Research ArticleA Statistical Mechanics Approach to the Study of EnergyUse Behaviour

Godwin Osabutey ,1 Alex Akwasi Opoku ,1 and Samuel Gyamfi2

1Mathematics and Statistics Department, University of Energy and Natural Resources, P. O. Box 214, Sunyani, Ghana2Energy and Engineering Department, University of Energy and Natural Resources, P. O. Box 214, Sunyani, Ghana

Correspondence should be addressed to Alex Akwasi Opoku; alex.opoku@uenr.edu.gh

Received 12 November 2019; Accepted 23 March 2020; Published 1 May 2020

Academic Editor: Abdel-Maksoud A. Soliman

Copyright © 2020 Godwin Osabutey et al. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work isproperly cited.

This paper proposes a statistical mechanical model for the energy conservation behaviour of energy users. The model was inspiredby the multipopulation Curie-Weiss model and the discrete choice with social interaction model. We demonstrate that the modelintegrates the sociopsychological, the economic, and the energy technologist viewpoints to energy management, allowing us tosubject these perspectives to an empirical study.

1. Introduction

Energy is one of the most important drivers of the economyof every country. Often, managing the energy resources hasbeen a problem. Electrical energy plays an indispensable rolein undertaking most of our daily activities from refrigeration,cooking, lighting, entertaining to powering of machines andrecord keeping in the industrial and commercial sectors [1].Electrical energy serves as a crucial resource for productiveand quality education, efficient healthcare delivery, transpor-tation systems, and the exploration of minerals such as ironand gold. These serve as the building block on which everysector of a nation’s economy thrives for development [1].The efficient use of resources available at a given time bythe energy end users has also been a challenge. The demandfor energy around the world keeps on increasing, and theexisting traditional forms of energy such as the fossil fuelsand coal, which are not renewable sources of energy, arestruggling to keep up with this demand over time. Moreover,the regular usage of the fossil fuels has a negative impact onglobal warming and the ecosystem. Adaptation of a betterand dynamic demand response program which promotesenergy conservation practices among energy users is highlysought after now than ever. Economic-engineering studieshave attempted to close this gap in energy efficiency through

the adoption of different technologies [2]. A model that cansimultaneously capture group dynamics, body use, cognitiveprocesses, and human-machine interactions is needed. Todate, progress towards such a model has been limited bythe theoretical preferences of the various disciplines involvedin energy research [2].

The main interest here is to import ideas from discretechoice theory and statistical physics to develop models forenergy conservation decision-making. Here, our focus is ona model that captures how individuals’ decision about energyconservation is influenced by their socioeconomic attributesand the choices of others. This work is structured as follows:Section 2 gives an overview of existing models and theories inrelation to human behaviour and energy use. In Section 3, wepropose a model from discrete choice theory as a model forinterdependent energy conservation decision-making. Sec-tion 4 discusses a more generalised form and multipopula-tion version of the Curie-Weiss model and provides anestimation procedure suitable for the model. Conclusion ofthe work is found in Section 5.

2. Modelling of Energy Use Behaviour

Modelling of individual behaviour has always been a chal-lenge in the context of energy conservation. There are various

HindawiJournal of Applied MathematicsVolume 2020, Article ID 7384053, 14 pageshttps://doi.org/10.1155/2020/7384053

schools of thought here. These are the economists, energytechnologists/analysts, and sociopsychologists schools ofthought. Their approaches to modelling human energy useand behaviour will be discussed below, and we will proposea model which can incorporate all the three approacheslater. This section will begin by discussing the economistsand energy technologists approach, and this will be followedby the sociopsychologist approach. These approaches arequalitative and form the theoretical framework of thisstudy.

2.1. Economists Approach. The price of energy and income ofindividuals is the main determinants of individuals energyconservation behaviour in economics studies. The theoryof utility maximization and consumer rationality in micro-economics modulates consumer’s demand. This theory isderived from the concept that consumers always want tomaximize their utility at the least possible cost, i.e., con-sumers try to weigh all other alternatives and actions to seethe one that will maximize their utility/satisfaction [3]. Inthe economics viewpoint, rationality is defined as a funda-mental axiom which control consumer behaviour. Consumerenergy conservation decisions must be rational, and effectivepolicies such as labels on electrical appliances and energydemand response programs (efficiency methods) are liableto weaken the economic theory of rationality by constrainingconsumers of want for options [4].

In [4], the authors examined the disparities betweenconsumer rationality and other alternatives to energy effi-ciency, and they found it useful to describe the link betweenconsumer rationality and energy demand behaviour. Thefirst viewpoint considered in [4] is attributed to ideologiesfrom neoclassical economics. According to a neoclassicaleconomic viewpoint on consumer rationality and energydemand behaviour as discussed in [4], the consumers ofenergy and suppliers/producers have a fixed preference thatthey both endeavour to fulfil by means of market mediation.The underlined preferences of consumers turn out to influ-ence their decisions to accept or reject energy-efficient appli-ances and equipment. These usually reflect a consumer’srational and relevant evaluation of costs and benefits from acertain technology or practice.

Economists always base their points of view on thenotion that individuals react to the “performance of the mar-ket system in terms of informal arguments based on intuitivenotions of rationality and preference” [5]. In this case, ifrational theory of choice is a literal system of describing con-sumer behaviour, then energy consumers, producers, andpolicy makers face the challenge of solving complex optimi-zation problems daily. These problems are not just for a dailycost minimization but also becomes an optimal control prob-lem and stochastic dynamic problems [4, 5].

2.1.1. Limitation to the Economists Approach. The economictheory of utility maximization and consumer rationality isused as a model for influencing consumers’ demand withprice. An experiment conducted by psychologists has shownthat individuals do not consistently make rational decisions[6] and their behaviour is not consistent with the standard

model of rational choice in economics [4]. Some of the fac-tors include time inconsistency, reference dependence, andbounded rationality. The economic theory of decision-making is incapable to fully explain human behaviour, to bespecific behaviours related to residential energy use. The the-ory of bounded rationality suggests that “individuals employheuristics” when making choices instead of the standard“strict rigid rules of optimization” [6].

2.2. Energy Technologists Approach. The second view pointdiscussed in [4] is also related to consumer rationality andother alternatives to energy efficiency and attributed toenergy technologists and behavioural scientists. Energy tech-nologists and behavioural scientists assert that individuals’choices for their preferences do not minimize their costs ofobtaining energy services. In the view of technological ana-lysts, consumers are not having accurate knowledge regard-ing energy-efficient technologies and lack the ability tomake accurate decisions even when they are given full andcomplete information [4]. Making accurate and precise deci-sions is based on relevant technical skills which are possessedby experts and specialists in economics, mathematics, andenergy-related disciplines. Making correct choices always assupposed by economists is highly impossible, since there willbe a need of high-speed computers to run complex optimiza-tion problems on energy technologies [4].

The work in [6] also contributes to understanding energyefficiency and stresses on the importance of human behav-iour as a factor that is always ignored. The author discussesthe following three basic routes that can lead to energysavings:

(1) Replacing the existing housing stock with low-energybuildings designed primarily to minimize heatingand cooling loads [6]

(2) Developing energy-efficient domestic equipment [6]

(3) Promoting and achieving “energy-conscious” behav-iour among end users [6]

Energy-efficient methods mostly follow the first tworoute, i.e., the design of buildings and energy-efficient appli-ances. The “physical-technical economic model” (PTEM) isone of the dominant models in energy analysis. The PTEMproposes that the energy conservation in buildings/house-holds relies almost always on the engineering design (physicalcharacteristics) of buildings and the efficiency of technologi-cal appliances. This model overlooks and considers humanbehaviour as an insignificant factor to reducing peak demandin the residential sector of energy consumption.

The PTEM focuses on energy pricing, technological tools,and improvement and considers social actions and noneco-nomic factors as an insignificant aspect of consumption [7].This is clearly contrary to what happens in the real world.Household energy consumption varies, and this can beattributed to variations in engineering design and economicfactors, demographic characteristics (gender, employmentstatus, household size, race, etc.), and the behaviour of thehousehold members.

2 Journal of Applied Mathematics

2.2.1. Limitation to the Energy Technologists Approach. ThePTEM focuses on energy pricing, technological tools, andimprovement and considers social actions and noneco-nomic factors as an insignificant aspect of consumption.On the contrary, social behaviours and noneconomic factorsare significant in the study of human energy consumptionactivities [7].

2.3. Sociopsychological Approach. Research has shown thatconsumers do not always act rationally as suggested byeconomist. Individual behaviour is bounded by time, andrationality and individuals are reference dependent [6].According to [8, 9], the consumers of energy do not just con-sider the minimization of energy cost in their day to dayactivities. Energy users are also consumers and thereforeshow higher preferences for things such as comfort andappearance than the monetary benefits obtained from usingenergy-efficient appliances. Consumers are in a social groupand therefore influenced by the things they hear fromneighbours or family more than information from distantenergy experts.

As stated earlier, the theory of bounded rationality sug-gests that “individuals employ heuristics to make a decisionrather than a strictly rigid rule of optimization” [6]. This isbecause consumers do not have the ability to compute theutility of all alternatives and the complexity of situations.Researchers have been stressing the importance of studyingresidential energy consumption from a social context. It isbelieved that despite the fact that encouraging changes inindividuals’ behaviour towards energy conservation is rele-vant, analysis from a social perspective will also provide abroad framework to understand residential energy conserva-tion behaviours of individuals [3].

2.3.1. Theory of Reasoned Action. The model proposed byIcek Ajzen which is known as the theory of reasoned actionmakes an attempt to explain human behaviour [10]. Accord-ing to [11], behavioural intention is simply what one intendto do or not to do, and it is the most appropriate determinantof behaviour. Behavioural intention is explained by the atti-tude (A) and subjective norm (SN) of an individual. Attitudeis simply an individual’s evaluation of behaviour, whereassubjective norm refers to the influence from the society orsurrounding on an individual to accept or not to accept a cer-tain norm or belief. In [10], the author treats the likelihood ofresponse by an individual as three self-standing notions,namely belief, attitude, and intention. The work in [12] alsoindicates that for any behaviour, individuals have severalbeliefs but have the ability to make use of only few at anygiven instant. Beliefs about likely consequences of attitudeand its evaluations are the major determinants of attitude[10, 11]. According to [10], it is the case that individualsestablishes/builds beliefs about an object by associating it toan underlying characteristics or attributes of the object inquestion. An example of how individuals form beliefs is asfollows; after a careful study of the exam result, it is obviousthat the students (object) in the class learned very hard orcopied each other or brought in foreign materials or stolethe questions (attributes) [10]. These attributes are evaluated

to be positive or negative by an individual concerning theobject. Hence, individuals immediately develop an attitudetowards the object. It is clear from these points that attitudeis a result of beliefs about an object, and how it is evaluatedby an individual. Suppose an individual has n beliefs(b1, b2,⋯, bn) and n subjective evaluations (e1, e2,⋯, en) ofthe beliefs. It follows from [10, 13] that the attitude A of theindividual is proportional to the sum of the product of thebeliefs and their evaluations, i.e.,

A∝ 〠n

i

biei: ð1Þ

Thus, attitude is proportional to the scalar product of thevectors b = ðb1, b2,⋯, bnÞ and e = ðe1, e2,⋯, enÞ of beliefsand subjective evaluations, respectively.

As indicated above, behaviour is explained by both theattitude and subjective norm of individuals. Subjective normis simply how an individual perceive others approval abouthis or her behaviour. The strength of normative belief si, fori = 1, 2,⋯, n, is concerned with the evaluation made by anindividual’s reference group, peers, colleagues, family, etc.relating to the approval or disapproval for the individual tohold belief bi. Subjective norm is determined by beliefs orthoughts others esteem as important about specific thingsone should do, and how much one is motivated to complyðm1,⋯,mnÞ with others on beliefs ðb1, b2,⋯, bnÞ [13]. Sub-jective norm is directly proportional to the sum of the prod-ucts of the strength of each normative belief (si) and anindividual’s motivation to comply (mi) to the belief bi acrossthe n different beliefs [10, 11, 13]. This can be mathematicallyexpressed as

SN ∝ 〠n

i

simi: ð2Þ

To properly explain behaviour, the theory of reasonedaction proposes that behaviour (B) is approximated bybehavioural intention (BI), and BI is proportional to thesum of attitude and subjective norm. Thus,

B ≈ BI = k1A + k2SN = k1 〠n

i=1biei + k2 〠

n

i=1simi: ð3Þ

2.3.2. Theory of Planned Behaviour. The theory of plannedbehaviour is also a sociopsychological theory which aims atexplaining human behaviour [13]. This theory is an exten-sion to the theory of reasoned action which looks at the pos-sible risky prediction that can be made with the theory ofreasoned action [11]. The theory of planned behaviourextends the theory of reasoned action by adding perceivedbehavioural control. Perceived behavioural control describesthe extent of control individuals think they have over theirbehaviour, and it comes as an additional factor to behav-ioural intention. It is usually measured by things such as“passing the exam is under my control” and “it is easy forme to pass the exam” [11]. According to [11], perceived

3Journal of Applied Mathematics

behavioural control is included in behavioural intention;hence, behavioural intention can be used directly to predictbehavioural achievement. The theory says that the mostappropriate determinant of one’s actual behaviour is simplythe motive an individual possesses to act in a certain manner.The core to the theory of reasoned action discussed aboveis intention. It can be noted that intention is central to thistheory of behaviour as well. It is assumed that intentionplays the role of capturing motivational factors that influ-ences the actual behaviour of an individual [10, 13, 14].Given the nature of a problem, behavioural intention is adescription of the willingness of individuals to attempt andan indication of extensive effort willing to exert in order toadopt a behaviour [10, 13, 14].

2.3.3. Goal-Framing Theory. The goal-framing theory dealswith cognitive process and studies environmental behav-iours. According to the work in [15], the goal-framing theoryis motivated by studies revolving around cognitive social psy-chology and aims at influencing one’s goals during cognitiveprocesses. Cognitive processes consider the mental functionof the mind related to logical reasoning and evaluations.For the theory to work, an individual is given a prior knowl-edge about certain innovations. Then, the receiver of theknowledge forms an attitude towards the innovation, decidesto reject or approve of the innovation based on the knowl-edge acquired, applies it, and confirms a decision. The frame-work identifies three goals that are relevant to the study ofenvironmental behaviour. These goals are the normative,hedonic, and gain goals, which are mostly relied on forproenvironmental behaviour studies [15]. In regard to thistheory, normative goal frame is defined to be the motivationan individual has “to act appropriately.”When an individualis obsessed with this goal, then an individual has the tendencyto act “according to one’s own or others norm’s” in the mostappropriate way and kindled to portray proenvironmentalbehaviour [15]. An individual inclined to hedonic goal isconcerned with things that enable him or her to “feel betterright now.” These include obsession in seeking personalpleasure, contentment, or arousing one’s personal interest.Individuals with this goal are very sensitive to things thataffect their satisfaction or pleasure [15]. An individual fallingunder the gain goal frame is one with the motivation to“guard and improve one’s resources” [15]. The frameworkdiscussed above is mostly used in analysing and developingpolicies and making decisions concerning proenvironmentalbehaviours. Considering the three goal frames discussed, aneffective strategy to foster interest in proenvironmentalbehaviours will be strengthening normative goals to reducethe adverse impact of one’s behaviour on nature [16]. Bydoing so, individuals would behave appropriately even whenhedonic or gain goals are focal. From this perspective, thenormative goal seekers’ behaviour is consistent with the rea-soned action behaviour if an individual’s environment istaken to be the reference group of the individual, and the rea-soned action behaviour is dominated by the subjective norm.

Sociopsychological research have indicated that humanbehaviour can better be explained by many factors (e.g.,norms, beliefs, values). Consider an individual with higher

value for the environment. This individual will more likelyrespond to environmental information than prices, while anindividual with higher value for energy security (blackout)will respond to information relating to practices that willensure no blackout than prices and environment [6].

The authors in [17] investigated the belief that being awitness to the behaviour of others significantly influencethe behaviour of an individual, and it can lead to peoplebelieving in things that are untrue, saying things that arefalse, or doing things that are wrong. In their work, they con-ducted two studies with the first study focusing on exploringrespondents’ stated reasons for engaging in energy conserva-tion [17], while the second study provided an initial test ofthe actual factors influencing participants’ conservationbehaviour [17]. In the first study, participant indicated thatthe behaviour of their neighbours (descriptive norm) hadthe least influence on them to conserve energy. Result fromthe second study indicates that descriptive norms is the mostinfluential factor to consider concerning individual energyconservation behaviour. They also found out that the envi-ronment and moral obligation to the society are the mostimportant factors to practicing energy conservation. In fact,results from the study suggest that energy users do not knowwhat actually motivates them to practice energy conservationand would not be able to forecast the most effective approachto conservation. It was also noted that normative influence isa powerful but underrated form of all social influence [17].

2.3.4. Limitation to the Sociopsychological Approach. Inenergy-related decisions, the sociopsychological models dis-cussed above provides qualitative interpretations to individ-uals’ behaviour. It lacks the ability to interpret the effect ofsocial interaction among group of individuals based on cer-tain socioeconomic attributes.

2.4. Logistic Regression Approach. Logistic approach is a well-known technique in predicting the behaviour of individuals.A study done in [14] employed ideas from the goal-framingtheory with the interest to determine the most dominant goalin energy efficiency practices among individuals with knowl-edge about energy conservation. The utility company Latve-nergo, in Latvia, was used as the survey area. The utilitycompany Latvenergo is the main supplier and distributor ofenergy in Latvia. The main idea behind their study was todelve into the attitude or behaviour of people that are wellvest in energy efficiency to investigate how they form attitudetowards energy efficiency practices. A correlation analysis wascarried out to study the relationship between the response ofparticipant being actively engaged in energy-saving activi-ties and all other answers of the questionnaire. The logisticapproach imported by the researchers was intended to iden-tify a linear combination of factors that could possiblyexplain the behaviour of the staff of Latvenergo to practiceor not to practice energy conservation behaviours. The logis-tic function used in the work is given by

log P Xð Þ1 − P Xð Þ� �

= β0 + β1x1 + β2x2+⋯+βkxk: ð4Þ

4 Journal of Applied Mathematics

Here, PðXÞ represents the probability for individual toengage in energy savings activity, βk is the coefficient ofthe kth independent variable xk, and β0 is the constant inthe model.

Their result from the correlation analysis indicated thatrespondents who are actively practising energy conservationbehaviours and those that are not, expressed, most of the time,normative view. It was also observed from their fitted regres-sionmodel that there exists no one clear goal dominating indi-vidual’s motivation in Latvenergo. The authors made use of afinal model they called “energy efficiency motivation,” thismodel combines all the three goal frames and with othervariables. This final model was able to explained 51.3685%deviance in the data and 13.0465% of adjusted deviance.

According to the authors, the final model had a loweradjusted deviance when compared with similar studies [18,19]. They believe that the broad survey setting could be a fac-tor and suggest a methodical improvement which will nar-row down to a specific energy efficiency practice such as theusage of LED appliances.

2.4.1. Limitation to the Logistic Model Approach. The logisticmodel studied above considers only socioeconomic attributesof individuals and makes no use of interaction among indi-vidual. Individuals are in a community and therefore influ-enced by the decisions of others within their referencegroup and beyond.

The studies conducted above bring to light various per-spectives of energy efficiency strategies and implementationsin support of demand response programs. The studies dis-cussed so far are all qualitative with the exception of Section2.4 and therefore forms a theoretical background of thispaper. These theories and models will culminate into a quan-titative model which will be used to analyse individual’senergy use behaviour in a community. The interest fromthe above viewpoints is a mathematical model that incorpo-rates all these ideas and how individuals’ decisions aboutenergy conservation are influenced by their socioeconomicattributes and interaction.

3. Quantitative Model for Human Behaviour

A quantitative model that integrates group interactions, bodyuse, cognitive processes, and human-machine interactionswill be desirable. Progress towards such a model has beenlimited by the theoretical preferences of the various disci-plines involved in energy research [2].

3.1. Modelling of Decision-Making and Energy Use. In thework of [2], the authors emphasized that theoreticalapproaches to decision-making have two main roles:

(1) To explain behaviour and identify important behav-ioural drivers for interventions to target

(2) To provide a framework for empirical research on theimpact of these interventions

Intervention in their work means any regulation, policy,program, measure, activity, or event that aims to influence

behaviour [2]. According to [7], social processes or influ-ences are quite complex and always ignored when modellingenergy conservation. Individual behavioural study is good,but individuals are in a society of which collective behaviourhas influence on their behaviour and choices. Families andlarge social groups are supposed to be considered in theanalysis of energy conservation [7]. The work in [7] sug-gests that research with a simple rational model on energyconservation decision-making is needed for both organisa-tional and residential sectors.

Our focus, in this paper, is to import ideas from dis-crete choice theory and statistical physics to develop modelsfor energy conservation decision-making. The interest hereis a model that will capture how individual’s choices aboutenergy conservation are influenced by their socioeconomicattributes and interaction. By socioeconomic attributes, weare referring to individual’s gender, income level, religion,and ethnicity.

3.2. Discrete Choice with Social Interaction. We propose, inthis section, a discrete choice with social interaction modelfor energy conservation behaviour. The discussion here fol-lows closely the presentations in [20, 21]. Imagine there areN-energy users who are to decide on whether to conserveor not to conserve energy at some common time. We encodethe choice made by the ith energy consumer as σi ∈ f−1, 1g,with σi = 1 representing conservation decision and σi = −1representing nonconservation decision. The decisions ofall the consumers in the reference group is an N-coordi-nate vector σ = ðσ1,⋯, σNÞ. In the sequel we denote byσ−i = ðσ1,⋯, σi−1, σi+1,⋯, σNÞ, the ðN − 1Þ-dimensionalvector of decisions made by all consumers less the deci-sion of consumer i. From an economic perspective, con-sumer i will decide on σi if it maximizes his/her utilityfunction VðσiÞ expressed as

V σið Þ = u σið Þ + S σi, μe σ−ið Þð Þ + ε σið Þ: ð5Þ

The quantities uðσiÞ and Sðσi, μeðσ−iÞÞ are, respectively,referred to as the private and social utilities/incentives the ith

consumer derives from the decision σi. The perception/beliefof consumer i about the choices σ−i made by the rest of theconsumers in the reference group is quantified by the proba-bility μeðσ−iÞ. Further, εðσiÞ is that part of the consumer’s util-ity that is unobserved by the experimenter/modeller, butconsumer i is aware of it at the time of taking the decisionσi. It is referred to as the random part of the utility. For ourapplication of the discrete choice theory to energy use behav-iour, the random utility may correspond to the consumer’sinability to properly optimize their utility. This may resultin the consumer making incorrect choice even when he/sheis provided with full and complete information. Further, itmay reflect the lack of information on the part of the con-sumer about new technologies.

The authors of [22] noted that the above discrete choicemodel is equivalent to a statistical mechanical model calledthe Curie-Weiss model when we have the following:

5Journal of Applied Mathematics

(i) uðσiÞ = hσi, for some h ∈ℝ. For this choice of theprivate utility h = 1/2½uð1Þ − uð−1Þ�, i.e., h is propor-tional to the difference between the private utilitiesof the two choices.

(ii) Sðσi, μeðσ−iÞÞ = Jσim, where m = 1/N − 1∑j≠iσi andJ ∈ℝ. This choice of our social utility belongs to alarge class utility functions that exhibit totalisticand constant strategic complementarity [23].

(iii) εð−1Þ − εð1Þ is logistically distributed, i.e.,

ℙ ε −1ð Þ − ε 1ð Þ ≤ xð Þ = 11 + exp −βxð Þ , β > 0: ð6Þ

It follows from the above considerations that if consumeri decides on σi = 1, then it is the case that Vð1Þ >Vð−1Þ. Itfollows from here that the probability that consumer i takesthe decision σi is given by

P σið Þ = exp β hσi + Jσi �mei½ �ð Þ

∑ηi∈ −1,1f g exp β hηi + Jηi �mei½ �ð Þ : ð7Þ

The parameter h quantifies an individual’s personalmotivation to make a choice, and J modulates an individual’smotivation to conform (not to conform) to social norm m.Positive h values indicate that at the private level an individ-ual will be motivated to choose 1, and the alternative choiceof -1 will be preferred for negative h values. J > 0 on the otherhand, will promote conformity to social norm and noncon-formity to social norm if J < 0. The parameter β describeshow much of the actual decision of the consumer is deter-mined by the deterministic part of the utility.

Under the assumption that energy consumers in a refer-ence group do not communicate or coordinate their deci-sions, the probability that the reference group makes achoice σ = ðσ1,⋯, σNÞ is given by

P σð Þ =exp β∑N

i=1 hσi + Jσim½ �� �

∑η1∈ −1,1f g ⋯∑ηN∈ −1,1f g exp β∑Ni=1 hηi + Jηim½ �

� �

=YNi=1

exp β hσi + Jσim½ �ð Þ∑ηi∈ −1,1f g exp β hηi + Jηim½ �ð Þ

ð8Þ

Note from (8) that σ1,⋯, σN are independent andidentically distributed Bernoulli random variables with com-mon mean

E σ1ð Þ = tanh βh + βJmð Þ: ð9Þ

The strong law of large numbers then tells us that asN tends to infinity the empirical average mN = 1/N∑N

i=1σialmost surely tends to tanh ðβh + βJmÞ. Further, it followsfrom [24] that the fix point equation (10) below has at leastone solution:

m∗ = tanh βh + βJm∗ð Þ: ð10Þ

Under the social equilibrium condition of [25, 26], wehave that

E σ1ð Þ =m = tanh βh + βJmð Þ: ð11Þ

This implies that the equilibrium mean choice level ofa consumer in the reference group is a solution to (10).Under the social equilibrium condition, the discrete choicemodel under consideration is equivalent to the Currie-Weissmodel [27]. This statistical mechanical model could exhibitmultiple equilibria due to the properties of the tanh function.The multiplicity of mean equilibrium levels of choicedepends sensitively on the strength of the social utilityand the magnitude of the bias towards one choice inducedby the private utility. In particular, the following result isproved in.

Proposition 1. For the solution(s) to (10) the following holds:

(1) If h = 0, then (10) has

(a) a unique solution for 0 < βJ ≤ 1

(b) three solutions for the case βJ > 1

(2) Suppose h ≠ 0, then there is a threshold H, dependingon β and h, such that (10) has

(a) a unique solution with the same sign as h if ∣βh∣>H

(b) three solutions, one of them has the same sign as h,and the others have opposite sign, if ∣βh∣ ≤H

3.3. Discrete Choice and Energy Use Behaviour Models. In thesociopsychological approach to modelling behaviour, beliefsare the building blocks. The beliefs about an object areformed base on an individual’s evaluation of the attributesof the object [10]. How this evaluation of attributes of anobject is done is silent in the sociopsychological models. Thisevaluation, we believe, is the starting point of the discretechoice theory. Here, it is natural to assume that individualsevaluate the attributes of an object based on their own attri-butes and the influences from their respective social environ-ments. Thus, individuals’ attributes and social environmentsare the determinants of their evaluation of attributes of anobject, which in turn inform their beliefs about the object.In particular, the objects forming the basis of the sociopsy-chological model are obtained from the discrete choice modelafter optimizing the utility function of the discrete choicemodel. Let us now look at how the various models for energyuse behaviour fit into the present scheme of discrete choicewith social interaction.

The discrete choice model offers a natural extension tothe theory of reasoned action. In that, upon optimizing the

6 Journal of Applied Mathematics

utility function, the optimal utility is still a random quantity,the optimal private utility now becomes the attitude and theoptimal social utility becomes the social norm part of behav-iour. Here, we are looking at the theory of reasoned actionwith random behaviour. The randomness here could bemodelling some intrinsic attributes of the individual thatcannot be observed by the modeller. Due to this randomnature of behaviour, we can no longer describe behaviourwith a value but rather with probability distribution on theset of all possible realisations of behaviour.

The theory of planned behaviour extends the theory ofreasoned action by adding perceived behavioural control.People can control their behaviour if they know themselvesvery well. It is this knowledge that determines the level ofcontrol an individual will have over his or her behaviour. Inthe context of the discrete choice theory, this control mayrefer to the extent to which individuals know about theparameters h and β, i.e., how individual evaluates their beliefsand the motivation for the individual to comply or not tocomply with the perceived evaluation by the rest of the refer-ence group. Thus, to get the discrete choice analogue of thetheory of planned behaviour, the parameters β and h needto be taken randomly with some a priori distribution, whichcould be updated as and when we get new information viaBayes rule. This leads us into disordered discrete choicemodels. For a statistical mechanical version of this model,we refer the reader to [28]. We shall not treat these class ofmodels here.

The discrete choice analogue for the normative and gaingoal frames are the case where the social utility component ofthe utility dominates the private utility of an individual. Overhere, one is preoccupied with the needs of neighbourhood orthe environment. The hedonic goal frame corresponds to thecase where the private utility part of an individual’s utilitydominates. We shall later on observe that the logistic modelconsidered in [14] results from the discrete choice modelwhen there is no social utility.

The random component of the utility function for thediscrete choice theory may corresponds to the individual’sinability to properly optimize their utility. This may resultin the individual making incorrect choice even when he/sheis provided with full and complete information. Further, itmay reflect the lack of information on the individual’s partabout new technologies. This connects the discrete choicetheory to the energy technologists’ model.

3.4. The Curie-Weiss Model. This section is devoted to thestatistical mechanical model that will be shown to be equiva-lent to the above discrete choice model. We will follow closelythe discussion and notation from [21]. The objects appearingin the discrete choice theory are obtained for the statisticalmechanical model after taking the thermodynamic limit,i.e., the limit as the number of consumers tend to infinity.The statistical mechanics model of interest is referred to asthe Curie-Weiss model is introduced as follows: consider apositive integer n and let Ωn = f−1, +1gn, be the set of spinconfigurations of the system. The Curie-Weiss model is givenby an energy function/Hamiltonian Hn onΩn, which for anyσ ∈Ωn takes the form

Hn σð Þ = −J2n 〠

n

i,j=1σiσj − h〠

n

i=1σi = −n

J2m

2n + hmn

� �−12 ,

ð12Þ

where

mn =1n〠n

i=1σi: ð13Þ

The quantity mn is called the empirical average, and itassumes values from the set f−1, −ðn − 1Þ/n,⋯, 0,⋯, ðn− 1Þ/n, 1g, for each positive integer n. The parameter h andJ are real valued, and they represent the interaction biasand interaction strength, respectively. h is the strength ofexternal field, and J measures the strength of the interactionamong pairs of spins. The first term in the Hamiltonian turnsto align pairs of spins if J > 0 and the pairs will not be alignedif J < 0. The second term also tends to align spins in the direc-tion of the external field h. Let α be the uniform probabilitymeasure on f−1, 1g, i.e., αð−1Þ = αð1Þ = 1/2 and put Pn =α⊗n, the product probability measure onΩn. The above Ham-iltonian defines a physical system whose equilibrium state isgiven by the probability measure

μn σð Þ = 1Zn

exp −βHn σð Þð ÞPn σð Þ, σ ∈Ωn, ð14Þ

The normalization term Zn is called the partition func-tion of the model and is defined as

Zn = 〠~σ∈Ωn

exp −βHn ~σð Þð ÞPn ~σð Þ: ð15Þ

The parameter β is the inverse temperature. The Curie-Weiss Hamiltonian can be written in terms of the empiricalaverage as follows:

Hn σð Þ = F mnð Þ = Jm2n

2 + hmn + o nð Þ, ð16Þ

where mn is the empirical measure of σ:Further, we write the μn expectation for any bounded

measurable function f defined on Ωn as follows:

<f>n,β,h =ðΩn

f mnð Þdμn σð Þ: ð17Þ

The specific empirical magnetization of the model is alsogiven as

mn β, hð Þ = 1n

ðΩn

〠n

i=1σi

!dμn σð Þ = 1

βn∂∂h

log Zn: ð18Þ

7Journal of Applied Mathematics

The second equality follows from (12), (14), and (15).The specific magnetization then becomes

m β, hð Þ = limn→∞

mn β, hð Þ: ð19Þ

To understand the form of the specific magnetization, theexponential growth rate of the partition function Zn will becrucial. Let us now lay some foundation to access this growthrate. Suppose n = n1 + n2 and recall from (13) that

mn =n1nmn1

+ n2nmn2

: ð20Þ

Using the convexity of the map x↦ x2 and (12), we get

Zn ≤ Zn1Zn2

: ð21Þ

Therefore

log Zn ≤ log Zn1+ log Zn2

: ð22Þ

It follows from the Fekete’s subadditive lemma [29] that

limn→∞

1nlog Zn, existsand lim

n→∞

1nlog Zn = inf

n

1nlog Zn:

ð23Þ

Next, we state a result taken from [24] that characterizesthe limiting μn expectation of bounded measurable functionsof the empirical average mnðβ, hÞ.

Theorem 2.

(1) Suppose f is a bounded and continuous function fromℝ to ℝ . Then,

where zðβ, hÞ is the global minimizer of the function

iβ,h zð Þ = −12βJz2 − βhz + 1 − z

2log 1 − zð Þ

+ 1 + z2

log 1 + zð Þð25Þ

on ∣z∣ ≤ 1 and zðβ,±Þ = limh→0±zðβ, hÞ.(2) The specific magnetization mðβ, hÞ of the Curie-

Weiss model is given by the unique global minimizerzðβ, hÞ of (25) for β > 0, h ≠ 0 and 0 < βJ ≤ 1, h = 0.In particular,

m β, ±ð Þ = limh→0±

m β, hð Þ

=z β, 0ð Þ = 0 for 0 < β J ≤ 1,

z β,±ð Þ ≠ 0 for β J > 1:

( ð26Þ

The above theorem says that whenever βJ > 1, a dramaticchange will be observed in the macroscopic behaviour of thesystem if a small change is made in h around the value h = 0.Observe that the specific magnetization for the Curie-Weissmodel, given in the above theorem, coincides with some ofthe social equilibrium average choice levels found as solutionto the self-consistency equation (10). Because the self-consistency equation (10) holds for the global minimizersof (25). Some of the solutions to the self-consistency equation

(10) may not be specific magnetization of the Curie-Weissmodel as they may be local minimizer and local maximizerof the function (25). In particular, for the case h ≠ 0, Theorem2 suggests that there is a unique specific magnetization thathas the same sign as h as opposed to the result in Proposition1 2, where it is observed that there are three self-consistentsolutions when β is high and h ≠ 0. In fact, the self-consistent solution that has the same sign as h becomes thespecific magnetization for the Curie-Weiss model, and theother two are, respectively, local minimum and maximumof the function (25). The local maximum and minimumare, respectively, the unstable and metastable self-consistentmean choice levels of a consumer. For the rest of the paper,we will concentrate on the self-consistent solutions that areconsistent with the statistical mechanical prescription, i.e.,global minimizers of (25). Thus, up to these self-consistentmean choice levels, the discrete choice model and theCurie-Weiss model are equivalent. In particular, the jointdistribution (8) of the decisions made by the consumers inthe reference group is obtained from the Curie-Weiss modelby taking n − dimensional marginal of the Curie-WeissGibbs measure μn, given in (14), as n tends to infinity.

4. Multipopulation Curie-Weiss Model

We present a multipopulation version of the Curie-Weissmodel in this section. This model will be our benchmarkmodel for modelling energy conservation behaviour of energyusers. Here, we follow closely the discussion in [30]. Our aimis to understand the collective behaviour of consumers in a

limn→∞

< f>n,β,h =f z β, hð Þð Þ forβ > 0, h ≠ 0and0 < βJ ≤ 1, h = 0

12 f z β,+ð Þð Þ + 1

2 f z β,−ð Þð Þ forβJ > 1, h = 0,

8><>: ð24Þ

8 Journal of Applied Mathematics

reference group from the intricate interactions among theconsumers [20, 22, 31]. The statistical mechanical modelintroduced in this section will help to achieve our aim.Already statistical mechanics has successfully been appliedto study the aggregate behaviour of individuals facing interde-pendence binary choice [27]. Statistical mechanics provides ageneral framework for understanding aggregate behavioursof interacting individuals. Here, our emphasis is on using amultipopulation Curie-Weiss model to study energy conser-vation behaviour of a community of energy users who areinteracting with themselves and their environment.

The Hamiltonian Hn for configurations (collective deci-sions) σ ∈ f−1, 1gn given by

Hn σð Þ = 12n 〠

n

i,l=1Jilσiσl + 〠

n

i=1hiσi ð27Þ

is made up of the interaction part controlled by the interac-tion strengths Jil and the external field part also modulatedby hi. The interactions tend to align spins/decisions σi andσl whenever Jil > 0, i.e., neighbouring consumers will preferto be either +1 (conserve energy) or -1 (not to conserveenergy). Negative interaction coupling strengths, i.e., Jil < 0,will result in neighbouring consumers taking different deci-sions. From statistical mechanics literature, the parameterhi is referred to as the external magnetic field applied to thesite i. From our socioeconomic applications point of view,hi is the intrinsic attributes of a consumer that determineshis/her behaviour. Positive values for hi favours +1 decisionwhile negative values of hi favours −1 decision. For social sci-ence applications of the model of interest, we refer the readerto [20, 27, 30].

Here, we study the energy conservation behaviour ofenergy consumers in a reference group, and how the behav-iour of an individual in the reference group is influenced byhis/her intrinsic socioeconomic attributes and social interac-tion with the rest of the members of the reference group.Notable socioeconomic attributes of energy consumers mayinclude gender, place of residence, level of education, ethnic-ity, beliefs, values, and norms. These attributes influence aconsumer’s evaluation of the attributes of energy alternativesand energy saving technologies. A key assumption that wewould employ in our study is that consumers with the samesocioeconomic attributes will tend to behave the same way,while consumers with different socioeconomic attributesbehave differently. This assumption reduces the number ofparameters in the Hamiltonian (27), allowing a systematicstatistical estimation procedure to be developed to estimatethe parameters of the model from data.

Suppose our reference group is made up of energy con-sumers with k socioeconomic attributes, and the group ismade up on n consumers. Therefore, consumer iwill be iden-tified by his/her attributes vector

ai = a 1ð Þi , a 2ð Þ

i ,⋯,a kð Þi

� �: ð28Þ

Here, aðjÞi ∈ Aj is the jth attribute of individual i, and Aj is

the set of attribute alternatives for attribute j = 1,⋯, k. LetdðjÞ = ∣Aj∣ be the number of attribute alternatives for attri-bute j, which we assume to be finite. Therefore, our referencegroup of size n can be partitioned into

d =Ykj=1

d jð Þ ð29Þ

nonover lapping subgroups, with each subgroup made upof individuals with the same socioeconomic attribute alter-natives. Suppose we identify these sub-groups with number1, 2,⋯, d, we denote by Ing and ∣Ing ∣ =ng the subset of indi-

viduals in sub-group g and the number of individuals insub-group g, respectively. Thus, n = n1+⋯+nd and for anyg ≠ g0 = 1, 2,⋯, d, Ing ∩ Ing0 =∅. Further, we assume that

for each g = 1,⋯, d,

γgn =ngn

and γg = limn→∞

γgn: ð30Þ

We assume that all individuals in subgroup g have thesame private utility �hg and for any pair of subgroups gand g0, Jil = �Jgg0 whenever i ∈ Ing and l ∈ Ing0 . It implies

from this assumption and equation (27) that

Hn σð Þ = 12n〠

d

g=1〠d

g0=1〠i∈Ing

〠l∈Ing0

Jilσiσl

0@

1A

+ 〠d

g=1〠i∈Ing

hiσi =12n〠

d

g=1〠d

g0=1�Jgg0ngm

gnng0m

gn′

+ 〠d

g=1�hgngm

gn = n

"〠d

g=1〠d

g0=1

�Jgg02 γgnm

gnγ

gn′mg

n′

+ 〠d

g=1�hgγ

gnm

gn

#= n

"〠d

g=1mg

n

〠d

g0=1

�Jgg02 γgnγ

gn′mg

n′

+ �hgγgn

!#= n〠

d

g=1mg

nUn,g:

ð31Þ

In the above, we have used that

mgn =

1ng

〠i∈Ing

σi, ð32Þ

is the empirical average decision for the individuals in sub-group g. The Hamiltonian of the model has now become afunction of the empirical averages of the decisions of indi-viduals in the subgroups. From our socioeconomic applica-tion of the model, Hn measures the level of satisfaction foreach joint decision vector of the individuals in the referencegroup. Here, �Jgg0 is the strength of the influence subgroup g

9Journal of Applied Mathematics

has on sub-group g0. It is also a measure of the socialincentive for subgroups g and g0 to interact. Positive �Jgg0values imply that the subgroups are satisfied if their empir-ical means have the same signs. On the other hand, if the�Jgg0 < 0, the empirical means of the different subgroups willprefer to have different signs, i.e., interaction between thetwo subgroups is discouraged. Figure 1 presents an artisticimpression of the �Jgg0 ’s when d = 4. The parameter �hg isthe private incentive of the group g. This describes howsubgroup g is satisfied with itself.

The finite volume Gibbs measure associated to our multi-population Hamiltonian Hn (31) takes the form

ωn σð Þ = eHn σð ÞPn σð ÞZn

, for σ ∈Ωn, ð33Þ

where

Zn = 〠~σ∈Ωn

eHn ~σð ÞPn ~σð Þ ð34Þ

is the normalization constant called the partition function ofthe model.

Recall from (32) the definition of the subgroup empiricalaverage mg

n and write mn = ðm1n,⋯,md

nÞ ∈ ½−1, 1�d for thevector of subgroup empirical averages. Define for any n ∈ℕ,

Fn mnð Þ = 〠d

g=1〠g0=1d

�Jgg02 γgnγ

gn′mg

nmgn′ + 〠

d

g=1�hgγ

gnm

gn: ð35Þ

Suppose a sequence ðmnÞn≥1 of empirical mean vectors

converges to m ∈ ½−1, 1�d , then

F mð Þ = 〠d

g=1〠d

g0=1

�Jgg02 γgγg′mgmg′

+ 〠d

g=1�hgγ

gmg = limn→∞

Fn mð Þ:ð36Þ

In the sequel, we shall write γ, �h, and �J for

γ = γ1,⋯, γd� �

, �h = �h1,⋯, �hd

� �∈ℝdand�J

= �Jg,g0�

1≤g,g0≤d:ð37Þ

The finite volume pressure of the multipopulation modelthen becomes

pn =1nlog Zn: ð38Þ

The large n behaviour of the model is controlled by theexponential growth rate of the partition function, i.e., thepressure. It follows from [32] that the thermodynamic limit

p γ, �h, �J�

= limn→∞

1nlog Zn ð39Þ

exists. An earlier proof of this claim for the case k = 1 andγgn = γg, for all n, is found in [33, 34].

Theorem 3. Suppose the parameters, γ , h, and �J , are given asabove. Then the limiting pressure has the following variationalrepresentation

p γ, �h, �J�

= supm∈ −1,1½ �d

F mð Þ − 〠d

g=1γgI mgð Þ

" #, ð40Þ

with

I mgð Þ = 1 −mg

2log 1 −mgð Þ + 1 +mg

2log 1 +mgð Þ ð41Þ

and F is defined in (36).

The maximizers m of the variational problem (40) havecoordinates mg satisfying the self-consistency equations

mg = tanh Ug

� , g = 1, 2,⋯, d: ð42Þ

Here, we have used that Ug is the limit of Un,g (31) as ntends to infinity, i.e.,

Ug = 〠g0=1d

�Jgg02 γgγg′mg′ + �hgγ

g: ð43Þ

Group

J11

I1

Group

J44

I4Group

J33

J32J23

J42

J13

J34

J43

J41J14

J31

J12

J21

J 24

I3

Group

J22

I2

Figure 1: A representation of the interaction network for the model(I1, I2, I3, and I4 are the subgroups of the reference group. Within I1(resp. I2, I3, I4), individuals feel a mean-field interaction withcoupling J11 (resp. J22, J33, and J44). Subgroup I1 influences theother subgroups with coupling strength (J12, J13, J14).

10 Journal of Applied Mathematics

In the sequel, we put Jgg0 = ð�Jgg0 /2Þγgγg′ and hg = γg�hg.Therefore, it follows from equation (43) that

Ug = 〠d

g0=1Jgg0m

g′ + hg, for g = 1,⋯, d: ð44Þ

The above theorem is proved in [35]. The proof for thecase k = 1 and γgn = γg for all n was earlier given in [33]. It fol-lows from [33, 35] and references therein that the marginaldistribution for the ith energy user in subgroup g to make achoice σi ∈ f−1,+1g is given by

ℙ σið Þ = eσiUg

eUg + e−Ug: ð45Þ

The expected value of the ith individual’s decision resultsin the self-consistency equations

mg = E σið Þ = eUg

eUg + e−Ug−

e−Ug

eUg + e−Ug

= tanh Ug

� , for g = 1, 2,⋯, d:

ð46Þ

Recall from (46) that mg is the average decision of theindividuals in group g. Again, with respect to the parametersJgg0 and hg, it follows from (44) that Ug is a linear regression

model with covariates mg′, for g0 = 1,⋯, d. Further, wemodel the private utility part hg as a linear regression of theattributes alternatives (28) as follows:

hg = α0 + 〠k

j=1〠d jð Þ

s=1αjs〠r∈L

δajg ,r: ð47Þ

Recall that our model considers k socioeconomic attri-butes. Here, αj

s measures the value an individual places onthe s attribute alternative of attribute j, and α0 is the uniformprivate utility for all the decision makers. ajg is the alternativeof attribute jth present in group g, and dðjÞ is the count of thedifferent alternative for attribute j. The set of all alternativesfor the different attributes is the alternatives set L, and δajg ,ris the Kronecker delta function. The form of our private util-ity hg (47) accounts for the influence of each of the attributealternatives on the private utility contrary to the case consid-ered in [34], where the influence of some of the attributealternatives was put to zero. Consequently, the linear regres-sion models for the Ug have Jgg0 , α

js, and α0 as their param-

eters that should be estimated.The case Jgg0 = 0 for every g and g0 is the model with no

interaction. The regression model for Ug becomes

Ug = hg, for g = 1,⋯, d: ð48Þ

Therefore, it follows from (45) that the probability for theith individual in subgroup g to decide on σi = 1 takes the form

ℙ σi = 1ð Þ = eUg

eUg + e−Ug= 11 + e−2Ug

: ð49Þ

Our model reduces to the logistic regression when theJgg0 = 0 for every g and g0. Therefore, taking logarithm ofthe quotient of the probabilities to choose σi = 1 againstσi = −1 results in the following interaction version of thelogistic regression

12 log ℙ σi = 1ð Þ

ℙ σi = −1ð Þ� �

= 〠d

g0=1Jgg0mg0 + hg: ð50Þ

As mentioned above, if Jgg0 = 0 for every g and g0, theabove logarithm of the quotient becomes the logisticregression model outlined in Section 2.4. This model wasused in [14] to study the energy conservation behaviour ofenergy users.

4.1. Parameter Estimation. In what follows, we will esti-mate mg (46) by mg

n (32). For large enough N , γgn willbe close to γg; hence, we drop the n dependence of γgn .Consequently, the parameters Jgg0 and hg become n indepen-dent. Two procedures will be discussed for the estimation ofthe parameters.

4.1.1. Estimation Procedure 1. Recall from (46) that tanh ðUgÞmodel’s prediction of mg and it is estimated by the empiricalaverage choice mg

n . Due to the monotonicity property of thetanh function, we estimate the parameters of the model vialooking for the parameter values that minimizes the sum ofsquares error

〠g

arctan h mgnð Þ −Ug

�2: ð51Þ

since the parameters to be estimated are linear in Ug. This

procedure is only possible if mgn stays away from being +1

or −1. In the regime where the interaction strength is non-vanishing the covariates of the regression model may be cor-related. Due to this, the partial least squares (PLS) estimationprocedure may be used [36–40]. This procedure extracts a setof independent predictors from a set of correlated predictorsthat jointly explain most of the variability in both the depen-dent and the independent variables. The key question here is,how many independent predictors should we use for the esti-mation? This question is answered with the help of the rootmean square error of prediction for models formed fromincreasing subsets of the extracted independent predictorsthat are sorted in decreasing order of importance. The rootmean square error of prediction for the models decreases ini-tially the more predictors are introduced one at a time. Thebest fitted model is the one beyond which the root meansquare error of prediction starts to rise. One could not accessconfidence intervals for PLS estimates. Bootstrap resampling

11Journal of Applied Mathematics

techniques are usually applied to the best fitted model to gen-erate a number of estimates that are used to construct confi-dence intervals for the estimates.

4.1.2. Estimation Procedure 2. This section outlines theparameter estimation procedure for multipopulation Curie-Weiss model [41, 42]. Recall from the expression for Ug in(44) and (31) that

Hn σð Þ = n〠d

g=1mgUg = n〠

d

g=1mg 〠

d

g0=1Jgg0m

g′ + hg

!

= n 〠d

g,g0=1Jgg0mgmg′ + 〠

d

g=1hgm

g

!

= n Jm,mh i + h,mh ið Þ

ð52Þ

where m = ðm1,⋯,mdÞ, h = ðh1,⋯, hdÞ and J is the interac-tion matrix

J =

J11 J12 ⋯ J1d

J12 J22 ⋯ J2d

⋮ ⋮ ⋮

J1d J2d ⋯ Jdd

0BBBBB@

1CCCCCA ð53Þ

From the self-consistency equations (46) we have that

mg J , hð Þ = E σið Þ = tanh 〠d

g0=1Jgg0m

g′ + hg

!,

for g = 1, 2,⋯, d:ð54Þ

If the self-consistency equations above admit a uniquethermodynamic stable solution mðJ , hÞ = ðm1ðJ , hÞ,⋯,md

ðJ , hÞÞ, then the following identity hold [33, 42],

limn→∞

ωn mgnð Þ =mg J , hð Þ, g = 1,⋯, d, ð55Þ

where ωn and mgn are defined as in (33) and (32), respec-

tively. Note that ωnðmgnÞ is the average value of the mag-

netization. According to [42], the susceptibility matrix ofthe model has the following entries:

χgg0 = limn→∞

∂∂hg0

ωn mgnð Þ, for g, g0 = 1,⋯, d

= ∂mg J , hð Þ∂hg0

= ∂∂hg0

tanh hg + 〠d

g0=1Jgg0m

g′ ! !

= 1 − mgð Þ2� �

δgg0 + 〠d

g0=1Jgg0χgg0

!,

ð56Þ

where δgg0 = 1 if g = g0 and zero otherwise is the Kro-necker delta . Hence, the susceptibility matrix is given as,

χ = P I + Jχð Þ, ð57Þ

where P = diag f1 − ðm1Þ2,⋯, 1 − ðmdÞ2g and I is anidentity matrix. Again, for each pair of groups g, g0 = 1,⋯, d,

∂∂hg0

ωn mg σð Þð Þ = ∂∂hg0

∑σ∈ΩnmgeHn σð Þ

∑σ∈ΩneHn σð Þ

!

= ng0 ωn mgmg′� �

− ωn mgð Þωn mg′� �� �

:

ð58Þ

It follows from (56) and (58) that the susceptibility matrixχ can be computed with the help of (57). With this, we obtainan expression for the interaction matrix J , which is related tothe averages and correlations of the magnetizations in thethermodynamic limit [42], i.e.,

J = P−1 − χ−1� : ð59Þ

The external field h is obtained by using the above esti-mate for the interaction matrix J and the self-consistencyequations (54), i.e.,

hg = tanh−1 mg J , hð Þð Þ − 〠d

g0=1Jgg0m

g′ J , hð Þ

= tanh−1 limn→∞

ω mgð Þ� �

− 〠d

g0=1Jgg0 limn→∞

ω mg′� �

,

for g = 1,⋯, d:ð60Þ

Upon estimating the hg’s, we then fit the followingregression model (47) to determine the socioeconomic attri-butes that contribute to the estimated private incentive hg foreach of the groups.

hg = α0 + 〠k

j=1〠d jð Þ

s=1αjs〠r∈L

δajg ,r: ð61Þ

5. Conclusion

This study incorporates ideas from economics, energymanagement, social psychology, discrete choice theory,and statistical physics. In the sociopsychological modelsdiscussed, it was noted that beliefs are their building blocksand that beliefs about an object are formed base on an indi-vidual’s evaluation of the attributes of the object [10, 13].How these evaluations are made, we argue, is the startingpoint of the discrete choice model. The study proposes a sta-tistical mechanical model to bridge the sociopsychological,economics, and energy models used in energy management.

12 Journal of Applied Mathematics

Our model is motivated by the equivalence between dis-crete choice theory with social interaction and the Curie-Weiss model from statistical mechanics [22]. This formula-tion provides a general framework for understanding howinterdependences modulate the aggregate behaviours ofindividuals.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The authors thank the staff at the Mathematics and StatisticsDepartment and Energy and Engineering Department ofUniversity of Energy and Natural Resources for their supportand kindness during the period this paper was written.

References

[1] E. N. Kumi, The Electricity Situation in Ghana: Challengesand Opportunities. CGD Policy Paper, Center for GlobalDevelopment, Washington DC, 2017, http://www.cgdev.org/publication/electicity-situation-ghana-challenges-and-opportunities.

[2] C. Wilson and H. Dowlatabadi, “Models of decision makingand residential energy use,” Annual Review of Environmentand Resources, vol. 32, no. 1, pp. 169–203, 2007.

[3] S. Gyamfi, S. Krumdieck, and T. Urmee, “Residential peakelectricity demand response—highlights of some behaviouralissues,” Renewable and Sustainable Energy Reviews, vol. 25,pp. 71–77, 2013.

[4] A. H. Sanstad and R. B. Howarth, “Consumer rationality andenergy efficiency,” in Proceedings of the ACEEE 1994 SummerStudy on Energy Efficiency in Buildings, Washington D.C.,1994.

[5] T. G. Cowing and D. L. McFadden, Microeconomic Modelingand Policy Analysis: Studies in Residential Energy Demand,Academic Press, Orlando, Florida, 1984.

[6] S. Gyamfi, “Demand response assessment and modelling ofpeak electricity demand in the residential sector: informationand communication requirements,” PhD Thesis, Universityof Canterbury, Department of Mechanical Engineering, 2010.

[7] L. Lutzenhiser, “Social and behavioral aspects of energy use,”Annual Review of Energy and the Environment, vol. 18, no. 1,pp. 247–289, 1993.

[8] E. Aronson, “Energy use: the human dimension,” in Commit-tee on Behavioral and Social Aspects of Energy Consumptionand Production, National Research Council, Freeman, NewYork, 1984.

[9] P. C. Stern, Improving Energy Demand Analysis. Panel onEnergy Demand Analysis, Committee on Behavioral and SocialAspects of Energy Consumption and Production, NationalResearch Council, National Academy Press, Washington, DC,1984.

[10] I. Ajzen, Attitudes, Personality and Behaviour, Open Univer-sity Press, Milton Keynes, Second edition, 1988.

[11] D. Trafimow, “The theory of reasoned action,” Theory & Psy-chology, vol. 19, no. 4, pp. 501–518, 2009.

[12] G. A. Miller, “The magical number seven plus or minus two:some limits on our capacity for processing information,” Psy-chological Review, vol. 63, no. 2, pp. 81–97, 1956.

[13] I. Ajzen, “The theory of planned behavior,” OrganizationalBehavior and Human Decision Processes, vol. 50, no. 2,pp. 179–211, 1991.

[14] U. Bariss, A. Dandens, L. Timma, A. Blumberga, andD. Blumberga, “How to assess involvement of electricity enduser in energy efficiency improvement - analysis of surveyresults,” Energy Procedia, vol. 72, pp. 270–277, 2015.

[15] S. Lindenberg and L. Steg, “Normative, gain and hedonic goalframes guiding environmental behavior,” Journal of SocialIssues, vol. 63, no. 1, pp. 117–137, 2007.

[16] L. Steg, J. W. Bolderdijk, K. Keizer, and G. Perlaviciute, “Anintegrated framework for encouraging pro-environmentalbehaviour: the role of values, situational factors and goals,”Journal of Environmental Psychology, vol. 38, pp. 104–115,2014.

[17] J. M. Nolan, P. W. Schultz, R. B. Cialdini, N. J. Goldstein, andV. Griskevicius, “Normative social influence is underdetected,”Society for Personality and Social Psychology, vol. 34, no. 7,pp. 913–923, 2008.

[18] M. Tonglet, P. S. Phillips, and M. P. Bates, “Determining thedrivers for householder pro-environmental behaviour: wasteminimisation compared to recycling,” Resources, Conservationand Recycling, vol. 42, no. 1, pp. 27–48, 2004.

[19] E. Vīgants, A. Blumberga, L. Timma, I. Ījabs, and D. Blumberga,“The dynamics of technological substitution: the case of eco-innovation diffusion of surface cleaning products,” Journal ofCleaner Production, vol. 132, pp. 279–288, 2016.

[20] W. A. Brock and S. N. Durlauf, “Discrete choice with socialinteractions,” The review of Economic Studies, vol. 68, no. 2,pp. 235–260, 2001.

[21] A. A. Opoku, K. O. Edusei, and R. K. Ansah, “A conditionalCurie Weiss model for stylized multi-group binary choice withsocial interaction,” Journal of Statistical Physics, vol. 171, no. 1,pp. 106–126, 2018.

[22] S. N. Durlauf, “How can statistical mechanics contributeto social science?,” Proceedings of the National Academyof Sciences of the United States of America, vol. 96, no. 19,pp. 10582–10584, 1999.

[23] R. Cooper and A. John, “Coordinating coordination failures inKeynesian models,” The Quarterly Journal of Economics,vol. 103, no. 3, pp. 441–463, 1988.

[24] R. S. Ellis, Entropy, Large Deviations and Statistical Mechanics,Classics in Mathematics, Springer, 1985.

[25] C. Manski, “Identification problems in the social sciences,” inSociological Methodology, P. Marsden, Ed., vol. 23, Basil Black-well, Cambridge, 1993.

[26] C. F. Manski, “Identification of endogenous social effects: thereflection problem,” Review of Economic Studies, vol. 60,no. 3, pp. 531–542, 1993.

[27] S. N. Durflar, Statistical Mechanics Approach to SocioeconomicBehavior, vol. 203 of NBER, technical working paper series,technical working paper, 1996.

[28] A. Bovier and V. Gayrard, “The thermodynamics of the Curie-Weiss model with random couplings,” Journal of StatisticalPhysics, vol. 72, no. 3-4, pp. 643–664, 1993.

13Journal of Applied Mathematics

[29] M. Fekete, “Über die Verteilung der Wurzeln bei gewissenalgebraischen Gleichungen mit ganzzahligen Koeffizienten,”Mathematische Zeitschrift, vol. 17, no. 1, pp. 228–249, 1923.

[30] A. A. Opoku, G. Osabutey, and C. Kwofie, “Parameter evalua-tion for a statistical mechanical model for binary choice withsocial interaction,” Journal of Probability and Statistics,vol. 2019, Article ID 3435626, 10 pages, 2019.

[31] D. Mcfadden, “Economic choices,” The American EconomicReview, vol. 91, no. 3, pp. 351–378, 2001.

[32] A. A. Opoku and G. Osabutey, “A multi-population Curie-Weiss model for binary choice with social interaction,” inpreparation.

[33] I. Gallo and P. Contucci, “Bipartite mean field spin systems,”in Existence and Solution, vol. 14, MPEJ, 2008.

[34] I. Gallo, “An equilibrium approach to modelling social interac-tion,” PhD Thesis, University of Bologna, Bologna, Italy, 2009.

[35] A. A. Opoku and G. Osabutey, “Multipopulation spin models:a view from large deviations theoretic window,” Journal ofMathematics, vol. 2018, Article ID 9417547, 13 pages, 2018.

[36] H. Abdi, “Partial least squares regression and projection onlatent structure regression (PLS Regression),” WIREs Compu-tational Statistics, vol. 2, no. 1, pp. 97–106, 2010.

[37] K. S. Ng, A Simple Explanation of Partial Least Squares, Draft,2013.

[38] H. Wold, “Estimation of principal components and relatedmodels by iterative least squares,” in Multivariate Analysis, P.R. Krishnaiaah, Ed., pp. 391–420, Academic Press, New York,1966.

[39] S. Wold, “Personal memories of the early PLS development,”Chemometrics and Intelligent Laboratory Systems, vol. 58,no. 2, pp. 83-84, 2001.

[40] K. J. Worsley, “An overview and some new developments inthe statistical analysis of PET and fMRI data,” Human BrainMapping, vol. 5, no. 4, pp. 254–258, 1997.

[41] R. Burioni, P. Contucci, M. Fedele, C. Vernia, and A. Vezzani,“Enhancing participation to health screening campaignsby group interactions,” Scientific Reports, vol. 5, no. 1, article9904, 2015.

[42] M. Fedele, C. Vernia, and P. Contucci, “Inverse problemrobustness for multi-species mean-field spin models,” Journalof Physics A: Mathematical and Theoretical, vol. 46, no. 6, arti-cle 065001, 2013.

14 Journal of Applied Mathematics