Patent Publication Number: US-10317864-B2

Title: Systems and methods for adaptively updating equipment models

Description:
CROSS-REFERENCE TO RELATED PATENT APPLICATION 
     This application is a continuation of U.S. patent application Ser. No. 14/579,736 filed Dec. 22, 2014, the entire disclosure of which is incorporated by reference herein. 
    
    
     BACKGROUND 
     The present description relates generally to equipment models used to predict the performance of building equipment. The present description relates more particularly to systems and methods for determining when an equipment model no longer accurately represents the corresponding building equipment. 
     Equipment models are used to predict performance metrics for various types of equipment. For example, an equipment model for a chiller in a HVAC system may be used to predict a variable of interest for the chiller (e.g., power consumption, efficiency, etc.) of the chiller as a function of one or more predictor variables (e.g., chiller load, temperature setpoint, etc.). The accuracy of equipment models can have a strong impact on the performance of any system that relies upon the modeled information. For example, a model-based control methodology may use equipment models to optimally distribute loads across multiple devices based on the modeled performance of each device. If the equipment or system is no longer accurately represented by the model, the prediction capabilities may diminish and the model-based control methodology may be suboptimal. It is difficult and challenging to identify when an equipment model has diminished prediction capability. 
     SUMMARY 
     One implementation of the present disclosure is a system for adaptively updating a predictive model for building equipment or a collection of building equipment. The system includes an operating data aggregator module configured to collect a first set of operating data for the building equipment corresponding to a first time period and a second set of operating data for the building equipment corresponding to a second time period. The system further includes a model generator module configured to generate a first set of model coefficients for the predictive model using the first set of operating data and a second set of model coefficients for the predictive model using the second set of operating data. The system further includes a test statistic module configured to generate a test statistic based on a difference between the first set of model coefficients and the second set of model coefficients and a critical value module configured to calculate a critical value for the test statistic. The system further includes a hypothesis testing module configured to perform a statistical hypothesis test. The statistical hypothesis test may include comparing the test statistic with the critical value to determine whether the predictive model has changed. The system further includes a model update module configured to adaptively update the predictive model in response to a determination that the test statistic exceeds the critical value. 
     In some embodiments, adaptively updating the predictive model includes generating a new set of model coefficients for the predictive model, determining whether the new set of model coefficients improves a fit of the predictive model to a set of operating data relative to a previous set of model coefficients used in the predictive model, and replacing the previous set of model coefficients with the new set of model coefficients in the predictive model in response to a determination that the new set of model coefficients improves the fit of the predictive model. In some embodiments, adaptively updating the predictive model includes retaining the previous set of model coefficients in the predictive model in response to a determination that the new set of model coefficients does not improve the fit of the predictive model. In some embodiments, adaptively updating the predictive model includes replacing the previous set of model coefficients with the new set of model coefficients if the fit of the new set of model coefficients is within a threshold. In this way, the new set of model coefficients may be used even if the fit is not improved, so long as the fit provided by the new set of model coefficients is within the threshold. 
     In some embodiments, calculating the critical value includes identifying a parameter representing a predetermined likelihood that the statistical hypothesis test improperly rejects a null hypothesis that the predictive model has not changed. Calculating the critical value may further include using an inverse cumulative distribution function for the test statistic to determine, based on the parameter, the critical value for the test statistic such that the test statistic has the predetermined likelihood of exceeding the critical value when the predictive model has not changed. 
     In some embodiments, the model generator module is configured to adjust at least one of the first time period and the second time period to define an adjusted time period based on a current time. The model generator module may further be configured to iteratively update at least one of the first set of model coefficients and the second set of model coefficients using a set of the operating data corresponding to the adjusted time period. 
     In some embodiments, the system includes a demand response module configured to use the updated predictive model to generate a control output for the building equipment using a model-based control methodology. 
     In some embodiments, the system includes an autocorrelation corrector configured to remove an autocorrelated model error from at least one of the first set of operating data and the second set of operating data prior to the model generator module determining the sets of model coefficients. In some embodiments, removing the autocorrelated model error includes determining a residual error representing a difference between an actual output of the building equipment and an output predicted by the predictive model, using the residual error to calculate a lag one autocorrelation for the model error, and transforming at least one of the first set of operating data and the second set of operating data using the lag one autocorrelation. 
     Another implementation of the present disclosure is system for detecting a fault in a predictive model for building equipment or a collection of building equipment. The system includes an operating data aggregator module configured to collect a first set of operating data for the building equipment corresponding to a first time period and a second set of operating data for the building equipment corresponding to a second time period. The system further includes a model generator module configured to generate a first set of model coefficients for the predictive model using the first set of operating data and a second set of model coefficients for the predictive model using the second set of operating data. The system further includes a test statistic module configured to generate a test statistic based on a difference between the first set of model coefficients and the second set of model coefficients and a critical value module configured to calculate a critical value for the test statistic. The system further includes a hypothesis testing module configured to perform a statistical hypothesis test. The statistical hypothesis test may include comparing the test statistic with the critical value to determine whether the predictive model has changed. The system further includes a fault detection module configured to generate a fault indication in response to a determination that the test statistic exceeds the critical value. 
     In some embodiments, generating the fault indication includes generating a fault event indicating that the predictive model has changed and appending to the fault event a statistical confidence that the predictive model has changed. The statistical confidence may be based on a parameter of the statistical hypothesis test. 
     In some embodiments, calculating the critical value includes identifying a parameter representing a predetermined likelihood that the statistical hypothesis test improperly rejects a null hypothesis that the predictive model has not changed. Calculating the critical value may further include using an inverse cumulative distribution function for the test statistic to determine, based on the parameter, the critical value for the test statistic such that the test statistic has the predetermined likelihood of exceeding the critical value when the predictive model has not changed. 
     In some embodiments, the model generator module is configured to adjust at least one of the first time period and the second time period to define an adjusted time period based on a current time. The model generator module may be configured to iteratively update at least one of the first set of model coefficients and the second set of model coefficients using a set of the operating data corresponding to the adjusted time period. 
     In some embodiments, the system includes a demand response module configured to use the updated predictive model to generate a control output for the building equipment using a model-based control methodology. 
     In some embodiments, the system includes an autocorrelation corrector configured to remove an autocorrelated model error from at least one of the first set of operating data and the second set of operating data prior to the model generator module determining the sets of model coefficients. 
     Another implementation of the present disclosure is a method for identifying changes in a predictive model for building equipment or a system including a collection of building equipment. The method includes collecting, by an operating data aggregator module of an equipment model adaptation system, a first set of operating data for the building equipment corresponding to a first time period and a second set of operating data for the building equipment corresponding to a second time period. The method further includes generating, by a model generator module of the equipment model adaptation system, a first set of model coefficients for the predictive model using the first set of operating data and a second set of model coefficients for the predictive model using the second set of operating data. The method further includes generating, by a test statistic module of the equipment model adaptation system, a test statistic based on a difference between the first set of model coefficients and the second set of model coefficients. The method further includes calculating, by a critical value module of the equipment model adaptation system, a critical value for the test statistic. The method further includes comparing, by a hypothesis testing module of the equipment model adaptation system, the test statistic with the critical value using a statistical hypothesis test to determine whether the predictive model has changed. The method further includes triggering an action performed by the equipment model adaptation system in response to the test statistic exceeding the critical value. The action includes at least one of generating a fault indication and adaptively updating the predictive model. 
     In some embodiments, the action includes adaptively updating the predictive model. Adaptively updating the predictive model may include generating a new set of model coefficients for the predictive model and determining whether the new set of model coefficients improves a fit of the predictive model to a set of operating data relative to a previous set of model coefficients used in the predictive model. Adaptively updating the predictive model may further include replacing the previous set of model coefficients with the new set of model coefficients in the predictive model in response to a determination that the new set of model coefficients improves the fit of the predictive model or retaining the previous set of model coefficients in the predictive model in response to a determination that the new set of model coefficients does not improve the fit of the predictive model. In some embodiments, adaptively updating the predictive model includes replacing the previous set of model coefficients with the new set of model coefficients if the fit of the new set of model coefficients is within a threshold. In this way, the new set of model coefficients may be used even if the fit is not improved, so long as the fit provided by the new set of model coefficients is within the threshold. 
     In some embodiments, the action includes generating the fault indication. Generating the fault indication may include generating a fault event indicating that the predictive model has changed and appending to the fault event a statistical confidence that the predictive model has changed. The statistical confidence may be based on a parameter of the statistical hypothesis test. 
     In some embodiments, the test statistic is an F-statistic based on the difference between the first set of model coefficients and the second set of model coefficients. 
     In some embodiments, calculating the critical value includes identifying a parameter representing a predetermined likelihood that the statistical hypothesis test improperly rejects a null hypothesis that the predictive model has not changed. Calculating the critical value may further include using an inverse cumulative distribution function for the test statistic to determine, based on the parameter, the critical value for the test statistic such that the test statistic has the predetermined likelihood of exceeding the critical value when the predictive model has not changed. 
     In some embodiments, the method includes removing an autocorrelated model error from at least one of the first set of operating data and the second set of operating data prior to determining the sets of model coefficients. 
     Those skilled in the art will appreciate that the summary is illustrative only and is not intended to be in any way limiting. Other aspects, inventive features, and advantages of the devices and/or processes described herein, as defined solely by the claims, will become apparent in the detailed description set forth herein and taken in conjunction with the accompanying drawings. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  is a drawing of a building equipped with a building management system (BMS) including various types of building equipment for monitoring and/or controlling conditions within the building, according to an exemplary embodiment. 
         FIG. 2  is a block diagram illustrating the BMS of  FIG. 1  in greater detail including a BMS controller and a plurality of building subsystems, according to an exemplary embodiment. 
         FIG. 3  is a block diagram of an equipment model adaptation system that may be implemented as part of the BMS controller of  FIG. 2 , according to an exemplary embodiment. 
         FIG. 4  is a block diagram illustrating a model generator module of the equipment model adaptation system of  FIG. 3  in greater detail, according to an exemplary embodiment. 
         FIG. 5  is a drawing of a timeline illustrating sliding data windows that may be used by the equipment model adaptation system of  FIG. 3  to iteratively define data sets, according to an exemplary embodiment. 
         FIG. 6  is a flowchart of a process for identifying changes in a predictive model for building equipment, according to an exemplary embodiment. 
         FIG. 7  is a flowchart of a process for adaptively updating an equipment model with a new set of model coefficients, according to an exemplary embodiment. 
     
    
    
     DETAILED DESCRIPTION 
     Referring generally to the FIGURES, systems and methods for adaptively updating equipment models are shown, according to various exemplary embodiments. Equipment models are used to predict performance metrics for various types of equipment. For example, an equipment model for a chiller in a HVAC system may be used to predict a variable of interest for the chiller (e.g., power consumption, efficiency, etc.) of the chiller as a function of one or more predictor variables (e.g., chiller load, temperature setpoint, etc.). The accuracy of equipment models can have a strong impact on the performance of any system that relies upon the modeled information. For example, a model-based control methodology may use equipment models to optimally distribute loads across multiple devices based on the modeled performance of each device. If the equipment or system is no longer accurately represented by the model, the prediction capabilities may diminish and the model-based control methodology may be suboptimal. The systems and methods described herein can be used to identify when an equipment model no longer accurately represents the modeled equipment. 
     In some embodiments, equipment models are generated using a set of operating data from the building equipment. The operating data may include values for one or more independent variables (i.e., predictor variables) provided as inputs to the equipment models and one or more dependent variables (i.e., variables of interest) predicted by the equipment models. A regression technique can be used to generate a set of model coefficients {circumflex over (β)} that fits the operating data. In some embodiments, a first set of operating data corresponding to a first time period is used to generate a first set of model coefficients {circumflex over (β)} 1  for an equipment model. A second set of operating data corresponding to a second time period may be used to generate a second set of model coefficients {circumflex over (β)} 2  for the same equipment model. Each set of model coefficients may be represented by a vector consisting of random variables. 
     In some embodiments, a test statistic is generated to determine whether the predictive model has changed. The test statistic may be an F-statistic F Δβ  based on a difference Δβ between the first set of model coefficients and the second set of model coefficients. Statistical hypothesis testing may be performed using the test statistic F Δβ  to determine whether the predictive model has changed. For example, a critical value f crit  may be calculated using an inverse cumulative distribution function for the test statistic. The critical value f crit  may be selected such that the test statistic F Δβ  has a predetermined statistical likelihood of being less than the critical value when the predictive model has not changed. In some embodiments, the systems and methods described herein formulate a null hypothesis that the predictive model has not changed. The null hypothesis may be rejected in response to a determination that the test statistic F Δβ  exceeds the critical value f crit . 
     In some embodiments, the systems and methods of the present disclosure generate a fault event in response to rejecting the null hypothesis. The fault event may indicate that the predictive model has changed and may include the predetermined statistical likelihood that the null hypothesis was properly rejected. The fault event can be reported to a user, stored in memory, or otherwise used in various fault detection and diagnostic processes. 
     In some embodiments, the systems and methods of the present disclosure adaptively update the equipment model in response to rejecting the null hypothesis. Adaptively updating the equipment model may include generating a new set of model coefficients for the equipment model based on a recent set of operating data. In some embodiments, the new set of model coefficients are automatically applied to the equipment model. In other embodiments, the systems and methods described herein determine whether the new set of model coefficients result in a better fit of the predictive model to the set of operating data relative to an existing set of model coefficients used by the model. If the new set of model coefficients result in a better fit, the new set of model coefficients may be applied to the equipment model (e.g., replacing the existing set of model coefficients). However, if the new set of model coefficients does not result in a better fit, the existing set of model coefficients may be retained in the equipment model. In some embodiments, the new set of model coefficients may be applied to the equipment model even if the fit is not improved, so long as the fit provided by the new set of model coefficients is within a threshold. 
     Referring now to  FIG. 1 , a perspective view of a building  100  is shown, according to an exemplary embodiment. A building automation system (BAS) serves building  100 . In general, a BAS includes hardware and/or software configured to control, monitor, and manage devices in or around a building or building area. BAS subsystems or devices can include heating, ventilation, and air conditioning (HVAC) subsystems or devices, security subsystems or devices, lighting subsystems or devices, fire alerting subsystems or devices, elevator subsystems or devices, other devices that are capable of managing building functions, or any combination thereof. Throughout this disclosure, BAS subsystems and devices are referred to as building equipment. 
     Referring now to  FIG. 2 , a block diagram of a building automation system  200  including a BAS controller  202  and a plurality of building subsystems  228  is shown, according to an exemplary embodiment. Building subsystems  228  are shown to include a building electrical subsystem  234 , an information communication technology (ICT) subsystem  236 , a security subsystem  238 , a HVAC subsystem  240 , a lighting subsystem  242 , a lift/escalators subsystem  232 , and a fire safety subsystem  230 . In various embodiments, building subsystems  228  can include fewer, additional, or alternative subsystems. For example, building subsystems  228  may also or alternatively include a refrigeration subsystem, an advertising or signage subsystem, a cooking subsystem, a vending subsystem, a printer or copy service subsystem, or any other type of building subsystem that uses controllable equipment and/or sensors to monitor or control building  100 . BAS controller  202  may be configured to achieve energy consumption and energy demand reductions by integrating the management of building subsystems  228 . 
     Each of building subsystems  228  may include any number of devices, controllers, and connections for completing its individual functions and control activities. For example, HVAC subsystem  240  may include a chiller, a boiler, any number of air handling units, economizers, field controllers, supervisory controllers, actuators, temperature sensors, and other devices for controlling the temperature within a building. As another example, lighting subsystem  242  may include any number of light fixtures, ballasts, lighting sensors, dimmers, or other devices configured to controllably adjust the amount of light provided to a building space. Security subsystem  238  may include occupancy sensors, video surveillance cameras, digital video recorders, video processing servers, intrusion detection devices, access control devices and servers, or other security-related devices. 
     Still referring to  FIG. 2 , BAS controller  202  is shown to include a communications interface  207  and a BAS interface  209 . Interface  207  may facilitate communications between BAS controller  202  and external applications (e.g., monitoring and reporting applications  222 , enterprise control applications  226 , remote systems and applications  244 , applications residing on client devices  248 , etc.) for allowing user control, monitoring, and adjustment to BAS controller  202  and/or subsystems  228 . Interface  207  may also facilitate communications between BAS controller  202  and client devices  248 . BAS interface  209  may facilitate communications between BAS controller  202  and building subsystems  228  (e.g., HVAC, lighting security, lifts, power distribution, business, etc.). 
     Interfaces  207 ,  209  can be or include wired or wireless communications interfaces (e.g., jacks, antennas, transmitters, receivers, transceivers, wire terminals, etc.) for conducting data communications with building subsystems  228  or other external systems or devices. In various embodiments, communications via interfaces  207 ,  209  may be direct (e.g., local wired or wireless communications) or via a communications network  246  (e.g., a WAN, the Internet, a cellular network, etc.). For example, interfaces  207 ,  209  can include an Ethernet card and port for sending and receiving data via an Ethernet-based communications link or network. In another example, interfaces  207 ,  209  can include a WiFi transceiver for communicating via a wireless communications network. In another example, one or both of interfaces  207 ,  209  may include cellular or mobile phone communications transceivers. In one embodiment, communications interface  207  is a power line communications interface and BAS interface  209  is an Ethernet interface. In other embodiments, both communications interface  207  and BAS interface  209  are Ethernet interfaces or are the same Ethernet interface. 
     Still referring to  FIG. 2 , BAS controller  202  is shown to include a processing circuit  204 . Processing circuit  204  may be communicably connected to BAS interface  209  and/or communications interface  207  such that processing circuit  204  and the various components thereof can send and receive data via interfaces  207 ,  209 . 
     Processing circuit  204  is shown to include a processor  206  and memory  208 . Processor  206  can be implemented as a general purpose processor, an application specific integrated circuit (ASIC), one or more field programmable gate arrays (FPGAs), a group of processing components, or other suitable electronic processing components. Memory  208  (e.g., memory, memory unit, storage device, etc.) may include one or more devices (e.g., RAM, ROM, Flash memory, hard disk storage, etc.) for storing data and/or computer code for completing or facilitating the various processes, layers and modules described in the present application. Memory  208  may be or include volatile memory or non-volatile memory. Memory  208  may include database components, object code components, script components, or any other type of information structure for supporting the various activities and information structures described in the present application. According to an exemplary embodiment, memory  208  is communicably connected to processor  206  via processing circuit  204  and includes computer code for executing (e.g., by processing circuit  204  and/or processor  206 ) one or more processes described herein. 
     In an exemplary embodiment, BAS controller  202  is integrated within a single computer (e.g., one server, one housing, etc.). In various other exemplary embodiments BAS controller  202  can be distributed across multiple servers or computers (e.g., that can exist in distributed locations). Further, while  FIG. 2  shows applications  222  and  226  as existing outside of BAS controller  202 , in some embodiments, applications  222  and  226  may be hosted within BAS controller  202  (e.g., within a memory device). 
     Still referring to  FIG. 2 , memory  208  is shown to include an enterprise integration layer  210 , an automated measurement and validation (AM&amp;V) layer  212 , a demand response (DR) layer  214 , a fault detection and diagnostics (FDD) layer  216 , an integrated control layer  218 , and a building subsystem integration later  220 . Layers  210 - 220  may be configured to receive inputs from building subsystems  228  and other data sources, determine optimal control actions for building subsystems  228  based on the inputs, generate control signals based on the optimal control actions, and provide the generated control signals to building subsystems  228 . The following paragraphs describe some of the general functions performed by each of layers  210 - 220  in BAS  200 . 
     Enterprise integration layer  210  may be configured to serve clients or local applications with information and services to support a variety of enterprise-level applications. For example, enterprise control applications  226  may be configured to provide subsystem-spanning control to a graphical user interface (GUI) or to any number of enterprise-level business applications (e.g., accounting systems, user identification systems, etc.). Enterprise control applications  226  may also or alternatively be configured to provide configuration GUIs for configuring BAS controller  202 . In yet other embodiments, enterprise control applications  226  can work with layers  210 - 220  to optimize building performance (e.g., efficiency, energy use, comfort, or safety) based on inputs received at interface  207  and/or BAS interface  209 . 
     Building subsystem integration layer  220  may be configured to manage communications between BAS controller  202  and building subsystems  228 . For example, building subsystem integration layer  220  may receive sensor data and input signals from building subsystems  228  and provide output data and control signals to building subsystems  228 . Building subsystem integration layer  220  may also be configured to manage communications between building subsystems  228 . Building subsystem integration layer  220  translate communications (e.g., sensor data, input signals, output signals, etc.) across a plurality of multi-vendor/multi-protocol systems. 
     Demand response layer  214  may be configured to optimize resource usage (e.g., electricity use, natural gas use, water use, etc.) and/or the monetary cost of such resource usage in response to satisfy the demand of building  100 . The optimization may be based on time-of-use prices, curtailment signals, energy availability, or other data received from utility providers, distributed energy generation systems  224 , from energy storage  227 , or from other sources. Demand response layer  214  may receive inputs from other layers of BAS controller  202  (e.g., building subsystem integration layer  220 , integrated control layer  218 , etc.). The inputs received from other layers may include environmental or sensor inputs such as temperature, carbon dioxide levels, relative humidity levels, air quality sensor outputs, occupancy sensor outputs, room schedules, and the like. The inputs may also include inputs such as electrical use (e.g., expressed in kWh), thermal load measurements, pricing information, projected pricing, smoothed pricing, curtailment signals from utilities, and the like. 
     According to an exemplary embodiment, demand response layer  214  includes control logic for responding to the data and signals it receives. These responses can include communicating with the control algorithms in integrated control layer  218 , changing control strategies, changing setpoints, or activating/deactivating building equipment or subsystems in a controlled manner. Demand response layer  214  may also include control logic configured to determine when to utilize stored energy. For example, demand response layer  214  may determine to begin using energy from energy storage  227  just prior to the beginning of a peak use hour. 
     In some embodiments, demand response layer  214  includes a control module configured to actively initiate control actions (e.g., automatically changing setpoints) which minimize energy costs based on one or more inputs representative of or based on demand (e.g., price, a curtailment signal, a demand level, etc.). Demand response layer  214  may further include or draw upon one or more demand response policy definitions (e.g., databases, XML files, etc.). The policy definitions may be edited or adjusted by a user (e.g., via a graphical user interface) so that the control actions initiated in response to demand inputs may be tailored for the user&#39;s application, desired comfort level, particular building equipment, or based on other concerns. For example, the demand response policy definitions can specify which equipment may be turned on or off in response to particular demand inputs, how long a system or piece of equipment should be turned off, what setpoints can be changed, what the allowable set point adjustment range is, how long to hold a high demand setpoint before returning to a normally scheduled setpoint, how close to approach capacity limits, which equipment modes to utilize, the energy transfer rates (e.g., the maximum rate, an alarm rate, other rate boundary information, etc.) into and out of energy storage devices (e.g., thermal storage tanks, battery banks, etc.), and when to dispatch on-site generation of energy (e.g., via fuel cells, a motor generator set, etc.). 
     In some embodiments, demand response layer  214  uses equipment models to determine an optimal set of control actions. The equipment models may be used to predict various performance metrics such as power or efficiency for BAS devices. Advantageously, BAS controller  202  may be configured to adaptively adjust the equipment models used by demand response layer  214  to improve the accuracy of the equipment models. This process is described in greater detail with reference to  FIGS. 3-4 . 
     Integrated control layer  218  may be configured to use the data input or output of building subsystem integration layer  220  and/or demand response later  214  to make control decisions. Due to the subsystem integration provided by building subsystem integration layer  220 , integrated control layer  218  can integrate control activities of the subsystems  228  such that the subsystems  228  behave as a single integrated supersystem. In an exemplary embodiment, integrated control layer  218  includes control logic that uses inputs and outputs from a plurality of building subsystems to provide greater comfort and energy savings relative to the comfort and energy savings that separate subsystems could provide alone. For example, integrated control layer  218  may be configured to use an input from a first subsystem to make an energy-saving control decision for a second subsystem. Results of these decisions can be communicated back to building subsystem integration layer  220 . 
     Integrated control layer  218  is shown to be logically below demand response layer  214 . Integrated control layer  218  may be configured to enhance the effectiveness of demand response layer  214  by enabling building subsystems  228  and their respective control loops to be controlled in coordination with demand response layer  214 . This configuration may advantageously reduce disruptive demand response behavior relative to conventional systems. For example, integrated control layer  218  may be configured to assure that a demand response-driven upward adjustment to the setpoint for chilled water temperature (or another component that directly or indirectly affects temperature) does not result in an increase in fan energy (or other energy used to cool a space) that would result in greater total building energy use than was saved at the chiller. 
     Integrated control layer  218  may be configured to provide feedback to demand response layer  214  so that demand response layer  214  checks that constraints (e.g., temperature, lighting levels, etc.) are properly maintained even while demanded load shedding is in progress. The constraints may also include setpoint or sensed boundaries relating to safety, equipment operating limits and performance, comfort, fire codes, electrical codes, energy codes, and the like. Integrated control layer  218  is also logically below fault detection and diagnostics layer  216  and automated measurement and validation layer  212 . Integrated control layer  218  may be configured to provide calculated inputs (e.g., aggregations) to these higher levels based on outputs from more than one building subsystem. 
     Fault detection and diagnostics (FDD) layer  216  may be configured to provide on-going fault detection of building subsystems, building subsystem devices, and control algorithms used by demand response layer  214  and integrated control layer  218 . FDD layer  216  may receive its inputs from integrated control layer  218 , directly from one or more building subsystems or devices, or from another data source. FDD layer  216  may automatically diagnose and respond to detected faults. The responses to detected or diagnosed faults may include providing an alert message to a user, a maintenance scheduling system, or a control algorithm configured to attempt to repair the fault or to work-around the fault. In other exemplary embodiments FDD layer  216  is configured to provide “fault” events to integrated control layer  218  which executes control strategies and policies in response to the received fault events. According to an exemplary embodiment, FDD layer  216  (or a policy executed by an integrated control engine or business rules engine) may shut-down systems or direct control activities around faulty devices or systems to reduce energy waste, extend equipment life, or assure proper control response. 
     FDD layer  216  may be configured to store or access a variety of different system data stores (or data points for live data). FDD layer  216  may use some content of the data stores to identify faults at the equipment level (e.g., specific chiller, specific AHU, specific terminal unit, etc.) and other content to identify faults at component or subsystem levels. FDD layer  216  may be configured to output a specific identification of the faulty component or cause of the fault (e.g., loose damper linkage) using detailed subsystem inputs available at building subsystem integration layer  220 . 
     FDD layer  216  may use any of a variety of fault detection and diagnostic techniques. For example, FDD layer  216  may be configured to use a statistical analysis of equipment model coefficients to identify faults in equipment operation. This process is described in greater detail with reference to  FIG. 3 . In other embodiments, FDD layer  216  can utilize pattern recognition methods, pattern classification methods, rule-based classification methods, outlier analysis, statistical quality control charting techniques, or the like to conduct its statistical analysis. 
     Automated measurement and validation (AM&amp;V) layer  212  may be configured to verify that control strategies commanded by integrated control layer  218  or demand response layer  214  are working properly (e.g., using data aggregated by AM&amp;V layer  212 , integrated control layer  218 , building subsystem integration layer  220 , FDD layer  216 , or otherwise). The calculations made by AM&amp;V layer  212  may be based on building system energy models and/or equipment models for individual BAS devices or subsystems. For example, AM&amp;V layer  212  may compare a model-predicted output with an actual output from building subsystems  228  to determine an accuracy of the model. In some embodiments, AM&amp;V layer  212  determines whether a new set of model coefficients for an equipment model fits a set of operating data better than a previous set of model coefficients. If the new set of model coefficients provides a better fit, AM&amp;V layer  212  may initiate a refitting operation. This process is described in greater detail with reference to  FIGS. 3-4   
     Referring now to  FIG. 3 , a block diagram illustrating an equipment model adaptation system  300  is shown, according to an exemplary embodiment. In some embodiments, system  300  is a component of BAS controller  202  (e.g., implemented as a memory module within BAS controller  202 ). In various other embodiments, system  300  may be a component of an integrated controller for building equipment (e.g., a single piece of equipment, a collection of equipment, etc.), a field controller, a supervisory controller, an enterprise-level controller, a client device (e.g., a computer terminal, a desktop computer, a laptop computer, a portable device, etc.) or any other system or device capable of performing the functions described herein. In an exemplary embodiment, system  300  is integrated within a single computer (e.g., one server, one housing, etc.). In various other exemplary embodiments, system  300  can be distributed across multiple servers or computers (e.g., that can exist in distributed locations). 
     System  300  may be configured to use statistical hypothesis testing to detect a change in a predictive model for building equipment or a collection of building equipment. For example, system  300  may use a first set of operating data to generate a first set of model coefficients for the predictive model and a second set of operating data to generate a second set of model coefficients for the predictive model. System  300  may generate a test statistic based on the sets of model coefficients and compare the test statistic with a critical value. In response to the test statistic exceeding the critical value, system  300  may determine that the predictive model has changed. Upon determining that the predictive model has changed, system  300  may generate a fault indication and/or adaptively update the predictive model with a new set of model coefficients. 
     Still referring to  FIG. 3 , system  300  is shown to include a communications interface  307 , a BAS interface  309 , and a processing circuit  304 . Communications interface  307  and BAS interface  309  may be the same or similar to interfaces  207  and  209 , respectively, as described with reference to  FIG. 2 . Processing circuit  304  may be communicably connected to BAS interface  309  and/or communications interface  307  such that processing circuit  304  and the various components thereof can send and receive data via interfaces  307 ,  309 . 
     Processing circuit  304  is shown to include a processor  306  and memory  308 . Processor  306  can be implemented as a general purpose processor, an application specific integrated circuit (ASIC), one or more field programmable gate arrays (FPGAs), a group of processing components, or other suitable electronic processing components. Memory  308  (e.g., memory, memory unit, storage device, etc.) may include one or more devices (e.g., RAM, ROM, Flash memory, hard disk storage, etc.) for storing data and/or computer code for completing or facilitating the various processes, layers and modules described in the present application. Memory  308  may be or include volatile memory or non-volatile memory. Memory  308  may include database components, object code components, script components, or any other type of information structure for supporting the various activities and information structures described in the present application. According to an exemplary embodiment, memory  308  is communicably connected to processor  306  via processing circuit  304  and includes computer code for executing (e.g., by processing circuit  304  and/or processor  306 ) one or more processes described herein. 
     Still referring to  FIG. 3 , memory  308  is shown to include an operating data aggregator  310 . Operating data aggregator  310  may serve as middleware configured to normalize communications or data received from interfaces  307 ,  309 . In some embodiments, operating data aggregator  310  is a middleware appliance or module sold by Johnson Controls, Inc. Operating data aggregator  310  may be configured to add operating data from building subsystems  228  to a BAS database  312 . Values for the operating data may be measured by sensors, derived from measured values, and/or calculated by various components of BAS  200  or system  300  based on the actual operation of building subsystems  228 . Operating data aggregator  310  may add timestamps to each of the stored values indicating a time at which the value was measured or otherwise obtained. The timestamps may be used to identify operating data corresponding to a particular time window for use in subsequent data processing and analysis. 
     BAS database  312  may be a data storage system or device configured to store values for the operating data. In various embodiments, BAS database  312  may be a component of memory  308  or a separate database. Operating data may include measured or calculated values for one or more independent variables (i.e., predictor variables x 1 , x 2 , x 3 , . . . x p ) that affect the performance of building equipment. For example, independent variables stored in BAS database  312  may include device setpoints, load variables, operating conditions, or any other controlled or uncontrolled variable that affects the performance of the building equipment. Operating data may further include values for one or more dependent variables (i.e., a performance metric y) that depend on the independent variables. Dependent variables stored in BAS database  312  may include, for example, power consumption, resource usage, operating efficiency, or any other variable of interest that depends on one or more of the independent variables. 
     The relationships between independent and dependent variables stored in BAS database  312  may be defined by equipment models  316 . Equipment models  316  are mathematical models that can be used by system  300  and/or BAS controller  202  to predict various performance metrics for the building equipment. Equipment models  316  may have many forms, one of which is a polynomial as shown in the following equation:
 
 y=β   0 +β 1   z+β   2   z   2 + . . . +β p   z   p +ε
 
where y is the dependent variable predicted by the equipment model (e.g., a performance metric such as power consumption, operating efficiency, etc.), z is the independent variable (e.g., a predictor variable such as device load), the β parameters are the model coefficients, and ε is the error between the actual value of y and a model predicted value. The model coefficients β 0  . . . β p  may be determined (e.g., by model generator module  314 ) by fitting the operating data from BAS database  312  to the equipment model (described in greater detail below). Note that the preceding equation is linear in the β parameters and therefore can be rewritten as:
 
 y=+β   0 +β 1   x   1 +β 2   x   2 + . . . +β p   x   p +ε
 
where x 1 =z, x 2 =z 2 , and x p =z p  
 
     Operating data aggregator  310  may collect values for each of the dependent and independent variables in BAS database  312  for a plurality of time steps i. Extending the previous equation from one data sample to n samples in time gives n equations of the form:
 
 y   i =β 0 +β 1   x   i1 +β 2   x   i2 + . . . +β p   x   ip +ε i  
 
where i=1 . . . n.
 
     The n equations can be arranged in matrix form as shown in the following equation: 
               Y   =       X   ⁢           ⁢   β     +   ɛ       ,     ɛ   ~     N   ⁡     (     0   ,       σ   2     ⁢   I       )                   where             Y   =     [           y   1               y   2             ⋮             y   n           ]                 X   =     [           x   11         …         x     1   ⁢   p               ⋮       ⋱       ⋮             x     n   ⁢           ⁢   1           …         x   np           ]                 β   =     [           β   1               β   2             ⋮             β   p           ]                 ɛ   =     [           ɛ   1               ɛ   2             ⋮             ɛ   n           ]           
The n×1 vector Y represents the n samples of the dependent variable y. The n×p matrix X contains the samples of the independent or predictor variables upon which the y values are based. The n×1 vector ε is the true model error and is normally distributed with zero mean. In some embodiments, the model errors are uncorrelated and not physically measurable. The p×1 vector β contains the model coefficients.
 
     Still referring to  FIG. 3 , memory  308  is shown to include a model generator module  314 . Model generator module  314  may generate an estimate {circumflex over (β)} for the model coefficients β that minimizes a function of the model error ε. In some embodiments, model generator module  314  generates the estimated model coefficients {circumflex over (β)} using a residual sum of squares (RSS) function as defined in the following equation:
 
RSS(β)=Σ( y   i   −x   i β) 2 =( Y−X β) T ( Y−X β)
 
     Model generator module  314  generate an ordinary least squares (OLS) estimate of the model coefficients {circumflex over (β)} by finding the model coefficient vector β that minimizes the RSS function. According to various exemplary embodiments, other methods than RSS and/or OLS may be used (e.g., weighted linear regression, regression through the origin, a principal component regression (PCR), ridge regression (RR), partial least squares regression (PLSR), etc.) to generate the model coefficients. 
     Model generator module  314  may find the vector {circumflex over (β)} that minimizes the RSS function using the following equation:
 
{circumflex over (β)}=( X   T   X ) −1   X   T   Y  
 
     Model generator module  314  may store the estimated model coefficients {circumflex over (β)} as coefficients of the corresponding equipment model in equipment models  316 . The operations performed by model generator module  314  to generate model coefficients {circumflex over (β)} and equipment models  316  are described in greater detail with reference to  FIG. 4 . 
     Once equipment models  316  have been updated and/or generated to include model coefficients {circumflex over (β)}, equipment models  316  can be used by system  300  and/or BAS controller  202  to predict the dependent variables in vector Ŷ. For example, a set of dependent variables Ŷ can be predicted for a current set of predictor variables X using the following equation:
 
 Ŷ=X {circumflex over (β)}
 
     In some embodiments, model generator module  314  determines a set of model coefficients for each of a plurality of different time periods. For example, model generator module  314  may use operating data (i.e., x values and y values) collected during a first time period T 1  to estimate a first set of model coefficients {circumflex over (β)} 1 . Model generator module  314  may use operating data collected during a second time period T 2  to estimate a second set of model coefficients {circumflex over (β)} 2 . In various embodiments, time periods T 1  and T 2  may be discrete time periods or may overlap at least partially. Model coefficients {circumflex over (β)} for different time periods can be analyzed to determine whether the predictive capabilities of the corresponding equipment model have degraded. Modules  318 - 324  provide the functionality for such an analysis. 
     Still referring to  FIG. 3 , memory  308  is shown to include an autocorrelation corrector  318 . One of the assumptions for a linear regression model of the form Y=X/3+ε is that the model error vector ε is uncorrelated. I.e.: 
               ɛ   =     [           ɛ   1               ɛ   2             ⋮             ɛ   n           ]       ,       cov   ⁡     (       ɛ   i     ⁢     ɛ   j       )       =       0   ⁢           ⁢   for   ⁢           ⁢   i     ≠   j             
However, operating data from real equipment may violate this assumption. The model error of equipment models  316  may be correlated for any of a variety of reasons including, for example, missing variables from the model form, system inertia, misspecification of the model form, and pure correlated errors, among others. When autocorrelation is present in the model errors, an ordinary least squares analysis may cause the variance of the coefficient estimates {circumflex over (β)} to be underestimated, which can adversely impact any statistical analysis of the coefficient estimates.
 
     Autocorrelation corrector  318  may be configured to remove an autocorrelated model error from the operating data in BAS database  312  prior to analyzing the model coefficients. In some embodiments, the errors can be assumed to have an AR(1) model of the form:
 
ε i =ρε i−1   +v  
 
where ε i  is the error at time step i, ε i−1  is the error at the previous time step i−1, and ν˜N(0,σ 2 I). Autocorrelation corrector  318  may access equipment models  316  to identify the model coefficient estimates {circumflex over (β)} and the corresponding residuals e. The residuals e are given by the following equation:
 
 e=Y−X {circumflex over (β)}
 
Autocorrelation corrector  318  may use the residuals e from equipment models  316  as inputs to the AR(1) model (e.g., substituting e for ε) and estimate a value for the lag one autocorrelation coefficient {circumflex over (ρ)} (e.g., using any of a variety of regression techniques).
 
     Autocorrelation corrector  318  may transform the independent variables x i  and the dependent variables y i  in BAS database  312  using the following transformations:
 
 y   i   *=y   i   −ρy   i−1  
 
 x   i   *=x   i   −ρx   i−1  
 
In various embodiments, the transformed variables x i * and y i * may be stored in BAS database  312  along with the pre-transformed variables x i  and y i  or may replace x i  and y i  in BAS database  312 .
 
     In some embodiments, model generator module  314  uses the transformed variables x i * and y i * to generate a new set of model coefficients {circumflex over (β)}*. The new set of model coefficients {circumflex over (β)}* may be generated in the same manner that the previous set of model coefficients {circumflex over (β)} was generated, using x i * and y i * in place of x i  and y i , respectively, in the applicable equations. For example, a transformed independent variable matrix X* and a transformed dependent variable vector Y* may be defined as follows: 
               Y   *     =     [           y   1   *               y   2   *             ⋮             y   n   *           ]                   X   *     =     [           x   11   *         …         x     1   ⁢   p     *             ⋮       ⋱       ⋮             x     n   ⁢           ⁢   1     *         …         x     n   ⁢           ⁢   p     *           ]           
The transformed vector Y* and matrix X* can be used to calculate the new set of model coefficients {circumflex over (β)}* using the following equation:
 
{circumflex over (β)}*=( X*   T   X *) −1   X*   T   Y*  
 
     Still referring to  FIG. 3 , memory  308  is shown to include a test statistic module  320 . Test statistic module  320  may be configured to generate a test statistic for use in comparing multiple sets of model coefficients (e.g., {circumflex over (β)} 1  and {circumflex over (β)} 2 ). The sets of model coefficients {circumflex over (β)} 1  and {circumflex over (β)} 2  may be generated by model generator module  314  using the equipment operating data stored in BAS database  312  for time periods T 1  and T 2 , respectively. Test statistic module  320  may use the original sets of model coefficients {circumflex over (β)} 1  and {circumflex over (β)} 2  based on the operating data aggregated by operating data aggregator  310  and/or the new sets of model coefficients {circumflex over (β)}* 1  and {circumflex over (β)}* 2  based on transformed operating data generated by autocorrelation corrector  318 . For simplicity, both types of model coefficients (e.g., based on transformed and non-transformed operating data) are referred to as {circumflex over (β)} 1  and {circumflex over (β)} 2  throughout the remainder of this disclosure. 
     If the equipment has not undergone any changes between time periods T 1  and T 2 , the two sets of model coefficients {circumflex over (β)} 1  and {circumflex over (β)} 2  may remain relatively constant. However, determining whether a statistically significant change in the model coefficients has occurred can be challenging since {circumflex over (β)} 1  and {circumflex over (β)} 2  are p×1 vectors consisting of random variables. For example, each vector {circumflex over (β)} may be distributed as:
 
{circumflex over (β)}˜ N (β,σ 2 ( X   T   X ) −1 )
 
Accordingly, the sum of squared error RSS based on {circumflex over (β)} divided by σ 2  may be represented by a chi-square distribution:
 
     
       
         
           
             
               RSS 
               
                 σ 
                 2 
               
             
             ~ 
             
               χ 
               
                 n 
                 - 
                 
                   ( 
                   
                     p 
                     + 
                     1 
                   
                   ) 
                 
               
               2 
             
           
         
       
     
     Test statistic module  320  may generate a test statistic that can be used to determine when the model coefficients have changed with statistical significance. The test statistic may be based on the difference Δβ between model coefficients {circumflex over (β)} 1  and {circumflex over (β)} 2 . The difference between model coefficients may be a normally distributed random vector as shown in the following equation:
 
Δβ=({circumflex over (β)} 1 −{circumflex over (β)} 2 )˜ N (0,σ 2 [( X   1   T   X   1 ) −1 +( X   2   T   X   2 ) −1 ])
 
where Δβ is the change in model coefficients between time periods T 1  and T 2  (e.g., Δβ=({circumflex over (β)} 1 −{circumflex over (β)} 2 )) and X 1  and X 2  are matrices of the independent variable samples in time periods T 1  and T 2 , respectively. The quadratic form of such a normally distributed random vector (i.e., a vector where the symmetric matrix defining the quadratic form is given by the inverse of the covariance matrix of the normal random vector) is itself a chi-square distributed random variable with degrees of freedom equal to the length of Δβ:
 
     
       
         
           
             
               
                 Δ 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   
                     
                       β 
                       T 
                     
                     ⁡ 
                     
                       [ 
                       
                         
                           
                             ( 
                             
                               
                                 X 
                                 1 
                                 T 
                               
                               ⁢ 
                               
                                 X 
                                 1 
                               
                             
                             ) 
                           
                           
                             - 
                             1 
                           
                         
                         + 
                         
                           
                             ( 
                             
                               
                                 X 
                                 2 
                                 T 
                               
                               ⁢ 
                               
                                 X 
                                 2 
                               
                             
                             ) 
                           
                           
                             - 
                             1 
                           
                         
                       
                       ] 
                     
                   
                   
                     - 
                     1 
                   
                 
                 ⁢ 
                 Δ 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 β 
               
               
                 σ 
                 2 
               
             
             ~ 
             
               χ 
               
                 p 
                 + 
                 1 
               
               2 
             
           
         
       
     
     The sum of two independent chi-square distributions is itself a chi-square distribution with degrees of freedom equal to the sum of the degrees of freedom of the two original chi-square distributions. Thus, the sum of the two sum of squared errors divided by the original variance is chi-square distributed, as: 
                   RSS   1     +     RSS   2         σ   2       ~     χ       n   1     +     n   2     -     2   ⁢     (     p   +   1     )         2           
where n 1  and n 2  are the number of data points used to estimate the model coefficients {circumflex over (β)} 1  and {circumflex over (β)} 2 , respectively (equal to the number of rows in X 1  and X 2 ) and p is the number of independent variables (equal to the number of columns in X 1  and X 2 ).
 
     The ratio of two chi-square distributions divided by their respective degrees of freedom is an F-distributed random variable: 
               F     Δ   ⁢           ⁢   β       =       (       Δ   ⁢           ⁢         β   T     ⁡     [       (       X   1   T     ⁢     X   1       )     +       (       X   2   T     ⁢     X   2       )       -   1         ]         -   1       ⁢   Δ   ⁢           ⁢   β         RSS   1     +     RSS   2         )     ⁢       (         n   1     +     n   2     -     2   ⁢   p       p     )     ~     F     p   ,       n   1     +     n   2     -     2   ⁢   p                     
where RSS 1  and RSS 2  are the residual sums of squares for model coefficients {circumflex over (β)} 1  and {circumflex over (β)} 2 , respectively, and F Δβ  is the test statistic. As Δβ moves away from the origin, F Δβ  increases. The maximum increase may occur in the direction of the least variance of the model coefficients and is scaled by the sum of squared errors. Thus, F Δβ  may be based on changes in model coefficients which can easily be related back to actual changes in the equipment. The F Δβ  statistic may also take into account random variation of the changes of the model coefficients even when the model is stationary. The F Δβ  statistic may further be converted into a standard normal variable Z Δβ  by the proper transformation function. In various embodiments, either F Δβ  or Z Δβ  may be used as the test statistic.
 
     Still referring to  FIG. 3 , memory  308  is shown to include a critical value module  322  and a hypothesis testing module  324 . Critical value module  322  may be configured to generate a critical value f crit  for the test statistic. Hypothesis testing module  324  may compare the critical value f crit  with the test statistic F Δβ  using a statistical hypothesis test to determine whether a change in the model coefficients Δβ is statistically significant. For example, hypothesis testing module  324  may formulate a null hypothesis that the model coefficients Δβ have not changed. Hypothesis testing module  324  may reject the null hypothesis in response to a determination that the test statistic F Δβ  exceeds the critical value f crit . 
     In some embodiments, critical value module  322  generates the critical value f crit  using an inverse cumulative distribution function for the test statistic F Δβ . For example, critical value module  322  may calculate the critical value f crit  using the following equation:
 
 f   crit   =F   p+1,n     1     +n     2     −2(p+1)   −1 (1−α)
 
where F −1  is the inverse cumulative distribution function for the test statistic and the parameter α represents the probability of a false positive (i.e., incorrectly rejecting the null hypothesis when in fact the model coefficients have not changed) for the statistical hypothesis test. Critical value module  322  may determine the parameter α by identifying an acceptable probability that the null hypothesis will be incorrectly rejected. In various embodiments, a value for the parameter α may be retrieved from memory, specified by a user, and/or calculated based on one or more stored values.
 
     Critical value module  322  may use the parameter α as an input to the inverse cumulative distribution function to determine a critical value f crit  that is greater than (1−α)% of values for the test statistic F Δβ  when the null hypothesis is valid. For example, if a value of α=0.05 is provided as an input to the inverse cumulative distribution function, the corresponding value for f crit  will be greater than (1−0.05)=0.95 or 95% of values for the test statistic F Δβ  when the null hypothesis is valid. 
     Hypothesis testing module  324  may compare the critical value f crit  with the test statistic F Δβ  to determine whether to reject the null hypothesis. If the value of the test statistic is larger than the critical value (i.e., F Δβ &gt;f crit ), hypothesis testing module  324  may reject the null hypothesis that the model coefficients have not changed. The value for the parameter α defines a statistical confidence (1−α) that the null hypothesis is properly rejected when f crit &gt;F Δβ . Thus, if the value of the test statistic is larger than the critical value, hypothesis testing module  324  may determine with (1−α)% confidence that the model coefficients have changed. If the value of the test statistic is not larger than the critical value (i.e., F Δβ ≤f crit ), hypothesis testing module  324  may fail to reject the null hypothesis and may determine that the model coefficients have not changed. 
     Still referring to  FIG. 3 , memory  308  is shown to include a fault detection module  326 . Fault detection module  326  may be configured to detect a fault in equipment models  316  and/or the equipment corresponding to equipment models  316  based on a result of the hypothesis testing performed by hypothesis testing module  324 . For example, fault detection module  326  may determine that a fault is detected in response to hypothesis testing module  324  rejecting the null hypothesis. In some embodiments, a detected fault indicates that the equipment is optimizing inefficiently (e.g., due to the equipment model no longer accurately predicting the performance of the equipment) and that the equipment model can be updated to better optimize the performance of the equipment. In some embodiments, a detected fault indicates that the equipment is controlling poorly (e.g., due to a physical change or defect in the equipment itself) and may be in need of repair or replacement. 
     Fault detection module  326  may automatically diagnose and respond to detected faults. The responses to detected or diagnosed faults may include providing an alert message to a user, a maintenance scheduling system, or a control algorithm configured to attempt to repair the fault or to work-around the fault. In some embodiments, fault detection module  326  is configured to provide “fault” events to various components of system  300 , BAS controller  202  (e.g., to integrated control layer  218  which can execute control strategies and policies in response to the received fault events) and/or to an external system or device (e.g., a client device, a user terminal, etc.) via communications interface  307 . Fault detection module  326  may report a detected fault and a statistical confidence with which the fault is detected. The statistical confidence may be a function of the α parameter (e.g., 1−α) used to reject the null hypothesis. In some embodiments, the statistical confidence is used by a fault diagnostic system or a user to determine a likelihood of the detected fault relative to other faults detected in the building automation system. 
     Still referring to  FIG. 3 , memory  308  is shown to include a model update module  328 . Model update module  328  may be configured to update equipment models  316  based on a result of the hypothesis testing performed by hypothesis testing module  324 . For example, model update module  328  may update one of equipment models  316  in response to hypothesis testing module  324  rejecting the null hypothesis with respect to the equipment model. Updating an equipment model may include replacing an existing set of model coefficients {circumflex over (β)} 1  used by the equipment model with a new set of model coefficients {circumflex over (β)} 2 . The new set of model coefficients {circumflex over (β)} 2  may be stored in equipment models  316  and used by various components of system  300  and/or BAS controller  202  (e.g., demand response layer  214 , integrated control layer  218 , etc.) to predict equipment performance and/or execute a model-based control strategy. 
     In some embodiments, model update module  328  refits the model coefficients in response to a determination that the model coefficients have likely changed. Refitting the model coefficients may include using operating data from BAS database  312  to retrain the predictive model and generate a new set of model coefficients {circumflex over (β)} 2 . The predictive model may be retained using new operating data (e.g., new variables, new values for existing variables, etc.) gathered in response to a determination that the model coefficients have likely changed or existing data gathered prior to the determination. In some embodiments, refitting the model coefficients includes using the set of model coefficients {circumflex over (β)} 2  generated for the second time period T 2  as the new set of model coefficients (i.e., without performing an additional refit). 
     In some embodiments, model update module  328  automatically applies the new set of model coefficients {circumflex over (β)} 2  to the corresponding equipment model in response to a determination that the model coefficients have likely changed. In other embodiments, model update module  328  determines whether the new set of model coefficients {circumflex over (β)} 2  is a better fit than a previous set of model coefficients {circumflex over (β)} 1  that would be replaced by the new set of model coefficients {circumflex over (β)} 2 . For example, model update module  328  may calculate a coefficient of variation of the root mean square error (CVRMSE) or any other metric that quantifies how well each set of model coefficients fits the equipment model. Model update module  328  may determine that the fit is improved when the new CVRMSE value is less than the previous CVRMSE value (e.g., CVRMSE 2 &lt;CVRMSE 1 ). Model update module  328  may be configured to apply the new set of model coefficients {circumflex over (β)} 2  to the equipment model only if the new set of model coefficients is a better fit than the previous set of model coefficients {circumflex over (β)} 1 . If the new set of model coefficients {circumflex over (β)} 2  is not a better fit than the previous set of model coefficients {circumflex over (β)} 1  (e.g., CVRMSE 2 &gt;CVRMSE 1 ), model update module  328  may discard the new set of model coefficients {circumflex over (β)} 2  and retain the previous set of model coefficients {circumflex over (β)} 1 . 
     In some embodiments, model update module  238  replaces the previous set of model coefficients {circumflex over (β)} 1  with the new set of model coefficients {circumflex over (β)} 2  if the fit of the new set of model coefficients is within a threshold (e.g., CVRMSE 2 &lt;threshold), even if the fit is not improved. Advantageously, this allows the equipment model to be updated with the new set of model coefficients {circumflex over (β)} 2  in the event that the new set of model coefficients {circumflex over (β)} 2  fit reasonably well (e.g., CVRMSE 2 =2.5, CVRMSE 2 &lt;threshold) but not as well as the previous set of model coefficients {circumflex over (β)} 1  (e.g., CVRMSE 1 =2). Updating the equipment model in this scenario may improve prediction accuracy and may be desirable, even if the fit is not improved. 
     Model update module  328  may be configured to cause other modules of memory  308  to perform their functions on a periodic basis (e.g., once per day, once per week, once per month, etc.) or on demand (e.g., in response to user input). For example, model update module  328  may trigger operating data aggregator  310  to collect operating data from building subsystems  228  for various time periods and store the operating data in BAS database  312 . Model update module  328  may cause model generator module  314  to generate one or more sets of model coefficients based on the collected operating data. 
     Model update module  328  may cause test statistic module  320  to generate a test statistic comparing multiple sets of model coefficients and may cause critical value module  322  to generate a critical value for the test statistic. Model update module  328  may trigger hypothesis testing module  324  to perform a statistical hypothesis test and to output a determination whether the null hypothesis is rejected. In various embodiments, model update module  328  may automatically refit the equipment model with a new set of model coefficients (e.g., in response to a determination that the null hypothesis is rejected) or conditionally refit the equipment model with the new set of model coefficients (e.g., refit only if the new set of model coefficients is a better fit than the previous set of model coefficients). 
     Referring now to  FIG. 4 , model generator module  314  is shown in greater detail, according to an exemplary embodiment. Model generator module  314  is shown to include a data clean-up module  402 . Data clean-up module  402  may be configured to receive operating data from BAS database and/or building subsystems  228  and pre-filter the data for data synchronization module  404  and the other modules of model generator module  314 . Data clean-up module  402  may include an outlier analysis module  410 , a data formatting module  412 , and a sorting module  414  for pre-filtering the data. Data clean-up module  402  may use sub-modules  410 - 414  to discard or format bad data by normalizing any formatting inconsistencies with the data, removing statistical outliers, or otherwise preparing the data for further processing. Data formatting module  412  may be configured to ensure that like data is in the same correct format (e.g., all time-based variables are in the same terms of hours, days, minutes, etc.). Sorting module  414  may be configured to sort data for further analysis (e.g., place in chronological order, apply timestamps, etc.). 
     Outlier analysis module  410  may be configured to test data points and determine if a data point is reliable. For example, if a data point is more than a threshold (e.g., three standard deviations, four standard deviations, or another set value) away from the an expected value (e.g., the mean) of all of the data points, the data point may be determined as unreliable and discarded. Outlier analysis module  410  may further calculate the expected value of the data points that each data point is to be tested against. Outlier analysis module  410  may be configured to replace the discarded data points in the data set with a NaN or another flag such that the new value will be skipped in further data analysis. 
     According to another exemplary embodiment, outlier analysis module  410  can be configured to conduct a cluster analysis. The cluster analysis may be used to help identify and remove unreliable data points. For example, a cluster analysis may identify or group operating states of equipment (e.g., identifying the group of equipment that is off). A cluster analysis can return clusters and centroid values for the grouped or identified equipment or states. The centroid values can be associated with data that is desirable to keep rather than discard. Cluster analyses can be used to further automate the data clean-up process because little to no configuration is required relative to thresholding. 
     Data clean-up module  402  may further include any other pre-filtering tasks for sorting and formatting the data for use by model generator module  314 . For example, data clean-up module  402  may include an integrator or averager which may be configured to smooth noisy data (e.g., a varying number of occupants in a building area). The integrator or averager may be used to smooth data over a desired interval (e.g., a 15 minute average, hourly average, etc.). 
     Still referring to  FIG. 4 , model generator module  314  is shown to include a data synchronization module  404 . Data synchronization module  404  may be configured to select a possible set of variables estimated to be significant to energy usage in the building. In some embodiments, data synchronization module  404  selects the possible set of variables (e.g., a preliminary set of variables) that are provided to stepwise regression module  434  for selection of the actual set of variables to use to generate the equipment model. According to various exemplary embodiments, the selection of some or all of the set of variables to use for equipment model generation may occur in data synchronization module  404 , stepwise regression analysis  434 , or a combination of both. Data synchronization module  404  may include sub-modules for calculating predictor variables and selecting one or more of the predicted variables to include in the possible set of variables. Data synchronization module  404  may further include sub-modules for selecting observed (e.g., measured) data points for the set of variables. 
     According to one exemplary embodiment, data synchronization module  404  is configured to calculate degree days and energy days (e.g., a predictor variable associated with heating or cooling of a building) and determine which of these predictors should be used to yield a better baseline model. The outputs of data synchronization module  404  (e.g., inputs provided to regression analysis module  406 ) may include the measurements or predictor variables to use, a period of time associated with the measurements or predictor variables, and errors associated with the data included in the measurements or predictor variables. 
     Data synchronization module  404  is shown to include an enthalpy module  416 , a balance point module  418 , a model determination module  420 , a regression period module  422 , an integration module  424 , a NaN module  426 , a missing days module  428 , a workdays module  430 , and an observed variable selection module  432 . Enthalpy module  416  may be configured to calculate an enthalpy given a temperature variable and a humidity variable. Enthalpy module  416  may combine an outdoor temperature variable and an outside air humidity variable via a nonlinear transformation or another mathematical function into a single variable. The single variable may then be used by model generator module  314  as a better predictor of a building&#39;s energy use than using both temperature and humidity values separately. 
     Balance point module  418  may be configured to find an optimal balance point for a calculated variable (e.g., a variable based on an enthalpy value calculated in enthalpy module  416 , an outdoor air temperature variable, etc.). Balance point module  418  may determine a base value for the variable for which the estimated variance of the regression errors is minimized. Model determination module  420  may be configured to determine a type of baseline model to use for measuring and verifying energy savings. The determination may be made based on an optimal balance point generated by balance point module  418 . 
     Regression period module  422  may be configured to determine periods of time that can be reliably used for model regression by model generator module  314  and data synchronization module  404 . In some embodiments, regression period module  422  uses sliding data windows to identify a plurality of data samples for use in a regression analysis. For example, regression period module  422  may define regression periods T 1  and T 2  and identify a plurality of data samples within each regression period. In some embodiments, regression period module  422  defines regression period T 2  as a fixed-duration time window (e.g., one month, one week, one day, one year, one hour, etc.) ending at the current time. Regression period module  422  may define regression period T 1  as a fixed-duration time window occurring at least partially prior to regression period T 2  (e.g., ending at the beginning of regression period T 2 , ending within regression period T 2 , etc.). As time progresses, regression period module  422  may iteratively redefine regression periods T 1  and T 2  based on the current time at which the regression is performed. Sliding data windows are described in greater detail with reference to  FIG. 5 . 
     Regression period module  422  may identify period start dates and end dates associated with calculated and measured variables for the data synchronization. Regression period module  422  may determine the start date and end date corresponding with the variable with the longest time interval (e.g., the variable for which the most data is available). For example, regression period module  422  may determine the period by finding the period of time which is covered by all variables, and providing the start date and end date of the intersection to data synchronization module  404 . Regression period module  422  may be further configured to identify data within the periods that may be erroneous or cannot be properly synchronized. 
     Integration module  424  may be configured to perform an integration over a variable structure from a given start and end time period (e.g., a time period from regression period module  422 ). According to an exemplary embodiment, integration module  424  uses a trapezoidal method of integration. Integration module  424  may receive an input from balance point module  418  or another module of data synchronization module  404  for performing an integration for a balance point determined by balance point module  418 . NaN module  426  may be configured to identify NaN flags in a variable structure. NaN module  426  may be further configured to replace the NaN flags in the variable structure via interpolation. NaN module  426  may receive an input from, for example, data clean-up module  402 , and may be configured to convert the outlier variables and NaNs determined in module  402  into usable data points via interpolation. 
     Missing days module  428  may be configured to determine days for which is there is not enough data for proper integration performance. Missing days module  428  may compare the amount of data for a variable for a given day (or other period of time) and compares the amount to a threshold (e.g., a fraction of a day) to make sure there is enough data to accurately calculate the integral. Workdays module  430  may be configured to determine the number of work days in a given interval based on the start date and end date of the interval. For example, for a given start date and end date, workdays module  430  can determine weekend days and holidays that should not figure into the count of number of work days in a given interval. Modules  428 ,  430  may be used by data synchronization module  404  to, for example, identify the number of days within a time interval for which there exists sufficient data, identify days for which data should not be included in the calculation of the baseline model, etc. 
     Observed variable selection module  432  may be configured to receive observed or measured data from the BMS and determine which observed data should be used to generate equipment models  316  based on the selection of calculated data in modules  418 - 420 . For example, when balance point module  418  determines a calculated variable, observed variable selection module  432  may be configured to determine if there is enough predictor variable data for the observed variable. According to an exemplary embodiment, the predictor variable data and observed variable data for a specific variable (e.g., temperature) may only be used when sufficient predictor variable data (e.g., degree days) for the observed variable data exists. For example, if the predictor variable data is available over a specified range (e.g., 20 days, 2 months, or any other length of time), then module  432  may determine there is enough predictor variable data such that the predictor variable data and observed variable data can be used for baseline model generation. 
     Still referring to  FIG. 4 , model generator module  314  is shown to include a regression analysis module  406 . Regression analysis module  406  may be configured to generate model coefficients for equipment models  316  using an ordinary least squares regression (OLSR) method or any other regression technique. Regression analysis module  406  is shown to include a baseline model generation module  438  for generating the baseline model and an OLSR module  436  for receiving data from data synchronization module  404 , applying the data to a OLSR method, and providing baseline model generation module  438  with a set of regression coefficients. 
     Baseline model generation module  438  may be configured to generate the baseline model. Baseline model generation module  438  may be configured to use OLSR module  436  to perform OLSR of the data and stepwise regression module  434  to determine the predictor variables for the baseline model and to eliminate insignificant variables. Module  438  may be configured to provide, as an output, baseline equipment models and may be configured to calculate various statistics for further analysis of the baseline model (e.g., computing the number of independent observations of data in the data set used, computing the uncertainty of the model, etc.). 
     Regression analysis module  406  is further shown to include stepwise regression module  434 . Stepwise regression module  434  may be configured to perform stepwise linear regression in order to eliminate statistically insignificant predictor variables from an initial set of variables selected by data synchronization module  404 . In other words, stepwise regression module  434  uses stepwise regression to add or remove predictor variables from a data set (e.g., the data set from data synchronization module  404 ) for further analysis. 
     Stepwise regression module  434  may be configured to add or remove predictor variables from a set for further analysis in a systematic way. At each step, module  434  may conduct statistical hypothesis testing (e.g., by computing a probability of obtaining a test statistic used to describe the similarity between data values) to determine if the variable should be added or removed. For example, for a particular variable, if the variable would have a zero (or near zero) coefficient if it were in the baseline equipment model, then the variable may be removed from consideration for the baseline equipment model. According to various alternative embodiments, other approaches to stepwise regression are used (e.g., factorial designs, principal component analysis, etc.). 
     OLSR module  436  may be configured to receive a subset of the variables from data synchronization module  404  which has been selected by stepwise regression module  434 , and to compute an ordinary least squares regression of the variables in order to generate a baseline model. According to various alternative embodiments, other methods (e.g., a principal component regression (PCR), ridge regression (RR), partial least squares regression (PLSR)) are also or alternatively used in the baseline model calculation instead of an OLSR method. 
     Still referring to  FIG. 4 , model generator module  314  is further shown to include a cross-validation module  408 . Cross-validation module  408  may be configured to validate the baseline model generated by regression analysis module  406  (e.g., validating that there is no overfitting of the model, validating that the order and number of variables in the model is correct, etc.) by applying data for a test period of time (in the past) to the model and determining whether the model provides a good estimate of energy usage. 
     Cross-validation of the baseline model may be used to verify that the model will fit or adequately describe varying data sets from the building. According to one exemplary embodiment, cross-validation module  408  may use a K-fold cross-validation method. The K-fold cross validation method is configured to randomly partition the historical data provided to model generator module  314  into K number of subsamples for testing against the baseline model. In other embodiments, a repeated random sub-sampling process (RRSS), a leave-one-out (LOO) process, a combination thereof, or another suitable cross-validation routine may be used by cross-validation module  408 . 
     In some embodiments, cross-validation module  408  is used to quantify a fit of a set of model coefficients. Cross-validation module  408  may determine whether a new set of model coefficients fits an equipment model better than a previous set of model coefficients based on one or more sets of operating data. Cross-validation module  408  may output a result of the determination to model update module  328 , which may be configured to update the model coefficients in equipment models  316  only if the new set of model coefficients is a better fit. 
     Referring now to  FIG. 5 , a timeline  500  illustrating sliding data windows that may be used by BAS controller  202  and/or system  300  are shown, according to an exemplary embodiment. Timeline  500  is shown to include a plurality of data samples  502 , represented by dots on timeline  500 . Data samples  502  include the operating data collected by operating data aggregator  310  and stored in BAS database  312 . Each of data samples  502  may include any number of data points at a particular time step i. For example, each data sample  502  may include values for one or more independent variables x i1 , . . . x ip  and/or one or more dependent variables y i1 , . . . y ip  in equipment models  316  at a particular time step i, where i=1 . . . n, n is the total number of data samples  502 , and p is the total number of data points in each data sample  502 . Data samples  502  may be received from building subsystems  228  (e.g., from sensors, from local controllers, from equipment, etc.), retrieved from memory  308 , provided by an external system or device (e.g., control setpoints provided by a client device or supervisory controller, etc.), and/or calculated based various data inputs. Data samples  502  may be stored in BAS database  312  and retrieved by operating data aggregator  310  for use in generating model coefficients β, as described with reference to  FIGS. 2-4 . 
     Timeline  500  is shown to include data windows T 1  and T 2 . Data windows T 1  and T 2  define sets of data samples  502  that are used by model generator module  314  to generate model coefficients β. For example, model generator module  314  may perform a first regression analysis using the data samples  502  within data window T 1  to generate the first set of model coefficients β 1 . Model generator module  314  may perform a second regression analysis using the data samples  502  within data window T 2  to generate the second set of model coefficients β 2 . Data windows T 1  and T 2  may be defined by regression period module  422 , as described with reference to  FIG. 4 . 
     In some embodiments, data windows T 1  and T 2  are fixed-duration windows having start times and end times based on the current time. For example, in timeline  500 , the end time of data window T 2  is shown as the current time. As time progresses (e.g., from Day 1, to Day 2, to Day 3, . . . , to Day m), the end time of data window T 2  may be iteratively updated (i.e., sliding forward in time) to match the current time. The start time of data window T 2  may be a predetermined duration (e.g., one month, one week, one year, one day, etc.) prior to the end time of data window T 2 . As time progresses, the start time of data window T 2  may also slide forward in time to maintain the fixed duration of the data window. In some embodiments, data window T 1  is also a fixed-duration window having a start time prior to the start time of data window T 2 . As shown in  FIG. 5 , data window T 1  may have an end time that coincides with or is immediately prior to the start time of data window T 2 . In other embodiments, data window T 1  may have an end time that occurs within data window T 2  such that data windows T 1  and T 2  partially overlap. Data window T 1  may also slide forward in time as time progresses. 
     Sliding data windows T 1  and T 2  forward in time changes the set of data samples  502  within each data window. In some embodiments, model generator module  314  iteratively determines the model coefficients β 1  and β 2  using updated sets of data samples  502  as data windows T 1  and T 2  move forward in time. With each iteration of the model coefficients β 1  and β 2 , an updated test statistic F Δβ  may be generated by test statistic module  320  and compared with a critical value f crit  by hypothesis testing module  324 . 
     In some embodiments, data window T 1  does not move forward in time along with data window T 2 , but rather defines a fixed set of data samples  502  upon which the model coefficients β 1  in an existing equipment model are based. Data window T 2  may slide forward in time to iteratively redefine a new set of data samples  502  that are used by model generator module  314  to generate a new set of model coefficients β 2 . By maintaining data window T 1  at the set of data samples  502  used to generate the model coefficients β 1  in the current equipment model, the statistical hypothesis test performed by hypothesis testing module  324  can determine whether the model coefficients β 1  currently used in the equipment model are significantly different from the iteratively-updated values β 2 . When the null hypothesis is rejected, a new set of model coefficients may be applied to the equipment model by model update module  328 . Upon updating the equipment model, data window T 1  may be moved forward in time to encompass the data samples  502  used to generate the new model coefficients applied to the equipment model. 
     Referring now to  FIG. 6 , a flowchart of a process  600  for identifying changes in a predictive model for building equipment is shown, according to an exemplary embodiment. In some embodiments, process  600  is performed by BAS controller  202  and/or equipment model adaptation system  300 , as described with reference to  FIGS. 2-3 . Process  600  may be used to determine, via statistical hypothesis testing, whether a predictive model for building equipment has changed. 
     The predictive model may be one of equipment models  316 , as described with reference to  FIG. 3 . For example, the predictive model may be a mathematical model having one or more independent variables x and one or more dependent variables y. In some embodiments, the predictive model is of the form:
 
 y=β   0 +β 1   x   1 +β 2   x   2 + . . . +β p   x   p +ε
 
where y is the dependent variable predicted by the model (e.g., a performance metric such as power consumption, operating efficiency, etc.), x 1  . . . x p  are the independent variables (e.g., predictor variable such as device load), the β parameters are the model coefficients, and ε is the error between the actual value of y and a model predicted value. The predictive model may be used by BAS controller  202  in a model-based control strategy to predict the performance of the building equipment. In various embodiments, the building equipment may be a single device, a collection of devices, a building subsystem, a subplant, or any other set of one or more devices for which a variable of interest can be predicted or estimated via a predictive model.
 
     Still referring to  FIG. 6 , process  600  is shown to include collecting a first set of operating data and a second set of operating data for the building equipment (step  602 ). Operating data may include, for example, measured or calculated values for the independent and dependent variables in the predictive model. Values for the operating data may be measured by sensors, derived from measured values, and/or calculated by various components of BAS  200  or system  300  based on the actual operation of building subsystems  228  during the first and second time periods. 
     The first set of operating data may correspond to a first time period T 1  and the second set of operating data may correspond to a second time period T 2 . Step  602  may include organizing the operating data into the first set and the second set using timestamps associated with various data points of the operating data. For example, any data point that has a timestamp within the first time period T 1  may be assigned to the first set of operating data, whereas any data point that has a timestamp within the second time period T 2  may be assigned to the second set of operating data. In some embodiments, step  602  includes defining the first time period T 1  and the second time period T 2 , as described with reference to  FIG. 5 . 
     Step  602  may include collecting the operating data from building subsystems  228 , storing the operating data in BAS database  312 , and/or retrieving the operating data from BAS database  312 . In some embodiments, step  602  includes one or more of the data clean-up operations described with reference to data clean-up module  402  (e.g., outlier analysis, data formatting, sorting, data scrubbing, steady state filtering, etc.). In some embodiments, step  602  includes one or more of the data synchronization operations described with reference to data synchronization module  404  (e.g., determining a balance point, determining an appropriate parameter order for the predictive model, selecting predictor variables to include in the predictive model, identifying data that is not a number, identifying missing days, identifying workdays, etc.). 
     Still referring to  FIG. 6 , process  600  is shown to include generating a first set of model coefficients for the predictive model using the first set of operating data (step  604 ) and generating a second set of model coefficients for the predictive model using the second set of operating data (step  606 ). Steps  604 - 606  may be performed as separate regression processes based on different sets of operating data. For example, step  604  may include using the first set of operating data to generate a first set of model coefficients {circumflex over (β)} 1  corresponding to the first time period T 1 . Step  606  may include using the second set of operating data to generate a second set of model coefficients {circumflex over (β)} 2  corresponding to the second time period T 2 . 
     In some embodiments, steps  604 - 606  are performed by model generator module  314 , as described with reference to  FIGS. 3-4 . For example, steps  604 - 606  may include generating an estimate {circumflex over (β)} for the model coefficients β that minimizes a function of the model error ε. In some embodiments, steps  604 - 606  include generating the estimated model coefficients {circumflex over (β)} using a residual sum of squares (RSS) function as defined in the following equation:
 
RSS(β)=Σ( y   i   −x   i β) 2 =( Y−X β) T ( Y−X β)
 
     Steps  604 - 606  may include generating an ordinary least squares (OLS) estimate of the model coefficients {circumflex over (β)} by finding the model coefficient vector β that minimizes the RSS function. Steps  604 - 606  may include finding the vector {circumflex over (β)} that minimizes the RSS function using the following equation:
 
{circumflex over (β)}=( X   T   X ) −1   X   T   Y  
 
According to various exemplary embodiments, other methods than RSS and/or OLS may be used (e.g., weighted linear regression, regression through the origin, a principal component regression (PCR), ridge regression (RR), partial least squares regression (PLSR), etc.) to generate the model coefficients.
 
     Still referring to  FIG. 6 , process  600  is shown to include generating a test statistic based on a difference between the first set of model coefficients and the second set of model coefficients (step  608 ). In some embodiments, step  608  is performed by test statistic module  320 , as described with reference to  FIG. 3 . The difference Δβ between the first set of model coefficients {circumflex over (β)} 1  and the second set of model coefficients {circumflex over (β)} 2  may be a normally distributed random vector as shown in the following equation:
 
Δβ=({circumflex over (β)} 1 −{circumflex over (β)} 2 )˜ N (0,σ 2 [( X   1   T   X   1 ) −1 +( X   2   T   X   2 ) −1 ])
 
where Δβ is the change in model coefficients between time periods T 1  and T 2  (e.g., Δβ=({circumflex over (β)} 1 −{circumflex over (β)} 2 )) and X 1  and X 2  are matrices of the independent variable samples in time periods T 1  and T 2 , respectively.
 
     Step  608  may include calculating the test statistic using the following equation: 
               F     Δ   ⁢           ⁢   β       =       (       Δ   ⁢           ⁢         β   T     ⁡     [       (       X   1   T     ⁢     X   1       )     +       (       X   2   T     ⁢     X   2       )       -   1         ]         -   1       ⁢   Δ   ⁢           ⁢   β         RSS   1     +     RSS   2         )     ⁢       (         n   1     +     n   2     -     2   ⁢   p       p     )     ~     F     p   ,       n   1     +     n   2     -     2   ⁢   p                     
where n 1  and n 2  are the number of data points used to estimate the model coefficients {circumflex over (β)} 1  and {circumflex over (β)} 2 , respectively (equal to the number of rows in X 1  and X 2 ), p is the number of independent variables (equal to the number of columns in X 1  and X 2 ), RSS 1  and RSS 2  are the residual sums of squares for model coefficients {circumflex over (β)} 1  and {circumflex over (β)} 2 , respectively, and F Δβ  is the test statistic.
 
     In some embodiments, the test statistic F Δβ  is an F-distributed random variable. As Δβ moves away from the origin, F Δβ  increases. The maximum increase may occur in the direction of the least variance of the model coefficients and may be scaled by the sum of squared errors. Thus, F Δβ  may be based on changes in model coefficients which can easily be related back to actual changes in the equipment. The test statistic F Δβ  also considers random variation of the changes of the model coefficients even when the model is stationary. In some embodiments, step  608  includes converting the test statistic F Δβ  into a standard normal variable Z Δβ  using a transformation function. In various embodiments, either F Δβ  or Z Δβ  may be used as the test statistic. 
     Still referring to  FIG. 6 , process  600  is shown to include calculating a critical value for the test statistic (step  610 ) and comparing the test statistic with the critical value using a statistical hypothesis test to determine whether the predictive model has changed (step  612 ). In some embodiments, steps  610  and  612  are performed by critical value module  322  and hypothesis testing module  324 , respectively, as described with reference to  FIG. 3 . In step  612 , the critical value f crit  may be compared with the test statistic F Δβ  using a statistical hypothesis test to determine whether a change in the model coefficients Δβ is statistically significant. For example, step  612  may include formulating a null hypothesis that the model coefficients Δβ have not changed and rejecting the null hypothesis in response to a determination that the test statistic F Δβ  exceeds the critical value f crit . 
     In some embodiments, step  610  includes generating the critical value f crit  using an inverse cumulative distribution function for the test statistic F Δβ . For example, step  610  may include calculating the critical value f crit  using the following equation:
 
 f   crit   =F   p+1,n     1     +n     2     −2(p+1)   −1 (1−α)
 
where F −1  is the inverse cumulative distribution function for the test statistic and the parameter α represents the probability of a false positive (i.e., incorrectly rejecting the null hypothesis when in fact the model coefficients have not changed) for the statistical hypothesis test performed in step  612 . Step  610  may include determining the parameter α by identifying an acceptable probability that the null hypothesis will be incorrectly rejected. In various embodiments, a value for the parameter α may be retrieved from memory, specified by a user, and/or calculated based on one or more stored values.
 
     Step  610  may include using the parameter α as an input to the inverse cumulative distribution function F −1  to determine, based on the parameter α, the critical value f crit  such that the test statistic F Δβ  has a predetermined likelihood of exceeding the critical value f crit  when the predictive model has not changed. The predetermined likelihood may be the same as the acceptable probability that the null hypothesis will be incorrectly rejected. The critical value f crit  output by the inverse cumulative distribution function F −1  may be greater than (1−α)% of values for the test statistic F Δβ  when the null hypothesis is valid. For example, if a value of α=0.05 is provided as an input to the inverse cumulative distribution function, the corresponding value for f crit  will be greater than (1−0.05)=0.95 or 95% of values for the test statistic F Δβ  when the null hypothesis is valid. 
     Step  612  may include comparing the critical value f crit  with the test statistic F Δβ  to determine whether to reject the null hypothesis that the predictive model has not changed. If the value of the test statistic is greater than the critical value (i.e., F Δβ  &gt;f crit ), step  612  may include rejecting the null hypothesis. The value for the parameter α defines a statistical confidence (1−α) that the null hypothesis is properly rejected when f crit &gt;F Δβ . Thus, step  612  may include determining with (1−α)% confidence that the predictive model has changed when the test statistic is greater than the critical value. If the value of the test statistic is not larger than the critical value (i.e., F Δβ ≤f crit ), step  612  may include failing to reject the null hypothesis and determining that the predictive model has not changed. 
     Still referring to  FIG. 6 , process  600  is shown to include determining whether the predictive model has changed (step  614 ). Step  614  may include determining that the predictive model has changed in response to rejecting the null hypothesis in step  612 . In some embodiments, step  614  includes identifying a statistical confidence (e.g., 1−α) with which the null hypothesis is rejected. The statistical confidence may represent a likelihood that the predictive model has in fact changed when the null hypothesis is rejected. Step  614  may include determining that the predictive model has not changed in response to failing to reject the null hypothesis in step  612 . 
     In response to a positive determination in step  614  (i.e., a determination that the predictive model has changed), process  600  may proceed to generating a fault indication (step  618 ) and/or adaptively updating the predictive model (step  620 ). In some embodiments, steps  618  and  620  are performed by fault detection module  326  and model update module  328 , respectively, as described with reference to  FIG. 3 . Steps  618 - 620  are shown occurring in parallel in  FIG. 6 . However, it is not necessary that both steps  618 - 620  are performed. In various embodiments, process  600  may include performing both steps  618 - 620  or omitting one or both of steps  618 - 620 . 
     Step  618  may include determining that a fault is detected in response to a determination that the predictive model has changed. In some embodiments, a detected fault indicates that the equipment is optimizing inefficiently (e.g., due to the predictive model no longer accurately predicting the performance of the equipment) and/or that the equipment model can be updated to better optimize the performance of the equipment. In some embodiments, a detected fault indicates that the equipment is controlling poorly (e.g., due to a physical change or defect in the equipment itself) and may be in need of repair or replacement. 
     Step  618  may include automatically diagnosing and responding to detected faults. The responses to detected or diagnosed faults may include providing an alert message to a user, a maintenance scheduling system, or a control algorithm configured to attempt to repair the fault or to work-around the fault. In some embodiments, step  618  includes providing “fault” events to various components of system  300 , BAS controller  202  (e.g., to integrated control layer  218  which can execute control strategies and policies in response to the received fault events) and/or to an external system or device (e.g., a client device, a user terminal, etc.). Step  618  may include reporting a detected fault and a statistical confidence with which the fault is detected. The statistical confidence may be a function of the α parameter (e.g., 1−α) used to reject the null hypothesis. The statistical confidence may be appended to the fault event and communicated along with the fault event. 
     Step  620  may include adaptively updating the predictive model in response to the determination in step  614  that the predictive model has changed. Step  620  may include refitting the predictive model with a new set of model coefficients. In some embodiments, the new set of model coefficients are generated using new operating data (e.g., new variables, new values for existing variables, etc.) gathered in response to a determination that the predictive model has changed. In other embodiments, the new set of model coefficients are generated prior to the determination. For example, step  620  may include using the set of model coefficients {circumflex over (β)} 2  generated in step  606  as the new set of model coefficients (i.e., without performing an additional refit). 
     In some embodiments, step  620  include automatically applying the new set of model coefficients to the predictive equipment model whenever the determination in step  614  indicates that the predictive model has changed. In other embodiments, step  620  includes determining whether the new set of model coefficients is a better fit than a previous set of model coefficients that would be replaced by the new set of model coefficients. For example, step  620  may include calculating a coefficient of variation of the root mean square error (CVRMSE) or any other metric that quantifies how well each set of model coefficients fits the equipment model. Step  620  may include determining that the fit is improved when the new CVRMSE value is less than the previous CVRMSE value (e.g., CVRMSE 2 &lt;CVRMSE 1 ). 
     In some embodiments, step  620  includes applying the new set of model coefficients to the predictive model only if the new set of model coefficients is a better fit than the previous set of model coefficients. If the new set of model coefficients is not a better fit than the previous set of model coefficients (e.g., CVRMSE 2 ≥CVRMSE 1 ), step  620  may include discarding the new set of model coefficients and retaining the previous set of model coefficients in the predictive model. 
     Still referring to  FIG. 6 , process  600  is shown to include adjusting the data windows used to define the first set of operating data and the second set of operating data (step  616 ). Step  616  may be performed in response to a negative determination in step  614  (i.e., a determination that the predictive model has not changed) and/or following steps  618 - 620 . Adjusting the data windows may include iteratively redefining the time periods T 1  and/or T 2  as time progresses. 
     In some embodiments, the time periods T 1  and T 2  are fixed-duration periods having start times and end times based on the current time. For example, the end time of time period T 2  may be the current time. As time progresses (e.g., from Day 1, to Day 2, to Day 3, . . . , to Day m), the end time of the second time period T 2  may be iteratively updated (i.e., sliding forward in time) to match the current time. The start time of the second time period T 2  may be a predetermined duration (e.g., one month, one week, one year, one day, etc.) prior to the end time of the second time period T 2 . As time progresses, the start time of time period T 2  may also slide forward in time to maintain the fixed duration of the time period. In some embodiments, time period T 1  is also a fixed-duration time period having a start time prior to the start time of time period T 2 . Time period T 1  may have an end time that coincides with or is immediately prior to the start time of time period T 2 . In other embodiments, time period T 1  may have an end time that occurs within time period T 2  such that time periods T 1  and T 2  partially overlap. Time period T 1  may also slide forward in time as time progresses. 
     In some embodiments, time period T 1  does not move forward in time along with time period T 2 , but rather defines a fixed set of data samples upon which the model coefficients β 1  in an existing equipment model are based. Time period T 2  may slide forward in time to iteratively redefine a new set of data samples that are used in step  606  to generate the second set of model coefficients β 2 . By maintaining time period T 1  at the set of data samples used to generate the model coefficients β 1  in the current equipment model, the statistical hypothesis test performed in step  612  can determine whether the model coefficients β 1  currently used in the equipment model are significantly different from the iteratively-updated values β 2 . When the null hypothesis is rejected, a new set of model coefficients may be applied to the equipment model in step  620 . Upon updating the equipment model, time period T 1  may be moved forward in time to encompass the data samples used to generate the new model coefficients applied to the equipment model. 
     After adjusting the data windows in step  616 , process  600  may return to step  602 . Process  600  may be repeated iteratively (e.g., on a periodic basis, on demand, etc.). With each iteration, steps  602 - 620  may be performed. Sliding time periods T 1  and T 2  forward in time changes the set of data samples within each time period. Accordingly, each iteration of process  600  may collect different sets of operating data in step  602  and use the different sets of operating data to generate the model coefficients β 1  and β 2 . 
     Referring now to  FIG. 7 , a flowchart of a process  700  for adaptively updating an equipment model with a new set of model coefficients is shown, according to an exemplary embodiment. In some embodiments, process  700  is performed by BAS controller  202  and/or equipment model adaptation system  300 , as described with reference to  FIGS. 2-3 . Process  700  may be used to accomplish step  620  of process  600  (i.e., adaptively updating a predictive model). 
     Process  700  is shown to include identifying an existing set of model coefficients currently used in a predictive model for building equipment (step  702 ). The existing set of model coefficients may be identified by accessing a stored predictive model and retrieving the model coefficients. In some embodiments, the existing set of model coefficients are a first set of model coefficients β 1  generated using operating data corresponding to a first time period T 1 . For example, the existing set of model coefficients β 1  may be generated in step  604  of process  600 . 
     Process  700  is shown to include generating a new set of model coefficients for the predictive model (step  704 ). In some embodiments, the new set of model coefficients are generated using new operating data (e.g., new variables, new values for existing variables, etc.) gathered in response to a determination that the predictive model has changed. In other embodiments, the new set of model coefficients are generated prior to the determination. For example, step  704  may include using the set of model coefficients {circumflex over (β)} 2  generated in step  606  as the new set of model coefficients (i.e., without performing another regression operation). 
     Still referring to  FIG. 7 , process  700  is shown to include determining whether the new set of model coefficients improve the fit of the predictive model relative to the existing set of model coefficients (step  708 ). In some embodiments, step  708  includes calculating a coefficient of variation of the root mean square error (CVRMSE) or any other metric that quantifies how well each set of model coefficients fits the equipment model. An existing CVRMSE value CVRMSE 1  may be calculated to represent the quality of the fit of the existing set of model coefficients and a new CVRMSE value CVRMSE 2  may be calculated to represent the quality of the fit of the new set of model coefficients, where lower CVRMSE indicate a better fit. Step  708  may include determining that the fit is improved when the new CVRMSE value is less than the existing CVRMSE value (e.g., CVRMSE 2 &lt;CVRMSE 1 ). 
     If the new set of model coefficients improve the fit of the predictive model (i.e., the result of the determination in step  708  is “yes”), process  700  may replace the existing set of model coefficients with the new set of model coefficients in the predictive model (step  710 ). The updated predictive model with the new model coefficients may then be used to predict the performance of the building equipment. If the new set of model coefficients does not improve the fit of the predictive model (i.e., the result of the determination in step  708  is “no”), process  700  may retain the existing set of model coefficients in the predictive model (step  712 ). 
     Still referring to  FIG. 7 , process  700  is shown to include adjusting the data windows (step  714 ). Step  714  may be performed following step  710  or  712  as a precursor to subsequent iteration of process  700 . Adjusting the data windows may include iteratively redefining the time periods T 1  and/or T 2  as time progresses. Step  714  may be substantially the same or similar to step  616  of process  600 . After adjusting the data windows in step  714 , process  700  may return to step  704 . Process  700  may be repeated iteratively (e.g., on a periodic basis, on demand, etc.). With each iteration, steps  704 - 714  may be performed. 
     The construction and arrangement of the systems and methods as shown in the various exemplary embodiments are illustrative only. Although only a few embodiments have been described in detail in this disclosure, many modifications are possible (e.g., variations in sizes, dimensions, structures, shapes and proportions of the various elements, values of parameters, mounting arrangements, use of materials, orientations, etc.). For example, the position of elements may be reversed or otherwise varied and the nature or number of discrete elements or positions may be altered or varied. Accordingly, all such modifications are intended to be included within the scope of the present disclosure. The order or sequence of any process or method steps may be varied or re-sequenced according to alternative embodiments. Other substitutions, modifications, changes, and omissions may be made in the design, operating conditions and arrangement of the exemplary embodiments without departing from the scope of the present disclosure. 
     The present disclosure contemplates methods, systems and program products on memory or other machine-readable media for accomplishing various operations. The embodiments of the present disclosure may be implemented using existing computer processors, or by a special purpose computer processor for an appropriate system, incorporated for this or another purpose, or by a hardwired system. Embodiments within the scope of the present disclosure include program products or memory comprising machine-readable media for carrying or having machine-executable instructions or data structures stored thereon. Such machine-readable media can be any available media that can be accessed by a general purpose or special purpose computer or other machine with a processor. By way of example, such machine-readable media can comprise RAM, ROM, EPROM, EEPROM, CD-ROM or other optical disk storage, magnetic disk storage or other magnetic storage devices, or any other medium which can be used to carry or store desired program code in the form of machine-executable instructions or data structures and which can be accessed by a general purpose or special purpose computer or other machine with a processor. Combinations of the above are also included within the scope of machine-readable media. Machine-executable instructions include, for example, instructions and data which cause a general purpose computer, special purpose computer, or special purpose processing machines to perform a certain function or group of functions. 
     Although the figures may show a specific order of method steps, the order of the steps may differ from what is depicted. Also two or more steps may be performed concurrently or with partial concurrence. Such variation will depend on the software and hardware systems chosen and on designer choice. All such variations are within the scope of the disclosure. Likewise, software implementations could be accomplished with standard programming techniques with rule based logic and other logic to accomplish the various connection steps, processing steps, comparison steps and decision steps.