1 is successively Dalian transgenic soybean, Dalian soybean meal, Zhengzhou strongwheat, Zhengzhou hard wheat, Shanghai aluminum, Shanghai copper and Shanghai naturalrubber (log2r-log2c) correlation dimension diagrams. The slope of the curve in a certain inter-val in the diagram tending to be stable is the correlation dimension. On the top of each figureis the original data of that the embedding dimension equals to two (three, four, five are notdrawn). The below four are the surrogate data when the embedding dimension equals to two,three, four or five. We can observe the difference of correlation dimension between the originaldata and the surrogate data. Concrete results can be seen in Table 1 and the critical value Z isall greater than 1 .96, it explains that the above species all have non-linear chaotic characteristicsto some degrees. Seen from the critical value Z, Shanghai natural rubber is the most sensitiveand its anti-jamming ability is relatively weak. According to the sensitive degree from strongto weak (anti-jamming ability from weak to strong), other varieties are successively Daliansoybean meal, Shanghai copper, Dalian transgenic soybean, Shanghai aluminum, Zhengzhoustrong wheat, Zhengzhou hard wheat. The variety which has an appropriately anti-jammingability has a strong ability of holding funds. Then, it will attract more funds into the market.It can provide better, more stable investment environment, reduce investment risk, avoid the. -5日-5个ODs -10母OD十SD+ SD-15+SD.. SD15 t- SD-205-2024681012log2r(a) Dalian transgenic soybean(b) Dalian soybean meal中国煤化工YHCNM HG1310Yong-xin LIN, Yu-shu CHEN, and Qing-jie CAO三~5-10g -10-母OD+ SD15 t士SD十sD-15-20 t-20-25 L-51(c) Zhengzhou strong wheat(d) Zhengzhou hard wheat0三-5--10 t甘OD曾”SD15 fSD-BD+ SI3D;D-8101214-2515log2rlog,r(e) Shanghai aluminumI ShanghcopperE! -10-+OD学-15」呻SD-25 I(g) Shanghai natural rubberFig.1 logr-log2C correlation dimension figure :behavior of manipulation and monopoly on the market to make the market healthily develop.Whereas the investing risk of the excessive sensitive varieties would be great, the varieties ofover-strongly anti-jamming ability are not beneficial to investment and price discovery. There-fore, seen from the critical value Z, Shanghai copper, Dalian transgenic soybean, Dalian soybeanmeal of seven varieties are relatively better (the appropriate anti-jamming ability), which is suit-able for investment and development. The above analysis is also consistent to the actual marketsituations in that period. Such a complex system is of time dependent. We should do a specificanalysis according to the specific situations.Thirdly, according to [14]-[16], we reduce noise of Dalian transgenic soy bean, Dalian soybean中国煤化工MHCNMH GNonlinear and chaotic analysis of a financial complex system1311meal, Zhengzhou strong wheat, Zhengzhou hard wheat, Shanghai aluminum, Shanghai copperand Shanghai natural rubber by the singular value decomposition technology and make Fig. 2.Figure 2 is respectively the diagrams of noise reduction by the singular value decomposi-tion technology of Dalian transgenic soybean, Dalian soybean meal, Zhengzhou strong wheat,Zhengzhou hard wheat, Shanghai aluminum, Shanghai copper and Shanghai natural rubber.On the top of each figure is the original data and the below is the data after reducing noise.Using the singular value decomposition technology to process the signal is a signal decompo-sition technology rather than a noise reduction technology. The characteristic of the singularvalue decomposition technology researches the feature of evolutionary trace through the recon-struction of phase space of the nonlinear dynamical system, thereby decomposing the signal.It makes a singular spectrum analysis on time series. Because different orders of singular valuespectrum entropy are selected, it is also different that the effect of reducing noise is reachedby anti-process through the singular value decomposition. The effect of the signal reductionof noise obtained is significantly different under the diferent order of singular value spectrumentropy. When the order of reducing noise of singular spectrum selected is low, the informationcontained by the signal of reducing noise is incomplete and even the phenomenon of wave distor-tion will happen. It is difficult to make an accurate reflection to characteristics of the effectiveinformation of original information. When the order of reducing noise of singular spectrumselected is higher, it still remains a portion of noise information in the signal after reducingnoise and cannot achieve the goal of fully reducing noise. We can determine effective embeddeddimension m of time series (m is the amount of essential eigenvalue which is greater than zeroin evidence) when we process these data with singular spectrum decomposition. The thresholdvalue can be regarded as the important evidence to determine the order of chaotic model.60005000-OD-DD4 000- DD差2 0003000000 t2000200 4060000 10000o1000150Time(a) Dalian transgenic soybean(b) Dalian soybean meal25002 500-0D-oDDD喜1500g 15001 0001 0000 t50010001500500500 1000 1500 2000 2500(c) Zhengzhou strong wheat(d) Zhengzhou hard wheat中国煤化工MYHCNM HG1312Yong-xin LIN, Yu shu CHEN, and Qing-jie CAO2500010000-0D20000- -DD8000- -DD .10000 nyd6000上1000E 4000/mmm50002000-。5001000150020005001000 1500 2 000Time(e) Shanghai aluminum(f) Shanghai copper30000DD200000150005010001 500(g) Shangheai natural rubberFig.2 Noise reduction by the singular value decomposition technologyThe signal got by the decomposition of the futures market data when this paper applies thesingular value decomposition technology in the second stage has random elements. It does notaffect our judgement and analysis on this kind of complicated systems, but it presents perfectlythe main forces of market (the hedger or investor is the main force). The purpose of applyingthe singular value decomposition technology here is to dissect the system, further do a chaoticdetermination on the main components of the system besides doing a chaotic test to the entiresystem by use of the phase-randomized method.To the concept of noise which is originating from oscillation of the construction machineryand so on, the noise often displays as a state with high frequency components beause thenoise is produced as the equipment or structure appears a malfunction or deformation. So thegeneral method of noise reduction is to reduce the high frequency components of the series toget the relatively smooth and stable data. It is significantly different from the former that thispaper applies the concept of noise to the capital market, such a complicated system. In thiskind of complicated financial system, the factors which are seemingly random can be seen asnoise.But this noise can not be regarded simply as so-called malfunction or nonsense. Thisis exactly the chaotic feature. It is superficially random and confusing but it has its own lawsin chaotic. Judging from the pictures of the singular value decomposition to seven futuresvarieties，the signal got after decomposing still has the high frequency components and isslightly stable compared with the signal before decomposing. So people will misunderstand anddoubt whether the noise is reduced. The fact is that it uses the singular value decompositiontechnology to decompose the signal of the chaotic system, thereby getting the data signalof the main influencing factors of complicated systems. Specifically, we detect and get theinformation of main forces in the futures market through the singular value decomposition中国煤化工MHCNM HGNonlinear and chaotic analysis of a financial complex system1313technology. Although the signal we got still has the high frequency components, we achieve thegoal.The noise reduction here refers to try to pursue the max-eigenvalue, reduce the other smalleigenvalue and zero eigenvalue, thereby extracting the data of main influencing factors. Somevarieties are easy to extract the signal of main forces in the actual practice, others are notbecause the data of difficult decomposition explains that the infuencing factors in this com-plicated system is unitary and the main force of market itself has an absolute leadership whenwe use the singular value decomposition technology to decompose the signal. Certainly, theconditions of the easily decomposed varieties are relatively complicated. If it has no obviouseigenvalue which is greater than zero and stable when the random signal applies the singularvalue decomposition technology and the chaotic signal can be found, this point can show thediference between the random signal and the chaotic signal.We can see that coupling of noise and signal of Shanghai aluminum is very strong when weuse the singular value decomposition technology to decompose the signal of all varieties. It alsoshows that hedgers of the market have an absolute leadership as main forces. So it is similar toDalian transgenic soybean and Zhengzhou hard wheat. To the other varieties of the Shanghaicopper and natural rubber, speculators are very active as main forces and there is much morespeculators in the market. Such a complex system is of time dependent. We should do a furtheranalysis according to the specific situations.Finally, we use the general constrained randomization handle the original data and the dataafter noise reduction of all varieties based on formula (13) and (14) for processing, and weobtain Tables 2 and 3.Table 2 Comparison of the general constrained randomization cost function value of the originaldatDalianZhengzhouZhengzhou Shanghai Shanghai Naturaltransgenic soybeanstronghard wheat aluminumcopperubbersoybeanmealwheatmin E({n})1.66E+07 1.34E+07 2.10E+07 2.60E+06 3.26E+08 1.28E+10 1.15E+091.41E+081.15E+083.17E+073.18E+07 3.50E+09 5.49E+10 6.66E+09min E({in}) 1.68E+08 1 .25E+086.49E+07 4.77E+07 4.36E+09 2.18E+11 1 .69E+10max E({xn}) 2.66E+08 2.11E+08 9.62E+07 6.47E+07 6.46E+09 2.53E+11 2.37E+10Table 3 Comparison of the general constrained randomization cost function value after reducingnoise by the singular value decompositioncoppermin E({in}) 2.67E+06 7.37E+05 1.63E+05 1.24E+06 4.73E+05 3.20E+08 7.33E+07max E({in}) 2.58E+07 4.32E+06 2.16E+06 1.44E+07 4.95E+06 2.09E+09 9.52E+07min E({正n})4.17E+07 1.42E+07 4.62E+06 2.42E+07 5.42E+06 6.74E+09 2.21E+09max E({in})4.60E+071.45E+076.86E+06 4.14E+07 9.29E+06 7.46E+09 2.23E+09Tables 2 and 3 are respectively the maximum and the minimum obtained by optimizingof the cost functions (13) and (14) when q equals to 1. The previous cost function is thesummation of the autocorrelation function difference of the original data and surrogate data.The following cost function is the summation of the autocorrelation function difference of twosets of random numbers obtained after disrupting the original data. We can determine thenon-linear chaotic characteristic by comparing the different intervals of the extreme value of中国煤化工MYHCNM HG1314Yong- xin LIN, Yu shu CHEN, and Qing-jie CAOtwo cost functions. The concrete results can be seen in Table 2 (the comparison of the generalconstrained randomization cost function value of the original data), Table 3 (the comparison ofthe general constrained randomization cost function value after reducing noise by the singularvalue decomposition). The calculating results indicate that the above seven varieties all havenon-linear chaotic characteristics. .Whether it exists, the chaotic attractor in the complicated systems is the key and premiseof whether can further apply the chaotic theory to research the complicated chaotic system. Sothe chaotic test on this kind of complicated system is very important. Classical testing methodssuch as BDS test, R\S analysis are relatively mature indeed, including the methods used inevery stage of this paper which is not fresh if the single method is adopted. However, can theabove methods can really test the chaotic? This paper advocates the determination on thechaotic existence of a complicated system from the physical characteristics and structures of acomplicated system. Firstly, it does the chaotic test from the entire system. Secondly, the maininfuencing forces of complicated system are got by the dissection of structure and the reductionof the other non- principally infuencing factors (noise) of complicated systems, thereby doing thechaotic determination again. It does a chaotic test on all kinds of possiblly contained conditionsof the complicated system to prevent from omission and reach the wrong answer through theapplication of methods of weak disturbance and strong disturbance and gradually, step by step,successively combined strategy. In addition, the chaotic exists in nonlinear systems but thenonlinear is not enough to explain that the chaotic exists in complicated system. And thispaper can test not only the nonlinear of system but also the existence of system mechanismsimplied in complicated systems through combined testing methods, thereby being more likelyto determine the chaotic existence of complicated systems. Certainly, the determining methodsand tactics remain to be improved with the research developing.4 Conclusions(i) The analyzing results indicate that the Chinese futures market of commodity is a com-plicated nonlinear system with an obvious nonlinear chaotic characteristic. The determinationof this non-linear chaotic characteristic is of essential theoretical and practical application valueto the future modeling.(i) We can see that coupling of noise and signal of Shanghai aluminum is very strong whenwe use the singular value decomposition technique to decompose the signal of all varieties. Italso shows that hedgers of the market have an absolute leadership as main forces. So is similarto Dalian transgenic soybean and Zhengzhou hard wheat. To the other varieties of the Shanghaicopper and natural rubber, speculators are very active as main forces and there is much morespeculators in the market. Such a complex system is of time dependent. We should do anfurther analysis according to the specific situations.(ii) The general constrained randomization method can test a null hypothesis caused by thenature of any complete set of observables of the original time series. The anti-jamming abilityof portion signal which is obtained after the singular value decomposition of seven varietiesis very strong. We suggest the strong disturbance method on the basis of the improved gen-eral constrained randomized method which can further conveniently determine the non- linearchaotic characteristic of system. The results indicate that such a complex system obviously hasnon- linear chaotic characteristics.(iv) Seen from the critical value Z, Shanghai natural rubber is the most sensitive ancits anti-jamming ability is relatively weak. According to the sensitive degree from strong toweak (anti-interference capability from weak to strong), other varieties are successively Daliansoybean meal, Shanghai copper, Dalian transgenic soybean, Shanghai aluminum, Zhengzhoustrong wheat, Zhengzhou hard wheat. The variety which has an appropriately anti-jammingability has a strong ability of holding funds. Then it will attract more funds into the market.中国煤化工MYHCNM HGNonlinear and chaotic analysis of a financial complex system1315It can provide better, more stable investment environment, reduce investment risk, avoid thebehavior of manipulation and monopoly on the market to make the market healthily develop.Whereas the investing risk of the excessive sensitive varieties would be great, over-strongly anti-jamming ability of the varieties are not beneficial to investment and price discovery. Therefore,seen from the critical value Z, Shanghai copper, Dalian transgenic soybean, Dalian soybeanmeal of seven varieties of are relatively better (the appropriate anti-jamming ability), which issuitable for investment and development. The above analysis is also consistent to the actualmarket situations. Such a complex system is of time dependent. We should do a specific analysisaccording to the specific situations.(v) The complexity of all varieties is similar from the view point of correlation dimension.The correlation dimension is between [1.55, 1 .95] before reducing noise. The correlation dimen-sion is between [1.60, 1.87] after we do signal decomposition to the original time series throughthe singular value decomposition technology. The correlation dimension which is between[1, 2] shows that it hides two types of main influencing factors in the trend of seven vari-eties of our commodity futures market and two forces are in action in the market. In addition,seen from the eigenvalue got in the process of singular value decomposition technology, there aretwo eigenvalues that are obviously greater than zero. This also explains that two kinds of forceshave main infuences in the futures market. These decide the complexity of the commodityfutures market and the internal infuencing factors contained in the trend of future market inshort term. The critical value Z is greater, which means that this variety is sensitive and thenonlinear are strong. The connection of the critical value and the correlation dimension canbe used for estimating the risk degree of all varieties and provide some reference for the riskinvestment of the investors and the risk control of capital markets.References[1] Abarbanel, H. D. L, Prediction in chaotic nonlinear systems: methods for timeseries with broad-band Fourier spectra. Phys. Rev. A 41(4), 1782 - 1807 (1990)[2] Engle, R. F. 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