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ax.plot(np.linspace(0,t_final,int(t_final/delta_t)), X, color = 'black')
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ax.set_xlabel('t', fontsize=16)
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ax.set_ylabel('x', fontsize=16)
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# %% To retrieve the moments, use:
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edges, moments = jd.moments(timeseries = X, bw = 0.35)
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# and don't forget that the Kramers─Moyal coefficient need `moments/delta_t`
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# %% Let us plot the first Kramers─Moyal coefficient 'moments[1,...]/delta_t'
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fig, ax = plt.subplots(1,1,figsize=(6,3))
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ax.plot(edges, moments[1,...]/delta_t, color = 'black', label = '1st Kramers─Moyal coefficient')
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ax.plot(edges, a(edges), '--', color = 'black', label = 'Theoretical curve a = -0.5*x')
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ax.set_xlim([-5,5]); ax.set_ylim([-5,5])
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ax.set_xlabel('x', fontsize=16)
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ax.set_ylabel('$D_1$(x)', fontsize=16)
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ax.legend(fontsize=13)
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# %% The second Kramers─Moyal coefficient 'moments[2,...]/delta_t'
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fig, ax = plt.subplots(1,1,figsize=(6,3))
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ax.plot(edges, moments[2,...]/delta_t, color = 'black', label = '2nd Kramers─Moyal coefficient')
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ax.plot(edges, (b(0)**2 + xi*lamb)*np.ones_like(edges), '--', color = 'black', label = 'Theoretical curve $b^2+λξ$')
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ax.set_xlim([-5,5]); ax.set_ylim([3,8])
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ax.set_xlabel('x', fontsize=16)
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ax.set_ylabel('$D_2$(x)', fontsize=16)
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ax.legend(fontsize=13)
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# %% And the fourth Kramers─Moyal coefficient 'moments[4,...]/delta_t'
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fig, ax = plt.subplots(1,1,figsize=(6,3))
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ax.plot(edges, moments[4,...]/delta_t, color = 'black', label = '4th Kramers─Moyal coefficient')
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ax.plot(edges, (3*(xi**2)*lamb)*np.ones_like(edges), '--', color = 'black', label = 'Theoretical curve $3λξ^2$')
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ax.set_xlim([-5,5]); ax.set_ylim([0,60])
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ax.set_xlabel('x', fontsize=16)
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ax.set_ylabel('$D_4$(x)', fontsize=16)
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ax.legend(fontsize=13)
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# %% Finally, we can use simply the 'jump_amplitude' and 'jump_rate' functions
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# to recover the ξ and λ parameters
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xi_est = jd.jump_amplitude(moments = moments, verbose = True)
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print(xi_est)
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lamb_est = jd.jump_rate(moments = moments, xi_est = xi, verbose = True)
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print(lamb_est/delta_t)
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# Don't forget that the jump rate λ needs to be divide by 'delta_t' to yield a
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# comparible result.
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# %% #########################################################################
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# To understand the usage of the q_ratio function, let us generate to sample
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# trajectories: one without jumps, denoted d_timeseries, and one with jumps
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# denoted j_timeseries.
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# integration time and time sampling
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t_final = 10000
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delta_t = 0.01
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# Drift function
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def a(x):
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return -0.5*x
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# Diffusion function
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def b(x):
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return 0.75
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# generate 2 trajectories
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d_timeseries = jd.jd_process(t_final, delta_t, a=a, b=b, xi=0, lamb=0)
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j_timeseries = jd.jd_process(t_final, delta_t, a=a, b=b, xi=2.5, lamb=1.75)
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# %% Subsequently we time a time scale to analyse, as
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lag = np.logspace(0, 3, 25, dtype=int)
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d_lag, d_Q = jd.q_ratio(lag, d_timeseries)
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j_lag, j_Q = jd.q_ratio(lag, j_timeseries)
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# %% we can then finally plot the results
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fig, ax = plt.subplots(1,1,figsize=(6,3))
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ax.loglog(d_lag, d_Q, '-', color = 'black', label='diffusion')
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ax.loglog(j_lag, j_Q, 'o-', color = 'black', label='jump-diffusion')
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# ax.set_xlim([-5,5]); ax.set_ylim([-5,5])
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ax.set_xlabel('lag', fontsize=16)
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ax.set_ylabel('$Q$-ratio', fontsize=16)
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ax.legend(fontsize=13)
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fig.tight_layout()
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fig.savefig('q_ratio.png', dpi=300)
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