This paper focuses on the application of an efficient implementation of the vector-valued kernel Ridge regression (KRR) to the uncertainty quantification (UQ) of the scattering parameters of a low-noise amplifier (LNA). Specifically, the performance of the proposed technique have been investigated for the statistical assessment of the mean value, variance and probability density function (PDF) of the S11 and S21 parameters of a 2-GHz LNA induced by 25 stochastic input parameters and compared with the corresponding reference results computed via a plain Monte Carlo (MC) simulation

Manfredi, Paolo

Soleimani, Nastaran

Trinchero, Riccardo

English

PORTO@iris (Publications Open Repository TOrino - Politecnico di Torino)

17 October 2023POLITECNICO DI TORINORepository ISTITUZIONALEEfficient Implementation of the Vector-Valued Kernel Ridge Regression for the Uncertainty Quantification of theScattering Parameters of a 2-GHz Low-Noise Amplifier / Soleimani, Nastaran; Manfredi, Paolo; Trinchero, Riccardo. -ELETTRONICO. - (2023), pp. 143-146. (Intervento presentato al convegno IEEE MTT-S International Conference onNumerical Electromagnetic and Multiphysics Modeling and Optimization (NEMO'2023) tenutosi a Winnipeg, MB, Canadanel 28-30 June 2023).OriginalEfficient Implementation of the Vector-Valued Kernel Ridge Regression for the UncertaintyQuantification of the Scattering Parameters of a 2-GHz Low-Noise AmplifierIEEE postprint/Author's Accepted ManuscriptPublisher:PublishedDOI:Terms of use:Publisher copyright©2023 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in anycurrent or future media, including reprinting/republishing this material for advertising or promotional purposes, creatingnew collecting works, for resale or lists, or reuse of any copyrighted component of this work in other works.(Article begins on next page)This article is made available under terms and conditions as specified in the  corresponding bibliographic description inthe repositoryAvailability:This version is available at: 11583/2980404 since: 2023-07-26T08:36:40ZIEEE MTT-S International Conference on Numerical Electromagnetic and Multiphysics Modeling andEfficient Implementation of the Vector-Valued Kernel Ridge Regressionfor the Uncertainty Quantification of the Scattering Parameters of a2-GHz Low-Noise AmplifierNastaran Soleimani, Paolo Manfredi and Riccardo TrincheroPolitecnico di Torino, Torino, Italy; e-mail: {nastaran.soleimani,paolo.manfredi,riccardo.trinchero}@polito.itAbstract—This paper focuses on the application of an efficientimplementation of the vector-valued kernel Ridge regression(KRR) to the uncertainty quantification (UQ) of the scatteringparameters of a low-noise amplifier (LNA). Specifically, theperformance of the proposed technique have been investigatedfor the statistical assessment of the mean value, varianceand probability density function (PDF) of the S11 and S21parameters of a 2-GHz LNA induced by 25 stochastic inputparameters and compared with the corresponding referenceresults computed via a plain Monte Carlo (MC) simulation.Index Terms—Machine learning, kernel machine, vector-valued KRR, stochastic analysis, amplifiers.I. INTRODUCTIONUncertainty quantification (UQ) represents an essentialtask in the early-stage design of microwave components anddevices, since it allows to account for the impact of processvariation in modern mass-production electronics. In the abovescenario, Monte Carlo (MC) simulation can be seen as themost straightforward way to deal with the inherently statisticalnature of the problem at hand. The MC sampling allowsanalyzing the impact of random input variables on a givenset of outputs of interest by considering the results of a largenumber of deterministic parametric simulations (usually inthe order of thousands), in which the value of the stochasticparameters is drawn according to their probability densityfunction (PDF). Such brute force approach turns out to beextremely robust and easy to implement within the simulationflow used by most of the commercial circuital and full-wavesolvers, but it is also characterized by a low efficiency andby a non-negligible computational cost [1].In the last decades, surrogate modeling techniques havebeen proposed for the UQ of the responses of electronicdevices and circuits as more efficient alternatives to directMC sampling [2]–[9]. A surrogate model is “a model of amodel” acting as a closed-form and fast-to-evaluate emulatorof the input-output map characterizing the true underlyingmodel, also known as computational model. A surrogatemodel is built by solving a regression problem based onthe information provided by a small set of training samplescalculated via the computationally expensive computationalmodel. After the training, the surrogate model can be usedas an efficient alternative to the plain computational modelin computationally expensive task such as the UQ and opti-mization. It is important to stress that the performance of theresulting surrogate model, in terms of accuracy with respect ofthe number of training samples, unavoidably depend on themodel structure and the regression technique used to buildit [1].In the above framework, kernel-based machine learning re-gressions have shown several advantages with respect to stan-dard regression techniques relying on basis expansion (e.g.,ordinary least-squares, ridge and LASSO regressions, poly-nomial chaos [7]), since they allow building non-parametricsurrogate models in which the number of unknowns is in-dependent from the number of input parameters consideredby the model [1], [5]. Moreover, they rely on a linear modelstructure in which the model unknowns can be estimated fromthe solution of a convex optimization problem, with severalbenefits in term of training time and accuracy, when a smallset of training samples is available [3], [6]. On the other hand,different from artificial neural network structures, state-of-the-art kernel machine regressions are limited to scalar-outputproblem, thus making their direct application to vector-valuedregression problems, usually encountered in microwave andelectronic applications, rather cumbersome [5]. A possibleway to overcome the above limitation is to combine single-output kernel machine regressions with data compressiontechniques, like the principal component analysis (PCA) [8].The underlying idea is to apply a compression technique onthe available data in order to reduce the number of outputcomponents to be modeled and thus the number of single-output regression to be trained. Such approach has shownpromising results in UQ problems [8], [9].As an alternative to the above techniques, this paper inves-tigates the performance in the UQ scenario of a generalizedvector-valued formulation of the kernel Ridge regression(KRR) able to directly account for the multi-output nature ofthe regression problem, without requiring any data manipula-tion or compression [4], [5], [10], [11], [12]. Specifically, thispaper deals with the development an efficient implementationof the vector-valued KRR based on a diagonalization processapplied on its constitutive equations [11]. The effectivenessand the performance of the proposed technique are inves-tigated for the UQ of the scattering parameters of a low-noise amplifier (LNA) affected by 25 stochastic parametersand compared with the ones obtained by a MC simulation.II. VECTOR-VALUED KERNEL RIDGE REGRESSIONLet us consider the problem of building a generic vector-valued surrogate model f̂ : X → Y, starting from theinformation available on the training set D = {(xl, y l}Ll=1,where xl ∈ X ⊆ Rp represents the generic l-th configurationof the input parameters and y l ∈ Y ⊆ RD represents the cor-responding vector-valued output. The above problem turns outto be equivalent to learn D scalar functions f̂ (d) : X → R withd = 1, . . . , D, such that f̂ = [f̂ (1), . . . , f̂ (D)]T , minimizingthe following empirical risk functional:f̂ = argminf̃∈HD∑d=1L∑l=1(y(d)l − f̃(d)(xl))2 + λ∥f̃ ∥2H (1)where λ is the regularizer hyperparameter providing a trade-off between the model flatness and accuracy on the trainingset, whilst y(d)l and f̃(d)(xl) represent the d-th componentof the l-th training output and the corresponding modelprediction, respectively.According to the representer theorem for vector-valuedregression problems presented in [12], any optimal solution f̂of (1) for a generic d′-th output component, takes the form:f̂ (d′)(x) =D∑d=1L∑l=1k ((x, d′), (xl, d)) cd,l, (2)where cd,l are unknown coefficients to be estimated dur-ing the model training and k((x, d′), (xl, d)) : Rp×p ×R{1,...,D}×{1,...,D} → R represents a new kernel functionacting on both the input parameters x ∈ Rp and the outputdimensions {1, . . . , D}. The above model can be rewritten inits matrix formulation, which writes:f̂ (x) =L∑l=1K(x,xl)cl, (3)where K(·, ·) : Rp×p → RD×D is the multi-output kernelmatrix and cl = [c1,l, . . . , cD,l]T ∈ RD are column vectorscollecting the regression unknowns.Without loss of generality, the matrix kernel functionK(x,x′) can be built as a separable kernel function definedas the product between two scalar kernels acting either on theinput space or output dimensions, such that:[K(x,x′)][d,d′] = k((x, d), (xl, d′))= kx(x,x′)ko(d, d′), (4)where kx and ko are scalar kernels acting on the input space(i.e., kx : X × X → R) and output dimensions (i.e., ko :{1, . . . , D} × {1, . . . , D} → R). In our implementation, aradial basis function (RBF) kernel is used for the scalar kernelkx and ko, respectively.By using (3) and (4), the empirical functional minimizationin (1) can be recast in terms of the following discrete-timeSylvester equation:KxCB+ λC = Y, (5)where Kx is a L × L Gram matrix computed from theinput samples {xl}Ll=1 (i.e., [Kx ]ij = kx(xi,xj)), B is aD × D Gram matrix computed on the output dimensions{1, . . . , D} (i.e., [B]ij = ko(di, dj)), C = [c1, . . . , cL]T ∈RL×D is a matrix collecting the model unknowns and Y =[y1, . . . , yL]T is a L × D matrix associated to the trainingoutput.By using the properties of the Kronecker product, thesolution of the discrete-time Sylvester in (5) can be obtainedas the solution of a linear system [13]. Unfortunately, thisapproach leads to a huge linear system with (LD) × (LD)equations, whose inversion would require a computationalcost proportional to O(L3D3).Such a prohibitive training cost can be heavily reduced byapplying a diagonalization procedure on the kernel matricesKx and B [11], i.e.,Kx = UΛUT and B = TMTT , (6)where U ∈ RL×L and T ∈ RD×D are matrices collecting theeigenvectors of the matrices Kx and B, respectively, whereasΛ ∈ RL×L and M ∈ RD×D are diagonal matrices collectingthe corresponding eigenvalues.Substituting (6) into the original Sylvester equation in (5),leads to the following formulation:ΛC̃M+ λC̃ = Ỹ (7)where C̃ = UTCT and Ỹ = UTYT are new transformedmatrices collecting a transformed version of regression un-knowns and source terms.Due to the diagonal structure of (7), a generic ij-entry ofthe unknown matrix [C̃]ij = cij can be suitably computedvia a scalar equation using the diagonal eigenvalue matricesΛ and M, as:c̃ij =ỹij[Λ]ii[M]jj + λ. (8)Once all the entries of the matrix C̃ij has been computedvia (8), the original unknown matrix C is reconstructed as:C = UC̃TT . (9)For the case of large multi-output kernel matrices, theabove diagonalization procedure for solving the discrete-timeSylvester equation turns out to more efficient than the equiv-alent vectorized solution based on the Kronecker formulationpresented in [4], [5]. Indeed, since the diagonalization isapplied on the matrices Kx and B separately, the overallcomputational cost required for the model training reducesfrom O(L3D3) to O(L3+D3+L2D+LD2), with a beneficialeffect on the training time when the product L×D is large.III. APPLICATION EXAMPLE: 2-GHZ LNAThe performance of the proposed technique has beenevaluated on the UQ of the scattering parameters of the2-GHz LNA shown in Fig. 1 by considering 25 Gaussianstochastic variables affecting the parasitic resistances, capac-itances and inductances of the BJT, its forward current gain,4.7 pF5.6 pF100 nF 100 ΩBFG425W15 kΩ5.6 pF 1 nF22 Ω 82 Ω2.7 pFRF in50Ω+4.5 VRF out50ΩTL1TL2TL3TL4 TL4Fig. 1. 2-GHz BJT LNA.TABLE ITRAINING TIME AND RELATIVE L2-ERROR COMPUTED FROM THEFREQUENCY-DOMAIN SAMPLES OF THE PREDICTIONS OF THESURROGATE MODELS TRAINED WITH AN INCREASING NUMBER OFTRAINING SAMPLES VIA THE PROPOSED IMPLEMENTATION OF THEVECTOR-VALUED KRR FOR THE S11 AND S21 PARAMETERS BYCONSIDERING 1000 TEST SAMPLES.ParameterL = 30 L = 50 L = 100ttrain L2-error ttrain L2-error ttrain L2-errorS11 74s 6.15% 115s 3.58% 205s 2.70%S21 78s 1.96% 118s 1.54% 216s 1.16%all the lumped components in the amplifier schematic, andthe widths of the microstrip lines, each with a 10% relativestandard deviation [7], [9]. The considered test case hasbeen implemented as a parametric small-signal ac analysisin HSPICE. Such implementation allows computing the two-port scattering parameters S11 and S21 of the LNA at 201frequency points for any configuration of the 25 random pa-rameters. The HSPICE simulations have been used to generatethree training sets with an increasing number of samples (i.e.,L = 30, 50, and 100) based on a latin hypercube samplingscheme and to run a 1000-sample MC simulation that is usedhereafter as a reference for the proposed statistical analysis.Two independent surrogate models, one for S11 and one forS21, have been trained via the proposed implementation ofthe vector-valued KRR presented in Section II.Table I reports the performance of the obtained surrogatemodels in terms of training time and relative L2-norm errorcomputed on a test set collecting the results of a 1000-sampleMC simulation for the parameters S11 and S21 by consideringan increasing number of the training samples (i.e., L = 30,50, and 100). The results show a constant reduction of themodel error (i.e., the relative L2-error) with respect to thenumber of training samples (i.e., L), thus highlighting thecapability of the proposed vector-valued KRR of learning theactual information provided by the training set. Concerningthe computational cost, the training time required to buildthe proposed vector-valued surrogate models is less than7 min for all the considered modeling scenarios. After theFig. 2. S11 parameter of the LNA in Fig. 1. Top panel: the gray lines showthe magnitude of the S11 computed from a 1000-sample MC simulation.The blue solid and the red dashed lines are the magnitude of the averageS11 obtained from the MC samples and the proposed model, respectively.Bottom panel: the blue solid and the red dashed lines are the magnitude ofthe S11 variance obtained from the MC samples and the proposed models,respectively.Fig. 3. S21 parameter of the LNA in Fig. 1. Top panel: the gray lines showthe magnitude of the S21 computed from a 1000-sample MC simulation.The blue solid and the red dashed lines are the magnitude of the averageS21 obtained from the MC samples and the proposed models, respectively.Bottom panel: the blue solid and the red dashed lines are the magnitude ofthe S21 variance obtained from the MC samples and the proposed models,respectively.training, the evaluation of the obtained model on the 1000test samples required less than 20 s, while the correspondingMC simulation requires 1131 s.Figures 2 and 3 compare the mean (top panels) and thevariance (bottom panels) of the absolute value of the S11 andS21 parameters predicted by the proposed surrogate modelstrained with L = 50 training samples against the corre-sponding results computed via the MC samples. The resultsFig. 4. Comparison of among the PDFs of the S11 and S21 parameters atthe frequency f0 = 2GHz computed from the predictions of the surrogatemodels built via the proposed vector-valued KRR evaluated on 1000 testsamples and the corresponding ones computed from a 1000-sample MCsimulation.show an excellent agreement between the mean value and thevariance predicted by the proposed surrogate model and theones computed from the MC samples, being the two sets ofcurves almost perfectly overlapped. Furthermore, Fig. 4 showsthe probability density function (PDF) of the S11 and S21parameters computed at f0 = 2GHz. Also in this case, thehistograms predicted by the proposed models are in perfectagreement with the corresponding ones calculated from theMC simulation, thus confirming the excellent performance ofthe proposed approach in the UQ scenario.IV. CONCLUSIONSThis paper investigated the effectiveness and the accuracyof a surrogate model built via an efficient implementationof the vector-valued KRR for the UQ of the scatteringparameter of a LNA. Specifically, the performance of theproposed technique have been investigated for the statisticalassessment of the mean value, variance, and PDF of the S11and S21 parameters of a 2-GHz LNA induced by 25 stochasticparameters and compared with the corresponding referenceresults computed via a MC simulation.REFERENCES[1] R. Trinchero and F. Canavero, “Machine Learning Regression Tech-niques for the Modeling of Complex Systems: An Overview,” IEEEElectromagnetic Compatibility Magazine, vol. 10, no. 4, pp. 71-79, 4thQuarter 2021.[2] R. Trinchero and F. G. Canavero, “Modeling of eye diagram height inhigh-speed links via support vector machine,” in Proc. of 2018 IEEE22nd Workshop on Signal and Power Integrity (SPI), Brest, 2018, pp.1–4.[3] N. Soleimani and R. 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Efficient Implementation of the Vector-Valued Kernel Ridge Regression for the Uncertainty Quantification of the Scattering Parameters of a 2-GHz Low-Noise Amplifier

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Efficient Implementation of the Vector-Valued Kernel Ridge Regression for the Uncertainty Quantification of the Scattering Parameters of a 2-GHz Low-Noise Amplifier

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