Is stochastic grey-box modelling suited for physical properties estimation of building components from on-site measurements?

Abstract

In current energy requirements, the thermal performance of buildings is assessed with simplified energy models. A performance label is calculated based on thermal properties of the constituent components of the building envelope. These properties, however, do not include factors such as workmanship issues, or moisture or airflow influences which might affect the thermal performance as designed. To have a better view on the actual thermal quality of building components, a reliable thermal characterisation method of building components on-site is required. The typically used semi-stationary measurement methods have an application that is seasonally bounded or can require long measurement periods. Because of these drawbacks, dynamic parameter estimation methods have gained interest. In this article, the physical interpretability of a typical stochastic grey-box model used to thermally characterise building components is assessed. The identifiability of this model structure is examined by observing the profile likelihood of its parameters for typical measurements. The results allow identification of the extent to which models can estimate the thermal properties of building components in a robust way. A comparison of both analyses allows to define indications for physically interpretable parameters.

Keywords

Profile likelihood data-based identifiability maximum likelihood estimation grey-box models in-situ measurements thermal performance building components

Introduction

Although the thermal performance of a building’s envelope is a main factor in the determination of the energy performance of the whole building, currently, the thermal properties of a building’s fabric are assessed without consideration of many important variables. For both new and existing buildings, the envelope’s thermal performance is calculated from the thermal properties of the constituent building components and their material layers. These properties, however, often are theoretical values obtained from standards and product information. They do not recognise the reality of the built environment where weather conditions, workmanship issues, moisture or undesired air flows in the construction, and so on can influence the thermal behaviour of the building materials and components (Hens et al., 2007; Lowe et al., 2007). These deviations between theory and reality on the material and component level propagate to the level of the whole building creating a gap between the expected and the actual thermal performance of buildings (Cesaratto and De Carli, 2013; Gorse et al., 2011; Jain and Ramallo-González, 2014). Ignoring the latter leads to building regulations being applied on theoretical rather than actual buildings. A reduction in this gap between theory and reality may be accomplished by learning from the thermal properties of building components as-built, that is, properties estimated from on-site measurements. Hence, by relying on measured rather than theoretical values, a major source of uncertainty in the assessment of the thermal quality of the building envelope could be eliminated. Moreover, knowledge of as-built thermal properties could be used to assure qualitative workmanship of the building process.

The methods that are most commonly used for in situ thermal characterisation of building components are described in ISO 9869:1994 (1994). They estimate the thermal resistance of building elements from measurements of the heat flow rate through the internal face of the component together with measurements of the air or surface temperatures on both sides of the component. The main methods proposed in ISO 9869 assume steady-state relationships between these quantities, although an alternative dynamic analysis method is also mentioned in Annex B of the standard. By assuming steady-state behaviour, the common average methods disregard the component’s dynamic response to the indoor and outdoor conditions. Hence, they rely on averaged values of the measured data to approach stationary conditions, which infers that their application is often seasonally bounded and/or requires long measurement periods in order to obtain reliable results (Deconinck and Roels, 2014; Naveros et al., 2012).

Because of these drawbacks, more advanced dynamic data analysis methods have gained interest in recent years (Baker and Van Dijk, 2008; Biddulph et al., 2014; Cucumo et al., 2006). In particular, parameter estimation methods seem promising. These methods estimate the parameters of a mathematical model by tuning the dynamic output behaviour of this model to the observed measurements of the studied system, both subject to the same boundary conditions. In essence, these methods construct data-driven models that mimic the input–output behaviour of the observed system. The parameters that are thereby estimated are basically scaling factors to adjust the model output to the measured output (Ljung, 1999). Nevertheless, by formulating an appropriate model structure based on prior physical knowledge, a direct physical interpretation can be attached to the model parameters. Such models are often referred to as grey-box models (Ghiaus, 2014; Ljung, 1999).

To guarantee the physical interpretability of a model’s parameters, the formulation of a physical model structure is, however, only a first step. Next to that, it is very important that the identified model is the only possible parametrisation explaining the input–output characteristic of the system. The question whether the parameters of a model can be uniquely defined is a question of identifiability. In the literature, the notion of identifiability is twofold (Raue et al., 2009). On one hand, the concept of structural identifiability, an intrinsic property of the model, assures that a unique parametrisation is theoretically possible (Bellman and Åström, 1970). On the other hand, practical identifiability addresses the ability of estimating the unknown parameters uniquely from the available measurement data. Hence, a structural identifiable model can still lead to practically unidentifiable parameters due to a limited amount and/or quality of the measured data. In sum, physical interpretability can only be guaranteed for the structurally and practically identifiable parameters of an appropriate physical model of the studied building component.

In this article, the physical interpretability of a grey-box model structure that is typically used to thermally characterise building components is assessed. Thereby, a stricter definition of grey-box models (Kristensen et al., 2004a, b; Melgaard, 1994) is further adopted. According to the definition, a grey-box model is a stochastic state space model consisting of a set of continuous time stochastic differential equations (SDEs), describing the dynamic behaviour of the physical object, and a set of discrete time measurement equations, describing the model output. Such a model formulation is useful, especially in the context of physical parameter estimation because insights into physical mechanisms can easily be incorporated into state-space models (Ljung, 1999). Next to that, the stochastic framework of the model allows for a parameter estimation in a prediction error setting. As opposed to an output error setting, this avoids the absorption of random effects in the parameters’ estimates. Furthermore, the stochastic framework allows the use of statistical tools for validation of the estimated models (Kristensen et al., 2004a). Typically, the estimation of stochastic grey-box models is solved by a maximum likelihood approach (Kristensen et al., 2004a, 2004b; Madsen and Holst, 1995; Meeker and Escobar, 1995).

This particular concept of stochastic grey-box modelling has been explored in the past for modelling the heat dynamics of buildings or building components. Some studies only focus on the short-term prediction capabilities of the models without concerning about the physical interpretation of the parameter estimates (Fux et al., 2014; Rabl, 1988; Reynders et al., 2014). Other studies do exploit the physical interpretability of the parameters and use grey-box modelling as a tool for thermal characterisation (Andersen et al., 2000, 2014; Bacher and Madsen, 2011; Baker and Van Dijk, 2008; Biddulph et al., 2014; Gutschker, 2008; Jiménez and Madsen, 2008; Jiménez et al., 2008; Madsen and Holst, 1995; Naveros et al., 2014; Reynders et al., 2014). Yet, most studies do not comment on the identifiability of their assumed model structures and its possible influence on the physical interpretation of the parameters.

In this article, the identifiability and interpretability of the thermal parameters of a typical grey-box model structure used to thermally characterise building components are assessed in the light of an example. Simulated measurement data of an insulated cavity wall in a moderate European climate are used to estimate models with a common physical model structure. As simulated data are used, the exact thermal properties of the studied wall are known. The presence of non-identifiability in the models is assessed as well as the influence of the non-identifiable parameters on the physical interpretation of the combined thermal parameters of interest, that is, the total thermal resistance and effective thermal capacity of the considered wall. Furthermore, in parallel to the identifiability analysis, a statistical evaluation of the models is performed. A comparison of both analyses allows to look for indications of non-identifiable parameters in the statistical evaluation criteria. Such indications would allow deciding on the identifiability of the model parameters based on the statistical validation criteria only. Even more, these would allow a more general approach to discuss the physical interpretability of the considered type of grey-box models for similar characterisation problems of building components on-site.

This article is organised as follows. In the first section, a short introduction to the theoretical concepts of maximum likelihood estimation and statistical model validation criteria is given. Furthermore, an approach to detect structurally and practically non-identifiable parameters based on the profile likelihood is introduced. The second section presents the simulated experimental data of the considered cavity wall for typical winter and summer periods. Also, the typical model structure used to estimate the wall’s thermal parameters is discussed. In the following section, the results of the identifiability and statistical evaluation analysis are represented and finally, conclusions are drawn in the last section.

Theoretical concepts

Maximum likelihood estimation

Maximum likelihood estimation is a parameter estimation method which uses the likelihood function as the objective function quantifying the agreement of observed measurements with model outputs. For a sequence of measurements Y _N = y₀, y₁, …, y_N the likelihood function is defined as

L (θ; Y_{N}) = p (Y_{N} | θ) = (\prod_{k = 1}^{N} p (y_{k} | Y_{k - 1}, θ)) p (y_{0} | θ)

(1)

with p( Y _N | θ ) the joint probability density of the output Y _N . The maximum likelihood estimates of a grey-box model’s parameters are those parameters maximising the likelihood function

\hat{θ} = a r g \max_{θ} (L (θ; Y_{N} | y_{0}))

(2)

Or, to improve the numerics of the optimisation, those parameters minimising the negative logarithm of the likelihood function. By assuming Wiener processes for the noise models in the SDEs of the grey-box model, the goal function can be optimised by linear methods using the extended Kalman filter (Kristensen et al., 2004b).

Profile likelihood approach

Identifiability issues are, in essence, related to the sensitivity of the goal function to variations of the model parameters. Therefore, identifiability is often examined by scanning the objective function for flat manifolds, multiple maxima and so on in the parameter space. In Raue et al. (2009), an approach to detect structural and practical non-identifiable parameters is proposed. The method observes the profile likelihood function for a selection of model parameters. This function explores the parameter space of each selected parameter in the direction of the least increase in the objective function. The main idea is that a flat manifold in the objective function indicates non-identifiability (Raue et al., 2014). For a likelihood objective function, the profile likelihood of a parameter θ_i is defined as the likelihood function which is, for each fixed value along a profile of parameter θ_i, re-optimised for all other parameters $θ_{j \neq i}$ . Hence, the profile likelihood returns the highest possible likelihood value for a range of fixed values of θ_i

P L (θ_{i}) = \max_{θ_{j \neq i}} (L (θ; Y_{N}))

(3)

The profile likelihood can be calculated for all the parameters of interest of the model. Furthermore, the generalised likelihood ratio test of θ_i can be calculated as

R (θ_{i}) = \frac{P L (θ_{i})}{L (\hat{θ}; Y_{N})}

(4)

with $L (\hat{θ}; Y_{N})$ the maximum of the likelihood function after optimisation of all model parameters. A general property of the likelihood ratio test states that −2log(R(θ_i)) is asymptotically χ²-distributed (Meeker and Escobar, 1995). Therefore, −2log(R(θ_i)) is further denoted as $χ_{PL}^{2}$ . Based on this property, likelihood-based confidence intervals (Meeker and Escobar, 1995; Venzon and Moolgavkar, 1988) can be determined for each parameter θ_i

{θ_{i} : χ_{P L}^{2} = - 2 \log (\frac{P L (θ_{i})}{L (\hat{θ}; Y_{N}}) < Δ_{α}}

with Δ_α the αth quantile of a χ²-distribution for 1 degree of freedom. Graphically, the confidence interval of parameter θ_i can be determined by the intersections of $χ_{PL}^{2}$ and the threshold Δα (see Figure 1). In Raue et al. (2009), structural and practical non-identifiability is linked with the notion of likelihood-based confidence intervals. A structurally unidentifiable parameter is defined as a parameter which keeps $χ_{PL}^{2}$ on a constant level below the desired threshold Δ_α, resulting in an infinite likelihood-based confidence interval. A practically unidentifiable parameter is defined as a parameter with an infinite or half-infinite confidence interval, meaning that $χ_{PL}^{2}$ stays below the threshold Δ_α in minimum one direction of θ_i, although $χ_{PL}^{2}$ may have a minimum. An identifiable parameter is defined as a parameter with finite confidence bounds, meaning that $χ_{PL}^{2}$ crosses the threshold Δ_α in both directions of θ_i. Identifiability issues can be visually assessed by plotting $χ_{PL}^{2}$ and Δ_α in function of θ_i as is schematically shown in Figure 1.

Statistical model evaluation tools

The estimation of a stochastic grey-box model in a prediction error setting enables the use of statistical tools for validation of the estimated model (Kristensen et al., 2004b). A summary of the most important criteria is given below.

To evaluate the overall model performance, or, in other words to evaluate the model’s ability to mimic the input–output behaviour of the studied system, the residuals between the measurements and model output can be examined. A model is said to have good predictive capabilities if the residuals show no significant autocorrelation, that is, if the residuals are not significantly different from white noise. This can be assessed in the time domain by looking at the autocorrelation function (ACF) of the residuals and in the frequency domain by looking at the cumulated periodogram (CP) (Madsen et al., 2015). Depending on the intended application of the model, the whiteness of the residuals should be examined in a one-step-ahead prediction setting (where the importance of short-term behaviour prevails) or in a pure simulation setting (where the importance of long-term behaviour prevails) (Kristensen et al., 2004a). In a one-step prediction setting, the predicted states of the model are updated at each time step, reckoning with the information obtained during the previous time steps. The updates result from the Kalman filter which weights the measured and calculated uncertainties to decide to which extent the measurements and model should be trusted. In a pure simulation setting, no intermediate updates of the modelled states are computed. Furthermore, the whiteness of the residuals can also be checked for a new data set rather than the data set the model was trained on. In general, if no white-noise residuals are obtained, the model is not able to explain all dynamics of the input–output characteristic. Hence, a higher model order or a different model structure is required (Madsen et al., 2015).

To select the optimal order model from a set of nested models, a likelihood ratio test can be used. This test concludes on whether the larger of two nested models has a significantly better model fit (Bacher and Madsen, 2011), hereby reckoning with the increasing number of parameters to be estimated for the larger model.

Also, on the level of the individual model parameters, statistical properties can be used to evaluate the model. Based on the assumption of the asymptotic Gaussianity of the estimator $\hat{θ}$ in equation (2), the inverse of the Hessian, evaluated at the minimum of the objective function, can be interpreted as the covariance matrix of the model parameters (Kristensen et al., 2004a, 2004b). Hence, an estimate of the uncertainty of the individual parameter estimates and their correlations can be derived. In essence, the latter are derived from the curvature of a quadratic approximation of the objective function. A model with highly correlated parameters implies that the estimated values of the correlated parameters are interdependent. Hence, high correlations may indicate an over-parametrised model.

Furthermore, the asymptotic Gaussianity of the estimator in equation (2) allows a significant testing of the individual model parameters (Kristensen et al., 2004a). A marginal t test is performed to examine whether the parameters are significantly different from zero. Different levels of significance can be regarded, however, typically, p values <0.05 indicate significant parameters. Insignificant parameters are seen as unnecessary parameters to adjust the model output to the measurements and are advised to be removed from the model description (Madsen et al., 2015).

Application

The goal of this article is to examine the physical interpretability of a typical stochastic grey-box model structure used to thermally characterise building components from on-site measurements. The physical interpretability of the main parameter of interest, that is, the total thermal resistance, is of most importance. Next to that, the physical interpretability of the effective thermal capacity of the wall is also of interest. As an illustration, the characterisation of a south-facing insulated cavity wall is investigated. The wall’s thermal parameters are estimated from a typical Belgian winter and summer data set. To assure an exact knowledge of the goal thermal properties of the considered cavity wall, no real experimental data, but simulated measurement data are used in this study. Hence, the estimated parameters can be easily compared to the true physical values of the system.

Case study

A south-facing insulated masonry wall is simulated for a Belgian climate with HAMFEM, a finite element program based on the standard partial differential equations of heat, air and moisture transfer in porous building materials (Janssen et al., 2007). For the considered case, only heat transfer in the wall is considered. To mimic realistic measurement results, both system and measurement noise are added to the simulated data. White noise is assumed for both noise types. Concerning the heat flux, white measurement noise is assumed with zero mean and a relative standard deviation of 5% of the measured heat flux. Concerning the surface temperatures, white measurement noise with zero mean and an absolute standard deviation of 0.25°C is assumed. The considered system noise leads to white noise on the heat flux with zero mean and a standard deviation of 1.78 W/m². To reproduce the behaviour of an actual wall being a distributed system, the wall is simulated with a fine mesh of 200 elements and 201 nodes. The surface temperatures and internal heat flux that result from this HAMFEM simulation will serve as the grey-box models’ inputs and output, respectively. The thermal properties used for the one-dimensional simulations are represented in Table 1. Constant thermal properties are assumed so that temperature and moisture dependencies of the wall’s thermal resistance are excluded. The simulations are performed over one year and with a calculation time step of one minute, but the resulting data is averaged to hourly values for the application of the parameter estimation. Averaging is a low-pass filter and as the thermal parameters of interest, that is, the total thermal resistance and the effective thermal capacity are low-frequency dynamics, this will not jeopardise the estimation of the thermal parameters. Off course, the added system and measurement noise diminish when sampled, since it is assumed to be white noise.

Table 1.

Thermal properties of the simulated cavity wall from outside to inside.

	D (m)	R (m² KW⁻¹)	C (J m⁻²K⁻¹)
Brick	0.090	0.113	3.57e5
Insulation	0.127	3.614	7.44e3
Brick	0.140	0.187	2.36e5
Plaster	0.015	0.088	1.80e5
Total		4.00	6.18e5

Selected data sets

The dynamic excitation of a wall by its adjacent environments has two characteristic appearances in moderate climates. On one hand, during winter periods, the indoor environment is typically heated and the indoor air temperature is at a rather constant temperature higher than the outdoor air temperature. On the other hand, during summer periods, the indoor air temperature is generally not controlled. No heating or cooling is applied so that the indoor air temperature is a free floating temperature. In the HAMFEM simulation of the cavity wall, the internal environment is heated to a set temperature of 20°C during winter. During summer, no heating nor cooling is applied resulting in a free-floating indoor air temperature. For the outdoor environment, the typical moderate climate of Uccle (Belgium) is assumed.

Two data periods of 10 and 9 days in winter and summer, respectively, are selected from the year of simulations of the cavity wall. Figure 2 depicts the simulated surface temperatures and the internal heat flux for these periods. The main difference between the winter and summer measurements for the considered conditions is the dynamic excitation. During the winter period, the variations of the wall’s surface temperatures and internal heat flux are induced only by the outdoor weather conditions, as the indoor air temperature is kept at a constant temperature, while, in summer, the variations of the signals are induced by both the outdoor and indoor conditions.

Selected model types

To model the dynamic thermal behaviour of the observed cavity wall, a set of continuous time SDEs is formulated based on prior physical knowledge. Typically, building components are described by a series of thermal resistances and capacitances connecting the input temperatures at the internal and external face of the wall (Biddulph et al., 2014; Ghiaus, 2013; Melgaard, 1994; Naveros and Ghiaus, 2015; Naveros et al., 2014). The number of capacitances in the model is defined by the model order. To construct a stochastic grey-box model, the set of SDEs is coupled to a discrete time measurement equation describing the model output that will be accorded to the measured heat flux. For the assumed model structure, the internal heat flux can be modelled as the difference between the internal surface temperature and the closest model state temperature divided by the corresponding model resistance.

The model structure is represented by the RC-networks in Figure 3 for a model order n. Note that the surface temperatures T_se and T_si are the known inputs of the model and that Q_hfm is the model output that will be accorded to the measured heat flux, that is, the simulated heat flux by HAMFEM.

The SDEs are described in equation (5) where the set of SDEs is summarised in one equation that has to be repeated for i = 1,…, n with n the model order. The measurement equation is described in equation (6)

d T_{i = 1, ··, n} = \frac{1}{C_{i} R_{i}} (T_{i - 1} - T_{i}) d t + \frac{1}{C_{i} R_{i + 1}} (T_{i + 1} - T_{i}) d t + σ_{i} d ω_{i}

(5)

Q_{h f m} = \frac{1}{R_{n + 1}} (T_{n + 1} - T_{n})

(6)

with T₀ = T_se the external surface temperature, T_n_+ 1 = T_si the internal surface temperature, T_i the model states representing internal wall temperatures, R_i the model resistances, C_i the model capacitances, t the time, ω_i a standard Wiener process and σ_i the incremental variance of the Wiener process. The total thermal resistance and capacity of the wall are calculated as R_tot = ∑R_i and C_tot = ∑C_i, respectively.

Results

Identifiability analysis

For a discussion of the physical interpretability of the cavity wall’s grey-box models, first till fourth-order models are estimated from winter and summer measurements. An identifiability analysis is performed for the first-, second- and third-order model to address the question whether the estimated model parametrisations are unique for the considered excitation conditions.

To visually assess the identifiability of the models, Figure 4 depicts $χ_{P L}^{2}$ in function of the different thermal model parameters for both excitation conditions. Winter and summer results are distinguished by a different colour, and the different order models are represented row wise by increasing model order. Note that the higher the model order, the more the model parameters are present. A threshold of Δ_0.95,df = 1 is the set for the evaluation of the profile likelihood-based confidence intervals and is represented in Figure 4 as the dotted horizontal line.

In winter, the results for the first-order model (Figure 4(a)) reveal practical non-identifiability for all thermal parameters of the model, except for the final internal thermal resistance R₂. As explained previously, the non-identifiability can be inferred from the fact that $χ_{PL}^{2}$ stays below the threshold Δ_α in minimum one direction of θ_i for C₁ and R₁. In contrast, for parameter R₂, the curve $χ_{PL}^{2}$ crosses the threshold in both directions. This parameter is identifiable, nevertheless, with a large confidence interval.

In summer, the dynamic excitations come from both sides of the wall. The additional variations in input and output signals improve the identifiability of the parameters of the first-order model. From Figure 4(a), it is now seen that all model resistances and capacitances are practically identifiable.

Similar observations are made for the higher order models (Figure 4(b) and (c)). In winter, almost all thermal parameters of the second- and third-order model are non-identifiable, except for some of the inner thermal resistances. In summer, the increased dynamics lead to more identifiable model parameters than in winter. For the second-order model, all thermal parameters are practically identifiable. The additional dynamic information is summer is, however, not sufficient to estimate all parameters of a third-order model. Some of the thermal parameters now appear to be practically non-identifiable. Nevertheless, more parameters are identifiable in summer than in winter.

In essence, the results reveal that winter data are not dynamically informative enough to estimate all model parameters separately. By contrast, the additional information embedded in summer data is sufficient to estimate all model parameters up till a second-order model.

Yet, non-identifiable parameters in a model do not automatically signify that the total thermal resistance R_tot = ∑R_i and capacity C_tot = ∑C_i of the wall cannot be estimated robustly. To verify this, the behaviour of the combined parameters R_tot and C_tot must be examined for the practically non-identifiable regions. Or, in other words, the variation of the sum of the model resistances or capacitances must be examined for all regions that optimise the objective function, that is, for the range of values of all other model parameters having $χ_{PL}^{2}$ lower than Δ_0.95,df = 1. These relations can be visualised by picturing the optimised remaining parameters $θ_{j \neq i}$ , retrieved during the profile likelihood calculation of θ_i, over the unidentifiable region of θ_i (Raue et al., 2009). The total thermal resistance or capacity can then be pictured as the sum of the optimised model resistances or capacitances, respectively.

Figure 5 depicts these functional relations for the third-order model in summer. For clarity, the upper part of Figure 5 repeats $χ_{PL}^{2}$ for all model parameters. The middle part represents the functional relations of the thermal resistances R_i and the total thermal resistance R_tot in function of the other model parameters in their non-identifiable region, that is, for values with $χ_{PL}^{2}$ lower than the threshold Δ_0.95,df = 1. Analogously, the lower part of Figure 5 represents the functional relations of the thermal capacitances C_i and the effective wall capacity C_tot in function of the other parameters in their non-identifiable region. Figure 5 shows clear interactions between the model resistances and the other model parameters. For example, for the unidentifiable region of C₁, it is seen that a change in C₁ evokes a change in the optimised parameter R₁. The other thermal resistances R₂, R₃ and R₄ are by contrast independent of the value of C₁. As another example, it is seen that for the unidentifiable region of C₃, the parameter R₂ is independent of the value of C₃. The parameters R₁, R₃ and R₄ do vary over the unidentifiable region of C₃. They vary, however, in between their confidence intervals, which are, only for the unidentifiable parameters, unlimited. The spiky behaviour of the parameter R₁ over the unidentifiable region of C₃ is a result of the interdependency with parameter C₁ which was mentioned in the previous example. A mirrored spiky behaviour can be observed for the parameter C₁. Nevertheless, despite the interactions between the model resistances in the unidentifiability regions, a horizontal line indicates that the total thermal resistance R_tot is not involved in the practical unidentifiabilities and remains unaffected. Similarly, clear interactions between the model capacitances in the unidentifiable regions of the other parameters are seen from Figure 5, but now, also the total thermal capacity C_tot appears to be affected by the practical unidentifiability. This means that the estimated wall capacity of the unidentifiable model may not be physically interpreted as its value depends on the estimated combination of the other model parameters. The fact that multiple model parameter combinations optimise the objective function can physically be explained by stating that time constants of the wall are estimated rather than separate R_i- and C_i-values. For the other unidentifiable models, similar results can be seen from Figures 8 to 10 (Appendix 1).

The previous findings can also be confirmed from Figure 6, which compares the estimated total thermal parameters with the system’s real thermal properties, that is, the properties used to simulate the measurement data. It can be seen that, for almost all models in winter and summer, the total thermal resistance is robustly estimated as the goal value lies within the 95% confidence intervals of the estimated parameters (see Figure 6(a)). Only the first-order model shows an unreliable thermal resistance estimation for the summer measurements. The statistical validation, however, will demonstrate that first-order models are not fit to explain all the present dynamics in the measurement data. Furthermore, it is noticed that the estimated thermal capacity only approaches realistic values for the two identifiable models, that is, the first- and second-order model in summer (see Figure 6(b)). Note that the estimated capacity slightly underestimates the goal value because the effective capacity is estimated rather than the total thermal capacity (Reynders et al., 2014). All other models, which correspond to the non-identifiable models, systematically overestimate the thermal capacity.

The main conclusions of the identifiability analysis reveal that, despite the presence of practically non-identifiable model parameters, the examined model structure always allows to estimate the total thermal resistance when a sufficient order model is selected. In contrast, the value of the total thermal capacity is affected by non-identifiable parameters in the model structure and loses its physical meaning. Hence, identifiable models are required for an estimation of the total thermal capacity. In the next section, it is examined whether these conclusions are reflected in the model validation criteria.

Statistical model evaluation

Before examining the individual parameter estimates, the overall model performance is studied in order to eliminate inadequate models. To assess the overall model performance, the residuals are examined in the frequency domain by looking at the CP. Figure 7(a) depicts the CP for the residuals of all models in a one-step prediction setting in winter and summer. All models show white noise residuals indicating an accurate imitation of the short-term input–output behaviour of the cavity wall. However, because the thermal parameters of interest are important for the long-term behaviour of the system, the whiteness of the residuals is also assessed in a simulation setting (Figure 7(b)). It is seen that the residuals of the first-order models are autocorrelated for both winter and summer conditions. Thus, the long-term input–output behaviour of the wall is not correctly modelled by a first-order model in winter nor in summer. Based on this information, the first-order models are falsified for winter and summer conditions and are no longer considered for the physical parameter estimation of the wall’s thermal properties.

Higher order models, however, do show white-noise residuals in a simulation setting. This shows that, even for the non-identifiable second- and third-order models with unreliable parameter estimations, the predictive capabilities are always guaranteed. A likelihood ratio test designates the third-order model as most optimal for both winter and summer periods. This shows that the likelihood ratio test selects the model that best explains the observed measurements and does not reckon with the model’s identifiability, as otherwise the identifiable second-order model would be preferred over the non-identifiable third-order model in summer.

In general, it can be said that the selection of the optimal model order for physical parameter estimation purposes is not straightforward. Based on the observations for the considered cavity wall, a minimal and maximal model order can be set. The minimal order is the lowest possible model order corresponding to a model with white-noise residuals. The maximum model order is the order selected by the likelihood ratio test, as a further increase in model order does not significantly improve the model’s predictive capabilities. The optimal model order within these boundaries is then advised to be the model order that provides the desired accuracy levels for the parameters of interest because, as will be seen in the next paragraph, the latter are good indications to what extent the parameter estimates can be trusted.

To find indications of non-identifiable parameters, the individual estimated model parameters are statistically evaluated. Therefore, the significant testing and the approximate standard deviations, which are estimated together with the parameters, are studied. Both values are summarised in Tables 2 to 7 for different order models in winter and summer. For comparison, the tables also mention which parameters were identifiable according to the profile likelihood analysis.

Table 2.

Statistical evaluation criteria for the first-order model in summer.

M1m	Statistical Evaluation criteria			Identifiability
M1m	Value	sd_{α = 0.68}	p value	Significance	Identifiable?
C ₁	2.5e+05	1.8e+04	2.2e−16	***	Yes
C_tot	2.5e+05	1.8e+04
R ₁	6.5e+00	1.6e+00	5.2e−05	***	Yes
R ₂	1.1e−01	2.9e−03	2.2e−16	***	Yes
R_tot	6.6e+00	1.6e+00

The p-values summarise the significance testing results. Parameters with a high significance level (p<0.001) are indicated as ***. Less significant parameters (0.01<p-value < 0.05) are indicated as * and insignificant parameters are indicated with No.

Table 3.

Statistical evaluation criteria for the second-order model in summer.

M2m	Statistical Evaluation criteria			Identifiability
M2m	Value	sd_{α = 0.68}	p value	Significance	Identifiable?
C ₁	7.0e+04	1.2e+04	5.4e−08	***	Yes
C ₂	2.1e+05	5.6e+03	2.2e−16	***	Yes
C_tot	2.8e+05	9.3e+03
R ₁	2.4e+00	3.1e−01	3.1e−13	***	Yes
R ₂	1.2e+00	2.2e−01	1.5e−07	***	Yes
R ₃	1.1e−01	1.8e−03	2.2e−16	***	Yes
R_tot	3.7e+00	2.1e−01

Table 4.

Statistical evaluation criteria for the third-order model in summer.

M3m	Statistical Evaluation criteria			Identifiability
M3m	Value	sd_{α = 0.68}	p value	Significance	Identifiable?
C ₁	2.0e+08	3.4e+10	1.0e+00	No	No (+∞)
C ₂	2.1e+05	3.7e+04	5.1e−08	***	Yes
C ₃	4.5e+04	3.8e+04	2.3e−01	No	No (−∞)
C_tot	2.0e+08	3.4e+07
R ₁	1.6e−04	2.8e−02	1.0e+00	No	No (−∞)
R ₂	3.7e+00	1.6e−01	2.2e−16	***	Yes
R ₃	8.3e−02	1.0e−02	2.4e−14	***	Yes
R ₄	6.7e−02	2.8e−02	1.6e−02	*	No (−∞)
R_tot	3.8e+00	1.6e−01

Table 5.

Statistical evaluation criteria for the first-order model in winter.

M1m	Statistical Evaluation criteria			Identifiability
M1m	Value	sd_{α = 0.68}	p value	Significance	Identifiable?
C ₁	3.5e+13	3.1e+16	1.0e+00	No	No (+∞)
C_tot	3.5e+13	3.1e+16
R ₁	3.6e−09	3.2e−06	1.0e+00	No	No (−∞)
R ₂	4.0e+00	6.8e−02	2.2e−16	***	Yes
R_tot	4.0e+00	6.8e−02

Table 6.

Statistical evaluation criteria for the second-order model in winter.

M2m	Statistical Evaluation criteria			Identifiability
M2m	Value	sd_{α = 0.68}	p value	Significance	Identifiable?
C ₁	5.2e+15	4.7e+17	9.9e−01	No	No (+∞)
C ₂	4.5e+05	1.3e+06	7.3e−01	No	No (+∞)
C_tot	5.2e+15	4.7e+17
R ₁	8.8e−12	8.0e−10	9.9e−01	No	No (−∞)
R ₂	1.1e−01	3.2e−01	7.3e−01	No	No (−∞)
R ₃	3.9e+00	3.2e−01	2.2e−16	***	Yes
R_tot	4.0e+00	6.8e−03

Table 7.

Statistical evaluation criteria for the third-order model in winter.

M3m	Statistical Evaluation criteria			Identifiability
M3m	Value	sd_{α = 0.68}	p value	Significance	Identifiable?
C ₁	3.8e+12	7.4e+13	9.6e−01	No	No (+∞)
C ₂	4.6e+05	1.6e+06	7.8e−01	No	No (±∞)
C ₃	1.4e+05	6.5e+05	8.3e−01	No	No (−∞)
C_tot	3.8e+12	7.4e+13
R ₁	4.9e−09	9.3e−08	9.5e−01	No	No (∞)
R ₂	8.1e−02	2.6e−01	7.6e−01	No	No (−∞)
R ₃	2.1e−01	8.6e−01	8.0e−01	No	No (−∞)
R ₄	3.7e+00	1.1e+00	1.0e−03	***	Yes
R _tot	4.0e+00	9.2e−03

From Tables 2 to 7, it can be seen that the non-identifiable parameters correspond to those model parameters that are insignificant or show a low significance level. For the first-order model in winter, for example, the only significant parameter is the identifiable parameter R₂ (Table 5). In essence, a significant test performs a marginal t test to check whether the estimated parameter is significantly different from zero (Kristensen et al., 2004a). Or, in other words, it is tested whether zero lies within the 95% confidence intervals of the estimated parameter. Hence, the results from the significance testing highly depend on the estimated standard deviations of the parameters.

The concept of standard deviations, or derived properties, as indications for (non-)identifiable parameters is in line with the interpretation of the likelihood-based confidence intervals. As stated previously, infinite or half-infinite likelihood-based confidence regions indicate non-identifiable parameters. However, the confidence regions that are estimated along with the parameters by most software programs are based on the Hessian of the goal function in equation (2) evaluated at its minimum. The estimated standard deviations are thus based on a quadratic approximation of the log-likelihood function. Hence, no infinite intervals will be estimated; however, their estimates appear to be sufficiently large, compared to the estimated parameter, to imply insignificancies. For example, the first-order model in winter (Table 5) showed a half-infinite profile likelihood confidence interval for the parameter C₁. The standard deviation of C₁, estimated along with the parameter, is derived from a second-order approximation of $χ_{PL}^{2}$ in Figure 4(a) for C₁ in winter. Due to the approximative character, the estimated confidence interval will be finite, but will be large. Moreover, the standard deviation is sufficiently large (3.1E + 16 Jm⁻²K⁻¹) compared to the estimated value (3.5E + 13 Jm⁻²K⁻¹) so that C₁ is tested insignificant. Similar observations can be made for the other unidentifiable parameters of the models in winter and summer. The concept of insignificant parameters as indications for non-identifiable parameters may not be an established rule; however, as it is valid for typical excitation conditions, it may be generalised for similar characterisation problems, that is, the thermal characterisation of building components from standard on-site measurements.

Yet, the individual parameter evaluations do not inform about the combined parameters R_tot and C_tot. Based on the standard deviations of the individual parameters and the correlations of the latter, combined standard deviations can be calculated for R_tot = ∑R_i and C_tot = ∑C_i following the general rule that the variance of the sum of correlated variables equals the sum of their covariances. From Tables 2 and 3 and Figure 6, it can be seen that the identifiable models in summer, the first- and second-order models, result in total effective capacities with a reliable confidence region. The third-order model in summer and all models in winter (Tables 4 to 7), which have non-identifiable parameters, show very large confidence intervals indicating unreliable C_tot-estimates. The estimated total thermal resistances show small confidence intervals for all appropriate order models in winter and summer indicating reliable estimates (see Figure 6(a) and Tables 3 to 7).

Generally, for the considered boundary conditions, it is seen that insignificant or less significant parameters are an indication of non-identifiabilities in the model. This is in line with the concept of infinite or half-infinite likelihood-based confidence intervals indicating non-identifiable parameters. The fact that the estimated total capacity is affected by the presence of these non-identifiable parameters and that the estimated total thermal resistance is not, can be deduced from the combined standard deviations for these properties. Hence, the confidence intervals of R_tot and C_tot are a good indication to what extent these parameter estimates can be trusted.

Discussion

The identifiability analysis in this article shows that the more the input signals are dynamically exciting the considered building component, the more the physical information can be distracted from the estimated models. In that sense, the current analysis was based on worst case scenarios by not controlling the indoor temperature in summer and by keeping it constant in winter. Real indoor boundary conditions would improve the identifiability. Even more, this understanding could be exploited by designing proper indoor conditions that optimise the identifiability of the considered model structures. For example, a Pseudo-Random Binary Sequence (PRBS-) or a Randomly Ordered Logarithmically Distributed Binary Sequence (ROLBS-) regulated heating input or air temperature could be applied as controlled indoor conditions. However, this would unnecessarily complicate the measurements and the simplicity of the actual measurement set-up, which requires no specific indoor conditions, would be lost. Besides, the analysis proved that the typical winter and summer measurements both allow to robustly estimate the main parameter of interest: the total thermal resistance.

Furthermore, the physical interpretation of the individual model parameters can be questioned when looking at the identifiable second-order model in summer. From a physical point of view, one would expect a model of an insulated cavity wall to have two thermal capacities with small resistances on the inside and outside representing the inner and outer brick wall and a large thermal resistance in between the capacitances representing the insulation layer. The estimated internal distribution of the second-order model’s resistances and capacitances is, however, different, as can be seen in Figure 6(a). Also, the identifiable parameter R₃ of the second-order model in winter is different from the estimated R₃ in summer. Thus, although a physical and identifiable model structure is assumed, the individual model parameters of a reduced order model cannot be clearly pinpointed to the separate layers of the wall. A more complicated relation between the physical properties and the model parameters is playing as a consequence of the simplification that is made by modelling a distributed system by a lumped system (Ramallo-González et al., 2013). Hence, the estimated internal distribution of the wall will be signal-dependent and difficult to interpret.

Next to that, note that the results of the identifiability analysis in this article slightly changed the interpretation of the statistical parameter evaluation criteria. Typically, insignificant or highly correlated parameters are seen as unnecessary parameters to adjust the model output to the measurements and are advised to be removed from the model. Based on the results of this article, these parameters are no longer seen as unnecessary for the considered model structure. They are instead regarded as nuisance parameters that have no specific physical meaning but are necessary tools for accurately estimating the total thermal resistance of the building component of interest.

Conclusion

This article addresses the physical interpretability of a typical data-driven model structure used to estimate the thermal properties of building components from standard on-site measurements. Physical interpretability can only be guaranteed when the parameters of interest of a well-selected physical model structure are structurally and practically identifiable. Therefore, an identifiability analysis of this model structure’s thermal parameters is performed in the light of an example. Different order models are estimated based on simulated measurement data of an insulated cavity wall in a moderate European climate. The wall’s total thermal resistance and effective thermal capacity are estimated based on a typical winter and summer data set.

In winter, the identifiability analysis learns that the standard measurements are not as dynamically informative as needed for robust parameter estimation. The model structure has several practically non-identifiable model parameters from the first-order model on. This means that multiple combinations of model parameters optimise the objective function. Typically, the inner thermal resistances are identifiable, whereas outer model resistances and capacitances are highly correlated and operate as time constants rather than separate parameters. Nevertheless, it is seen that the estimated total wall resistance is not affected by the presence of non-identifiable parameters and can be accurately estimated. By contrast, the estimated total wall capacity loses its physical meaning.

In summer, the increased dynamic excitation enables a robust estimation of all thermal parameters up till a second-order model. The identifiable models allow a robust estimation of both the total thermal resistance and the effective thermal capacity. A third-order model, however, shows again some non-identifiabilities.

A comparison of the identifiability results with the statistical evaluation of the model parameters learns that the non-identifiable parameters correspond to the insignificant or less significant parameters. Their significance level is highly dependent on the approximate estimation of the standard deviation. The concept of standard deviations indicating non-identifiabilities is in line with the interpretation of the likelihood-based confidence intervals where infinite or half-infinite confidence regions indicate non-identifiable parameters. Due to the approximative character of the confidence intervals estimated along with the parameters, no infinite intervals can be estimated; however, their estimates appear to be sufficiently large, compared to the estimated parameter, to imply insignificancies. The finding that non-identifiable parameters correspond to insignificant or less significant parameters is no established rule; however, as it is proven for typical excitation conditions, it might be generalised for similar characterisation tasks with this model structure, that is, the thermal characterisation of building components from standard on-site measurements. The physical interpretability of the combined total thermal resistance and capacity can also be judged based on their standard deviation.

In general, the identifiability analysis learns that an identifiable model is required to estimate the effective thermal capacity of the wall. By contrast, non-identifiabilities do not affect the estimation of the total thermal resistance. In essence, the non-identifiable parameters can be seen as nuisance parameters that have no physical meaning but are necessary tools for accurately estimating the total thermal resistance of the building component of interest.

Footnotes

Appendix 1 Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: Research funded by a Ph.D. grant (grant number 121167) of the Agency for Innovation by Science and Technology (IWT).

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