SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Ido Greenberg
Netanel Yannay
ELTA Systems
natiy4@gmail.com
Shie Mannor
Technion, Nvidia Research
In non-linear filtering, it is traditional to compare non-linear architectures such as neural networks to the standard linear Kalman Filter (KF). We observe that this mixes the evaluation of two separate components: the non-linear architecture, and the parameters optimization method. In particular, the non-linear model is often optimized, whereas the reference KF model is not. We argue that both should be optimized similarly, and to that end present the Optimized KF (OKF). We demonstrate that the KF may become competitive to neural models – if optimized using OKF. This implies that experimental conclusions of certain previous studies were derived from a flawed process. The advantage of OKF over the standard KF is further studied theoretically and empirically, in a variety of problems. Conveniently, OKF can replace the KF in real-world systems by merely updating the parameters. Our experiments are published in Github, and the OKF in PyPI.
The Kalman Filter (KF) [Kalman, 1960] is a celebrated method for linear filtering and prediction, with applications in many fields including tracking, navigation, control and reinforcement learning [Zarchan and Musoff, 2000, Kirubarajan, 2002, Kaufmann et al., 2023]. The KF provides optimal predictions under certain assumptions (namely, linear models with i.i.d noise). In practical problems, these assumptions are often violated, rendering the KF sub-optimal and motivating the growing field of non-linear filtering. Many studies demonstrated the benefits of non-linear models over the KF [Revach et al., 2022, Coskun et al., 2017].
Originally, we sought to join this line of works. Motivated by a real-world Doppler radar problem, we developed a dedicated non-linear Neural KF (NKF) based on the LSTM sequential model. NKF achieved significantly better accuracy than the linear KF.
Then, during ablation tests, we noticed that the KF and NKF differ in both architecture and optimization. Specifically, the KF's noise parameters are traditionally determined by noise estimation [Odelson et al., 2006]; whereas NKF's parameters are optimized using supervised learning. To fairly evaluate the two architectures, we wished to apply the same optimization to both. To that end, we devised an Optimized KF (OKF, Section 3). KF and OKF have the same linear architecture: OKF only changes the noise parameters values. Yet, unlike KF, OKF outperformed NKF, which reversed the whole experimental conclusion, and made the neural network unnecessary for this problem (Section 4).
Our original error was comparing two different model architectures (KF and NKF) that were not optimized similarly. A review of the non-linear filtering literature reveals that this methodology is used in many studies. Specifically, for a baseline KF model, the parameters are often tuned by noise estimation [fa Dai et al., 2020, Aydogmus and Aydogmus, 2015, Revach et al., 2022]; by heuristics [Jamil et al., 2020, Coskun et al., 2017, Ullah et al., 2019]; or are simply ignored [Gao et al., 2019, Bai et al., 2020, Zheng et al., 2019], often without public code for examination. Hussein [2014]
Figure 1: Since the KF assumptions are often violated, noise estimation does not optimize the MSE. Instead, our method (OKF) optimizes the MSE directly. In particular, neural network models should be tested against OKF rather than the non-optimized KF – in contrast to the common practice in the literature.
even discusses the (Extended-)KF sensitivity to its parameters, and suggests a neural network with supervised learning – yet never considers the same supervised learning for the KF itself. In all these 10 studies, the non-linear model's added value cannot be inferred from the experimental evidence.
So far, OKF is presented as a methodological contribution: for comparison with non-linear methods, it forms a more coherent baseline than KF. In addition, OKF provides a practical contribution: it achieves more accurate filtering than the KF, using identical architecture. This is demonstrated extensively in Section 5 and Appendix B – in different domains, over different problem variations, using different KF baselines, with different data sizes, and even under distributional shifts.
Discrepancy of objectives: The advantage of OKF over KF may come as a surprise: KF's standard noise estimation is known to already obtain the MSE-optimal parameters! However, this optimality relies on unrealistic assumptions, often considered "fairy tales for undergraduates" [Thomson, 1994]. When violated, a conflict emerges between noise estimation and MSE optimization. We study this in detail: (a) Section 5 analyzes the conflict theoretically under certain assumption violations; (b) Appendix B.1 shows that even oracle noise estimation cannot optimize the MSE; (c) Appendix B.2 shows that when using noise estimation, the MSE may degrade with more data. In this light, all our findings can be summarized as follows (also see Fig. 1):
Contribution: (a) We observe that in most scenarios, since the KF assumptions do not hold, noise estimation is not a proxy to MSE optimization. (b) We thus present the Optimized KF (OKF), also available as a PyPI package. We analyze (theoretically and empirically) the consequences of neglecting to optimize the KF: (c) the standard KF tuning method leads to sub-optimal predictions; (d) the standard methodology in the literature compares an optimized model to a non-optimized KF, hence may produce misleading conclusions. Note that we do not argue against the non-linear models; rather, we claim that their added value cannot be deduced from flawed experiments.
Scope: We focus on the supervised filtering setting, where training data includes both observations and the true system states (whose prediction is usually the objective). Such data is available in many practical applications. For example, the states may be provided by external accurate sensors such as GPS fa Dai et al. [2020]; by manual object labeling in computer vision [Wojke et al., 2017]; by controlled experiments of radar targets; or by simulations of known dynamic systems.
As demonstrated in the 10 studies cited above, this supervised setting is common in non-linear filtering. In linear Kalman filtering, this setting seems to be solved by trivial noise estimation; thus, the literature tends to overlook it, and instead focuses on settings that do not permit trivial noise estimation, e.g., learning from observations alone. Nevertheless, we argue that even in the supervised setting, noise estimation is often not a proxy to MSE optimization, and thus should often be avoided.
Consider the KF model for a dynamic system with no control signal [Kalman, 1960]:
$$
X _ {t + 1} = F _ {t} X _ {t} + \omega_ {t} \quad (\omega_ {t} \sim \mathcal {N} (0, Q)), \quad Z _ {t} = H _ {t} X _ {t} + \nu_ {t} \quad (\nu_ {t} \sim \mathcal {N} (0, R)). \tag {1}
$$
$X_{t}$  is the system state at time  $t$ , and its estimation is usually the goal. Its dynamics are modeled by the linear operator  $F_{t}$ , with random noise  $\omega_{t}$  whose covariance is  $Q$ .  $Z_{t}$  is the observation, modeled by the operator  $H_{t}$  with noise  $\nu_{t}$  whose covariance is  $R$ . The notation may be simplified to  $F, H$  in the stationary case.
Figure 2: The KF algorithm. The prediction step is based on the motion model  $\tilde{F}_t$  with noise  $\hat{Q}$ , whereas the update step is based on the observation model  $\tilde{H}_t$  with noise  $\hat{R}$ .
The KF represents  $X_{t}$  via estimation of the mean  $\hat{x}_t$  and covariance  $\hat{P}_t$ . As shown in Fig. 2, the KF alternately predicts the next state (prediction step), and processes new information from incoming observations (update or filtering step). The KF relies on the matrices  $\tilde{F}_t, \tilde{H}_t, \hat{Q}, \hat{R}$ , intended to represent  $F_t, H_t, Q, R$  of Eq. (1). Whenever  $F_t, H_t$  are known and stationary, we may simplify the notation to  $\tilde{F}_t = F$ ,  $\tilde{H}_t = H$ .
The KF estimator  $\hat{x}_t$  is optimal in terms of mean square errors (MSE) – but only under a restrictive set of assumptions [Kalman, 1960]:
Assumption 1 (KF assumptions).  $\tilde{F}_t = F_t$ ,  $\tilde{H}_t = H_t$  are known and independent of  $X_{t}$  (linear models); each sequence  $\{\omega_t\}, \{\nu_t\}$  is i.i.d; the covariances  $\hat{Q} = Q$ ,  $\hat{R} = R$  are known; and  $\hat{x}_0, \hat{P}_0$  correspond to the mean and covariance of the initial  $X_0$ .
Theorem 1 (KF optimality; e.g., see Jazwinski [2007], Humpherys et al. [2012]). Under Assumption 1, the KF estimator  $\hat{x}_t$  minimizes the MSE w.r.t.  $X_t$ .
The KF accuracy strongly depends on its parameters  $\hat{Q}$  and  $\hat{R}$  [Formentin and Bittanti, 2014]. As motivated by Theorem 1, these parameters are usually identified with the noise covariance  $Q$ ,  $R$  and are set accordingly: "the systematic and preferable approach to determine the filter gain is to estimate the covariances from data" [Odelson et al., 2006]. In absence of system state data  $\{x_{t}\}$  (the "ground truth"), many methods were suggested to estimate the covariances from observations  $\{z_{t}\}$  alone [Mehra, 1970, Zanni et al., 2017, Park et al., 2019, Feng et al., 2014]. We focus on the supervised setting, where the states  $\{x_{t}\}$  are available in the training-data (but not in inference).
Definition 1 (Supervised data). Consider  $K$  trajectories of a dynamic system, with lengths  $\{T_k\}_{k=1}^K$ . We define their supervised data as the sequences of true system states  $x_{k,t} \in \mathbb{R}^{d_x}$  and observations  $z_{k,t} \in \mathbb{R}^{d_z}$ :  $\{\{(x_{k,t}, z_{k,t})\}_{t=1}^{T_k}\}_{k=1}^K$ .
If  $F_{t}, H_{t}$  are known, the supervised setting permits a direct calculation of the sample covariance matrices of the noise [Lacey, 1998]:
$$
\hat {Q} := \operatorname {C o v} \left(\left\{x _ {k, t + 1} - F _ {t} x _ {k, t} \right\} _ {k, t}\right), \quad \hat {R} := \operatorname {C o v} \left(\left\{z _ {k, t} - H _ {t} x _ {k, t} \right\} _ {k, t}\right). \tag {2}
$$
Since Theorem 1 guarantees optimality when  $\hat{Q} = Q$ ,  $\hat{R} = R$ , and Eq. (2) provides a simple estimator for  $Q$  and  $R$ , Algorithm 1 has become the gold-standard tuning method for KF from supervised data.
Algorithm 1 (KF noise estimation). Given supervised data  $\{(x_{k,t},z_{k,t})\}$ , return  $\hat{Q}$  and  $\hat{R}$  of Eq. (2).
While Algorithm 1 is indeed trivial to apply in the supervised setting, we show below that when Assumption 1 is violated, it no longer provides optimal predictions. Violation of Assumption 1 can be partially handled by certain variations of the KF, such as Extended KF (EKF) [Sorenson, 1985] and Unscented KF (UKF) [Wan and Van Der Merwe, 2000].
3 Optimized Kalman Filter
Estimation of the KF noise parameters  $\hat{Q},\hat{R}$  has been studied extensively in various settings; yet, in our supervised setting it is trivially solved by Algorithm 1. However, once Assumption 1 is violated,
such noise estimation is no longer a proxy to MSE optimization – despite Theorem 1. Instead, in this section we propose to determine  $\hat{Q}$  and  $\hat{R}$  via explicit MSE optimization. We rely on standard optimization methods for sequential supervised learning; as discussed below, the main challenge is to maintain the Symmetric and Positive Definite (SPD) structure of  $\hat{Q},\hat{R}$  as covariance matrices.
Formally, we consider the KF (Fig. 2) as a prediction model  $\hat{x}_{k,t}(\{z_{k,\tau}\}_{\tau=1}^{t}; \hat{Q}, \hat{R})$ , which estimates  $x_{k,t}$  given the observations  $\{z_{k,\tau}\}_{\tau=1}^{t}$  and parameters  $\hat{Q}, \hat{R}$ . We define the KF optimization problem:
$$
\underset {Q ^ {\prime}, R ^ {\prime}} {\operatorname {a r g m i n}} \sum_ {k = 1} ^ {K} \sum_ {t = 1} ^ {T _ {k}} \operatorname {l o s s} \left(\hat {x} _ {k, t} \left(\left\{z _ {k, \tau} \right\} _ {\tau = 1} ^ {t}; Q ^ {\prime}, R ^ {\prime}\right), x _ {k, t}\right), \quad \text {s . t .} Q ^ {\prime} \in S _ {+ +} ^ {d _ {x}}, R ^ {\prime} \in S _ {+ +} ^ {d _ {z}}, \tag {3}
$$
where  $S_{++}^{d}\subset \mathbb{R}^{d\times d}$  is the space of Symmetric and Positive Definite matrices (SPD), and  $\mathrm{loss}(\cdot)$  is the objective function (e.g.,  $\mathrm{loss}(\hat{x},x) = ||\hat{x} -x||^2$  for MSE). Prediction of future states can be expressed using the same Eq. (3), by changing the observed input from  $\{z_{k,\tau}\}_{\tau = 1}^{t}$  to  $\{z_{k,\tau}\}_{\tau = 1}^{t - 1}$ .
A significant challenge in solving Eq. (3) is the SPD constraint. Standard numeric supervised optimization methods (e.g., Adam [Diederik P. Kingma, 2014]) may violate the constraint. While the SPD constraint is often bypassed using diagonal restriction [Li et al., 2019, Formentin and Bittanti, 2014], this may significantly degrade the predictions, as demonstrated in the ablation tests in Appendix B.4. Instead, to maintain the complete expressiveness of  $\hat{Q}$  and  $\hat{R}$ , we use the Cholesky parameterization [Pinheiro and Bates, 1996].
Algorithm 2: Optimized Kalman Filter (OKF)
1 Input: training data  $\{(x_{k,t},z_{k,t})\}_{k = 1}^{K}$  (Definition 1); batch size  $b$  ; loss function (e.g., MSE); optimization_step function (e.g., Adam)
2  $d_{x}\gets \mathrm{len}(x_{1,1}),\quad d_{z}\gets \mathrm{len}(z_{1,1})$
3 Initialize  $\theta_{Q}\in \mathbb{R}^{\frac{1}{2} d_{x}(d_{x} + 1)}$ $\theta_{R}\in \mathbb{R}^{\frac{1}{2} d_{z}(d_{z} + 1)}$
4 while training not finished do // Get  $Q,R$  using Eq. (4)
5  $\hat{Q}\gets L(\theta_{Q})L(\theta_{Q})^{\top},\quad \hat{R}\gets L(\theta_{R})L(\theta_{R})^{\top}$
6  $\mathcal{K}\gets$  sample(  $\{1,\dots ,K\}$  ，size  $= b)$
7  $C\gets 0$
8 for  $k$  in  $\mathcal{K}$  do
9 Initialize  $\hat{x}\in \mathbb{R}^{d_x}$
10 for  $t$  in  $1:T_{k}$  do //KF steps (Fig. 2)
11  $\hat{x}\gets \mathrm{KF\_predict}(\hat{x};\hat{Q})$
12  $\hat{x}\gets \mathrm{KF\_update}(\hat{x},z_{k,t};\hat{R})$
13  $C\gets C + \mathrm{loss}(\hat{x},x_{k,t})$
14  $\theta_{Q},\theta_{R}\gets$  optimization_step(C,  $(\theta_{Q},\theta_{R}))$
15 Return  $\hat{Q}$ ,  $\hat{R}$
The parameterization relies on Cholesky decomposition: any SPD matrix  $A \in \mathbb{R}^{d \times d}$  can be written as  $A = LL^{\top}$ , where  $L$  is lower-triangular with positive entries along its diagonal. Reversely, for any lower-triangular  $L$  with positive diagonal,  $LL^{\top}$  is SPD. Thus, to represent an SPD  $A \in \mathbb{R}^{d \times d}$ , we define  $A(L) \coloneqq LL^{\top}$  and parameterize  $L(\theta)$  to be lower-triangular, have positive diagonal, and be differentiable in the parameters  $\theta$ :
$$
(L (\theta)) _ {i j} := \left\{ \begin{array}{l l} 0 & \text {i f} i <   j, \\ e ^ {\theta_ {d (d - 1) / 2 + i}} & \text {i f} i = j, \\ \theta_ {(i - 2) (i - 1) / 2 + j} & \text {i f} i > j, \end{array} \right. \tag {4}
$$
where  $\theta \in \mathbb{R}^{d(d + 1) / 2}$ .
Both Cholesky parameterization and sequential optimization methods are well known tools. Yet, for KF optimization from supervised data, we are not aware of any previous attempts to apply them together, as noise estimation (Algorithm 1) is typically preferred.
We wrap the optimization process in the Optimized KF (OKF) in Algorithm 2, which outputs optimized parameters  $\hat{Q},\hat{R}$  for Fig. 2. Note that Algorithm 2 optimizes the state estimation at current time  $t$ . By switching Line 12 and Line 13, the optimization will instead be shifted to state prediction at the next time-step (as  $\hat{x}$  becomes oblivious to the current observation  $z_{k,t}$ ).
Lack of theoretical guarantees: If Assumption 1 cannot be trusted, neither noise estimation (Algorithm 1) nor OKF (Algorithm 2) can guarantee global optimality. Still, OKF pursues the MSE objective using standard optimization tools, which achieved successful results in many non-convex problems [Zhong et al., 2020] over millions of parameters [Devlin et al., 2019]. On the other hand, noise estimation pursues a conflicting objective, and guarantees significant sub-optimality in certain scenarios, as analyzed in Section 5.
Figure 3: A sample trajectory and the corresponding predictions (projected onto XY plane), in the Freemotion benchmark. The standard KF provides inaccurate predictions in certain turns.
(a) Free-motion benchmark
(b) No-turns benchmark
Figure 4: Test errors and  $95\%$  confidence intervals, over targets with different accelerations. The middle acceleration range coincides with the training accelerations (24-48 in (a) and 8-16 in (b)), and the other ranges correspond to out-of-distribution generalization.
4 OKF vs. Neural KF: Is the Non-Linearity Helpful?
In this section, we demonstrate that comparing an optimized neural network to a non-optimized baseline may lead to incorrect conclusions: the network may seem superior even if the complicated architecture has no added value. The implied message is not against neural networks, but rather that evaluating them against a non-optimized baseline carries a crucial flaw.
The Doppler radar problem: We consider a variant of the classic Doppler radar problem [Barton, 1988, Roy and Mitra, 2016], where various targets trajectories are tracked in a homogeneous 3D space, given regular observations of a Doppler radar. The state  $X = (x_{x}, x_{y}, x_{z}, x_{ux}, x_{uy}, x_{uz})^{\top} \in \mathbb{R}^{6}$  consists of 3D location and velocity. The goal is to minimize the MSE over the 3 location coordinates. While the true dynamics  $F$  are unknown to the KF, a constant-velocity model  $\tilde{F}$  can be used:
$$
\tilde {F} = \left( \begin{array}{c c c c} 1 & 1 & 1 & 1 \\ & 1 & 1 & 1 \\ & & 1 & 1 \\ & & & 1 \end{array} \right). \tag {5}
$$
An observation  $Z \in \mathbb{R}^4$  consists of the location in spherical coordinates (range, azimuth, elevation) and the radial velocity (the Doppler signal), with an additive i.i.d Gaussian noise. After transformation to Cartesian coordinates, the observation model can be written as:
$$
H = H (X) = \left( \begin{array}{c c c c} 1 & & & \\ & 1 & & \\ & & \frac {x _ {x}}{r} & \frac {x _ {y}}{r} \end{array} \frac {x _ {z}}{r}\right), \tag {6}
$$
where  $r = \sqrt{x_x^2 + x_y^2 + x_z^2}$ . Since  $H = H(X)$  relies on the unknown location  $(x_{x},x_{y},x_{z})$ , we instead substitute  $\tilde{H}\coloneqq H(Z)$  in the KF update step in Fig. 2.
Neural KF: The Neural Kalman Filter (NKF) incorporates an LSTM model into the KF framework, as presented in Appendix C and Fig. 16. We originally developed NKF to improve the prediction of the non-linear highly-mechanvering targets in the Doppler problem (e.g., Fig. 3), and made honest efforts to engineer a well-motivated architecture. Regardless, we stress that this section demonstrates a methodological flaw when comparing any optimized filtering method to the KF; this methodological argument stands regardless of the technical quality of NKF. In addition, Appendix C presents similar results for other variants of NKF.
Experiments: We train NKF and OKF on a dataset of simulated trajectories, representing realistic targets with free motion (as displayed in Fig. 3). As a second benchmark, we also train on a dataset of simplified trajectories, with speed changes but with no turns. The two benchmarks are specified in detail in Appendix B.1, and correspond to Fig. 11d and Fig. 11e. We tune the KF from the same datasets using Algorithm 1. In addition to out-of-sample test trajectories, we also test generalization to out-of-distribution trajectories, generated using different ranges of target accelerations (affecting both speed changes and turns radiuses).
Fig. 4 summarizes the test results. Compared to KF, NKF reduces the errors in both benchmarks, suggesting that the non-linear architecture pays off. However, optimization of the KF (using OKF) reduces the errors even further, and thus reverses the conclusion. That is, the advantage of NKF in this problem comes exclusively from optimization, and not at all from the expressive architecture.
5 OKF vs. KF:
Section 4 presents the methodological contribution of OKF for non-linear filtering, as an optimized baseline for comparison, instead of the standard KF. In this section, we study the advantage of OKF over the KF more generally. We show that OKF consistently outperforms the KF in a variety of scenarios from 3 different domains. This carries considerable practical significance: unlike neural models, shifting from KF to OKF merely requires change of the parameters  $\hat{Q}$ ,  $\hat{R}$ , hence can be deployed to real-world systems without additional overhead, complexity or latency on inference.
Recall that by Theorem 1, the KF is already optimal unless Assumption 1 is violated. Thus, the violations are discussed in depth, and the effects of certain violations are analyzed theoretically.
5.1 Doppler Radar Tracking
Theorem 1 guarantees the optimality of Algorithm 1 (KF). Yet, in Section 4, OKF outperforms the KF. This is made possible by the violation of Assumption 1: while the Doppler problem of Section 4 may not seem complex, the trajectories follow a non-linear motion model (as displayed in Fig. 3).
Imagine that we simplified the problem by only simulating constant-velocity targets, making the true motion model  $F$  linear. Would this recover Assumption 1 and make OKF unnecessary? The answer is no; the adventurous reader may attempt to list all the remaining violations before reading on.
The simulated targets move mostly horizontally, with limited elevation changes. This is not expressed by the KF's initial state distribution  $(\hat{x}_0,\hat{P}_0)$ . To remedy this, one may simulate motion uniformly in all directions. A third violation comes from the observation noise. While the radar noise is i.i.d in spherical coordinates (as mentioned in Section 4), it is not i.i.d in Cartesian coordinates (see discussion in Appendix A.2). To overcome this, one may simulate a radar with (physically-impossible) Cartesian i.i.d noise. This results in the unrealistically-simplified problem visualized in Fig. 5.
Figure 5: The original Doppler problem (left) is simplified to a toy problem (right), with linear motion, isotropic flying directions and physically-impossible radar. After all the simplifications, Assumption 1 still does not hold, thus Algorithm 1 is still sub-optimal and outperformed by OKF.
Despite the simplifications, it turns out that Assumption 1 is still not met, as the observation model in Eq. (6) is still not linear (i.e.,  $H = H(X)$  is not constant). As shown by Proposition 1, this single violation alone results in a significant deviation of Algorithm 1 from the optimal parameters.
We first define the simplified problem.
Problem 1 (The toy Doppler problem). The toy Doppler problem is the filtering problem modeled by Eq. (1), with constant-velocity dynamics  $F$  (Eq. (5)), Doppler observation  $H$  (Eq. (6)), and
$$
Q = \mathbf {0} \in \mathbb {R} ^ {6 \times 6}, \qquad R = \left( \begin{array}{c c c} \sigma_ {x} ^ {2} & & \\ & \sigma_ {y} ^ {2} & \\ & & \sigma_ {z} ^ {2} \\ & & \sigma_ {D} ^ {2} \end{array} \right),
$$
where  $\sigma_x, \sigma_y, \sigma_z, \sigma_D > 0$ .
Recall that  $H = H(X)$  in Eq. (6) depends on the state  $X$ , which is unknown to the model. Thus, we assume that  $\tilde{H} = H(\tilde{X})$  is used in the KF update step (Fig. 2), with some estimator  $\tilde{X} \approx X$  (e.g.,
$\tilde{H} = H(Z)$  in Section 4). Hence, the effective noise is  $\tilde{R} \coloneqq Cov(Z - \tilde{H}X) \neq Cov(Z - HX) = R$ . Proposition 1 analyzes the difference between  $\tilde{R}$  and  $R$ . To simplify the analysis, we further assume that the error  $\tilde{X} - X$  within  $\tilde{H}$  (e.g.,  $Z - X$ ) is independent of the target velocity.
Proposition 1. In the toy Doppler Problem 1 with the estimated observation model  $\tilde{H}$ , the effective observation noise  $\tilde{R} = Cov(Z - \tilde{H}X)$  is:
$$
\tilde {R} = \left( \begin{array}{c c c} \sigma_ {x} ^ {2} & & \\ & \sigma_ {y} ^ {2} & \\ & \sigma_ {z} ^ {2} & \\ & & \sigma_ {D} ^ {2} + C \end{array} \right) = R + \left( \begin{array}{c c c} 0 & & \\ & 0 & \\ & & 0 \end{array} _ {C}\right), \tag {7}
$$