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Unsupervised learning of observation functions in state space models by nonparametric moment methods

  • *Corresponding author: Fei Lu

    *Corresponding author: Fei Lu 

MM, YGK and FL are partially supported by DE-SC0021361 and FA9550-21-1-0317. FL is partially funded by the NSF Award DMS-1913243

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  • We investigate the unsupervised learning of non-invertible observation functions in nonlinear state space models. Assuming abundant data of the observation process along with the distribution of the state process, we introduce a nonparametric generalized moment method to estimate the observation function via constrained regression. The major challenge comes from the non-invertibility of the observation function and the lack of data pairs between the state and observation. We address the fundamental issue of identifiability from quadratic loss functionals and show that the function space of identifiability is the closure of a RKHS that is intrinsic to the state process. Numerical results show that the first two moments and temporal correlations, along with upper and lower bounds, can identify functions ranging from piecewise polynomials to smooth functions, leading to convergent estimators. The limitations of this method, such as non-identifiability due to symmetry and stationarity, are also discussed.

    Mathematics Subject Classification: Primary: 62G05, 68Q32, 62M15.

    Citation:

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  • Figure 1.  Empirical densities from the data trajectories of the process $ (X_{t_l}) $ in (4.2) and the observation processes $ (Y_{t_l}) $ with $ {{f_*}} = f_i $, where $ f_i $'s are the three observation functions in (4.3). Since we do not have data pairs between $ (X_{t_l}^{(m)},Y_{t_l}^{(m)}) $, these empirical densities are the available information from data. Our goal is to find the function $ {{f_*}} $ in the operator that maps the densities of $ \{X_{t_l}\} $ to the densities of $ \{Y_{t_l}\} $

    Figure 2.  Learning results of Sine function $ f_1(x) = \sin(x) $ with model (4.2)

    Figure 3.  Learning results of Sine-Cosine function $ f_2(x) = 2\sin(x) + \cos(6x) $ with model (4.2)

    Figure 4.  Learning results of Arch function $ f_3 $ with model (4.2)

    Figure 5.  Learning results of Arch function $ f_3 $ with model (4.2) and i.i.d Gaussian observation noise

    Figure 6.  Learning results of $ {{f_*}}(x) = \sin(x) $ with the state space model being $ X_t = B_t+ X_0 $ where $ X_{0}\sim \mathrm{Unif}(0,1) $. Due to the symmetry with respect to the line $ x = \frac{1}{2} $, the estimator $ \widehat{f}(x) $ and its reflection $ \widehat{f}(1-x) $ are indistinguishable by the loss functional and they lead to similar prediction of the distribution of $ \{Y_{t_l}\} $

    Figure 7.  Learning results of $ {{f_*}}(x) = \sin(x) $ with stationary Ornstein-Uhlenbeck process. Due to limited information from the moments, the estimator is inaccurate

    Figure 8.  The selection of the dimension and the degree of B-spline basis functions in the case of Sine-Cosine function. In (a), the 2-Wasserstein distance reaches minimum among all cases when the degree is 2 and the knot number is 15, at the same time as the $ L^2({\overline \rho_T}^L) $ error reaches the minimum. Figure (b) shows the cross-validating error indicator $ g $ (defined in (B.3)) for selecting the dimension range $ N $, suggesting an upper bound $ N = 60 $ with the threshold

    Table Algorithm 1.  Estimating the observation function by nonparametric generalized moment methods

    Input: The state space model and data $ \{Y_{t_0:t_L}^{(m)} \}_{m=1}^M $ consisting of multiple trajectories of the observation process.
    Output: Estimator $ \widehat f $.
    $\;\;$1:$\;\;$Estimate the empirical density $ {\overline \rho_T}^L $ in (2.16) and find its support $ [R_{min}, R_{max}] $.
    $\;\;$2:$\;\;$Select a basis type, Fourier or B-spline, with an estimated dimension range $ [1,N] $ (by Algorithm 2), and compute the basis functions as described in Section 2.3 using the support of $ {\overline \rho_T}^L $.
    $\;\;$3:$\;\;$for $ n =1:N $ do
    $\;\;$4:$\;\;\;\;\;\;$Compute the moment matrices in (2.6)-(2.7) and the vectors $ b_{k,l}^M $ in (2.11).
    $\;\;$5:$\;\;\;\;\;\;$Find the estimator $ \widehat c_n $ by optimization with multiple initial conditions. Compute and record the values of the loss functional and the 2-Wasserstein distances.
    $\;\;$6:$\;\;$Select the optimal dimension $ n $ (and degree if B-spline basis) that has the minimal 2-Wasserstein distance in (B.5). Return the estimator $ \widehat f = \sum_{i = 1}^{n} c^i_{n} \phi_i $.
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    Table Algorithm 2.  Cross-validating Estimation of Dimension Range (CEDR) for hypothesis space

    Input: The state space model and data $ \{Y_{t_0:t_L}^{(m)} \}_{m=1}^M $.
    Output: A range $ [1,N] $ for the dimension of the hypothesis space for further selection.
    $\;\;$1$\;\;$stimate the empirical density $ {\overline \rho_T} $ in (2.16) and find its support $ [R_{min}, R_{max}] $.
    $\;\;$2:$\;\;$Set $ n=1 $ and $ g(n)=0 $. Estimate the threshold $ \tau $ in (B.4).
    $\;\;$3:$\;\;$While $ g(n)\leq \tau $ do
    $\;\;$4:$\;\;\;\;\;\;$Set $ n\leftarrow n+1 $. Update the basis functions, Fourier or B-spline, as in Section 2.3.
    $\;\;$5:$\;\;\;\;\;\;$Compute normal matrix $ \overline{A}_1 $ in (2.6) by Monte Carlo. Also, compute $ b $ and $ b' $ in (B.1).
    $\;\;$6:$\;\;\;\;\;\;$Eigen-decomposition of $ \overline{A}_1 $ as in (B.2); return $ \overline{A}_1 =\sum_{i=1}^n u_i \sigma_i u_i^T $ with $ u_i^\top B u_j= \delta_{i,j} $.
    $\;\;$7:$\;\;\;\;\;\;$Compute the Picard projection ratios: $ r_i = \frac{|u_i^\top (b-b')|}{\sigma_i} $ for $ i=1,\ldots,n $ and $ g(n)= \sum_{i=1}^n r_i^2 $.
    $\;\;$8:$\;\;$Return $ N=n $.
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