Molecular Circuits for Dynamic Noise Filtering

Molecular Circuits for Dynamic Noise Filtering SI Appendix

Christoph Zechner, Georg Seelig, Marc Rullan and Mustafa Khammash

S.1 Optimal Filtering of Stochastic Biochemical Networks S.1.1 Mathematical modeling We model extrinsic noise as a stochastic birth-death process ρ

φ

∅− *Z− * ∅,

(1)

with ρ and φ as the respective rate constants. The dynamics of Z(t) can be described by a master equation of the form d P (z, t) = AP (z, t) = ρP (z − 1, t) + φ [z + 1] P (z + 1, t) − [ρ + φz] P (z, t), dt

(2)

with P (z, t) := P (Z(t) = z) as the probability of finding the system in state z at time t and A as the infinitesimal generator of Z(t). One can show that the stationary probability distribution of this process is given by   ρ P∞ (z) = Poiss z; , (3) φ i.e., a Poisson distribution with rate parameter λ = ρ/φ. We assume that we can observe the process Z(t) through an artificial reaction (i.e., the sensor reaction) c

Z(t)

Y ∅ −− −−* ?,

(4)

with cY a rate constant determining the speed of this reaction. Mathematically, the corresponding reaction counter is given by the random time-change model [1], i.e.,  Z t  Y (t) = RY cY Z(s)ds , (5) 0

with RY as a unit Poisson process. The firing-times τ1 , τ2 , . . . associated with Y (t) can be understood as noisy measurements of the hidden state Z(t). For instance, if the time between two consecutive reactions is large, the current value of Z(t) is probably low. In contrast, if many reactions are observed within a small amount of time, a large Z(t) is more likely. The theory of optimal (or stochastic) filtering [2] provides a mathematical framework for inferring the hidden process Z(t) conditional on the sensor history until time t denoted by Yt . In the particular case considered here, the sensor history corresponds to the set of jump times of the sensor process until t, i.e.,  Yt = τ1 , τ2 , . . . , τY (t) . The reader shall note that Yt is a continuous-time observation because information about Z(t) is revealed also if no reaction happens within a certain amount of time.

1

S.1.2 The Kushner-Stratonovich equation Inferring Z(t) from sensed measurements Yt requires knowledge about the so-called filtering distribution P (Z(t) = z | Yt ). In case of continuous-time dynamics, this distribution can be shown to admit a stochastic differential equation – often referred to as the Kushner-Stratonovich equation [2]. While this equation is general, we consider only its particular form for the considered scenario. Following [19], the conditional distribution π(z, t) = P (Z(t) = z | Yt ) satisfies dπ(z, t) = [Aπ(z, t) − cY (z − M (t))π(z, t)] dt +

z − M (t) π(z, t)dY (t), M (t)

(6)

with M (t) = E [Z(t) | Yt ]. Note that as long as no reaction happens (i.e., dY (t) = 0) the solution of π(z, t) evolves according to an ordinary differential equation. If a reaction happens at t, (i.e., dY (t) = 1), the distribution changes instantaneously by the term accompanying dY (t). Using a sufficiently large state-space truncation, this equation can be integrated numerically to obtain a solution of π(z, t) (see Fig. S.1). However, for the purpose of synthetic biology, this equation is rather impractical since it does not allow for any realistic reasonably-sized molecular implementation. However, estimating Z(t) does not necessarily require the whole distribution of Z(t), but only a few ˆ summary statistics derived from it. Typically, one chooses Z(t) = M (t) to be an estimator, because it ˆ can be shown to minimize the mean squared error (MSE) between the estimated- and true values Z(t) and Z(t), respectively. Expanding eq. (6) in terms of central moments, we obtain   V (t) dY (t) dM (t) = D1 (t) − cY V (t) dt + M (t)   S 3 (t) dV (t) = D2 (t) − cY S 3 (t) dt + dY (t) M (t) (7) .. . h i S N +1 (t) dY (t), dSN (t) = DN (t) − cY S N +1 (t) dt + M (t) with V (t) as the variance, S N (t) as the N th order central moment and X DN (t) = (z − M (t))N Aπ(z, t)

(8)

z∈Z

as the N th order central moment P dynamics of the original (unconditional) process Z(t) for any N > 1. For N = 1, we have D1 (t) = z∈Z zAπ(z, t). Again, we are left with a generally infinite-dimensional system of equations. However, closed moment-dynamics can be obtained under certain assumptions about cY . In the following we will derive two approximate filter equations based on the assumption of either small or large cY .

S.1.3 Derivation of the Poisson filter We first assume that cY is small compared to the rate constants ρ and φ. This means that the interevent times between two consecutive sensor reactions is large on average, giving the distribution π(z, t) enough time to recover stationarity. With dY (t) = 0, the distribution π(z, t) admits the differential equation d (9) π(z, t) = ρπ(z − 1, t) + φ [z + 1] π(z + 1, t) − [ρ + φz] π(z, t) − cY (z − M (t))π(z, t). dt Using the Ansatz π(z, t) = Poiss(z; M (t)) and inserting it into (9) yields e−M (t) (z − M (t))M (t)z−1 M 0 (t) e−M (t) M (t)z−1 e−M (t) M (t)z+1 =ρ +φ z! (z − 1)! z! e−M (t) M (t)z e−M (t) M (t)z − [ρ + φz] − cY (z − M (t)) . z! z!

2

(10)

Dividing eq. (10) by e−M (t) (z − M (t))M (t)z−1 (z!)−1 and simplifying the resulting equation we obtain d M (t) = ρ − (φ + cY )M (t). (11) dt Therefore, as long as no sensor reaction happens, the filtering distribution π(z, t) is given by a Poisson distribution with rate M (t) given by eq. (11). After a sufficient amount of time, the distribution converges to π∞ (z, t) = Poiss(z, ρ(φ + cY )−1 ). As indicated before, we assume that π(z, t) is close to the stationary distribution before the next sensor reaction happens. In this case, the distribution changes instantaneously by the term accompanying dY (t) in eq. (6), i.e., π(z, t+) = π(z, t−) + =

z − M (t−) π(z, t−) M (t−)

z π(z, t−), M (t−)

(12)

with t− and t+ indicating the time immediately before and after the reaction occurs. We realize that the distribution π(z, t+) differs from a Poisson distribution only by a factor proportional to z such that z π(z, t+) = π(z, t−) ≈ Poiss(z, M (t+)), (13) M (t−) with M (t+) =

∞ X

zπ(z, t+) =

z=0

∞ X

z

z=0

z π(z, t−) = M (t−) + 1. M (t−)

(14)

This way, the filtering distribution is approximated by a Poisson distribution with a mean that is incremented by one. The overall filter dynamics can thus be described by a Poisson distribution π(z, t) = Poiss(z, M (t)) with M (t) satisfying dM (t) = ρ − (φ + cY )M (t) + dY (t).

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Note that as a consequence of the Poissonian assumption, the variance is the same as the mean, i.e., V (t) = M (t).

S.1.4 Derivation of the Gamma filter It has been noted in [19] that if cY is large, the filtering distribution π(z, t) can be approximated by a Gamma-distribution. This can be justified through a law of large numbers argument but for simplicity we follow a slightly different strategy here. In particular, it is based on the fact that a process that is Gamma-distributed a-priori, will also be Gamma-distributed a-posteriori when conditioning on the Poissonian sensor data. This is for instance the case if Z(t) is a Cox-Ingersoll-Ross (CIR) process governed by the SDE p dZ(t) = θ(µ − Z(t))dt + σ Z(t)dW (t), (16) with W (t) as a standard Brownian motion. Instead of approximating the conditional distribution π(z, t) directly, we try to cast the original birth-death process into a CIR formulation such that the corresponding analytical solution of π(z, t) applies. We first rewrite eq. (16) as p dZ(t) = (ρ − φZ(t)) dt + σ Z(t)dW (t), (17) with ρ := θµ and φ := θ. The diffusion parameter σ is then chosen such that the stationary variances of the two processes match. The equations describing mean and variance of the CIR process are known to be d M (t) = ρ − φM (t) dt d V (t) = σ 2 M (t) − 2φV (t), dt

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50 Sensor reactions

40 Ground-truth Z(t) Poisson-filter Gamma-filter Kushner-Stratonovitch

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Time t

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Figure S.1: Optimal filtering for a birth-death process with a Poissonian sensor. Blue regions indicate the µ ± 2σ credible intervals around the MMSE (green line) obtained by integrating the Kushner-Stratonovitch equation on a truncated grid (i.e., Z(t) ∈ [0, 40]). If cY is small, all filters achieve similar results. For large cY , the Gamma filter converges to the MMSE and the true signal Z(t), whereas the Poisson filter keeps fluctuating around it. The reason for this is that the increments of that filter are fixed to be one and cannot adapt to the uncertainty in the data (i.e., to cY ). As a consequence, the uncertainty about Z(t) does not go to zero when cY approaches infinity (contrary to the Gamma and MMSE filters). However, for small and intermediate values of cY , the Poisson filter achieves accurate estimates of Z(t) and can even outperform the Gamma filter. with a stationary mean and variance of M∞ = φρ and V∞ = σ 2 2φρ2 , respectively. If we set σ 2 = 2φ, we achieve for the CIR process a stationary variance of V∞ = φρ , being consistent with the original (Poissonian) birth-death process. Overall, we obtain p dZ(t) = (ρ − φZ(t))dt + 2φZ(t)dW (t), (19) which reminds us of the classical diffusion approximation of a birth-death process [1, 5]. We can now exploit the analytical formula for the filtering moments that applies for a CIR-process observed through a Poissonian sensor (see e.g., [19, 10]). In particular, we obtain V (t) dY (t) dM (t) = (ρ − φM (t) − cY V (t)) dt + M (t)   V (t)2 V (t)2 dV (t) = 2φM (t) − 2φV (t) − 2cY dt + dY (t). M (t) M (t)2

(20)

Two exemplary results of the Poisson- and Gamma filter are shown in Fig. S.1 and compared to the exact solution obtained by numerically integrating the Kushner-Stratonovich equation. Furthermore, the MSEs achieved using both filters are shown and compared to the MMSE in Fig. S.2.

S.1.5 Biochemical Implementation Conceptually, the practical implementation of a filter consists of two parts. The first one is the construction of the deterministic (ODE) part of the filter through biochemical reactions. This is not always straightforward because real physical systems underly certain constraints. In particular, they are always

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MSE of Z(t)

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MMSE Poisson filter Gamma filter

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Poisson filter Gamma filter

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Sensor rate

Figure S.2: Evaluation of the Poisson and Gamma filters. (a) Reconstruction accuracy of Z(t). The MSE between the estimators M (t) and the true signal Z(t) was computed at t = 1200 using N = 5000 samples for ρ = 0.01, φ = 0.001 and different values of cY . For small cY (i.e., weakly informative measurements), the Poisson filter is close to the MMSE estimator and outperforms the Gamma filter. It then reaches a plateau from which it cannot be improved further. In contrast, the Gamma filter approaches the MMSE for intermediate and large cY . (b) Accuracy as a function of parameter mismatch. Filters were equipped with log-normally 2 distributed birth- and death-rates (i.e., ρ and φ), i.e., LN (ln θ, σM ), with θ as the true value and σM as the logarithmic standard deviation. MSE values were computed using N = 5000 samples for different values of σM . Note that the interval ln θ ± 0.5 corresponds to one order of magnitude in θ in the linear domain. positive and also their temporal changes underly specific rate laws derived from first principles (e.g., the law of mass-action). The second part deals with the integration of the sensor reactions dY (t). The corresponding reaction can be understood as a listener that informs the filter about the arrival of a new sensor reaction by changing its abundance by a certain value. Subsequently, the filter processes this sensor reaction while waiting for the next one and so forth. One challenge is that the original process Z(t) is assumed to be stochastic, whereas the filter equation should be as deterministic as possible. However, since the latter relies on the same rate constants as Z(t) it is possibly just equally stochastic. In order to circumvent this, we need to scale up the filter to high copy numbers. For instance, if we aim to implement a rescaled version of the Poisson filter, we would have dM n (t) = nρ − (φ + cY )M n (t) + ndY (t),

(21)

n

with n as the scaling factor and M (t) = M (t)n as the rescaled number of copies in the reaction volume. Note that also the increments of the filter dY (t) have to be rescaled by n. Accordingly, a single sensor reaction should yield a change in molecules of n, i.e., c

Y Z −− * Z + n × M.

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At present, we see two possibilities to achieve such an amplification. The first one is to make use of large polymers, that dissociate upon addition of a certain factor. This is relatively straight-forward to implement using long double-stranded DNA molecules. An interesting and seemingly practical alternative is to run multiple copies of the same filter and sensor reaction with the original rate constants ρ, φ and cY – for instance using multiple equivalent plasmids. While the individual filters are just as stochastic as the original process Z(t), they will add up to a mixture process with smaller noise. This idea is related to the concept of ensemble learning [12] where multiple poor algorithms are combined to build a single accurate predictor. This is described in detail in Section S.2. In the following we describe possible implementations of both the Poisson- and the Gamma filter as chemical reaction networks (CRNs). Poisson filter. The construction of the Poisson filter is relatively straightforward because it is already in the form of a valid reaction rate equation. In particular, it corresponds to a linear birth-death

5

process with birth-rate ρ and death-rate φ + cY . Note that the rate constant ρ corresponds now to a deterministic rate-law. The corresponding stochastic rate constant relates to ρ through ρS = nρ. The overall implementation is then given by c

Y Z −− *Z+n×M

φ+c

nρ

(23)

Y ∅ −−−−* M −−−− * ∅.

The reader should note that both reactions are defined on the level of copy-numbers (i.e., stochastic description), whereas the latter will be practically deterministic if n is chosen sufficiently large. Gamma filter. Implementing the Gamma filter as a CRN is slightly more challenging. In particular, the input dY (t) enters in a complicated (i.e., state-dependent) fashion and furthermore, the ODE-part shows terms that are inconsistent with physical rate-laws. Here we propose to apply a variable transform such that the resulting equations become implementable. Let us consider the nonlinear transformation !   M (t)2 A(t) V (t) = F (M (t), V (t)) = M (t) (24) B(t) , V (t) whose dynamics can be computed using Ito’s lemma for counting processes. In particular, if we have a stochastic process of the form dX(t) = a(X(t))dt + b(X(t))dY (t),

(25)

the transformed dynamics are given by dF (X(t)) = ∂X F (X(t))a(X(t))dt + [F (X(t) + b(X(t))) − F (X(t))] dY (t).

(26)

In case of the Gamma filter and the particular choice of F , this formula yields dA(t) = [2ρB(t) − 2φA(t)B(t)] dt + dY (t) B(t)2 d B(t) = cY + φB(t)(1 − 2B(t)) + ρ dt A(t)

(27)

and for high abundance regimes, the latter simplifies to dA(t) = [2ρB(t) − 2φA(t)B(t)] dt + dY (t) B(t)2 d B(t) = cY − 2φB(t)2 + ρ . dt A(t)

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However, we are left with one non-mass-action type of rate-law, i.e., a production of B that is inversely proportional to A. To account for this, we define an auxiliary variable R(t) = B(t)/A(t), which can be realized through a birth-death process d R(t) = k1 B(t) − k1 A(t)R(t), dt

(29)

d where k must be chosen as large as to achieve quasi-steady-state in R(t), i.e., dt R(t) ≈ 0 on the time-scale of B(t) and A(t). In summary, we obtain the following mass-action-type of rate equation

dA(t) = [2ρB(t) − 2φA(t)B(t)] dt + dY (t) d B(t) = cY − 2φB(t)2 + ρB(t)R(t) dt d R(t) = k1 B(t) − k1 A(t)R(t). dt

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(30)

Since the transformed filter does not directly reveal estimates of Z(t), we require a suitable mechanism to transform the system back into its original form. In case of the mean M (t) = A(t)B(t)−1 , this can be achieved through another birth-death process d M (t) = k2 A(t) − k2 B(t)M (t), dt

(31)

which – for sufficiently large k2 – is given by M (t) ≈ A(t)B(t)−1 . In case of the variance V (t) = A(t)B(t)−1 , we proceed analogously, whereas only the death-rate changes to k2 B(t)2 V (t). Including a scaling by a factor n, the Gamma filter can be implemented by the following mass-action network c

Y Z −− *Z+n×A

2ρ

− * B− ) −− − −A+B 2φ/n nc

Y ∅ −−− *B

φ/n

2B −−* ∅ ρ/n

B + R −−* 2B + R k

(32)

1 B+R B −*

k1 /n

A + R −−−* A k

2 A −* A+M+V

k2 /n

B + M −−−* B k2 /n2

2B + V −−−−* 2B. Note that the third-order reaction 2B + V → 2B could be realized as two bimolecular reactions 2B → C and C + V → 2B.

S.2 Ensemble Filtering As discussed previously, the sensor reaction needs to produce n molecules to achieve deterministic behavior of the filter. Especially in living cells, however, this might be challenging to achieve. In the following, we describe an alternative approach that appears better compatible with available biological mechanisms. The main idea is to use n statistically independent replicates of the same filter with original (unscaled) dynamics. While a single filter is likely to be as stochastic as the original process Z(t), fluctuations will be repressed at the overall ensemble output. The second consequence is that each of the n individual filters will be equipped with its own one-molecule sensor as opposed to the n-molecule sensor of the original filter. In the following, we derive the mathematical equations that describe these modified filters (referred to as ensemble filters). For illustration, we restrict the following analysis to the Poisson filter but a more general perspective shall be subject of future work. We consider an ensemble of n identical filters M (j) (t) for j = 1, . . . , n. This time, we need to account for molecular fluctuations in M (j) that are due to the random birth- and death events of the filter. We (j) (j) define by Rb (t) and Rd (t) the number of birth- and death reactions until time t, respectively. Those can be decomposed as (j) ˜ (j) (t), Rb (t) = ρt + R (33) b Z t (j) ˜ (j) (t), Rd (t) = (φ + cY ) M (j) (s)ds + R (34) d 0

˜ (j) (t) and R ˜ (j) (t) as zero-mean processes. Each Poisson filter is composed as a birth-, death- and with R b d a sensor reaction, i.e., (j) (j) M (j) (t) = Rb (t) − Rd (t) + Y (j) (t). (35)

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The differential version of this relation yields (j)

(j)

dM (j) (t) = dRb (t) − dRd (t) + dY (j) (t) ˜ (j) (t) − (φ + cY )M (j) (t)dt − dR ˜ (j) (t) + dY (j) (t) = ρdt + dR b d   (j) (j) ˜ (j) (t) − dR ˜ (j) (t) . = ρ − (φ + cY )M (t) dt + dY (t) + dR b d | {z } | {z } intrinsic noise

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ideal PF dynamics

The first part correctly resembles the desired Poisson filter dynamics but it is corrupted by intrinsic molecular noise (second term). The amplitude of that noise is determined by the parameters ρ, φ and cY such that unsatisfactory results are likely. To control the amount of noise, we want to P make use of n ˆ (t) = n M (j) (t), identical replicates of these filters. In particular, we consider the summary process M j=1 which satisfies ˆ (t) = dM

n X

dM (j) (t)

j=1 n  n n    X X X ˜ (j) (t) ˜ (j) (t) − dR = ρ − (φ + cY )M (j) (t) dt + dY (j) (t) + dR b d j=1

j=1

j=1

n n n     X X X ˜ (j) (t) − dR ˜ (j) (t) = nρ − (φ + cY ) M (j) (t) dt + dY (j) (t) + dR b d j=1

j=1

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j=1

n n     X X ˆ (t) dt + ˜ (j) (t) − dR ˜ (j) (t) = nρ − (φ + cY )M dY (j) (t) + dR b d j=1

j=1

Next, we make use of the decomposition dY (j) (t) = cY Z(t)dt + dY˜ (j) (t), which yields n  n   X    X ˜ (j) (t) − dR ˜ (j) (t) ˆ (t) = nρ − (φ + cY )M ˆ (t) dt + cY Z(t)dt + dY˜ (j) (t) + dR dM b d j=1

j=1

n  n    X X ˜ (j) (t) − dR ˜ (j) (t) . ˆ (t) + ncY Z(t) dt + dR = nρ − (φ + cY )M dY˜ (j) (t) + b d j=1

j=1

|

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{z

sensor noise

}

|

{z

intrinsic noise

}

One can see that the term accompanied by dt scales with n while the noise terms do not. Relative to ˆ (t), the random fluctuations will vanish as n → ∞. Formally, this can be shown the total abundance M by rescaling the summary process by n (i.e., ensemble average), i.e., 1 ˆ dM (t) n n    1 ˆ 1 X  ˜ (j) ˜ (j) (t) − dR ˜ (j) (t) = ρ − (φ + cY ) M (t) + cY Z(t) dt + dY (t) + dR b d n n j=1

dM (t) =

  1 = ρ − (φ + cY )M (t) + cY Z(t) dt + n

n  X

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 ˜ (j) (t) − dR ˜ (j) (t) . dY˜ (j) (t) + dR b d

j=1

The ensemble output for n → ∞ therefore resembles the ordinary differential equation d M (t) = ρ − (φ + cY )M (t) + cY Z(t) = ρ − φM (t) + cY (Z(t) − M (t)). dt

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The dependency of the estimator M (t) on the scaling variable n for both the Poisson and ensemble Poisson filters is depicted in Fig. S.3. We want to point out an interesting property of the ensemble Poisson filter. In particular, each individual M (j) (t) contains a birth- and death- channel that is equivalent to those of the original process Z(t) (in

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Figure S.3: Illustration of intrinsic fluctuations in M (t). The filtering output was simulated stochastically for both the Poisson- and Ensemble Poisson filters for n = {1, 10, 100}. For n = 1, the filters yield very stochastic and imprecise estimates of Z(t). The accuracy improves for increasing n, although the ensemble Poisson filter generally outperforms the the Poisson filter. the sense that they have the same parameters ρ and φ). Practically, one could use a replicate of Z(t) (e.g., copies of the involved genes) and extend it by a sensor reaction Z → Z + M and one additional degradation reaction with rate cY . In this implementation, the ensemble Poisson filter readily applies to arbitrary nonlinear Z(t) because the problem of non-closed moment dynamics is bypassed. In other words, the ensemble is used to produce a Monte Carlo average of the desired mean dynamics of M (t). For large n, this generalized ensemble Poisson filter yields estimates of the form d M (t) = D1 (t) + cY (Z(t) − M (t)), dt

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with D1 (t) as the mean dynamics of process Z(t).

S.2.1 Error analysis S.2.1.1 Poisson Filter The scaled Poisson filter including intrinsic noise is given by dM n (t) = dRb (t) − dRd (t) + ndY (t), (42)     Rt Rt with dRb (t) = dN (nρt), dRd (t) = dN (φ + cY ) 0 M n (s)ds , dY (t) = dN cY 0 Z(s)ds and dN (t) as a unit Poisson firing process. For the desired estimator M (t), we then obtain dM (t) =

1 1 1 dM n (t) = dRb (t) − dRd (t) + dY (t). n n n

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Similarly, the signal Z(t) can be decomposed into a birth- and a death-part dZ(t) = dZb (t) − dZd (t),

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 R  t with dZb (t) = dN (ρt), dZd (t) = dN φ 0 Z(s)ds . We next define by e(t) = M (t) − Z(t) the error of the estimator. The dynamics of e(t) are given by 1 1 dRb (t) − dRd (t) + dY (t) − dZb (t) + dZd (t) n n    1 ˜ ˜ d (t) − (ρ − φZ(t)) dt − dZ˜b (t) − dZ˜d (t) = (ρ − (φ + cY )M (t)) dt + dY (t) + dRb (t) − dR n    1 ˜ ˜ d (t) − dZ˜b (t) − dZ˜d (t) = (−(φ + cY )M (t) + φZ(t)) dt + dY (t) + dRb (t) − dR n    1 ˜ ˜ d (t) − dZ˜b (t) − dZ˜d (t) dRb (t) − dR = (−(φ + cY )M (t) + φZ(t)) dt + cY Z(t)dt + dY˜ (t) + n    1 ˜ ˜ d (t) − dZ˜b (t) − dZ˜d (t) ˜ = (−(φ + cY )(M (t) − Z(t)) + φZ(t)) dt + dY (t) + dRb (t) − dR n    1 ˜ ˜ d (t) − dZ˜b (t) − dZ˜d (t) + dY˜ (t), = (−(φ + cY )e(t)) dt + dRb (t) − dR | {z } {z } sensor {z } | |n noise

de(t) = dM (t) − dZ(t) =

signal noise

intrinsic noise

(45) where symbols with a tilde denote centered (i.e., zero-mean) quantities. Taking the expectation, all these terms become zero and thus, d E [e(t)] = −(φ + cY )E [e(t)] , (46) dt which means that the expected error asymptotically tends to zero at rate (φ + cY ). In order to compute the mean squared error, we use Ito’s lemma to derive an equation for e(t)2 and take its expectation subsequently. Note that in the equation de(t) =

1 1 dRb (t) − dRd (t) + dY (t) − dZb (t) + dZd (t), n n

(47)

the individual counting processes never jump simultaneously (i.e., with probability zero). Therefore, each of the terms can be transformed separately. Similar as in eq. (26), an individual term needs to be transformed as dF (N (t)) = [F (N (t) + b(N (t)) − F (N (t))] dN (t). (48) For example, a term dN(t) = −n−1 dRd (t) maps to " # 2 1 − 2N (t)n 1 dN (t)2 = N (t) − − N (t)2 dN (t) = dRd (t). n n2

(49)

Altogether, we obtain for the squared error de(t)2 =

2e(t)n + 1 1 − 2e(t)n dRb (t) + dRd (t) + (2e(t) + 1) (dZd (t) + dY (t)) + (1 − 2e(t))dZb (t). n2 n2 (50)

It is straightforward to show that inserting the individual terms and taking the expectation yields       d  ρ 1 E e(t)2 = −2(φ + cY )E e(t)2 + 2ρ + cY 1+ . dt φ n

(51)

At steady state, we obtain for the MSE   ρ(2φ + cY ) M SEP F = lim E e2 (t) = t→∞ 2φ(φ + cY )

 1+

1 n

 =

ρ cY ρ 1 ρ(2φ + cY ) + + . φ + cY 2φ (φ + cY ) n 2φ(φ + cY )

(52)

We can see that the MSE consists of three additive contributions. One stems from intrinsic fluctuations of the estimator and it will go to zero for large n. However, two additional terms persist regardless of n. We will revisit eq. (52) in the next section when it is compared to the MSE obtained under the ensemble Poisson filter.

10

S.2.1.2 Ensemble Poisson Filter For a given n the estimator obtained using the ensemble Poisson filter takes the form dM (t) =

n  1 X  (i) (i) dRb (t) − dRd (t) + dY (i) (t) . n i=1

(53)

The corresponding error dynamics are then given by de(t) =

n  1 X  (i) (i) dRb (t) − dRd (t) + dY (i) (t) − dZb (t) + dZd (t) n i=1

= −(φ + cY )e(t)dt +

n n    X 1 X  ˜ (i) ˜ (i) (t) − dZ˜b (t) − dZ˜d (t) + 1 dRb (t) − dR dY˜ (i) (t) d n i=1 n | {z } i=1 {z } {z } | | signal noise intrinsic noise

(54)

sensor noise

and hence, d E [e(t)] = −(φ + cY )E [e(t)] . (55) dt Therefore, the expected mismatch between Z(t) and M (t) coincides with that from the Poisson filter. However, we can see that the sensor noise is scaled by a factor n−1 , indicating that it will tend to zero as n → ∞. We proceed as before and compute the transformed dynamics of e(t)2 , i.e., ! n n 2e(t)n + 1 X 1 − 2e(t)n X (i) (i) 2 (i) de(t) = dR (t) + dY (t) + dRd (t) b n2 n2 i=1 i=1

(56)

+ (2e(t) + 1)dZd (t) + (1 − 2e(t))dZb (t). Inserting the individual terms and taking the expectation and letting t → ∞ shows that M SEEP F =

ρ ρ + . φ + cY φn

(57)

The equation shows that if n → ∞, the ensemble Poisson filters outperforms the Poisson filter. In particular, eq. (52) shows in addition to the term to ρ(φ + cY )−1 a second positive term that is independent of n. From the inequality ρ ρ ρ 1 ρ(2φ + cY ) ρ cY < + , + + φ + cY φn φ + cY 2φ (φ + cY ) n 2φ(φ + cY )

(58)

we find that 1 n



(2φ + cY ) φ + cY − φ 2φ



(2φ + cY ) φ + cY − φ 2φ



(2φ + cY ) φ + cY 2 − cY cY