Distinguishing Chaos from 1/f Motions in Soil Failure

An ASABE – CSBE/ASABE Joint Meeting Presentation Paper Number: 1899139

Distinguishing Chaos from 1/f Motions in Soil Failure Kenshi Sakai1*, Shrini K. Upadhyaya2 ,Nina Sviridova1 1:Department of Agricultural and Environmental Engineering, Tokyo University of Agriculture and Technology,Tokyo,18-8509 *Corresponding author:[email protected] 2:Department of Biological and Agricultural Engineering, University of California ,Davis,CA,95619

Written for presentation at the 2014 ASABE and CSBE/SCGAB Annual International Meeting Sponsored by ASABE Montreal, Quebec Canada July 13 – 16, 2014 Abstract. Nonlinear time series analysis (NTA) was employed to identify determinism in soil failure patterns observed in tillage (soil cutting). As soil consists of large numbers of particles, one may expect soil failure to be described adequately as a stochastic process. However, clear spatial patterns are sometimes observed in tillage operation, associated with a predominant spatial frequency in the power spectrum of the soil cutting force data (Glancey et al., 1989, Glancey and Upadhyaya, 1995). We applied NTA to soil failure data to distinguish deterministic chaos from stochastic processes such as (1/f) fluctuations. We established four different soil conditions by controlling soil compactness and moisture. Deterministic chaos was identified only in tilled and dry soil. We reached this conclusion, by combining nonlinear deterministic prediction, Wayland test, Lyapunov spectrum and surrogate algorithms. We also recognized the difficulty of distinguishing deterministic chaos from (1/f) fluctuations, and improved the discrimination by introducing a comparison between the autocorrelation and NDP functions. Keywords. Nonlinear Dynamics, Deterministic Chaos, Tillage, Soil Failure, Tlanslation error, Inner Product

Introduction In soil tillage research, the existence of spatial oscillations in soil failure patterns has been discussed. Numerous observations of such oscillations have been reported (Upadhyaya etl.al, 1987, Upadhyaya et.al, 1999. Wolf,etl.al, 1981, Bowers, C. G., Jr. 1989). Glancy and Upadhyaya reported clear peak of frequency analysis with FFT (Glancey et al., 1989, Glancey and Upadhyaya, 1995, Glancey etl.al,1996). However, as soil consists of large numbers of soil elements that interact, it is difficult to see how a limit cycle (periodic) can be

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induced by a tillage process in which the soil is broken by a simple tine ripper. In fact, all frequency peaks observed so far were accompanied by significant broad band signals. Do the broad band components represent noise or are they caused by deterministic chaos? There are three possible explanations for the phenomena, namely a noise-induced limit cycle, stochastic motion (1/f) α or colored noise, and deterministic chaos. In this paper, we analyzed the soil failure patterns obtained under four different soil conditions and identify their nonlinear nature by nonlinear time series analysis.

Material and Method Field tests were conducted on a Yolo loam field located near the Western Center for Agricultural Equipment, University of California at Davis. The experiment followed a split-plot design where the main treatments were tilled and undisturbed, and the sub-treatments were irrigated/wet and dry soil. The tilled treatment consisted of ripping the soil to a depth of 60 cm and subsequently disking the surface to a depth of 15 cm. In the undisturbed treatment, the land laid fallow during the previous fall and winter seasons. The irrigated treatment consisted of 30 hours of sprinkler irrigation until the soil was saturated to a depth of 40 cm. In each subplot an instrumented tine (Texture Soil Compaction Index [TCI] sensor, developed at UC Davis; Liu et al., 1996) was used to obtain soil cutting force data at two different speeds – 0.22 m/s and 1.1 m/s. The experiment was replicated in four blocks. TD, TW,UD and UW are corresponding to tillage dry, tillage wet ,untilled dry and un-tilled wet, respectively. At the time of field tests, cone index measurements (ASAE, 1992) were obtained using a standard tractormounted, hydraulically driven, self-recording cone penetrometer that was capable of obtaining cone index data to a depth of 53 cm. Moreover, soil moisture contents and density profiles from 5 to 60 cm were obtained in increments of 5 cm depth using a strata gauge.

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0 10 20 30 40 50 0 10 20 30 40 50 The general trends of the soil cutting forces under the four different soil conditions are shown in Fig. 3 in the m cycles/m in the tilled dry condition time and the spatial frequency domains. Clear peaks displacement were observed at, 0.9 for fast and slow speed, respectively. The period looking motions were also observed in time domain. In the other three soil conditions, it was difficult to identify periodic motions as TD. General trends of soil cutting forces on 4 levels 2014 ASABE – CSBE/SCGAB Annual International Meeting Paper

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Fig. 2. Power spectrum As the soil body is structured by a large number of soil particles, it might be natural to assume that the soil failure process is stochastic, and to expect a (1/f) -type of motions. However, we found cyclic patterns in the time series associated with similar spatial patterns of crack development on the soil surface. This apparent periodicity could represent a limit cycle or deterministic chaos. Additionally, (1/f)  motions are common in stochastic processes. Nonlinear Time Series Analysis Deterministic chaos is defined as a random-looking motion generated by a deterministic process not by a stochastic process. Nonlinear time series analysis has been developed to distinguish deterministic chaos from stochastic motion (Takens, F 1981, Grassberger &Procaccia,1983, Thompson J.M.T. & Stewart, H.B.,1986, Moon F,1987,Abarabanel et.al,1993,Sakai, 2001). The main features of deterministic processes are 1) the existence of deterministic dynamics, 2) orbital instability, and 3) the fractal nature of their reconstructed attractors. According to the conventional nonlinear time series analysis scheme, we reconstructed the possible dynamics from observed soil cutting forces {xt}, with a time-delayed embedding technique. The possible phase trajectory is described in the reconstructed phase space (Packard et.al,1980) by

X(t)={x(t),x(t-), x(t-2),……., x(t-(m-1))}

(1)

where  is the time lag and m is the embedding dimension. The main features of deterministic chaos such as the underling deterministic dynamics and orbital instability can be characterized by nonlinear deterministic prediction and Lyapunov exponents(Grassberger&Procaccia,1983,Sano&Sawada, 1985, Wolf et. al,1985,Eckmann&Ruelle,1992). However, it is known that these conventional tools sometimes misidentify colored noise and 1/f noise as deterministic chaos. We demonstrate possible ways to overcome this difficulty by employing transition error (Wayland et.al,1993), the ratio of nonlinear deterministic prediction to auto-correlation function, and inner products of nearest neighbors.

Results and Discussion Nonlinear Deterministic Prediction Nonlinear deterministic prediction is a useful tool for the identification of deterministic chaos(Lorenz,1969, Farmer &Sidorowich, 1987, Sugihara&May,1990,Sakai&Aihara,1994,Sakai &Aihara,1999,Sakai et.al, 2008). As the existence of dynamics, the state dynamics should be predicted in the short term, as sufficiently reliable predictions are unlikely for the long term because of orbital instability. Using the characteristics of the two main features (determinism and orbital instability), nonlinear deterministic prediction can distinguish deterministic from stochastic processes. Here we employed the method of analogues proposed by Lorenz 2014 ASABE – CSBE/SCGAB Annual International Meeting Paper

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(1969). Let X(t) be the predictee to find the M of nearest neighbors  of X(t) denoted y(t,0),y(t,1),…,y(t,M) in the deconstructed phase space. The p-step forward prediction, X (t  p) , is determined as

Correlation coefficient

 1 X (t  p)  X (t )  M

M

 ( y(t  p, i)  X (t ))

(2)

i i

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Limit cycle

Limit cycle + noise Deterministic Chaos

White Noise

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Fig. 3 Concept of deterministic nonlinear prediction (DNP)

To evaluate the prediction performance, we used correlation coefficients (CC) and relative root mean squares of errors (RRMSE) as functions of prediction time step p: n

CC ( p) 

 ( X (t  p) 

  X (t  p) )( X (t  p)  X (t  p) )

t 1

n

 ( X (t  p) 

X (t  p) )

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 ( X (t  p)  X (t  p)) t 1 n

 ( X (t  p) 

(3)

   ( X (t  p)  X (t  p) ) 2 n

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(4)

X (t  p) )

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Deterministic nonlinear prediction was conducted on the reconstructed phase trajectory. The time lag is given as about 1/4 of the period of the predominant component in the power spectrum. In this nonlinear deterministic prediction, all data exhibited sensitivity to the initial conditions with declining C.C. and increasing RRMSE. UW showed the best performance of prediction. However, as shown in Figure 6, it cannot necessarily be concluded that the prediction in UW is successful. Figure 6 compares predictions,

 X (t  p) , with actual data, X (t  p) , at p = 5 for UW and TD. For UW, a pronounced time lag of about 5 is  found between prediction and actual value. Therefore the prediction X (t  p) is closer to X (t ) than to X (t  p) . This phenomenon can be recognized as the deference vectors v(t , i)  y(t  p, i)  X (t ), i  1,2,..., M (5)

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are randomly distributed so that the 2nd term of equation (2),

1 M

M

 ( y(t  p, i)  X (t )) , approaches zero. i i

This suggests that the prediction for UW is failed. In contrast for TD, the prediction

 X (t  p) is closer to

X (t  p) than to X (t ) . Additionally, as shown in Figure 6, the autocorrelation function of UW shows the highest correlation coefficient in the short time step range because of its coherent trend. This is the mechanism underlying the high prediction index (high correlation coefficient and low RRMSE) for UW in Figures 4 and 5. Based on these results, we propose the NDP/AFC ratio as a measure of the success of the prediction. It should be stressed that the NDP/AFC ratio is >1 only for TD (Fig. 7), suggesting that NDP/AFC can be employed to distinguish deterministic chaos from colored noise.

Fig. 4 Nonlinear deterministic prediction for correlation dimension and RRMSE (Fast)

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Fig.5 5 step forward prediction for TD and UW


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Fig. 7 NDP/ACF ratio vs prediction time step p Two Measures of Determinism Transition Error Wayland (1993) proposed a method to measure determinism in time series based on the Kaplan-Glass method. If the data is deterministic, we can assume the parallelness of a certain vector field and the M of translation vectors, v(t,i)=y(t,i)-x(t,i), i=1,2,…,M, to be very close to each other. The translation error is defined as

etrans(t ,1)  where

1 M v(t , j )  v  k  1 i1 v

(5)

v is the average of the translation vectors v(t , j ) .

Etrans is the median of

etrans(t ,1) , a measure of the translation error of the time series.

Here, we expand this translation error to include evolution time step p, so that Etrans(p) is the median of etrans(t , p) :

etrans(t , p) 

1 M v(t , i)  v  k  1 i1 v


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Evolution time step p Fig. 8 Translation error

Etrans( p) for four soil conditions

For reference, we calculated

Etrans( p) on Lorenz63 (time series data derived from Lorenz model which is

Etrans(1) were 0.002 and 0.844, respectively. At the prediction time step p =1, all translation errors, Etrans( p) , are larger than 1.0, which is larger than expected for white noise. With increasing p, Etrans( p) of TD, TW, UD and UW approach 0.0479, 0.1299, typical example of deterministic chaos) white noise, and their

0.4412, and 0.2764, respectively. In logarithmic expression, TD is smaller more than 1 decade of TW. The procedure enable to reduce the effect of induce noise and manifested flow of the dynamics at TD. Inner Products Based on the concept of translation errors, we tried to use inner products of translation vectors v(t , j ) to quantify parallelness. IP1 is defined as the averaged inner product of M translation vectors and their mean. IP2 is defined as the averaged inner product of M translation vectors and the translation vector of the predicted X (t ) . As with the translation error, IP1 and IP2 are implemented as functions of prediction time step p:

IP1(t , p) 

1 M 1 M   v ( t , i ) , v , IP 2 ( t , p )    v(t, i), X(t  p) - X(t) k  1 i 1 k  1 i1

TD

(7)

TW

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UD

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q

Fig. 9 Testing determinism with translation error

etrans(t , p) and two inner products, IP1(t , p) and

IP2(t , p) , at the low speed condition. The prediction time step p is defined as 2q-1.


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Conclusion Soil failure patterns were investigated by nonlinear time series analysis. The possible dynamics was reconstructed with the time delay embedding technique. Deterministic nonlinear prediction (DNP) was applied to cutting force data (two velocities; 1.1 and 0.22 m/s) from soil in four different soil conditions. The DNP/ACF ratio appears to be useful for the distinction between deterministic and stochastic processes. Translation errors and inner products (IP1 and IP2) were defined as functions of prediction time step p. As the effects of induced noise on the dynamics were eliminated by increasing p, the procedure enables the detection of determinism underlying soil failure patterns.

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