Giulio Bottegal

Estimation of building occupancy levels through environmental signals deconvolution

We address the problem of estimating the occupancy levels in rooms using the information available in standard HVAC systems. Instead of employing dedicated devices, we exploit the significant statistical correlations between the occupancy levels and the CO2 concentration, room temperature, and ventilation actuation signals in order to identify a dynamic model. The building occupancy estimation problem is formulated as a regularized deconvolution problem, where the estimated occupancy is the input that, when injected into the identified model, best explains the currently measured CO2 levels. Since occupancy levels are piecewise constant, the zero norm of occupancy is plugged into the cost function to penalize non-piecewise constant inputs. The problem then is seen as a particular case of fused-lasso estimator by relaxing the zero norm into the ℓ1 norm. We propose both online and offline estimators; the latter is shown to perform favorably compared to other data-based building occupancy estimators. Results on a real testbed show that the MSE of the proposed scheme, trained on a one-week-long dataset, is half the MSE of equivalent Neural Network (NN) or Support Vector Machine (SVM) estimation strategies.

Regularized Deconvolution-Based Approaches for Estimating Room Occupancies

We address the problem of estimating the number of people in a room using information available in standard HVAC systems. We propose an estimation scheme based on two phases. In the first phase, we assume the availability of pilot data and identify a model for the dynamic relations occurring between occupancy levels, CO2 concentration and room temperature. In the second phase, we make use of the identified model to formulate the occupancy estimation task as a deconvolution problem. In particular, we aim at obtaining an estimated occupancy pattern by trading off between adherence to the current measurements and regularity of the pattern. To achieve this goal, we employ a special instance of the so-called fused lasso estimator, which promotes piecewise constant estimates by including an ℓ1 norm-dependent term in the associated cost function. We extend the proposed estimator to include different sources of information, such as actuation of the ventilation system and door opening/closing events. We also provide conditions under which the occupancy estimator provides correct estimates within a guaranteed probability. We test the estimator running experiments on a real testbed, in order to compare it with other occupancy estimation techniques and assess the value of having additional information sources. Note to Practitioners - Home automation systems benefit from automatic recognition of human presence in the built environment. Since dedicated hardware is costly, it may be preferable to detect occupancy with software-based systems which do not require the installation of additional devices. The object of this study is the reconstruction of occupancy patterns in a room using measurements of concentration, temperature, fresh air inflow, and door opening/closing events. All these signals are information sources often available in HVAC systems of modern buildings and homes. We assess the value of such information sources in terms of their relevance in detecting occupancy in small and medium-sized rooms. The proposed estimation scheme is composed of two distinct phases. The first is a training phase where the goal is to derive a mathematical model relating the number of occupants with the concentration. It is required to record the actual occupants in the room for a time period spanning few days, a task that can be performed either with manual logging or with temporary dedicated hardware counting systems. In a second phase, we use the derived model to design an online software which collects measurements of the environmental signals and provides the number of people currently in the room. The estimated occupancy levels can then be employed to enhance the efficiency of the HVAC system of the building. We notice that, in modern residential buildings composed by structurally equal flats, the training phase can be run in one flat only, since the obtained model will be reasonably valid for the other flats.

Regularized spectrum estimation using stable spline kernels

Abstract This paper presents a new regularized kernel-based approach for the estimation of the second order moments of stationary stochastic processes. The proposed estimator is defined by a Tikhonov-type variational problem. It contains few unknown parameters which can be estimated by cross validation solving a sequence of problems whose computational complexity scales linearly with the number of noisy moments (derived from the samples of the process). The correlation functions are assumed to be summable and the hypothesis space is a reproducing kernel Hilbert space induced by the recently introduced stable spline kernel. In this way, information on the decay to zero of the functions to be reconstructed is incorporated in the estimation process. An application to the identification of transfer functions in the case of white noise as input is also presented. Numerical simulations show that the proposed method compares favorably with respect to standard nonparametric estimation algorithms that exploit an oracle-type tuning of the parameters.

A kernel-based approach to Hammerstein system identification

Abstract In this paper, we propose a novel algorithm for the identification of Hammerstein systems. Adopting a Bayesian approach, we model the impulse response of the unknown linear dynamic system as a realization of a zero-mean Gaussian process. The covariance matrix (or kernel) of this process is given by the recently introduced stable-spline kernel, which encodes information on the stability and regularity of the impulse response. The static nonlinearity of the model is identified using an Empirical Bayes approach, i.e. by maximizing the output marginal likelihood, which is obtained by integrating out the unknown impulse response. The related optimization problem is solved adopting a novel iterative scheme based on the Expectation-Maximization method, where each iteration consists in a simple sequence of update rules. Numerical experiments show that the proposed method compares favorably with a standard algorithm for Hammerstein system identification.

Robust EM kernel-based methods for linear system identification

Recent developments in system identification have brought attention to regularized kernel-based methods. This type of approach has been proven to compare favorably with classic parametric methods. However, current formulations are not robust with respect to outliers. In this paper, we introduce a novel method to robustify kernel-based system identification methods. To this end, we model the output measurement noise using random variables with heavy-tailed probability density functions (pdfs), focusing on the Laplacian and the Students t distributions. Exploiting the representation of these pdfs as scale mixtures of Gaussians, we cast our system identification problem into a Gaussian process regression framework, which requires estimating a number of hyperparameters of the data size order. To overcome this difficulty, we design a new maximum a posteriori (MAP) estimator of the hyperparameters, and solve the related optimization problem with a novel iterative scheme based on the Expectation-Maximization (EM) method. In the presence of outliers, tests on simulated data and on a real system show a substantial performance improvement compared to currently used kernel-based methods for linear system identification.

A note on generalized factor analysis models

An interesting generalization of dynamic factor analysis models has been proposed recently by Forni, Lippi and collaborators. These models, called generalized dynamic factor analysis models describe observations of infinite cross-sectional dimension. Quite surprisingly the inherent non-uniqueness of factor analysis models does not occur in this generalized context. We attempt an explanation of this fact by restricting the analysis to static generalized factor models. We show that there is a natural interpretation of generalized factor analysis models in terms of Wold decomposition of stationary sequences. A stationary sequence admits a (unique) generalized factor analysis decomposition if and only if two rather natural conditions are satisfied.

Blind identification strategies for room occupancy estimation

We propose and test on real data a two-tier estimation strategy for inferring occupancy levels from measurements of CO2 concentration and temperature levels. The first tier is a blind identification step, based either on a frequentist Maximum Likelihood method, implemented using non-linear optimization, or on a Bayesian marginal likelihood method, implemented using a dedicated Expectation-Maximization algorithm. The second tier resolves the ambiguity of the unknown multiplicative factor, and returns the final estimate of the occupancy levels. The overall procedure addresses some practical issues of existing occupancy estimation strategies. More specifically, first it does not require the installation of special hardware, since it uses measurements that are typically available in many buildings. Second, it does not require apriori knowledge on the physical parameters of the building, since it performs system identification steps. Third, it does not require pilot data containing measured real occupancy patterns (i.e., physically counting people for some periods, a typically expensive and time consuming step), since the identification steps are blind.

Outlier robust system identification: a Bayesian kernel-based approach

In this paper, we propose an outlier-robust regularized kernel-based method for linear system identification. The unknown impulse response is modeled as a zero-mean Gaussian process whose covariance (kernel) is given by the recently proposed stable spline kernel, which encodes information on regularity and exponential stability. To build robustness to outliers, we model the measurement noise as realizations of independent Laplacian random variables. The identification problem is cast in a Bayesian framework, and solved by a new Markov Chain Monte Carlo (MCMC) scheme. In particular, exploiting the representation of the Laplacian random variables as scale mixtures of Gaussians, we design a Gibbs sampler which quickly converges to the target distribution. Numerical simulations show a substantial improvement in the accuracy of the estimates over state-of-the-art kernel-based methods.

Variance analysis of linear SIMO models with spatially correlated noise

In this paper we address the identification of linear time-invariant single-input multi-output (SIMO) systems. In particular, we assess the performance of the prediction error method by quantifying ...

On the Zero-Freeness of Tall Multirate Linear Systems

In this technical note, tall discrete-time linear systems with multirate outputs are studied. In particular, we focus on their zeros. In systems and control literature zeros of multirate systems are defined as those of their corresponding time-invariant systems obtained through blocking of the original multirate systems. We assume that blocked systems are tall, i.e., have more outputs than inputs. It is demonstrated that, for generic choice of the parameter matrices, linear systems with multirate outputs generically have no finite nonzero zeros. However, they may have zeros at the origin or at infinity depending on the choice of blocking delay and the input, state and output dimensions.