- Email: [email protected]
Scaling properties in temporal patterns of schizophrenia
PHYSICA ELSEVIER
Physica A 230 (1996) 544-553
Scaling properties in temporal patterns of schizophrenia R.M. Diinki a,,, B. Ambiihl b a Computer Assisted Physics, University of Ziirich, Winterthurerstrasse 190, CH-8057 Ziirich, Switzerland bpsychiatric University Hospital Bern, Switzerland and Cantonal Psychiatric Hospital, Rheinau (ZH) Received 16 January 1996
Abstract
Investigations into the patterns of schizophrenia reveal evidence of scaling properties in temporal behaviour. This is shown in the spectral properties of mid-range and long-range (up to two years) daily recordings from a sample of patients drawn at the therapeutic dwelling SOTERIA (Amb/ihl et al., in: Springer Series in Synergetics, Vol. 58, eds. Tschacher et al. (Springer, Berlin, 1992) pp. 195-203 and references therein) of the Psychiatric University Hospital in Bern. The therapeutic setting is unique in that it tries to avoid treatment by medication. Power law behaviour has been found within fractal walk analysis and Fourier spectra for the daily fluctuations. A simple dynamic principle, based on a generic intermittency model, is put in relation to these time series thus predicting an additional scaling law for the distribution P(T) of time spans T between successive hospitalizations. Testing this hypothesis with our data shows only insignificant deviations. A possible role of this dynamic principle in the risk assignment of psychotic phases is explored with the help of an example.
Keywords: Scale-invariance; Schizophrenia; Intermittency; Temporal patterns; Power-laws
1. Introduction
Since the pioneering work of Mandelbrot [ 1] it has become a tradition to search for power laws in patterns of temporally fluctuating systems. This behaviour can have its origin in both deterministic laws or in stochastic dynamics [ 3 - 8 ] . This search spread out across the disciplines outside the physical sciences eventually reaching systems with increasing complexity. This is reflected within the biological and medical sciences [ 9 - 1 3 ] (see also [14] and references therein). Hence it is natural to continue the search for power-law phenomena even on higher levels of organization, e.g. within patterns * Corresponding author. Fax: ( + + ) 1 257 57 04; e-mail: [email protected]. 0378-4371/96/$15.00 Copyright @ 1996 Elsevier Science B.V. All rights reserved PH S03 78-43 71 ( 9 6 ) 0 0 0 9 7 - 0
R.M. DiinkL B. Ambiihl/Physica A 230 (1996) 544~53
545
of human behaviour [15, 16]. The variety of behavioural patterns of humans becomes especially broad when mental diseases are included. Among those, schizophrenia is important because of both the richness of its symptoms and the seemingly unordered temporal structure of the disorder [17]. In light of this fact, the existence of scaling laws within temporal patterns of schizophrenia would contradict the view of a disorder with no temporal structure. In addition, this could add to the solution of the problem of risk-assessment. If a deterministic origin can be ascribed to such behaviour, one can try to make forecasts through knowledge of a model, or at least a dynamic principle. An important class of such psychiatric models are based on threshold values [ 18]. Because of the problem of interpreting parameters and variables in direct psychological terms, attempts will be basically made on two basic requirements: (1) the model should have as few free parameters as possible; (2) the temporal phenomenology itself should allow to deduce as many of these parameters as possible. In view of these considerations, we search for scaling laws and explore the possibility of risk-assessment through a generic intermittency model. The rest of this paper is organized as follows: we present first an overview of our sample and calculations. Then we show the results from the analysis of daily fluctuations. In the third section a generic model is selected which allows for a link between daily fluctuations and time sequences between hospitalisations. This model consists essentially of a slowly evolving amplitude. This smooth behaviour becomes eventually disrupted by a burst when the amplitude passes a threshold value. The time interval between the bursts is referred to as laminar phase. The distribution of the laminar phase lengths exhibits power-law behaviour and is thus highly skewed. This model's suitability to deduce an approximate risk function for the next psychotic phase is explored. We then finish with our final section presenting our conclusions.
2. Data collection and analysis Investigated cases were of first-admitted schizophrenic patients from SOTERIA, a therapeutic group dwelling near Bern. It is a special feature of the therapeutic concept that one tries to avoid medication. If medication is applied, it is given in low dosages only. Staff members evaluate the fluctuations of psychotic symptoms on a daily basis with a seven item rating scale [2], measuring progressive lessening of reality [19]. The items are 1: relaxed and balanced; 2: unquiet, anxious, tense; 3: strong withdrawal or aggressive; 4: confusion, disorientation; 5: derealization, depersonalization; 6: delusions; 7: hallucinations, catatonic behaviour. Ratings higher than three are regarded to correspond to psychotic states, ratings not exceeding three are regarded to correspond to nonpsychotic states. The time series of 10 patients were judged to be long enough for analysis. These patients remained hospitalized between 4 mon and 2 yr. The corresponding time series have been analyzed by means of Fourier analysis and with M ( z ) , the mean distance a walker spanned within time r. Fourier analysis was performed with a method based on a technique described by Welch [20] to estimate
R.M. Diinki, B. Ambiihl/Physica A 230 (1996) 544-553
546
the power spectrum S(f). A Hamming window [21] with a length of 128 samples has been applied if not stated otherwise. This length was mainly determined through the length limits of the time series. Denoting the rating of day k as X(k) we define t=i
W[i] : : ~-~X(t)- X
(1)
t=0
from which we get the walks M ( r ) :---- (11w(i) -
w(i + z)l]).
In addition, we obtained data on the time intervals Ti, i = 1. . . . . 204, between successive hospitalisations, originating from a rehabilitation study [22]. No one of these 204 time intervals was exceeding 5000 days. They built the basis for time interval binning. A time interval T,. was collected into bin l if: T t-I ~
3. Analysis of daily fluctuations The daily recordings of our sample have been analyzed by means of power spectra
S(f) and walks M(z). Power spectra revealed almost linear behaviour in double logarithmic representation as shown in Fig. 1. Hence S(f) follows approximately f - ~ for a large range of frequencies. In fact, 94% of the variance in the log[f]-log[S(f)] plot are explained by the linear fit returning ~ = 0.87. The original time series is shown in Fig. 4. For the sample as a whole, we have found more or less agreement with this behaviour, though there are individual differences in both exponent and explained variance. This becomes evident when analyzing the whole sample: (~ = 0.92). The overall fit was found to explain 47% of the variance in l o g [ S ( f ) ] when we took the spectra
547
R.M. Diinki, B. Ambiihl/Physica A 230 (1996) 544-553 Power Spectrum of Daily Maximum Fluctuations of Patient A i
i
i
i
i
1.=~ rl+
+
++
0.5
+
+
-0.5 -h+,p
-3
-2'5
-1'5
log[f]
-0'5
Fig. 1. Spectral properties of a single patient (Patient A). Best fit was found with c< = 0.87. A Hamming window with length of 256 samples was applied. Base 10 logarithms are used. of all 10 time series and did one simultaneous fit to these 10 spectra. However, the explained variance increased to 88% when we first built the spectra's mean (log[S(f)]) and then fitted to the mean (not shown). Thus a major source of the total variance comes from interindividual differences. Fitting individually to each spectrum confirms this: The mean of the percentage of explained variance increases to 64%. A similar picture is found when regarding the walks instead of the power spectra: A scaling exponent /~ could be ascribed to these walks i.e. M ( r ) ~ r -/~, as is seen again for patient A (/~ = 0.79). We mention in particular that the scaling behaviour for this patient extends down to a few days, i.e. to the order of the minimal time scale available for these time series (Fig. (2(a)). Interindividual differences become evident again when studying the walk of the sample (~ = 0.89): The scaling region shrinks, but does not drop below 1 decade for the evolution of the mean. As indicated in Fig. 2(b), individual differences are responsible for this phenomenon: Not all patients display a scaling region as broad as patient A. Psychological scales cannot be treated in a strict sense as amplitudes of an intervalscaled variable. We therefore applied consistency checks, to check for spurious results due to amplitude spacing, by repeating the analysis with rescaled amplitudes. Two of the most frequent distortion effects of psychological scales are (a) overestimation of the spacing among the low items and (b) overestimation of the spacing among
R.M. DfinkL B. Ambiihl/Physica ,4 230 (1996) 544-553
548
the high items [23]. In view of this, we repeated the analysis taking the following transformations into account: (a) the logarithm of the ratings and (b) the exponential function of the ratings. The results remained very similar to the original ones. This contradicts the assumption that the chosen psychological scale exhibits distortion effects. We also checked for spurious results due to unfortunate effects of time windows involved in the analysis: The three longest series allowing for a window length of 256 days had been split into two halves and each half was then analysed with a window of 128 days. The halves did not behave differently than did the sample. This contradicts the assumption, that time window effects might play an important role.
4. A dynamic principle On the one hand, psychiatrists found that one important way to model the nonperiodic evolution of psychotic states is boolean kinetics [18]. This method assumes psychotic states to depend upon threshold values. On the other hand, mathematicians know about several different dynamical principles all of which leading to power-law 1.8
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,
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1i log(T)
112
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Fig. 2. Evolution of the walk M(z). A Hrlder (or Hurst) type scaling exponent fl can be ascribed to these walks. (a) M(r) for patient A; fl = 0.79. Scaling behaviour is found down to a few days, i.e. to the order of the minimal time scale available for these time series. (b) Walk evolution of the sample; ~ = 0.89. Shown are the mean (x) and 4- standard deviation (o). Base 10 logarithms are used.
549
R.M. Diinki, B. Ambiihl/Physica A 230 (1996) 544 553 .
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log(T) Fig. 2. Continued.
behaviour. Amongst these is also a class of principles according to which threshold values are involved: An ever increasing drift eventually becomes disrupted by passing above a certain threshold value, thus driving the system from one state into another (e.g. from laminar flow into turbulence): a class of intermittent signals. Systems of the type
xt+l = (xt + x 7 + ~) mod 1
(2)
mimic such laminar-to-turbulence behaviour [24] and lead to power laws in an appropriate low frequency range ([5, chap. 4] and references therein). In particular, the number of occurrences of large laminar durations T (i.e. time spans between two disruptions) are distributed as T -~'~z). We attempt to interprete our data as intermittent phenomena. Accordingly, we identify the regions where the daily ratings are high as the chaotic phases and the regions where they are low as the laminar phases. Hence the time intervals between hospitalisations (Ti) are regarded as long laminar phases. Consistent with the intermittency view, we hypothesize a power law within the distribution of the number of occurrences of the value of 7",.. Testing this hypothesis leads indeed to not rejecting the model distribution (Ti -- T e ) - 7 ; with Te = --80 and 7 = 1.21 (Z 2 = 32.5, d f = 24, P(Z 2) > 0.05; see Fig. 3). Here bins with low expectation value have been pooled for analysis. A negative
550
R.M. Dfinki, B. Ambiihl/Physica A 230 (1996) 544-553 35
,
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Fig. 3. Distributionsof time spans Ti. Actual hight (R) and model P(Ti) ~ (Ti - Te)-'/ (+), with Te = -80 and 7 = 1.21. Locationof bin is marked with an asterisk. Base 10 logarithms are used. excess time Te is expected because, normally, the duration of hospitalisation exceeds the duration of psychotic phases. It is inherent within this view that a sequence of low ratings (i.e. "laminar phases" ) within the rating series is also a possible sequence of time spans Ti, T/+l .... of an individual, if properly rescaled. In view of the intermittency approach we explored the extent to which (2) could serve as a basis for guessing the probability of a future psychotic phase to appear within a given time from the present (a kind of risk-analysis). Though the modulo operation in (2) prevents long range predictions of laminar phases, forecasting of the next following psychotic phase could still be allowed. We thereby followed a Monte Carlo scheme: (a) Select an appropriate level to dichotomize the time series; (b) Split the time series into a learn and a prediction region; (c) Generate a random population of initial parameter estimates for a model fit; (d) Define an appropriate measure monitoring the goodness of fit; (e) Fit the dichotomized time series within the learn region; (f) Upon acceptance of the fit, extend the simulation to the prediction region until the simulation exhibits the end of the laminar phase. This is the number of iterations needed to reach the chaotic phase; (g) Repeat (e) and (f) for all initial parameter estimates generated in (c). (h) Collect the ends of the first simulated laminar phase in the prediction region to get its statistics (as described in Section 2). This statistics allows for guessing a probability function for the end of the laminar phase.
551
R.M. Diinki, B. AmbiihlIPhysica A 230 (1996)544-553
'
tI
'I','
'
._1 O'J
0
100
200
300
400
500 Day
600
700
800
900
1000
Fig. 4. Attempt to predict patient A; Upper part: Within hospitalisation prognosis. Probability of the "laminar" phase to end before the indicated day: +: 2%; *: 15%; ®: 50%; o: 85%; x: 98 %. Lower part: Between hospitalisation prognosis; hospitalisation times (high) and estimated 50% risk of the end of the "laminar" phase.
Attempting to predict patient A, we used the following setting: Level for dichotomization coincides with turning point psychotic/nonpsychotic. We define the fit quality to consist of the contribution of two terms. One term is the squared difference between fit and time series as usual. The second term is introduced, because if an individual match is not found we are interested rather in regions behaving similarly. Accordingly, a sliding window technique was applied. The average of that part of the time series within the window was built for both, the fit and the simulated time series. The squared difference between fit and time series of this sliding average is included as the second term into the goodness-of-fit criterion. Fig. 4 shows an application onto the time series of patient A, where we selected the learning region to be day 1 to 300 (upper part) and day 1 to 723 when the patient left the hospital, respectively (whole curve, lower part). For the former, the "laminar phase" ended at day 346 (appearance of new bursts). Our guess of the risk entering into the psychotic state assigned the 50% probability close to this day and day 346 is well within the inner 70% interval (i.e. from the estimated 15% to the estimated 85% probability.). This region extends from day 327 to day 440. For the second time series, the laminar phase ended at day 988 (new hospitalisation). This can again be compared
552
R.M. Diinki, B. Ambiihl/Physica A 230 (1996) 544-553
with the date of the estimated 50% probability. We found this to be around day 850. Poor quality fitting problems within the parameter optimization procedure did not allow for a reasonable estimation of the inner 70% interval. But the available results suggest this quantity to be larger than in the former case, thus still leading to an acceptable guess. Though the shape of the risk probability can only be estimated approximately this way, the results are promising and suggest a way for further testing the hypothesis: one needs (1) faster and more accurate methods and (2) a population of probands suitable for this kind of analysis. This will enable us to analyse the outcomes with conventional statistic methods, e.g. by means of ~2 analysis between prediction and actual occurrences of psychotic symptoms. From the time scales involved in the evolution of these symptoms, it becomes obvious that requirement (2) remains a task on the long run.
5. Conclusion We have analysed the time evolution of schizophrenia of two different samples. The one based on daily ratings, the other from time spans between successive hospitalisations. Within both samples we found power-laws to describe various distributions. We propose an interpretation of these laws in terms of an intermittency model, interrelating the data from the two samples to each other. Using such a model for risk assessment is encouraging, but further methodological and clinical work is required to test the hypothesis stated in here, namely, that certain subtypes of schizophrenia might manifest intermittent behaviour thus implying a power-law behaviour inherent to time evolution of these subtypes.
Acknowledgements We thank Prof. L. Ciompi, University of Bem, for stimulating suggestions on the ideas expressed within this paper. The project was supported by the Swiss National Science Foundation.
References [1] B.B. Mandelbrot, The Fractal Geometry of Nature (W.H. Freeman, San Francisco, 1982). [2] B. Ambiihl, R.M. D/inki and L. Ciompi, Dynamical Systems and the Development of Schizophrenic Symptoms - an approach to a formalization, in: Self-Organization and Clinical Psychology, Vol. 58, eds. Tschacher, Schiepek and Brunner, Springer Series in Synergetics (Springer, Berlin, 1992) pp. 195-203. [3] I. Procaccia, and H.G. Schuster, Functional Renormalization-group theory of universal l/f noise in dynamical systems, Phys. Rev. A28 (1983) 1210-1212. [4] M. Warden, Deterministic Origin of 1If noise in magnetic resonance, Phys. Rev. E48 (1993) R639-R642. [5] H.G. Schuster, Deterministic Chaos, 2nd ed. (VCH-Publishers, Weinheim (Germany), 1989). [6] B.B Mandelbrot, Formes nouvelles du hasard dans les sciences, Economie appliqure 26 (1973) 307-319.
R.M. Diinki, B. Ambiihl/Physica A 230 (1996) 544 553
553
[7] B.B. Mandelbrot and J.R. Wallis, Noah, Joseph and operational hydrology, Water Resources Res. 4 (1968) 909-918. [8] B.B. Mandelbrot and J.R. van Ness, Fractional brownian motions, fractional noises and applications, SIAM Rev. 10 (1968) 422-437. [9] D.R. Morse, Fractal Dimension of Vegetation and the Distribution of arthropod body lengths, Nature 314 (1985) 731-733. [t0] C.-K.Peng, S.V. Buldyrev, S. Goldberger, S. Havlin, F. Sciortino, M. Simons and H.E. Stanley, Longrange correlations in nucleotide sequences, Nature 356 (1992) 168 170. [I 1] R. Voss, Evolution of long range fractal correlations and 1/f noise in DNA base sequences, Phys. Rev. Lett. 68 (1992) 3805-3808. [12] C.-K. Peng, J.E. Mietus, J.M. Hausdorff S. Havlin, H.E. Stanley and A.L. Goldberger Long-range anticorrelations and non-gaussian behaviour of the heartbeat, Phys. Rev. Lett. 70 (1993) 1343-1346. [13] B. Suki, A.-L. Barabasi, Z. Hantos F. Pet~ik, H.E. Stanley, Avalanches and power-law behaviour in lung inflation, Nature 368 (1994) 615 618. [14] A. Bunde and S. Havlin ed., Fractals in Science (Springer, Berlin, 1994). [15] E. Canessa, and A. Calmetta, Physics of a random biological process, Phys. Rev. E50 (1994) R47 R49. [16] K.-D. Kniffki, W. Mandel, P. Tran-Gia, Temporal Fluctuations in Biorythms: Expression of selforganized criticality, Fractals 1 (1993)380-387. [17] L. Ciompi and C. M/iller, Lebensweg und Alter der Schizophrenen. Eine katamnestische Langzeitstudie (Springer, Berlin, 1976). [18] L. Ciompi and B. Amb/ihl and R.M. D/Jnki, Schizophrenie und Chaostheorie, System Familie 5 (1992) 133 147. [19] E. Aebi K. Ackermann and D. Revenstorf, Ein Konzept der sozialen Unterstuetzung fuer akut Schizophrene. Zeitreihenanalysen taeglicher Fluktuationen psychotischer Merkmale, Zeitschr. f. Klin. Psychol. Psychopath. Psychother. 41 (1993) 18-30. [20] P.D. Welch, The use of fast fourier transform for the estimation of power spectra: a method based on time averaging over short modified periodograms, IEEE Trans. Audio Elect. 15 (1967) 70-73. [21] M.B. Priestley, Spectral Analysis and Time Series, 6th ed., (Academic Press, London, 1989). [22] L. Ciompi, personal communication, 1992. [23] J. Angst, and C. Scharfetter and H.H. Stassen, Classification of schizo-affective patients by multidimensional scaling and cluster analysis, Psychiatrica Clinica 16 (1983) 254-264. [24] Y. Pomeau and P. Manneville, Intermittent Transition to Turbulence in Dissipative Dynamical Systems, Commun. Math. Phys. 74 (1980) 189 197.
Physica A 230 (1996) 544-553
Scaling properties in temporal patterns of schizophrenia R.M. Diinki a,,, B. Ambiihl b a Computer Assisted Physics, University of Ziirich, Winterthurerstrasse 190, CH-8057 Ziirich, Switzerland bpsychiatric University Hospital Bern, Switzerland and Cantonal Psychiatric Hospital, Rheinau (ZH) Received 16 January 1996
Abstract
Investigations into the patterns of schizophrenia reveal evidence of scaling properties in temporal behaviour. This is shown in the spectral properties of mid-range and long-range (up to two years) daily recordings from a sample of patients drawn at the therapeutic dwelling SOTERIA (Amb/ihl et al., in: Springer Series in Synergetics, Vol. 58, eds. Tschacher et al. (Springer, Berlin, 1992) pp. 195-203 and references therein) of the Psychiatric University Hospital in Bern. The therapeutic setting is unique in that it tries to avoid treatment by medication. Power law behaviour has been found within fractal walk analysis and Fourier spectra for the daily fluctuations. A simple dynamic principle, based on a generic intermittency model, is put in relation to these time series thus predicting an additional scaling law for the distribution P(T) of time spans T between successive hospitalizations. Testing this hypothesis with our data shows only insignificant deviations. A possible role of this dynamic principle in the risk assignment of psychotic phases is explored with the help of an example.
Keywords: Scale-invariance; Schizophrenia; Intermittency; Temporal patterns; Power-laws
1. Introduction
Since the pioneering work of Mandelbrot [ 1] it has become a tradition to search for power laws in patterns of temporally fluctuating systems. This behaviour can have its origin in both deterministic laws or in stochastic dynamics [ 3 - 8 ] . This search spread out across the disciplines outside the physical sciences eventually reaching systems with increasing complexity. This is reflected within the biological and medical sciences [ 9 - 1 3 ] (see also [14] and references therein). Hence it is natural to continue the search for power-law phenomena even on higher levels of organization, e.g. within patterns * Corresponding author. Fax: ( + + ) 1 257 57 04; e-mail: [email protected]. 0378-4371/96/$15.00 Copyright @ 1996 Elsevier Science B.V. All rights reserved PH S03 78-43 71 ( 9 6 ) 0 0 0 9 7 - 0
R.M. DiinkL B. Ambiihl/Physica A 230 (1996) 544~53
545
of human behaviour [15, 16]. The variety of behavioural patterns of humans becomes especially broad when mental diseases are included. Among those, schizophrenia is important because of both the richness of its symptoms and the seemingly unordered temporal structure of the disorder [17]. In light of this fact, the existence of scaling laws within temporal patterns of schizophrenia would contradict the view of a disorder with no temporal structure. In addition, this could add to the solution of the problem of risk-assessment. If a deterministic origin can be ascribed to such behaviour, one can try to make forecasts through knowledge of a model, or at least a dynamic principle. An important class of such psychiatric models are based on threshold values [ 18]. Because of the problem of interpreting parameters and variables in direct psychological terms, attempts will be basically made on two basic requirements: (1) the model should have as few free parameters as possible; (2) the temporal phenomenology itself should allow to deduce as many of these parameters as possible. In view of these considerations, we search for scaling laws and explore the possibility of risk-assessment through a generic intermittency model. The rest of this paper is organized as follows: we present first an overview of our sample and calculations. Then we show the results from the analysis of daily fluctuations. In the third section a generic model is selected which allows for a link between daily fluctuations and time sequences between hospitalisations. This model consists essentially of a slowly evolving amplitude. This smooth behaviour becomes eventually disrupted by a burst when the amplitude passes a threshold value. The time interval between the bursts is referred to as laminar phase. The distribution of the laminar phase lengths exhibits power-law behaviour and is thus highly skewed. This model's suitability to deduce an approximate risk function for the next psychotic phase is explored. We then finish with our final section presenting our conclusions.
2. Data collection and analysis Investigated cases were of first-admitted schizophrenic patients from SOTERIA, a therapeutic group dwelling near Bern. It is a special feature of the therapeutic concept that one tries to avoid medication. If medication is applied, it is given in low dosages only. Staff members evaluate the fluctuations of psychotic symptoms on a daily basis with a seven item rating scale [2], measuring progressive lessening of reality [19]. The items are 1: relaxed and balanced; 2: unquiet, anxious, tense; 3: strong withdrawal or aggressive; 4: confusion, disorientation; 5: derealization, depersonalization; 6: delusions; 7: hallucinations, catatonic behaviour. Ratings higher than three are regarded to correspond to psychotic states, ratings not exceeding three are regarded to correspond to nonpsychotic states. The time series of 10 patients were judged to be long enough for analysis. These patients remained hospitalized between 4 mon and 2 yr. The corresponding time series have been analyzed by means of Fourier analysis and with M ( z ) , the mean distance a walker spanned within time r. Fourier analysis was performed with a method based on a technique described by Welch [20] to estimate
R.M. Diinki, B. Ambiihl/Physica A 230 (1996) 544-553
546
the power spectrum S(f). A Hamming window [21] with a length of 128 samples has been applied if not stated otherwise. This length was mainly determined through the length limits of the time series. Denoting the rating of day k as X(k) we define t=i
W[i] : : ~-~X(t)- X
(1)
t=0
from which we get the walks M ( r ) :---- (11w(i) -
w(i + z)l]).
In addition, we obtained data on the time intervals Ti, i = 1. . . . . 204, between successive hospitalisations, originating from a rehabilitation study [22]. No one of these 204 time intervals was exceeding 5000 days. They built the basis for time interval binning. A time interval T,. was collected into bin l if: T t-I ~
3. Analysis of daily fluctuations The daily recordings of our sample have been analyzed by means of power spectra
S(f) and walks M(z). Power spectra revealed almost linear behaviour in double logarithmic representation as shown in Fig. 1. Hence S(f) follows approximately f - ~ for a large range of frequencies. In fact, 94% of the variance in the log[f]-log[S(f)] plot are explained by the linear fit returning ~ = 0.87. The original time series is shown in Fig. 4. For the sample as a whole, we have found more or less agreement with this behaviour, though there are individual differences in both exponent and explained variance. This becomes evident when analyzing the whole sample: (~ = 0.92). The overall fit was found to explain 47% of the variance in l o g [ S ( f ) ] when we took the spectra
547
R.M. Diinki, B. Ambiihl/Physica A 230 (1996) 544-553 Power Spectrum of Daily Maximum Fluctuations of Patient A i
i
i
i
i
1.=~ rl+
+
++
0.5
+
+
-0.5 -h+,p
-3
-2'5
-1'5
log[f]
-0'5
Fig. 1. Spectral properties of a single patient (Patient A). Best fit was found with c< = 0.87. A Hamming window with length of 256 samples was applied. Base 10 logarithms are used. of all 10 time series and did one simultaneous fit to these 10 spectra. However, the explained variance increased to 88% when we first built the spectra's mean (log[S(f)]) and then fitted to the mean (not shown). Thus a major source of the total variance comes from interindividual differences. Fitting individually to each spectrum confirms this: The mean of the percentage of explained variance increases to 64%. A similar picture is found when regarding the walks instead of the power spectra: A scaling exponent /~ could be ascribed to these walks i.e. M ( r ) ~ r -/~, as is seen again for patient A (/~ = 0.79). We mention in particular that the scaling behaviour for this patient extends down to a few days, i.e. to the order of the minimal time scale available for these time series (Fig. (2(a)). Interindividual differences become evident again when studying the walk of the sample (~ = 0.89): The scaling region shrinks, but does not drop below 1 decade for the evolution of the mean. As indicated in Fig. 2(b), individual differences are responsible for this phenomenon: Not all patients display a scaling region as broad as patient A. Psychological scales cannot be treated in a strict sense as amplitudes of an intervalscaled variable. We therefore applied consistency checks, to check for spurious results due to amplitude spacing, by repeating the analysis with rescaled amplitudes. Two of the most frequent distortion effects of psychological scales are (a) overestimation of the spacing among the low items and (b) overestimation of the spacing among
R.M. DfinkL B. Ambiihl/Physica ,4 230 (1996) 544-553
548
the high items [23]. In view of this, we repeated the analysis taking the following transformations into account: (a) the logarithm of the ratings and (b) the exponential function of the ratings. The results remained very similar to the original ones. This contradicts the assumption that the chosen psychological scale exhibits distortion effects. We also checked for spurious results due to unfortunate effects of time windows involved in the analysis: The three longest series allowing for a window length of 256 days had been split into two halves and each half was then analysed with a window of 128 days. The halves did not behave differently than did the sample. This contradicts the assumption, that time window effects might play an important role.
4. A dynamic principle On the one hand, psychiatrists found that one important way to model the nonperiodic evolution of psychotic states is boolean kinetics [18]. This method assumes psychotic states to depend upon threshold values. On the other hand, mathematicians know about several different dynamical principles all of which leading to power-law 1.8
,
,
,
,
,
,
,
,
,
1.6 1.4 1.2 1 . ~ 0.8 0.6 0.4 0.2 0 -0.~
(a)
0
012
014
016
018
1i log(T)
112
114
116
1.18
2
Fig. 2. Evolution of the walk M(z). A Hrlder (or Hurst) type scaling exponent fl can be ascribed to these walks. (a) M(r) for patient A; fl = 0.79. Scaling behaviour is found down to a few days, i.e. to the order of the minimal time scale available for these time series. (b) Walk evolution of the sample; ~ = 0.89. Shown are the mean (x) and 4- standard deviation (o). Base 10 logarithms are used.
549
R.M. Diinki, B. Ambiihl/Physica A 230 (1996) 544 553 .
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behaviour. Amongst these is also a class of principles according to which threshold values are involved: An ever increasing drift eventually becomes disrupted by passing above a certain threshold value, thus driving the system from one state into another (e.g. from laminar flow into turbulence): a class of intermittent signals. Systems of the type
xt+l = (xt + x 7 + ~) mod 1
(2)
mimic such laminar-to-turbulence behaviour [24] and lead to power laws in an appropriate low frequency range ([5, chap. 4] and references therein). In particular, the number of occurrences of large laminar durations T (i.e. time spans between two disruptions) are distributed as T -~'~z). We attempt to interprete our data as intermittent phenomena. Accordingly, we identify the regions where the daily ratings are high as the chaotic phases and the regions where they are low as the laminar phases. Hence the time intervals between hospitalisations (Ti) are regarded as long laminar phases. Consistent with the intermittency view, we hypothesize a power law within the distribution of the number of occurrences of the value of 7",.. Testing this hypothesis leads indeed to not rejecting the model distribution (Ti -- T e ) - 7 ; with Te = --80 and 7 = 1.21 (Z 2 = 32.5, d f = 24, P(Z 2) > 0.05; see Fig. 3). Here bins with low expectation value have been pooled for analysis. A negative
550
R.M. Dfinki, B. Ambiihl/Physica A 230 (1996) 544-553 35
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Fig. 3. Distributionsof time spans Ti. Actual hight (R) and model P(Ti) ~ (Ti - Te)-'/ (+), with Te = -80 and 7 = 1.21. Locationof bin is marked with an asterisk. Base 10 logarithms are used. excess time Te is expected because, normally, the duration of hospitalisation exceeds the duration of psychotic phases. It is inherent within this view that a sequence of low ratings (i.e. "laminar phases" ) within the rating series is also a possible sequence of time spans Ti, T/+l .... of an individual, if properly rescaled. In view of the intermittency approach we explored the extent to which (2) could serve as a basis for guessing the probability of a future psychotic phase to appear within a given time from the present (a kind of risk-analysis). Though the modulo operation in (2) prevents long range predictions of laminar phases, forecasting of the next following psychotic phase could still be allowed. We thereby followed a Monte Carlo scheme: (a) Select an appropriate level to dichotomize the time series; (b) Split the time series into a learn and a prediction region; (c) Generate a random population of initial parameter estimates for a model fit; (d) Define an appropriate measure monitoring the goodness of fit; (e) Fit the dichotomized time series within the learn region; (f) Upon acceptance of the fit, extend the simulation to the prediction region until the simulation exhibits the end of the laminar phase. This is the number of iterations needed to reach the chaotic phase; (g) Repeat (e) and (f) for all initial parameter estimates generated in (c). (h) Collect the ends of the first simulated laminar phase in the prediction region to get its statistics (as described in Section 2). This statistics allows for guessing a probability function for the end of the laminar phase.
551
R.M. Diinki, B. AmbiihlIPhysica A 230 (1996)544-553
'
tI
'I','
'
._1 O'J
0
100
200
300
400
500 Day
600
700
800
900
1000
Fig. 4. Attempt to predict patient A; Upper part: Within hospitalisation prognosis. Probability of the "laminar" phase to end before the indicated day: +: 2%; *: 15%; ®: 50%; o: 85%; x: 98 %. Lower part: Between hospitalisation prognosis; hospitalisation times (high) and estimated 50% risk of the end of the "laminar" phase.
Attempting to predict patient A, we used the following setting: Level for dichotomization coincides with turning point psychotic/nonpsychotic. We define the fit quality to consist of the contribution of two terms. One term is the squared difference between fit and time series as usual. The second term is introduced, because if an individual match is not found we are interested rather in regions behaving similarly. Accordingly, a sliding window technique was applied. The average of that part of the time series within the window was built for both, the fit and the simulated time series. The squared difference between fit and time series of this sliding average is included as the second term into the goodness-of-fit criterion. Fig. 4 shows an application onto the time series of patient A, where we selected the learning region to be day 1 to 300 (upper part) and day 1 to 723 when the patient left the hospital, respectively (whole curve, lower part). For the former, the "laminar phase" ended at day 346 (appearance of new bursts). Our guess of the risk entering into the psychotic state assigned the 50% probability close to this day and day 346 is well within the inner 70% interval (i.e. from the estimated 15% to the estimated 85% probability.). This region extends from day 327 to day 440. For the second time series, the laminar phase ended at day 988 (new hospitalisation). This can again be compared
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with the date of the estimated 50% probability. We found this to be around day 850. Poor quality fitting problems within the parameter optimization procedure did not allow for a reasonable estimation of the inner 70% interval. But the available results suggest this quantity to be larger than in the former case, thus still leading to an acceptable guess. Though the shape of the risk probability can only be estimated approximately this way, the results are promising and suggest a way for further testing the hypothesis: one needs (1) faster and more accurate methods and (2) a population of probands suitable for this kind of analysis. This will enable us to analyse the outcomes with conventional statistic methods, e.g. by means of ~2 analysis between prediction and actual occurrences of psychotic symptoms. From the time scales involved in the evolution of these symptoms, it becomes obvious that requirement (2) remains a task on the long run.
5. Conclusion We have analysed the time evolution of schizophrenia of two different samples. The one based on daily ratings, the other from time spans between successive hospitalisations. Within both samples we found power-laws to describe various distributions. We propose an interpretation of these laws in terms of an intermittency model, interrelating the data from the two samples to each other. Using such a model for risk assessment is encouraging, but further methodological and clinical work is required to test the hypothesis stated in here, namely, that certain subtypes of schizophrenia might manifest intermittent behaviour thus implying a power-law behaviour inherent to time evolution of these subtypes.
Acknowledgements We thank Prof. L. Ciompi, University of Bem, for stimulating suggestions on the ideas expressed within this paper. The project was supported by the Swiss National Science Foundation.
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