1.1 Background

The United States of America is the second largest consumer of energy with only 4.3% of the world population [1,2]. The energy consumption in the US can be mainly attributed to following sectors: residential use (21%), commercial use (18%), transportation use (29%) and industrial use (32%) [3,4]. Given how individual mobility and activity participation influences energy use, it is not surprising that energy consumption in residential, commercial and transport sectors is intertwined. For instance, households that pursue longer commutes are likely to expend larger energy for transportation and are likely to expend lesser energy at their residence. Similarly, individuals working longer hours at office would contribute to increased energy consumption at commercial buildings and reduced energy use (at least from one individual) in the residence. The intricate relationship among these three sectors became prominent with the ongoing COVID-19 pandemic. Residential energy use increased by 8% during COVID-19 lockdown and/or mobility restrictions (between April to August 2020), while commercial and transportation related energy usage decreased 8% and 21%, respectively [5,6].

With the growing adoption of electric vehicles (EVs), the intricate relationship between energy consumption across sectors will be further strengthened [7,8]. The uptake of EVs and the potential energy source diversification (such as solar and wind energy) would result in a transformation of energy consumption and distribution patterns across the world [8]. The demand for charging the electrical vehicles at home, work and other potential locations is also likely to influence the spatio-temporal nature of the existing electricity demand. It is possible that the current demand on the grid could be rapidly altered with higher residential and commercial demand. There is a growing need for the development of modeling frameworks that provide insights on energy use and potential future energy demand evolution. A major bottleneck for model framework development is the unavailability of “perfect” data.

Recent technological advances and their adoption including sensing technology, smart energy sensors, connected and autonomous vehicles, shared mobility (bike sharing, scooter sharing and transportation network companies), naturalistic driving studies, and location-based smartphone data have resulted in large volumes of data being collected. This data explosion has shifted research challenges in multiple fields from modeling with limited data to developing modeling approaches that support effective utilization of the abundant data. The current research effort is focused on improving effective use of the multiple disparate sources of data available for energy use modeling by proposing a novel maximum likelihood based probabilistic data fusion approach.

Data fusion algorithms refers to the techniques of integrating two or more distinct data sources into a fused data that offers enriched information (additional explanatory variables) compared to the individual data sources [9]. The algorithms can be simple merging efforts across multiple datasets. Let us consider the compilation of a typical residential energy demand dataset. Utility companies compile energy use data using a smart energy sensor system with detailed information on energy demand in continuous time while also compiling residential unit characteristics (such as floor area and the number of bedrooms). The data also has unique information in terms of the residential unit location. Employing the location information, the dataset can be augmented with a Weather and Geographic Information System (GIS) file that provides location specific characteristics such as temperature and precipitation. The merging of data described here is a simple, deterministic fusion. Given the location, using GIS and appropriate weather data, the analyst can query or cross-reference for weather characteristics and append them to the energy demand record. The data fusion described is typically devoid of uncertainty (as long as the appropriate data processing steps are employed) and well defined as there are attributes that can be used to match data across these multiple datasets. Any data analysis in recent years includes such simple data fusion procedures.

The proposed research is geared towards fusing databases that are not relatable because of the inherent differences across these datasets. For these uniquely unmatched datasets, there is a significant need for a behavioral data fusion approach across various domains including energy demand analysis [10–14], mobility pattern analysis [15–17], freight movement modeling [18–20]; disaster evacuation planning [21] and traffic safety [22]. With increasing share of energy use for mobility (with EVs), it is important to examine how transportation mobility needs can influence energy use. The current research recognizes the potential relationship between energy and transportation datasets and provides an algorithm to enhance energy data modeling using information from transportation datasets. The proposed approach is general and can be applied across domains. With emerging advances in information technology and communication devices data from smartphone location data or cell phone OD data are ideal complements to traditional data by offering improved spatiotemporal coverage [23,24]. At the same time, these data are not usually available with person or household level characteristics. Thus, adoption of these data at a decision maker level would require an effective algorithm that can fuse this information with travel survey data.

1.2 Research approach

The data fusion algorithm developed in the current research is targeted toward datasets that contain information that is not uniquely matchable. Consider data from a Residential Energy Consumption Survey (RECS) data compiled by US Energy Information Administration (EIA) that provides energy use information by fuel type (such as electricity and natural gas) at a residential unit resolution along with household level information. To understand the determinants of energy use by fuel type, a linear regression model can be estimated using the set of independent variables available in the RECS dataset including household level characteristics: housing type, housing characteristics such as number of stories and bedrooms [25,26]; location characteristics: census region, division, located in urban/rural area [27,28]; and climatic characteristics: number of cooling and heating days [29–31]. However, the RECS data—source dataset—does not have any information on the number of employed individuals and household vehicle ownership. It is possible that these two variables are contributing factors for energy use. Employment status and vehicle ownership are indicative of the mobility needs of the household influencing energy consumption at the residence and for transportation needs. The proposed research develops methods that bring in this relevant information from another dataset–a donor dataset. The National Household Travel Survey (NHTS) administered by Federal Highway Administration (FHWA) surveys travel behavior patterns. NHTS dataset provides information on employed individuals and vehicle ownership–information that might assist in better understanding energy use and its prediction. With a focus on improving the model for the dependent variable of interest from the RECS dataset (energy use by fuel type in the example), our proposed research provides a probabilistic mechanism for appropriately fusing records from the NHTS dataset with records in the RECS dataset. For each RECS record, the algorithm considers a select set of records from the NHTS dataset with some common attributes (such as census region or household size) as a starting point for matching consideration. A weight function is defined that optimizes the weight for each RECS record while improving dependent variable model fit (energy use by fuel type). As the weight is unobserved to the analyst, the weight function proposed is analogous to the latent segmentation weight for a discrete outcome variable. In our research, the weight function is scored based on the similarity/dissimilarity of the source and donor records for common unmatched attributes (such as number of adults). The weight score is expected to be higher for source and donor records with more similarity. Across the selected donor records for a single source record, the weight sums to one. The donor records selected will provide additional useful variables missing for the source record.

The proposed fusion approach is illustrated using RECS and NHTS datasets for energy use by fuel type analysis. The model developed offers improved data fit for the dependent variables of interest. The main motivation behind our matching approach is to augment RECS data with NHTS data that contains detailed socio- demographics (gender, age), travel patterns (what mode is used for daily travel) and location information that could significantly affect energy usage. For instance, households situated in high population density locations typically have reduced floor area per capita and hence are likely to use less electricity for heating and cooling. Further, in recent years, energy consumption patterns are affected along two directions. First, the emergence of electric vehicles (EV) will transform the energy-transportation relationship. In the future, in households with EVs the energy consumption will be directly associated with vehicle ownership variables (how many electric cars) and vehicle usage dimensions. Second, during the COVID pandemic, a large number of workers facilitated by advances in information technology started to work from home influencing residential energy consumption. Currently RECS data does not provide any information on these important variables. NHTS data on the other hand can fill this gap as information on the number of vehicles in the HH, the corresponding vehicle types (fuel/electric) and the number of people working from home are available. Thus, the proposed fusion algorithm enables us to merge these two distinct datasets and create an enriched data source for analyzing energy consumption. Using the fused data, the association between additional categories of exogenous variables with residential energy demand can be tested. Thus, the model developed with the fused database will have additional explanatory power relative to the model developed solely using RECS data.

The rest of the paper is organized as follows: Section 2 provides a brief review of previous research on the application of data fusion algorithms in transportation field and highlights the contribution of the current study. Section 3 briefly outlines the methodological framework used in the analysis while a detailed description about the experimental setup of the study is presented in section 4. In section 5, we describe the model findings and finally, concluding thoughts are presented in section 6.

2.1 Literature on energy usage

Residential energy demand has been extensively researched in the energy analysis literature. However to conserve on space, we will provide a brief summary of these studies (see [31] for details on these studies). From our literature review, it is observed that earlier research focused on electricity and natural gas consumption [25,26,29–36] while very limited attention has been devoted to other forms of energies including fuel oil and LPG [31,32,37]. Interestingly, RECS is the most used database in United States for analyzing the usage of various energy sources [29–34]. Within these studies, the most prevalent form of energy usage considered is the continuous representation of energy use including energy consumption in BTU, or natural logarithm of energy consumption [29,30,33,34] while a handful of research efforts focused on the choice of energy source [30–32,34]. Given the continuous nature of the choice variable, it is not surprising earlier research adopted the regression framework for examining the energy usage. In particular, work in this area has ranged from simple linear regression [29,30,33,34] or discrete continuous models [30,34] to more advanced models such as the Multiple Discrete Continuous Extreme Value (MDCEV) model [31,32] for predicting the residential dwelling energy usage. In terms of the predictors, previous studies identified the following factors significantly affecting the residential energy usage: household level characteristics (HH income, race, household size, education) [25,31,36]; location characteristics (census region, type of location) [26,32], housing characteristics (such as year of construction, housing type, type of unit, square footage, and number of stories) [31,35,37], appliance use (such as appliances used in the housing unit) [31,38] and climatic characteristics (such as heating degree days and cooling degree days) [29–33,35].

2.2 Literature on data fusion techniques in energy

Data fusion algorithms have been widely researched and applied in various fields including statistics, business analysis, chemical engineering, energy demand, navigation industry and transportation [9,11,19,22,39,40]. For the current research effort, we have confined our attention to the studies adopting data fusion techniques in energy demand sector.

Energy efficiency (in building) is a heavily researched area where data fusion is applied at various resolutions. However, unlike transportation field, data fusion algorithms in energy demand literature mainly focused on appliance, sensor and semantic level fusion as opposed to data level fusion [14]. Example includes system identification combined with Kalman filtering [41], and deep learning-based techniques [11,42] that integrate data from multiple sources. These techniques have been applied to various types of data, including weather, occupancy, and equipment usage patterns. Multi-information fusion models, such as those using convolutional neural network (CNN) and long short-term memory (LSTM) networks, have also been used to enhance the accuracy of energy forecasting [43,44]. Based on the dimension of crucial interest, these studies can be broadly classified into two groups: 1) examine the occupancy status of the building and 2) understand the energy consumption pattern. The reader would note that data fusing algorithms have also been developed to minimize the variance of the fused data, which is beyond the scope of the current study (see [45,46] for details).

The first group of studies mainly adopted different data fusion algorithms for analyzing the occupancy status of a building, a crucial component in energy efficiency and energy consumption analysis [10,11,47–49]. For instance, Wang and his colleagues [47] considered K-Nearest Neighbour (KNN), Support Vector Machine (SVM) and Artificial Neural Network (ANN) algorithms to fuse the environmental data with WI-FI data for predicting the building occupancy. Another research effort by Nesa and Banerjee [48] presented Internet of Things (IoT) based real time sensor data fusion using the data collected from various sensors within office space to predict the occupancy status of the office spaces. Varlamis and his colleagues [10] fused sensor-based energy data with the historical data and user feedback to generate recommendations for smart homes and offices. Wang et al.,[11] used Long Short-Term Memory (LSTM) networks to fuse data from various utilities to predict internal heat gains for office buildings—a major component in heating, ventilation, and air conditioning (HVAC) operations. He et al. [49] proposed the fusion of LSTM and Back Propagation Neural Network (BPNN) algorithms to predict air conditioning load in buildings. Tan and his colleagues [43] employed rule-based decision-making algorithms to combine data from multiple sensors, such as motion, door, and light sensors to improve occupancy detection accuracy in residential buildings.

The second line of inquiry is focused on analyzing the energy consumption patterns of buildings by applying data fusion techniques [12,13,50,51]. Gouveia [13] fused the electricity consumption data from smart meters with door-to-door surveys to understand the energy patterns of the households. Wijayasekara and Manic [51] used ANN based data fusion method to increase the temporal resolution of building energy consumption data. Similar approach was also used by De Silva and his colleagues [50] to understand the energy consumption patterns in buildings. Gurino et al.,[12] compared the existing climatic databases with the simulated historical weather data aimed to generate a fused dataset by using various climate change models. This fused database was used to predict the consumption of energy requirements for office buildings.

2.3 Current study in context

The literature review clearly highlights the prevalence of data fusion algorithm across various energy sectors. However, all these studies focused on combining two/more datasets based on a common identifier (such as fusing information to a house based on its ID) or by employing black box approaches to data fusion. Furthermore, the data fusion approaches are geared towards compiling dependent variables of interest not available in one of the datasets. In our research, the focus is on providing additional independent variables for accurately representing the dependent variable of interest. The preceding discussion also makes it clear that data fusion algorithms in energy demand literature are primarily focused on semantic, sensor, and appliance level fusion, as opposed to observation level probabilistic fusion approach proposed in our study [14]. To the best of the authors’ knowledge, this is the first attempt (in both transportation and energy demand literature) to develop a behavioral fusion algorithm to combine two different datasets without any common identifier. A recent paper by Zhang and his colleagues (60) adopted a fusion approach to predict credit risks for small and medium-sized businesses (SMEs) in supply chain financing by merging behavioral and demographic data. However, the work also focused on deterministic fusion as both these data were matched based on the common entity of SMEs in supply chain finance.

The current approach is focused on a data fusion approach that augments RECS data (source) with additional variables from NHTS dataset (donor) with a focus on improving the data fit of the dependent variable of interest (energy use by fuel type) in the source dataset. The source and donor dataset can have common attributes such as census region, household size, household ownership, number of adults, and area (urban/rural). Ideally, selecting all or the majority of the common attributes for matching would provide the most precise fusion. However, the reader would recognize that selecting all or a large number of common attributes as matching variables can potentially reduce viable matching candidates or result in zero candidates. This would have resulted in the loss of records and potentially introduced bias, as significant portions of the dataset might be excluded from the analysis. Hence, we employ an approach where we choose a subset of common attributes for matching. As the matching between source and donor sets are being considered across different datasets, we hypothesize that fusing multiple candidates (as opposed to one record) would allow for a more useful and representative fused dataset. At the same time, as we fuse multiple records (say K) from the donor dataset (NHTS) with the source dataset (RECS), the source record will need to be duplicated K times to generate fused records. To address this duplication, a simple deterministic weight (1/K) is applied to ensure for each source record, the multiple matched rows of data represent only one new record. The proposed fusion approach makes several variables that are not available in the original dataset accessible for modeling. The benefit from these additional variables can be evaluated in a straightforward manner. If these additional variables contribute to improving the data fit of the dependent variable, then the fused dataset offers improved analysis of the dependent variable of interest. The improvement in data fit is compared using the log-likelihood and Bayesian Inference Criteria metrics that are well established in the literature.

The deterministic matching approach will work effectively with a small set of matching variables. As the number of potential matching variables increases, the number of exact matches could reduce very quickly. Therefore, we propose a matching approach with a probabilistic weight that penalizes differences between the source record and the donor record. So, in this approach, we allow for some variable mismatch and evaluate its impact on matching process by estimating a weight for each donor record that is fused with a source record. Specifically, the weight is parameterized as a function of the discrepancy for variables in both datasets. The contribution is influenced by similarity (or dissimilarity) across the common attributes between source and donor datasets. This weighting process effectively translates to estimating the weight contribution of the donor record to improve data fit of the dependent variable of interest (as opposed to using a uniform 1/K weight). The records with smaller mismatch are likely to have a weight higher than the deterministic weight (1/K) and records with higher mismatch are likely to have a weight lower than the deterministic weight. The parameters estimated as part of the weight function will inform us about the ranking of the various matching factors on their impact on the dependent variable of interest. For instance, household ownership status might not be as important as number of children in explaining household energy consumption patterns. In this case, the weight function coefficient for difference in the number of children variable will be larger in magnitude.

To better illustrate the data fusion process, an example is presented in Fig 1. The RECS Survey has four HHs with information on household size, household ownership status, number of adults in the HHs, number of rooms in the HH and the dependent variable: consumption of electricity (in millions of Btu). The NHTS data, in addition to household size, ownership status and number of adults, provides information on vehicle ownership and number of workers in the HH. The common variables across these two datasets are household size, ownership status, and the number of adults. Initially, we begin the fusion using all three matching attributes. In this process, we are able to find matches for all households except the third household. If we proceed with this fusion, then the third household would need to be excluded from the analysis, thereby compromising 25% of the records (1 household out of 4 households in RECS). To address this issue, we relax the matching assumption by considering two variables (household size, and household ownership status) as our matching attributes while use the remaining variable (number of adults) in the weight function. Based on this, we find three matches for the first HH, two matches for the second household, one match for the third household, and three matches for the fourth household. Now, using the matched records, a fused dataset is created with three repetitions of HH 1, two repetitions of HH2, 1 HH3 and three repetitions of HH4 with NHTS data columns including number of adults, vehicle ownership and number of workers in the HH (see Fig 1). As mentioned earlier, a weight function is used in the data to ensure that all the repetitions together represent one household in the RECS data. For the deterministic weight method, we assign an equal weight, that is 1/K for K repetitions. For example, for HH 1, which has three repetitions, each repetition would be assigned a weight of 1/3 (approximately 0.33). For the probabilistic weight method, we will calculate the difference in the number of adults variable (available in source and donor datasets but not matched) across the two datasets and use these differences to parameterize the weight function (details on this process is discussed in the methodology section). The probabilistic weight variable provides a higher weight when the difference is lower (or 0. For example, for HH 2 (see Fig 1), the first matched record has the same number of adults as the RECS dataset, resulting in a higher weight of 0.7. In contrast, the second matched record does not have the same number of adults, resulting in a lower weight of 0.3. Please note that the numbers provided in Fig 1 are for illustration purposes and will be estimated in our model within a maximum likelihood setting.

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Fig 1. RECS and NHTS data fusion illustration.

https://doi.org/10.1371/journal.pone.0309509.g001

In summary, the current study contributes to the energy and data science literature both empirically and methodologically. Empirically, the proposed fusion algorithm enables us to merge these two distinct datasets and create an enriched data source for analyzing energy consumption. Using the fused data, the association between additional categories of exogenous variables with residential energy demand can be tested. Thus, the model developed with the fused database will have enhanced explanatory and predictive power relative to the model developed solely using RECS data. Further, this enriched dataset, and the resulting model can significantly inform policy decisions. For example, understanding the impact of EV ownership and working-from-home trends on residential energy consumption can guide policymakers in designing targeted incentives for energy-efficient technologies and infrastructure. Methodologically, the study presents an innovative behavioral data fusion technique to combine two datasets without a common identifier. Further, our approach strategically selects variables for initial matching and incorporates the remaining ones into a weight function, ensuring an optimal balance between sample size and important variables. This type of behavioral fusion is introduced for the first time in this paper (to the best of the authors’ knowledge) and can be widely applied to various fields.

4.1 Selecting variables fusion

In the current analysis, we tried several combinations of these factors for linking the two datasets and for each combination, we calculate the improvement in average (we consider N = 15 samples) log-likelihood (LL) relative to the simple linear regression model that is estimated using the RECS data only. Finally, we select the corresponding combination that provides the superior improvement. The average LL improvement measures across each variable/variable groups are plotted in Fig 4. From this plot, we can clearly see the relatively higher average LL improvement when household from both datasets are fused based on census division and location classified as urban or rural. We select this variable group for linking the two datasets and proceed to the next step.

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Fig 4. Model fit summary across different variable group used for fusion.

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4.2 Selecting number of matching records for fusion

Based on the result obtained in the first step, we linked the two datasets based on similar HH location and created N = 15 fused databases using multiple matching records of K including 3,5, 10, 15, 20, 30, 40 and 50 (see Fig 5). We compute the improvement in average LL measures for different values of K. From the Fig 5, we can clearly see there is significant improvement in average LL as K increases in the initial stages. After a K value of 15, only marginal changes to average LL improvement are noticed. However, with increased K value, the model estimation times will continue to increase as the number of effective records increase with K. Thus, from the perspective of model improvement and run times, we select K = 15 as the optimal value. Thus, for each sample, 15 records from NHTS will be added to the RECS sample.

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Fig 5. Model fit summary across different number of fusion.

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4.3 Check parameter estimates stability

After selecting the variables and the number of records to be used for fusion, the next step is to evaluate the stability of the parameters of the energy demand model estimated using the fused data. As described, multiple samples were generated for the fused dataset, and it is important to confirm that the parameter estimates from all these samples offer consistent results. To undertake this evaluation, we propose an approximate t-statistic measure for each sample parameter estimate as follows: (1)

Where B_m is the average estimate value across all N samples (); B_s is the estimate for the s^th sample; SD_m is the average standard error for all N samples () and SD_s is the standard error for the sth sample. If the computed t-statistic value is greater than 1.65 it indicates that the parameter estimate is quite different from the average parameter across the samples. The t-statistic across all parameters and samples can be computed and used to measure the number of outliers. The presence of outliers will indicate that significant parameter variability across the samples and hence the results are less likely to be stable in this case. In our study context, we computed the approximate t-statistic for fused model parameters in the energy use component and parameters in the weight component. The results are plotted in Fig 6. The boxplots clearly illustrate significant stability in the parameters estimated. In fact, the computed approximate t-statistic does not reach 1.65 for even one parameter across all samples. The highest single value obtained is under 0.3, while the mean values range around 0.1. The results clearly indicate that for the fused dataset, we have obtained a reasonably stable estimate for all parameters.

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Fig 6. Test statistics (t-statistics) for parameter estimates across samples for each variables and models.

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6.1 Model fit

The experimental set up and the corresponding results establish the best model estimated using the fused dataset. We estimate multiple models to serve as a benchmark for the proposed models. First, we estimate a simple linear regression model (SLR) employing the RECS survey (with 4,000 HHs) data without fusing any record from the NHTS database. Second, we employ the fused dataset with K = 15 and N = 15 and estimate a linear regression model with equal weights (EWLR) allocation i.e. each fused record is weighted at (1/15). Finally, these two models are compared with the fused latent weight linear regression (LWLR) model. The models are estimated for two use cases: electricity energy use and natural gas energy use.

The performance of these models is compared based on the log-likelihood (LL) at convergence, the number of parameters estimated, and Bayesian Information Criterion (BIC). For the electricity demand model, the BIC (LL) values at convergence are: 1) SLR model (with 16 parameters)– 6,126.73 (-2997.01); 2) EWLR model (with 21 parameters)– 5,859.04 (-2814.00); and 3) LWLR model (with 23 parameters)– 5,806.38 (-2776.67). For the natural gas demand model, the values are: 1) SLR model (with 9 parameters)– 9,882.92 (-4891.95); 2) EWLR model (with 12 parameters)– 9,685.34 (-4,776.60); and 3) LWLR model (with 14 parameters)– 9,635.35 (-4740.66). Two important observations can be made from the model fit measures. First, models incorporating additional variable information from the NHTS dataset always provide improved performance irrespective of the dependent variable (electricity and natural gas usage). Second, within the models using fused dataset, the LWLR model outperforms the EWLR model as indicated by the lower BIC value associated with the LWLR model. This result clearly supports our proposed approach that a donor record’s contribution can be optimized using the weight function based on the similarity/dissimilarity of the common attributes. Overall, the model fit measures provide strong evidence for model improvement via fusion as well as weighted contribution estimation.

6.2 Estimation results

This section offers a discussion of the exogenous variable effects on energy usage for electricity and natural gas. Results obtained from the final model are presented in Table 2. It should be noted that the final specification of the model development was based on removing the statistically insignificant (90% significance level) variables from the model. A positive (negative) sign in the Table 2 indicates the increased (decreased) energy usage for the corresponding source (electricity/natural gas). The results are presented by variable groups.

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Table 2. Latent Weight Linear Regression (LWLR) model estimation results.

https://doi.org/10.1371/journal.pone.0309509.t002

6.2.1 RECS variables.

From our analysis, we find significant impacts of several RECS variables on energy consumption, as indicated in Table 1. To better illustrate these impacts for the readers, we present our findings graphically in Fig 7.

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Fig 7. Graphical representation of RECS variables’ impact on energy consumption.

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Constant: The constant parameter does not have any interpretation after incorporating other variables.

HH Characteristics: In terms of household characteristics, several attributes influence the usage of electricity and natural gas in residential dwellings. For instance, housing unit size (total square footage) reveals a positive impact on energy mix indicating a higher usage of electricity and natural gas in larger houses. This is intuitive as capital costs for installation for non-electricity sources might be high for smaller houses. On the other hand, in bigger houses, a mix of energy sources might be economical in the long run (see [30,32] for similar results). Further, higher number of bedrooms contribute to increased energy usage (both electricity and natural gas) as indicated by the positive coefficient in Table 2. In addition to the bedrooms, we also explored the impact of total number of rooms in a household on energy demand. Interestingly, we find that the variable has a significant positive impact on electricity consumption only. The reader would note that though all these variable seem to be influenced by each other, we did not find any significant correlation across them and thus are simultaneously considered in the model.

The results associated with housing type show significant impact on energy usage. Electricity consumption is likely to be higher in mobile homes while a lower usage of natural gas usage is observed in apartments. The results perhaps indicate inefficient cooling and heating in mobile homes resulting in increased electricity usage [52]. Further, building construction period is also found to have a significant impact on energy consumption. Specifically, we find an increased electricity usage in houses constructed after 1980 relative to the older houses (before 1980) while the natural gas usage is gradually declining in newer houses (after year 2000) as indicated by the negative sign in Table 2. The result is consistent with the overall trend of natural gas consumption in US. Newer buildings are associated with improved insulation, building materials and efficient heating systems contributing to lower benefits from employing natural gas consumption compared to the benefits of natural gas in to older buildings [32,53]. The growing adoption of all-electric homes in recent years is another important factor affecting natural gas consumption [54]. Finally, the income variable highlights a higher natural gas consumption in high-income households (greater than 120k).

Appliance Use: The intensity of appliance use in residential buildings potentially contributes to the overall energy usage. As expected, all of the appliance related attributes (use of ac and space heating; number of refrigerators, computers and smart phones in HH) positively impacted the electricity usage in a house [31] except the variable that corresponds to the use of humidifier. This result (humidifier) while counterintuitive at first glance, is presumably capturing the indirect relationship with the cooling and heating behaviour in a household. For instance, humidifier helps in creating a soothing environment by adding moisture in the air appropriately both in summer and winter season, thus minimizing the need of raising/lowering the temperature in a household [52] and hence possibly reducing electricity consumption.

Climatic Variables: The results related to climatic variables highlight the important role of weather in household energy usage. For representing the climatic variables, we considered heating and cooling degree days (please see [31] for detail) in a household that quantifies the demand for energy needed for heating and cooling requirements of a building respectively. Higher heating and cooling degree days directly refer to the cold and hot weather respectively. As expected, we find electricity usage to be positively associated with cooling degree days revealing an increased electricity consumption during hot days, perhaps alluding to the higher usage of AC during those times [55]. Contrastingly, natural gas consumption is higher during cold weather as evidenced by the positive sign specific to the heating degree days variable. Households in colder regions usually have higher space heating needs and natural gas is one of the predominant sources of fuel for space heaters. Similar findings are also observed in earlier research [31,32].

6.2.2 NHTS variables.

In the fused dataset, several variables fused from NHTS are tested in our analysis. Fig 8 provides a quick mechanism for the reader to understand the impact of different NHTS related variables on energy consumptions.

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Fig 8. Graphical representation of NHTS variables’ impact on energy consumption.

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The findings clearly highlight the reduced electricity usage in densely populated areas, perhaps indicative of the lower exposed floor area per capita [55]. In general, it appears that household with more females tend to use more electricity and natural gas relative to other households. This effect is perhaps the manifestation of the link between female and different activities in home including cooking, water heating, nurturing and cleaning [55]. Further, the estimated results show that the number of vehicles in a household is positively associated with household electricity consumption while a negative relationship is observed between the usage of natural gas and number of drivers in the household. The negative effect of the number of drivers in the household on its natural gas consumption may be attributed to the lesser time spent in houses as the ability to drive might encourage activities outside the home [56]. Interestingly, average age of a household (considering all members) reveals a negative effect on overall electricity consumption suggesting a reduced electricity use in a unit with older individuals. While this might seem counter intuitive on first glance as you would expect senior individuals to spend more time at home. However, the use of certain appliances such as deep freezer, dishwasher, tumble dryer and computers (and other devices) are relatively lower in houses with senior individuals and thus contribute to reduced electricity use [57,58]. Finally, average annual miles driven variable is found to be positively associated with natural gas consumption. This result is quite interesting and warrants further research. Overall, the findings are consistent with expectations and speak to the important role played by different factors in affecting residential energy demand.

6.3 Validation analysis

The model estimation results clearly illustrate the improved performance of the proposed model. In this section, we conduct a validation exercise, to evaluate the performance of the proposed LWLR model on the records not used for model estimation (hold-out sample). In the validation exercise, the performance of the fused LWLR model (with additional variables from NHTS and latent weight) is compared with the simple SLR model (employed with data form RECS only without fusing any record from the NHTS database) and equal weight EWLR model (with additional variables from NHTS and equal weight). The comparison exercise across the three models is conducted based on the predictive log-likelihood (LL) and BIC values.

The validation exercise is initially conducted with the 4000 record RECS estimation sample and 1686 record RECS validation sample. However, we realize that sample size in estimation could play a critical role in model performances [59] and hence we considered the influence of different sample sizes in model estimation by estimating the two model systems for different samples. Subsequently, to account for the impact of RECS sample size, we also conduct the validation exercise for different estimation and validation samples. In particular, from the RECS data, we randomly draw samples with 1,000; 2,000; 3,000; 4,000 and 5,000 households for estimation and for each estimation sample, the remaining households are considered as the hold-out samples. For example, RECS survey data provides information on 5,686 households. Out of these, for the first scenario, we considered 1,000 households as our estimation sample and the remaining 4,686 households are used for our validation exercise. For all these estimation and hold-out samples, we fused 15 records (K-15) from the NHTS dataset to the RECS dataset based on similar census division and location of the household. For the fused dataset, SLR, EWLR and LWLR models are estimated, and their performances based on predictive LL is compared. Further, as discussed earlier, for each record in the RECS data, there could be several potential matching records from the NHTS database and selecting 15 randomly out of these might introduce bias. Therefore, within each estimation and hold-out samples, we create 15 fused datasets (N), estimate (for estimation sample)/predict (for validation sample) the LL for each dataset across each model and finally compare the two models based on the average LL measures. The validation results are presented in Table 3.

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Table 3. Model validation results.

https://doi.org/10.1371/journal.pone.0309509.t003

Table 3 presents the validation results for two energy use for electricity and natural gas. For each sample size, the average log-likelihood over N = 15 samples for SLR, EWLR, LWLR model and the improvement (computed as 2*(LL _EWLR- LL_SLR) and 2*(LL _LWLR- LL_EWLR) are presented. In all cases, the LWLR model shows clear improvement. The improvement is consistent i.e. the improvement is higher as the dataset size increases for estimation and validation samples. We compare these improvements to the critical chi-square values for the models. For electricity EWLR model, we have 5 additional variables compared to SLR model providing a critical 95% chi-square value of 11.070. The improvements values presented are clearly higher than the critical value. Further, the LWLR model with 2 additional variables outperformed the EWLR model as indicated by the higher log-likelihood ratio value relative to the corresponding critical chi-square value (5.991 for 2 variables). Similar findings are also observed in the natural gas model. The EWLR model (3 additional variables from SLR model for natural gas) improvement for all the samples are also well over the critical chi-square value. The LWLR model provides superior performance for majority of the samples (7 out of 10 samples) compared to the EWLR model in predicting the natural gas consumption. So, from the results, we can conclude that model improvement via fusion and latent weight is consistent across estimation and validation samples. The validation results clearly highlight how new variables from the NHTS dataset contribute to improvement in predicting energy consumption. In summary, the results clearly provide evidence that the proposed algorithm offers enhanced explanatory power and predictive capability. The reader would note the adoption of other metrics such as BIC offer similar results and are not included for the sake of brevity.