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Non-parametric estimation of a Langevin model driven by correlated noise

  • Regular Article - Statistical and Nonlinear Physics
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Abstract

Langevin models are widely used to model various stochastic processes in different fields of natural and social sciences. They are adapted to measured data by estimation techniques such as maximum likelihood estimation, Markov chain Monte Carlo methods, or the non-parametric direct estimation method introduced by Friedrich et al. (Phys Lett A 271(3):217, 2000). The latter has the distinction of being very effective in the context of large data sets. Due to their \(\delta \)-correlated noise, standard Langevin models are limited to Markovian dynamics. A non-Markovian Langevin model can be formulated by introducing a hidden component that realizes correlated noise. For the estimation of such a partially observed diffusion a different version of the direct estimation method was introduced by Lehle et al. (Phys Rev E 97(1):012113, 2018). However, this procedure requests that the correlation length of the noise component is small compared to that of the measured component. In this work, we propose a direct estimation method without this restriction. This allows one to effectively deal with large data sets from a wide range of examples. We discuss the abilities of the proposed procedure using several synthetic examples.

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Data Availability Statement

This manuscript has no associated data or the data will not be deposited. [Authors’ comment: The data that support the findings of this study are available from the corresponding author upon reasonable request.]

Notes

  1. In the case that \(B^k\) contains a fixed point, the mean \(\langle Y_t | X_t\in B^k \rangle \) is approximately zero (thanks to an anonymous referee for this suggestion) and consequently \(\langle X_{t+\tau } - X_t | X_t\in B^k \rangle \approx D^{(1)}_x(x^k) = 0\). Thus, all bins \(B^k\) where \(\langle X_{t+\tau } - X_t | X_t\in B^k \rangle \approx 0\) holds are candidates for fixed points. We suppose that this is even an equivalence relation, yet we cannot justify this rigorously. This additional information can be useful for determining the initial guess during optimization (cf. Sect. 3). A single fixed point provides an indication for the parameter \(X^{*}\) of the linear model. If multiple fixed points are to be expected, their location could be a basis for an initial guess of higher order than linear.

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Author information

Authors and Affiliations

Authors

Contributions

CW developed the estimation procedure for the non-Markovian Langevin model and worked out the validation examples. OK supervised the work. CW wrote the manuscript in consultation with OK.

Corresponding author

Correspondence to Clemens Willers.

Appendices

Appendix A: Analytic solution of the linear model

Here, we calculate the stationary solution of the SDE model defined by the equations (\(\alpha ,\beta ,\theta >0\))

$$\begin{aligned} \dot{X}^{(1)}_t&= -\frac{1}{\alpha } X^{(1)}_t + \sqrt{\beta }\, X^{(2)}_t \end{aligned}$$
(A.1a)
$$\begin{aligned} \dot{X}^{(2)}_t&= -\frac{1}{\theta }\,X^{(2)}_t + \sqrt{\frac{1}{\theta }}\,\eta _t \end{aligned}$$
(A.1b)

in an analytic way. In general, the solution of the initial value problem defined by the Langevin equation

$$\begin{aligned} dX_t = -B X_t\, dt + \sigma \, dW_t, \end{aligned}$$
(A.2)

and the initial value \(X_0 = x_0\) is given by [2, 3]

$$\begin{aligned} X_t = e^{-Bt}x_0 + \int _0^t e^{-B(t-t')}\sigma \, dW_{t'}. \end{aligned}$$
(A.3)

In our model, we have \(X_t = (X^{(1)}_t, X^{(2)}_t)^\top \),

$$\begin{aligned} W_t = (W^{(1)}_t, W^{(2)}_t)^\top , B = \left( \begin{array}{cc} \frac{1}{\alpha } &{} -\sqrt{\beta }\\ 0 &{} \frac{1}{\theta } \end{array}\right) , \hbox {and} \ \sigma = \left( \begin{array}{cc} 0 &{} 0\\ 0 &{} \sqrt{\frac{1}{\theta }} \end{array}\right) . \end{aligned}$$

As we are interested in the stationary solution that emerges for large values of t, we end up with \(X_t^{\mathrm{st}} = \int _0^t e^{-B(t-t')}\sigma \, dW_{t'}\) and need to evaluate the matrix exponential. Here, we have to distinguish two cases: \(\alpha \ne \theta \) and \(\alpha =\theta \).

In the case \(\alpha \ne \theta \), we evaluate the matrix exponential via the diagonalization of B,

$$\begin{aligned} B = T^{-1}DT, \end{aligned}$$
(A.4)

where \(T=\left( \begin{array}{cc} 1 &{} \sqrt{\beta }\frac{\alpha \theta }{\alpha -\theta }\\ 0 &{} 1 \end{array} \right) \) and \(D=\left( \begin{array}{cc} \frac{1}{\alpha } &{} 0\\ 0 &{} \frac{1}{\theta } \end{array} \right) \):

$$\begin{aligned} e^{-B(t-t')} = T^{-1}e^{-D(t-t')}T. \end{aligned}$$
(A.5)

We obtain

$$\begin{aligned} X^{\mathrm{st},(1)}_t = \frac{\alpha \sqrt{\beta \theta }}{\alpha - \theta } \int _0^t e^{-\frac{1}{\alpha }(t-t')} - e^{-\frac{1}{\theta }(t-t')} \, dW_{t'}^{(2)} \end{aligned}$$
(A.6)

for the first component of the solution and

$$\begin{aligned} X^{\mathrm{st},(2)}_t = \frac{1}{\sqrt{\theta }} \int _0^t e^{-\frac{1}{\theta }(t-t')} \, dW_{t'}^{(2)} \end{aligned}$$
(A.7)

for the second component.

In the case \(\alpha =\theta \), we calculate the matrix exponential via the Jordan–Chevalley decomposition of B,

$$\begin{aligned} B = A + N, \end{aligned}$$
(A.8)

where \(A=\left( \begin{array}{cc} \frac{1}{\alpha } &{} 0\\ 0 &{} \frac{1}{\alpha } \end{array}\right) \) is diagonal, \(N=\left( \begin{array}{cc} 0 &{} -\sqrt{\beta }\\ 0 &{} 0 \end{array} \right) \) is nilpotent, and \(AN=NA\):

$$\begin{aligned} e^{-B(t-t')} = e^{-A(t-t')} e^{-N(t-t')}. \end{aligned}$$
(A.9)

We obtain

$$\begin{aligned} X^{\mathrm{st},(1)}_t = \sqrt{\frac{\beta }{\alpha }} \int _0^t (t-t') \, e^{-\frac{1}{\alpha }(t-t')}\, dW_{t'} \end{aligned}$$
(A.10)

for the first component of the solution. The second component is the same as in the latter case.

Appendix B: Variance and ACF of the linear model

At first, we calculate the autocovariance function \(\text {Cov}(X_t,X_s)\) for the first component of the stationary solution of the linear model in the case \(\alpha \ne \theta \) (cf. Appendix A; for the sake of readability, we write \(X_t\) instead of \(X_t^{(1)}\) and \(W_t\) instead of \(W_t^{(1)}\)):

$$\begin{aligned} \text {Cov}(X_t, X_s) = \bigl \langle (X_t - \langle X_t\rangle ) (X_s - \langle X_s\rangle ) \bigr \rangle . \end{aligned}$$
(B.11)

Due to \(\langle X_t \rangle =0\), we have

$$\begin{aligned}&\text {Cov}(X_t, X_s) = \bigl \langle X_t X_s \bigr \rangle \nonumber \\&\quad = \frac{\alpha ^2\beta \theta }{(\alpha - \theta )^2} \bigl \langle (I_{0,t}^{t,\alpha } - I_{0,t}^{t,\theta }) (I_{0,s}^{s,\alpha } -I_{0,s}^{s,\theta }) \bigr \rangle \nonumber \\&\quad = \frac{\alpha ^2\beta \theta }{(\alpha - \theta )^2} \Bigl (\bigl \langle I_{0,t}^{t,\alpha }I_{0,s}^{s,\alpha }\bigr \rangle -\bigl \langle I_{0,t}^{t,\alpha }I_{0,s}^{s,\theta }\bigr \rangle \nonumber \\&\quad \qquad \qquad \qquad - \bigl \langle I_{0,t}^{t,\theta }I_{0,s}^{s,\alpha }\bigr \rangle + \bigl \langle I_{0,t}^{t,\theta }I_{0,s}^{s,\theta } \bigr \rangle \Bigr ), \end{aligned}$$
(B.12)

where we define

$$\begin{aligned} I_{a,b}^{c,d} := \int _a^b e^{-\frac{1}{d}(c-t')}\, dW_{t'}. \end{aligned}$$
(B.13)

Representative for the four summands, we calculate \(\langle I_{0,t}^{t,\alpha }I_{0,s}^{s,\theta }\rangle \). Hereby we use the relation \(I_{a,b}^{c,d} = I_{a,b'}^{c,d} + I_{b',b}^{c,d}\) for \(a<b'<b\). Further, \(I_{a,b}^{c,d}\) and \(I_{a',b'}^{c',d'}\) are stochastically independent, if \(a<b\le a'<b'\), and \(\langle I_{a,b}^{c,d} \rangle = 0\). Without any loss of generality, we assume \(t>s\). All in all, we obtain

$$\begin{aligned} \langle I_{0,t}^{t,\alpha }I_{0,s}^{s,\theta }\rangle&= \bigl \langle (I_{0,s}^{t,\alpha } + I_{s,t}^{t,\alpha }) I_{0,s}^{s,\theta }\bigr \rangle \nonumber \\&= \bigl \langle I_{0,s}^{t,\alpha }I_{0,s}^{s,\theta } \bigr \rangle + \bigl \langle I_{s,t}^{t,\alpha }\bigr \rangle \bigl \langle I_{0,s}^{s,\theta }\bigr \rangle = \bigl \langle I_{0,s}^{t,\alpha }I_{0,s}^{s,\theta } \bigr \rangle . \end{aligned}$$
(B.14)

Now, we replace the Itô integrals by approximating sums. With a partition \(0=t_0<t_1<...<t_N=s\), we have

$$\begin{aligned} \bigl \langle I_{0,s}^{t,\alpha }I_{0,s}^{s,\theta } \bigr \rangle&=\lim _{N\rightarrow \infty } \Biggl \langle \left( \sum _{i=0}^{N-1} e^{-\frac{1}{\alpha }(t-t_i)}(W_{t_{i+1}}-W_{t_i})\right) \nonumber \\&\qquad \left( \sum _{i=0}^{N-1}e^{-\frac{1}{\theta }(s-t_i)} (W_{t_{i+1}}-W_{t_i})\right) \Biggr \rangle . \end{aligned}$$
(B.15)

\((W_{t_{i+1}}-W_{t_i})\) and \((W_{t_{j+1}}-W_{t_j})\) are stochastically independent, if \(i\ne j\). Further, \(\langle (W_{t_{i+1}}-W_{t_i}) \rangle = 0\). Consequently, when expanding the sums, the mixed terms vanish and we end up with

$$\begin{aligned}&\bigl \langle I_{0,s}^{t,\alpha }I_{0,s}^{s,\theta } \bigr \rangle \nonumber \\&\quad = \lim _{N\rightarrow \infty } \Biggl \langle \sum _{i=0}^{N-1} e^{-\frac{1}{\alpha }(t-t_i)}e^{-\frac{1}{\theta }(s-t_i)} (W_{t_{i+1}}-W_{t_i})^2 \Biggr \rangle \nonumber \\&\quad =\lim _{N\rightarrow \infty } \sum _{i=0}^{N-1}e^{-\frac{1}{\alpha }(t-t_i)} e^{-\frac{1}{\theta }(s-t_i)}(t_{i+1} - t_i). \end{aligned}$$
(B.16)

The latter expression is an approximation of the Riemann integral

$$\begin{aligned}&\int _0^s e^{-\frac{1}{\alpha }(t-t')}e^{-\frac{1}{\theta }(s-t')}\, dt' \nonumber \\&\qquad = \frac{\alpha \theta }{\alpha +\theta } \left( e^{-\frac{1}{\alpha }(t-s)} -e^{-\frac{1}{\alpha }t}e^{-\frac{1}{\theta }s} \right) , \end{aligned}$$
(B.17)

which, in the stationary case \(t\rightarrow \infty \) and \(s\rightarrow \infty \), yields \(\frac{\alpha \theta }{\alpha +\theta } e^{-\frac{1}{\alpha }(t-s)}\). All in all,

$$\begin{aligned} \langle I_{0,t}^{t,\alpha }I_{0,s}^{s,\theta }\rangle =\frac{\alpha \theta }{\alpha +\theta } e^{-\frac{1}{\alpha }(t-s)}. \end{aligned}$$
(B.18)

Consequently, for the covariance function we obtain

$$\begin{aligned} \text {Cov}(X_t, X_s) = \frac{\alpha ^2\beta \theta }{2(\alpha +\theta )(\alpha -\theta )} \left[ \alpha e^{-\frac{1}{\alpha }(t-s)} - \theta e^{-\frac{1}{\theta }(t-s)} \right] . \end{aligned}$$
(B.19)

From this, we can infer that

$$\begin{aligned} \text {Var}(X_t) = \text {Cov}(X_t, X_t) = \frac{\alpha ^2\beta \theta }{2(\alpha +\theta )}. \end{aligned}$$
(B.20)

Further, the autocorrelation function \(\text {ACF}(\tau )\), which is the normed autocovariance function (\(\text {ACF}(\tau ) =\text {Cov}(X_{\tau },X_0)/\text {Var}(X_0)\)), reads

$$\begin{aligned} \text {ACF}(\tau ) = \frac{1}{\alpha -\theta } \left[ \alpha e^{-\frac{1}{\alpha }\tau } - \theta e^{-\frac{1}{\theta }\tau } \right] . \end{aligned}$$
(B.21)

In the case \(\alpha =\theta \), we have (again we write \(X_t\) instead of \(X_t^{(1)}\) and \(W_t\) instead of \(W_t^{(1)}\))

$$\begin{aligned} \text {Cov}(X_t,X_s) = \frac{\beta }{\alpha } \bigl \langle J_{0,t}^t J_{0,s}^s \bigr \rangle , \end{aligned}$$
(B.22)

where

$$\begin{aligned} J_{a,b}^c := \int _a^b (c-t')e^{-\frac{1}{\alpha }(c-t')}\, dW_{t'}. \end{aligned}$$
(B.23)

With similar calculations as in the latter case, we obtain

$$\begin{aligned} \text {Cov}(X_t,X_s)&= \frac{\beta }{\alpha } \bigl \langle J_{0,s}^t J_{0,s}^s \bigr \rangle \nonumber \\&= \frac{\beta }{\alpha } \int _0^s (t-t')e^{-\frac{1}{\alpha } (t-t')}(s-t')e^{-\frac{1}{\alpha }(s-t')}\, dt'\nonumber \\&= \frac{\alpha ^2\beta }{4} e^{-\frac{1}{\alpha }(t-s)} +\frac{\alpha \beta }{4} e^{-\frac{1}{\alpha }(t-s)}(t-s) \end{aligned}$$
(B.24)

for the stationary case. From this, we can infer the relations

$$\begin{aligned} \text {Var}(X_t)&= \frac{\alpha ^2\beta }{4}\end{aligned}$$
(B.25)
$$\begin{aligned} \text {ACF}(\tau )&= e^{-\frac{1}{\alpha }\tau } + \frac{1}{\alpha }\tau e^{-\frac{1}{\alpha }\tau }. \end{aligned}$$
(B.26)

Appendix C: Taylor length \(\varLambda \) and correlation length L of the linear model

As introduced in Sect. 3, we obtain the Taylor length by a second-order series expansion of the autocorrelation function \(\text {ACF}(\tau )\) (cf. Appendix B) at \(\tau =0\) and the correlation length L by its integral. First, we regard the case \(\alpha \ne \theta \). From

$$\begin{aligned} \text {ACF}(\tau )&= \frac{1}{\alpha -\theta } \left[ \alpha e^{-\frac{1}{\alpha }\tau } - \theta e^{-\frac{1}{\theta }\tau } \right] \nonumber \\&\approx \frac{1}{\alpha -\theta } \Bigl [ \alpha \left( 1-\frac{1}{\alpha } \tau +\frac{1}{2\alpha ^2}\tau ^2\right) \nonumber \\&\quad - \theta \left( 1-\frac{1}{\theta }\tau +\frac{1}{2\theta ^2}\tau ^2\right) \Bigr ] \nonumber \\&= 1 - \frac{1}{2}\frac{\tau ^2}{\alpha \theta }, \end{aligned}$$
(C.27)

we obtain

$$\begin{aligned} \varLambda = \sqrt{\alpha \theta }. \end{aligned}$$
(C.28)

The correlation length L can be calculated as follows:

$$\begin{aligned} L&= \int _0^{\infty } \text {ACF}(\tau )\, d\tau \nonumber \\&= \frac{1}{\alpha -\theta } \left[ \alpha \int _0^{\infty } e^{-\frac{1}{\alpha }\tau }\, d\tau - \theta \int _0^{\infty } e^{-\frac{1}{\theta }\tau }\, d\tau \right] \nonumber \\&= \frac{\alpha ^2 - \theta ^2}{\alpha - \theta } \nonumber \\&= \alpha + \theta . \end{aligned}$$
(C.29)

Similarly, we obtain \(\varLambda = \alpha \) and \(L=2\alpha \) in the case \(\alpha =\theta \).

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Willers, C., Kamps, O. Non-parametric estimation of a Langevin model driven by correlated noise. Eur. Phys. J. B 94, 149 (2021). https://doi.org/10.1140/epjb/s10051-021-00149-0

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