Methods for machine learning via Gaussian processes

Abstract

Method for machine learning via Gaussian processes. A computer-implemented method for machine learning via Gaussian processes, GP, having the steps of selecting an accuracy target, selecting a priori GP family parameterized by hyperparameters, obtaining a training data set, selecting a GP parameterization to model, training the GP by optimizing a PAC Bayesian bound using the training data set, the prior GP family, and the accuracy target, and predicting the next expected value via the trained GP.

Description

Technical Field

[0001] The present invention relates to a method for data modeling using Gaussian processes; a system trained using the method; a computer program, the computer program comprising instructions, the instructions being configured to implement the method when the computer program is implemented on a computer; a machine-readable storage medium having the computer program stored thereon; and a computer configured to implement the method. Background Art

[0002] As is known, for example, from M. Seeger's "PAC-Bayesian Generalization Error Bounds for Gaussian Process Classification" (Journal of Machine Learning Research 3, 233-269 (2002)), Gaussian processes (GP) can be used for modeling in supervised learning. Large amounts of data can be handled with these Gaussian processes, but their use in safety-critical applications is not optimal, since good performance cannot be guaranteed. GP is useful in particular due to its nonparametric nature, and although GP, as a model based on the principle of probability, has an inherent degree of fuzziness, this fuzziness allows the efficiency of GP, for example the quality of modeling, to be inferred without necessarily having to be drawn in the case of hitherto unprocessed data.

[0003] For example, when fitting a large number of hyperparameters by optimizing marginal likelihood, so-called overfitting can be observed. Although the full Bayesian equation, i.e., by marginalizing the hyperparameters, reduces this risk, it results in unacceptable runtimes, since the relevant prediction distributions are generally no longer tractable analytically. Furthermore, this method does not automatically provide safety or generalization guarantees. Summary of the invention

[0004] Advantages of the present invention

[0005] The method with the features of independent claim 1 is a method for training a GP and sparsely approximating the GP by directly optimizing the PAC Bayesian bound in view of the generalization performance of the PAC Bayesian bound. The method is robust and yields significantly better generalization guarantees than other conventional GP approaches. In particular, since no marginalization of (hyper) parameters is required, the novel method is more efficient than methods with such marginalization.

[0006] The method therefore does not use a full Bayesian approach to avoid overfitting, but instead minimizes the generalization bound in order to ensure that overfitting does not occur. Computationally intensive marginalization is thereby avoided.

[0007] A new type of learning objective for GP models is disclosed, which allows rigorous and quantitatively good performance guarantees for future predictions. Such rigorous guarantees are the object of statistical learning principles. However, since the classical uniform learning guarantees for GP as non-parametric models do not provide assertions, such guarantees on learning cannot be used in the case of these models. Instead, traditional optimization objectives are empirical risk minimization (English (regularized) empirical risk minimization (ERM) ((regularized) empirical risk minimization)), maximum probability estimation (MLE) or variational inference (VI).

[0008] Better non-uniform guarantees are developed within the PAC Bayesian approach (PAC = English: Probably approximately correct). The guarantees are tailored specifically to probabilistic methods such as GP and can provide narrow generalization bounds, such as in GP classification, probabilistic support vector machine methods (SVM), linear classifiers or random neural networks.

[0009] So far, PAC Bayesian bounds have mostly been used for inferential evaluation of generalization performance, while learning via optimization of PAC Bayesian bounds has rarely been examined.

[0010] The present disclosure uses PC Bayesian bounds for training GPs, e.g., for regression cases, for classification, or other types of supervised learning. In particular, as a basic approach for ensuring good generalization, it is proposed to learn a complete and sparse GP predictor Q by directly minimizing a PAC Bayesian upper bound B(Q) to the true future risk R(Q) of the predictor.

[0011] Here, the KL divergence (Kullback-Leibler divergence) KL(Q‖P) in the PAC Bayesian principle can be analytically evaluated for many GP pairs P, Q when they have the same hyperparameters. This especially concerns common sparse GP variables, such as, for example, "Deterministic Training Conditional (DTC)", "Fully Independent Training Conditional (FITC)" and "Variational Free Energy (VFE)". They can therefore be used for new PAC Bayesian learning methods, in which the advantages in calculating sparse GPs are combined with principle guarantees. Here, only some of the different possible types of parameters (hyperparameters, induced points, observation noise, free form parameters) cause an increase in the PAC Bayesian bounds. According to the disclosure, GP learning is directly based on the inverse binary KL divergence and not on the looser bounds used so far, such as those resulting from the Pinsker inequality, for example.

[0012] GP learning methods can be used, as mentioned above, for example in regression tasks, whereas the PAC Bayesian bound has so far only been used in classification situations. However, since the PAC Bayesian bound requires a sub-Gaussian assumption about the (unknown) data distribution for regression with a possibly unrestricted loss function, a general restricted loss function for regression is used in order to keep the distribution free as in the regular PAC principle.

[0013] The learning objective exhibits robust optimization properties and scales to larger data sets similarly to other GP methods. In practical applications, better risk bounds are demonstrated by the disclosed method, in many cases improving assurance by a factor of two compared to and relative to conventional methods, where the number of sensing points is increased.

[0014] The strength of GP is that it can adapt to different data sizes due to its non-parametric nature.

[0015] Disclosure of the Invention

[0016] In the general PAC Bayesian approach and in standard supervised learning, we use a set of N training data (x i ,y i )∈X×Y(i=1,...,N) so that in the hypothesis space The training is performed in a hypothesis space that is a subset of the space of functions X→Y. This allows a learning algorithm that, instead of a single hypothesis h, outputs a probability distribution about the hypothesis The distribution Q of . In order to quantify how well the hypothesis h describes the data, a restricted loss function l:Y×Y→[0,1] is assumed to be given, which is assumed to be normalized to the interval [0,1] without limiting the generality. Measurement prediction How to better convert the actual output parameter y * Approximation of input parameter x * Assume that the empirical risk R of h is S (h) is then defined as the average training loss As in the traditional PAC principle, assume an (unknown) distribution μ=μ(x, y) based on the data set X×Y and define the (true) risk as R(h):=∫dμ(x, y)l(y, h(x)). In order to quantify the quality of the stochastic learning algorithm with respect to the assumed distribution Q, the empirical risk and the true risk are defined as follows:

[0017] [Equation A:]

[0018] [Equation B:]

[0019] The average loss for the training distribution or the actual distribution, also known as the Gibbs risk, is expressed using the above equation, where the hypothesis h is obtained from the distribution Q before prediction.

[0020] In the following we discuss the regression case, i.e. is a set of real numbers. The loss function for this case is, for example, The function r ± Specify an interval and predict outside this interval is considered insufficient. Similar to ε support vector regression, in this case, for example, r ± (y * ):=y * ±ε, where the desired accuracy target ε>0 is specified before training. In any case, the expected value is reduced to a one-dimensional integral via h~Q in equations A and B, since h(x * ) at each x * are real-valued random variables.

[0021] Instead of the random predictor h(x * ) where h ~ Q, sometimes for a deterministic Bayesian predictor In GP regression, this deterministic Bayesian predictor is simply equal to the input value x * The predicted mean at The corresponding Bayesian risk is defined as Although the PAC Bayesian principle does not directly apply to R Bay(Q) but only obtains a bound on R(Q), but it is easy to see that if exist The approximate convex surface and The distribution of is symmetric about its mean (e.g. Gaussian distribution), then R Bay (Q)≤2R(Q). Therefore, The following upper bound B(Q) on R(Q) implies an upper bound on R Bay Non-trivial bounds for (Q)≤2B(Q)<1.

[0022] In the following, the GP Q is to be trained by minimizing a suitable risk bound. Due to the probabilistic nature of the GP, a generalization bound for a random predictor is used, which is often better guaranteed than a generalization bound for a deterministic predictor. In this case, the PAC Bayes bound is used.

[0023] The PAC Bayesian principle is: for each loss function l with a value in the interval [0, 1], for each distribution μ, for all Regarding the hypothesis set For each distribution P of and for each δ∈(0,1], the following equation with probability at least 1-δ is suitable for the training data S~μ N The set:

[0024] [Equation C:]

[0025] The empirical risk R of the random predictor Q based on the training data set S S (Q) gives an upper bound (generalization guarantee) on the probability of the true risk R(Q) for a random predictor Q. This requires that we determine the probability of the hypothesis space before using the set S of training data. The distribution P in , and is suitable for The true risk R(Q) for each distribution Q on . In the following, according to the traditional glossary, in the PAC Bayesian environment P is also called the prior distribution (i.e. the previous distribution) and Q is also called the posterior distribution (i.e. the later distribution). The bound contains the following term, which can be interpreted as the complexity of the assumed distribution Q, namely the KL divergence (Kullback-Leibler divergence) where the values ​​are in [0, +∞]. The bounds also include the binary KL divergence For q, p∈[0,1], it is defined, more precisely, its (upper) negation kl with respect to the second argument (for q∈[0,1], ε∈[0,∞])) -1 :

[0026] [Equation D:]kl -1(q, ε):=max{p∈[0,1]:kl(q||p)≤ε},

[0027] The second argument is equal to the explicitly determined p∈[q,1], which satisfies kl(q||p)=ε.

[0028] The right side of equation D can be found upwards through Limit, however, this formula is greater than 1 and can thus represent a trivial bound. In addition, the full PAC Bayesian principle also yields a lower bound on R(Q), which is however not important since the objective here is to minimize the upper risk limit. Although further refinements of the bounds are possible (but only marginal), the parameter-free bound from equation C is also considered here.

[0029] If a countable family P parameterized by θ∈Θ θ For GP learning with hyperparameters, the above analysis can be generalized by determining the probability distribution p with respect to Θ θ And the prior distribution is defined as P: =∑ θ p θ P θ If Θ is a finite set, then the uniform distribution It is a typical choice. If you use the following situations: Applicable to each θ∈Θ, the PAC Bayesian principle is obtained about S~μ N With a probability of at least 1-δ, the following equation applies:

[0030] [Equation E:]

[0031] The bound from equation E applies simultaneously to all P θ and all Q. Therefore, we can optimize not only with respect to θ but also with respect to Q in order to obtain the best generalization guarantee with a confidence of at least 1-δ. B(Q) is used here for training.

[0032] An exemplary application of the present invention is, for example, the modeling of emissions of a vehicle in real driving operation (RDE emissions). In this case, a reliable guarantee for the emissions can be obtained, that is, it can be ensured that the emissions do not exceed the limit value G with a high probability.

[0033] Another application is, for example, the evaluation of autonomous vehicles. Instead of emissions, a quantitative measure (Maβ) of the critical state is evaluated in this case, which limit values ​​must not be exceeded. This measure indicates, for example, what state the individual systems of the vehicle have, for example a minimum distance to the edge of the road (e.g. 0.1 m) or a minimum distance to any other vehicle (e.g. 0.2 m) must not be exceeded.

[0034] Another application is, for example, the prediction of the charge state of a battery or the remaining operating time of a battery based on historical data, such as, for example, usage since the last charging cycle, or environmental data, such as, for example, temperature, load or charging current.

[0035] Another application is, for example, the production of windshield wipers, where one of the predictors Q is trained using a large amount of data from the production line. Thus, it is possible to predict with high reliability how stably the production line will work in the next period of time (for example, in the next 30 minutes).

[0036] Therefore, in one aspect of the invention, a method is proposed for determining emission values ​​of a motor vehicle that is at least partially driven by an internal combustion engine during actual driving operation (English: Real Driving Emissions (RDE)). In this case, a trajectory is provided that characterizes the driving cycle of the motor vehicle. The trajectory can be actually recorded by means of sensors or can be computer-generated. It is now decisive for the reliable determination of the emission values ​​to provide a Gaussian process that has been trained by means of the described learning method. This makes it possible to guarantee the determined emission values.

[0037] For this purpose, the corresponding emission values ​​of the motor vehicle when driving through the provided trajectory are determined by means of the GP, and the emission values ​​in the actual driving operation are determined based on the determined emission values. That is, the provided trajectory is preferably a statistically suitable sample of the trajectories that are actually possible in the actual driving operation, so that the emission values ​​in the actual driving operation can be determined particularly reliably.

[0038] On the other hand, a method for determining the reliability of a technical device is proposed. In this case, an operating cycle of the technical device, that is, a time series of operating states of the technical device, is provided. Reliability can be, for example, a parameter that characterizes the failure safety of the technical device, or a parameter that characterizes the operational safety of the technical device. Now, decisive for reliably determining the reliability is to provide a Gaussian process that has been trained with the aid of the described learning method. The Gaussian process is configured to determine the following parameters based on the trajectory supplied to it, which parameters characterize the reliability of the technical device when it travels over the corresponding trajectory supplied to the technical device. With the aid of the GP, parameters are determined when traveling over the provided trajectory and reliability is determined based on the determined parameters. Due to the training method used, it is possible to make safety assertions about the reliability determined in this way.

[0039] In another aspect, a method for operating a technical installation can then be provided in that at least one of the provided trajectories is a trajectory actually traveled by the technical installation and the technical installation is actuated accordingly depending on the determined reliability.

[0040] If it is determined that the determined reliability indicates a fear of a malfunction of the technical installation, for example because the determined value characterizing the reliability leaves a predefinable range characterizing “safety”, the technical installation can be switched into a safe operating mode. BRIEF DESCRIPTION OF THE DRAWINGS

[0041] Embodiments of the present invention are shown in the drawings and are explained in detail in the following description.

[0042] Figure 1 A flow chart 100 of a method for training a Gaussian process according to the present invention is shown. DETAILED DESCRIPTION

[0043] According to the above explanation, in the regression A few examples of loss functions in :

[0044]

[0045] for several selected ε>0; and

[0046] For several selected functions When classifying, an exemplary loss function is (In the binary case y∈{-1,+1}).

[0047] It is advantageous here if it is possible to calculate or efficiently approximate, for example analytically or with sufficient numerical accuracy (for all fixed x, y) and its univariate Gaussian distribution with respect to v∈ The derivative of with respect to the mean a and variance b of . This can be achieved by using the error function for all the above loss functions.

[0048] according to Figure 1 , when using GP modeling, through the positive definite kernel K: And the mean function m on the input set X: Specifying a prior distribution In the traditional full GP regression, the initial training data is assumed to be With independent and identically distributed Gaussian probabilities Yes N =(f(x 1 ), ..., f(x N )) with a perturbation version, and then choose the learned distribution Q as the Bayesian posterior distribution. Under this assumption, Q is GP again:

[0049] [Equation F:]

[0050]

[0051] in and m N =(m(x 1 ), ..., m(x N )). Use equation F so that for new input values ​​x * ∈X for f(x * )'s (random) prediction.

[0052] In the context of the PAC Bayesian bound, it is important that for each complete GP with a prior distribution P and a corresponding posterior distribution Q according to equation F, the KL divergence KL(Q||P) appearing in the above principle and in equation F can be determined from a finite-dimensional (N-dimensional) matrix. This enables efficient determination of the PAC Bayesian bound and training of the GP based on this by optimization of the PAC Bayesian bound. More precisely, it can be easily shown that such P and Q have the same conditional distribution P(f|f N )=Q(f|f N ), so we can directly calculate This is different from the equation F where Q(f) and y N or n No. Therefore, it applies

[0053] [Equation G:] KL(Q||P)=KL(Q(f N )Q(f|f N )||P(f N )P(f|f N ))=KL(Q(f N )||P(f N ))

[0054] [Equation H:] In the final step, the normal distribution is used and The known formula for the KL divergence between and performs a simple simplification.

[0055] The training goal of the full GP is to The family is parameterized by the hyperparameter θ and the noise level σ n Choose "good" values. These values ​​are then used to use the corresponding posterior distribution from equation F Make predictions. The quadratic exponential kernel can be used in this case, where is the signal variance and l i is the length scale and sets the mean function to zero. The hyperparameters are (SE-ARD-core) or when all length scales l 1 =…=l d ≡l is assumed to be equal (non-ARD)

[0056] The method now consists in that the parameters θ and σ n By making the upper limit from equation E Minimize to train, and thus select the GP predictor with the best generalization performance guarantee The generalization performance is guaranteed by the PAC Bayesian bound. In this case, σ n is not a hyperparameter, because the prior distribution P θ With σ n is irrelevant, and σ n Therefore, enhancing the expression ln|Θ| does not contribute; rather, the parameter σ n is the posterior distribution The two terms are free parameters in and (from Eq. H) and its derivatives can be computed efficiently, so that gradient-based optimization can be used to find n The right side of equation F minimize.

[0057] The trained hyperparameter θ can be selected from a discrete set θ, which must be specified independently of the training data S. This is achieved by: ln θ = (ln θ 1 , ..., ln θ T Each of the T components of ) is discretized into a grid-like set of (G+1) elements after gradient-based minimization. The next point makes the optimization objective Use in The SE-ARD kernel includes a T=d+1 parameter, while the standard SE kernel includes a T=2 parameter. For example, each component of ln Θ can be rounded to two decimal places in the range [-6, +6], i.e., L=6, G=1200. Such discretization is often useful for has a negligible effect on the prediction quality of , while coarser discretization (i.e., smaller |Θ|) often improves neither the bound nor the optimization significantly.

[0058] Examples of kernel functions, mean functions, or discrete sets of hyperparameters are as follows, where Where the input dimension d∈-m(x)=0 (standard mean function).

[0059] -m(x) through the relationship The physical model is given.

[0060] The hyperparameters are And ||·|| means The Euclidean distance (or Mahalanobis distance) in .

[0061] The hyperparameters are

[0062] -For discretization Θ, each component θ i It can be discretized to θ as follows:

[0063] Where L>0 and G∈, for example L=6, G=1201, such that |Θ|≤7.1×(the number of components of θ), where |Θ| denotes the size (cardinality) of the discretization set Θ.

[0064] It should be noted that the choices made so far should be taken before the training data set is known in order to guarantee consistent learning results. Therefore, parts of the training data set that should not be used for training later can be considered. The choices made can also be related to S, but the learned hypothesis may not obey the consistent guarantees suggested by the training objective.

[0065] Despite the fact that the bound in Eq. E is fitted with confidence 1-δ to each P from the a priori GP family θ and each distribution Q, but in the previous explanation we used P from equation F θ and the corresponding After that, the upper limit is only about the parameters θ, σ n Optimization. Efficiently compute KL(Q||P) (which depends on the property Q(f|f N )=P(f|f N ) and P(f N ) and Q(f N ) (see Eqs. G and H)). However, based on these two requirements, a general pair P, Q of GPs can be considered with an efficiently computable KL(Q||P), allowing for a wider range of application of learning methods, in particular also for sparse GP methods.

[0066] Instead of the point x obtained from the training set S as above 1 , ..., x N , we can select any number M of points from the input space These points are also called induced points. In addition, each Gaussian distribution According to the function value f M :=(f(z 1 ), ..., f(z M)) to select, each of which And the positive semidefinite matrix About f M The distribution Q of can be obtained by using the conditional distribution Q(f|f) from the prior distribution M )=P(f|f M ) and extends to all input parameters from X (see above). This gives the following predictive GP:

[0067] [Equation I:]

[0068]

[0069] in and m M :=(m(z 1 ), ..., m(z M )). With similar reasoning as for equations G and H we obtain:

[0070] [Equation J:]

[0071]

[0072] Therefore, the prior distribution P can be effectively optimized in Equation E θ and the associated posterior distribution The method is: the number M of sensing points and the position z 1 , ..., z M And parameter a M and B MM The optimization can be organized using the Sherman-Morrison-Woodbury formula to make it O(N) with respect to the full GP as described above. 3 ) and O(N 2 ), each gradient step takes time O(NM 2 +M 3 ) and memory O(NM+M 2 ). For M<<N, a large saving occurs.

[0073] Several traditional sparse GP methods are special cases of the above formula, in which a is selected based on the training data set S. M and B MM , so that only the sensing point z can be freely selected 1 , ..., z M and several other parameters, such as

[0074] [Equation K:] in in and Λ=diag(λ 1 , ..., λ N ) is a diagonal N×N matrix whose entries are If α=1 is set, this corresponds to the FITC approximation, while α=0 leads to a method based on VFE and DTC. Alternatively, it is also possible to interpolate linearly between the two options using α. Another form of sparse GP corresponds to B MM = 0, the potential function value f in the sparse GP M is fixed and not marginalized by it, however this leads to a divergent KL(Q||P)=∞ via equation J and thus to trivial bounds in equations C and E.

[0075] The learning method for sparse GPs then follows the same steps as described above: for the hyperparameters θ appropriately discretized after optimizing E, the augmentation term must be estimated It should be noted here that θ only contains the prior hyperparameters but does not include the parameter z from equation K 1 , ..., z M , a M , B MM , σ n , α, the equation K is also optimized. The number M can also vary and determine the computing resources. In addition, the optimization can be carried out not only discretely or continuously. MM When optimizing, you can use parameterized B MM =LL T Associated lower triangular matrix

[0076] An example of an optimization objective and an optimization method is for example a gradient-based optimization of B(Q) with respect to continuous parameters θ, ρ, wherein θ must nevertheless be specified in a predefined set θ. In this case, there is no need to discretize the obtained values ​​ρ. Other selection methods for θ, ρ can also be applied: each choice leads to an effective upper bound on the generalization risk R(Q) (see equation B). The overall example for the above example of the loss function and its derivatives is R(Q). S (Q) is useful. It is also possible to calculate or analyze the derivative kl -1 (for gradient-based optimization). Alternatively, a relaxed (i.e., larger) target such as B can be used. Pin (Q), not B(Q).

[0077] In the application of the present invention, when modeling the emissions of a vehicle in real driving operation (RDE emissions), a reliable guarantee of the emissions is obtained, for example, under the following parameters, i.e., it is ensured that the emissions do not exceed the limit value G with a high probability:

[0078] The above method can be combined with the loss function , in order to give a strict guarantee of RDE emissions in a typical driving cycle. The RDE emissions may vary about a selected threshold G. In this case, a number D of driving cycles are required in a first step, which are consistent with the driving cycle distribution (with an accuracy of δ TV , for example over the entire variation distance) is typical for which the RDE validation is to be performed. These driving cycles can originate from a previously recorded set of driving cycles and / or be generated therefrom. The method enables the RDE validation to be continued without having to measure the emissions during actual test drives (actual test drives can only be carried out to a limited extent for each specific engine arrangement). The method benefits from the availability of a plurality of typical driving cycles in step 1 and in step 2 when simulating the emissions.

[0079] The GP is trained using a training set, which is specific to the engine arrangement to be tested, which may include steps 110 to 150 of the method. The GP Q is output, which may output the (related) emissions for each driving cycle x. ε represents the GP modeling accuracy (loss function ) and δ∈(0, 1] represents the confidence parameter used for modeling.

[0080] In step 1, a number D of driving cycles x are now selected, for example from a previously recorded set or generated by a driving cycle generator. 1 , ..., x D , the driving cycle is typical for the RDE driving cycle distribution with accuracy δ TV . Then, a threshold value G is selected, such as a target emission value.

[0081] In step 2, which may include multiple implementations of step 160, for a driving cycle x 1 , ..., x D Each of the GPs uses GP to predict emissions

[0082] In step 3, including the analysis and determination of the guarantee, using statistical standard techniques (in particular Clopper-Pearson Lemma) it is possible to determine from a number d of predicted emissions exceeding the value G-ε D, δ, δ TV and other confidence levels δ CPA strict guarantee (upper limit) for the portion of a typical driving cycle that violates the RDE requirement is given in TV , δ CP For hours, this guarantee can exist with high probability.

[0083] When evaluating the autonomous vehicle, a quantitative measure of the critical state is evaluated, which must not exceed a limit value. This indicates, for example, what state the individual systems of the vehicle have, such as not falling below a minimum distance to the edge of the road (e.g. 0.1 m) or a minimum distance to any other vehicle (e.g. 0.2 m).

[0084] In this case, in contrast to the above, the emission is replaced by a quantitative measure of the risk (ie the actual emission value y, the predicted emission value and emission threshold G), which quantitative measure indicates, for example, how close the autonomous system (or one of its subsystems) is to a fault state. Such a measure must be selected taking into account the specific application, such as a minimum distance to the edge of the road, which must not fall below a certain threshold, such as 0.1 meters, in each driving cycle. Or a minimum distance to any other vehicle on the road, which must not fall below a certain threshold, such as 0.2 meters, in each driving cycle.

[0085] The risk value can be predicted for a number D of selected driving cycles, so that a strict guarantee for the risk value can be obtained by the GP method (up to ±ε and with probability ≥1-δ). The guarantee can then also be obtained by statistical standard techniques (especially Clopper-Pearson Lemma).

[0086] The GP is trained using a system-specific training set for the autonomous driving to be tested, which can include steps 110 to 150 of the method. A GP Q is output, which can output a risk value for each driving cycle x. ε represents the GP modeling accuracy (loss function ), and δ∈(0, 1] represents the confidence parameter used for GP modeling.

[0087] In step 1, a number D of driving cycles x are now selected, for example from a previously recorded set or generated by a driving cycle generator. 1 , ..., x D , the driving cycle is a typical driving cycle with an accuracy of δ TV Subsequently, a threshold G is selected, for example representing a risk measure.

[0088] In step 2, which may include multiple implementations of step 160, for a driving cycle x 1 , ..., x D Each of the GPs in order to predict the risk value

[0089] In step 3, including the analysis and determination of the guarantee, using statistical standard techniques (in particular Clopper-Pearson Lemma) it is possible to determine from a number d of predicted emissions exceeding the value G-ε D, δ, δ TV and other confidence levels δ CP A strict guarantee (upper limit) is given in for the part of the typical driving cycle that exceeds the dangerous value G. When δ, δ TV , δ CP For hours, this guarantee can exist with high probability.

[0090] The method can also be used to predict the charge state of a battery pack or its remaining operating duration based on historical data such as usage since the last charging cycle or environmental data such as temperature, load or charging current.

[0091] To this end, the GP Q is trained based on a plurality (N) of measurements of input values ​​x (e.g. battery pack history, environmental parameters) and corresponding output values ​​y (e.g. state of charge, remaining operating time), which can include steps 110 to 150 of the method. As a loss function, With accuracy target ε and desired confidence parameter δ. This training step is only performed at the beginning. Subsequent steps can be implemented as a loop until the battery pack operation duration ends.

[0092] In step 1, in a specific application such as, for example, a battery-operated vehicle, an input value x is read in, for example, from a sensor or a memory device. Such an input value is received repeatedly during the method runtime in a conventional manner.

[0093] In step 2, the output value is predicted from the input value x by the trained GP

[0094] In step 3, if the predicted If a previously selected threshold value G is exceeded or fallen below, a signal (e.g. a warning light) is given in order to warn the user, who can change the battery load by, for example, switching off consumers (such as a radio or air conditioning system) or switching to or connecting to an alternative energy source (such as an internal combustion engine in a hybrid vehicle).

[0095] The nature of the GP learning algorithm performed before step 1 ensures that, for each prediction The maximum error value of the algorithm with the minimum probability 1-δ is ±ε. Therefore, a statistical guarantee can be given so that if the actual battery state (i.e., the actual output value y) is lower than G-ε, the process triggered in step 3 is not performed, and conversely, if the battery state is higher than G+ε, the process is triggered in step 3. Therefore, it can be ensured that the system works as specified.

[0096] In the production of windshield wipers, a predictor Q is trained with a large amount of data from the production line. From this, it is possible to predict with high reliability how stable the production line will work in the next time period (e.g., in the next 30 minutes). Therefore, the method described above for GP modeling (which provides provable statistical guarantees about the prediction performance in the presence of new data) can be used to train a GP predictor Q from a large amount of production line data (from, for example, windshield wiper production) to predict how stable the production line will work in the upcoming time period (e.g., the next 30 minutes). When the production line is predicted to work sufficiently stably (i.e., with a sufficiently large spacing ε from the GP loss function and a sufficiently large confidence value 1-δ from the PAC Bayesian target), the monitoring personnel can reduce their attention or maintenance efforts and track other operations or insert pauses.

[0097] Since standstills are expensive, reducing the monitoring effort is only considered when there are strict guarantees about the correct way the production line is running. This is achieved by the learning method described above. In this case, the asymmetry in the tasks should be noted: if the production line fails, a reduction in monitoring can result in potentially high one-time costs, while continuous monitoring has much less catastrophic consequences even if the production line is running perfectly.

[0098] To this end, the GP Q is trained in a first step based on a plurality (N) of measurements of input values ​​x (e.g. temperature T and parameters κ related to the production machine) and corresponding output values ​​y (e.g. quality of the produced component), which can include steps 110 to 150 of the method. As a loss function, it is possible to use With an accuracy target ε and a desired confidence parameter δ. This training step is only performed at the beginning, but can be repeated later in a loop with a larger data base if necessary.

[0099] In step 2, the trained GP is trained on the current predicted temperature development and for different configurable parameters κ i is analyzed in order to find the optimal (good) parameter setting κ for the next M glass sheets, so as to achieve (i) a low predicted defect rate (GP means) and (ii) a low guaranteed loss for the trained GP (ie, applicable predictions).

[0100] In step 3 , the (human) operator is advised parameters κ for the next M glass sheets to be produced. In addition, the operator is given access to a confidence estimate as to how reliably the GP prediction can achieve an estimated low defect rate in the time window M.

[0101] If the confidence estimate reaches or exceeds a pre-set value, for example a green light may be illuminated. The operator can now decide, based on the confidence estimate, whether the machine can be left unattended.

[0102] In step 4, after the machine is operating as set and producing new parts, the quality of the new parts can be measured, and the measurements y can be used as additional data to re- or further train the GP with the enlarged training data. Thus, the GP predictions get better over time, especially in parts of the parameter space that the GP did not initially predict accurately or reliably.

[0103] Then, either step 1 can be followed, or steps 2 and 3 can be followed again in a loop.

Claims

1. A computer-implemented method for determining emission values ​​of a motor vehicle driven at least partially by an internal combustion engine during actual driving operation, wherein a trajectory is provided which characterizes a driving cycle of the motor vehicle, and wherein a Gaussian process GP is provided, which determines the emissions of the motor vehicle as a function of the trajectory supplied to it when the motor vehicle travels over the corresponding trajectory supplied to the motor vehicle, and wherein the corresponding emission value of the motor vehicle is determined as a function of the emission value determined during the actual driving operation, characterized in that The GP is trained from the GP parameterization by optimizing a PAC Bayesian bound (150) using a training dataset, a priori GP family, and an accuracy target, wherein optimization of the PAC Bayesian bound comprises minimizing an upper bound on the true risk of the GP predictor.

2. A computer-implemented method for determining the reliability of a technical device, wherein a trajectory is provided that characterizes an operating cycle of the technical device, and wherein a Gaussian process GP is provided, wherein the Gaussian process determines the following parameters based on the trajectory supplied to it, which parameters characterize the reliability of the technical device when traveling over the corresponding trajectory supplied to the technical device, and wherein the parameters are determined with the aid of the GP in each case based on the provided trajectory, and wherein the reliability is determined based on the determined parameters, characterized in that The GP is trained from the GP parameterization by optimizing a PAC Bayesian bound (150) using a training dataset, a priori GP family, and an accuracy target, wherein optimization of the PAC Bayesian bound comprises minimizing an upper bound on the true risk of the GP predictor.

3. A computer-implemented method according to claim 2, wherein the technical device is a motor vehicle assembly and the motor vehicle assembly includes a system for at least semi-autonomous operation of the motor vehicle and / or a battery pack for feeding an electric motor of the motor vehicle.

4. The computer-implemented method of claim 2, wherein the technical equipment is a production machine. 5 . The computer-implemented method of claim 2 , wherein the operating cycle of the technical device is a driving cycle of a motor vehicle.

6. The computer-implemented method (100) according to any one of claims 1 to 5, wherein the method (100) further comprises the steps of: Select an accuracy target (110) before training the GP; Selecting a priori GP family (120) before training the GP, wherein the prior GP family is parameterized by a hyperparameter; Obtaining a training data set (130) before training the GP; Selecting a GP parameterization for modeling (140) before training the GP; In response to a new input value x * The next expected value is obtained by predicting (160) the trained GP.

7. A computer-implemented method (100) according to any one of claims 1 to 5, wherein the accuracy target includes a loss function and a confidence parameter, wherein the loss function is related to a modeling task, wherein the modeling task is one of classification, regression or other tasks of supervised learning.

8. The computer-implemented method (100) of claim 6, wherein the GP parameterization used for modeling (140) is one of a full GP, a free sparse GP, or a special parameterization of a sparse GP.

9. The computer-implemented method (100) of claim 8, wherein the specific parameterization of the sparse GP is a deterministic training condition DTC, a fully independent training condition FITC or a variational free energy VFE.

10. The computer-implemented method (100) of claim 6, wherein the hyperparameters parameterize the kernel function and the mean function and the possible values ​​of the hyperparameters are in a predefined table.

11. The computer-implemented method (100) of claim 6, wherein the value predicted by the GP is one of: The vehicle's emissions during actual driving operation, Quantitative value of dangerous state of autonomous vehicle, a value relating to the state of charge of the battery pack or the remaining operating duration of the battery pack, and The value involved in production.

12. The computer-implemented method (100) of claim 11, wherein the production is the production of windshield wipers.

13. A method for operating a technical device, wherein the reliability of the technical device is determined using a method according to any one of claims 2 to 5, wherein at least one of the provided trajectories includes a trajectory actually traveled by the technical device, and wherein the technical device is controlled accordingly based on the determined reliability. 14 . The method according to claim 13 , wherein the technical installation is switched into a safe operating mode if the determined reliability indicates that there is a fear of a failure of the technical installation.

15. Computer program product comprising instructions arranged to implement the method according to any one of claims 1 to 14 when the computer program is executed on a computer. 16 . A machine-readable storage medium having a computer program stored thereon, the computer program comprising instructions, the instructions being configured to implement the method according to claim 1 when the computer program is executed on a computer.

17. Apparatus arranged to carry out the method according to any one of claims 1 to 14.

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