System, method, and computer readable medium for process control using ergodic analysis

Abstract

A computer implemented system for assessing stability of an operational environment is modeled by at least one dynamic variable during operations A computer is continuously storing, in the computer memory, data retrieved from the operational environment and corresponding to the dynamic variables of the operations. Using the software and the data retrieved from the operational environment, the computer calculates entropy distribution and ergodic properties within the operations. Monitoring the entropy distribution and the ergodic properties with the computer allows for identifying entropy deviations of the entropy distribution or ergodic deviations of the ergodic properties during a period of time; and determining if the entropy deviations or the ergodic deviations are within a threshold to quantify a stability profile for the operations.

Description

[0001] System, Method, and Computer Readable Medium for Process Control Using Ergodic Analysis

[0002] CROSS REFERENCE TO RELATED APPLICATIONS

[0003] The present application claims benefit of priority under 35 U. S. C § 119 (e) from U. S. Provisional Application Serial No. 63 / 744,595, filed January 13, 2025, the disclosure of which is hereby incorporated by reference herein in its entirety.

[0004] BACKGROUND

[0005] Modern operation environments of all kinds, including control systems and automated activities conducted with computers, networks and even artificial intelligence, always need to be monitored for stability. There exists a need in the modern age for faster and more reliable statistical analysis of data gathered in situ during operations of all kinds so that anomalistic behavior of systems and operations can be identified and remedied.

[0006] The invention itself, together with further objects and attendant advantages, will best be understood by reference to the following detailed description, taken in conjunction with the accompanying drawings.

[0007] These and other objects, along with advantages and features of various aspects of embodiments of the invention disclosed herein, will be made more apparent from the description, drawings and claims that follow.

[0008] Additional descriptions of aspects of the present disclosure will now be provided with reference to the accompanying drawings. The drawings form a part hereof and show, by way of illustration, specific embodiments or examples.

[0009] BRIEF SUMMARY

[0010] In an embodiment, a computer implemented system for assessing stability of an operational environment, modeled by at least one dynamic variable during operations includes a computer comprising a processor connected to computer memory storing software that when executed, causes the following steps to be performed. First the computer is continuously storing, in the computer memory, data retrieved from the operational environment and corresponding to the dynamic variables of the operations. Using the software and the data retrieved from the operational environment, the computer calculates entropy distribution and ergodic properties within the operations; monitoring the entropy distribution and the ergodic properties with the computer; identifying entropy deviations of the entropy distribution or ergodic deviations of the ergodic properties during a period oftime; and determining if the entropy deviations or the ergodic deviations are within a threshold to quantify a stability profile for the operations.

[0011] The computer implemented system includes aspects wherein the stability profile of the operations is accessible by the computer to identify shifts in an order of the operations, inefficiencies in the operations, disruptions in the operations, or anomalies in the operations.

[0012] The computer implemented system includes aspects further comprising retrieving data with sensors connected to the computer over a network connected to the operational environment.

[0013] The computer implemented system includes aspects of retrieving the data from graphical user interfaces connected to the computer over a network connected to the operational environment, wherein the graphical user interfaces provide human machine interfaces to the computer for data entry.

[0014] The computer implemented system includes aspects of calculating the ergodic properties of the operations with a space average of the data over a discrete epoch of time for a single active component within the operational environment and a time average over a longer period of time, greater than the discrete epoch of time, for a plurality of active units within the operational environments.

[0015] The computer implemented system includes aspects of using the software to assess distributional stationarity of the dynamic variables within the operational environment as a function of a variation between the space average of the operational environment and the time average of the operational environment.

[0016] The computer implemented system includes aspects of calculating the entropy distribution of the data retrieved from the operational environment as a function of probability distributions for events corresponding to the dynamic variables.

[0017] The computer implemented system includes aspects of calculating using the software to compare the relative probability of the events at specified times.

[0018] The computer implemented system includes aspects of calculating a Shannon entropy value for the data on a continuous basis and identifying differences in observed and expected Shannon entropy.The computer implemented system includes aspects of using the Shannon entropy to quantify average surprisal of time stamped events within the operations.

[0019] The computer implemented system includes aspects of assessing deviations in the entropy as indications of non-ergodic behavior of the operations.

[0020] The computer implemented system includes aspects of representing the dynamic variables of the operations in a state space comprising dimensions that correspond to respective dynamic variables.

[0021] The computer implemented system includes aspects of using the state space for the dynamic variables to identify equations to describe how respective dynamic variables change over time within the operations.

[0022] The computer implemented system includes aspects of encoding operational restraints on the equations and applying a maximum entropy principle to predict operations upon presence of a new constraint on the dynamic variables.

[0023] The computer implemented system includes aspects of using the entropy to quantify surprisal of time stamped events within the operations, and factoring surprisal calculations into the equations to account for data drift in the data collected from the operational environment.

[0024] BRIEF DESCRIPTION OF THE DRAWINGS

[0025] The foregoing and other objects, features and advantages of the present invention, as well as the invention itself, will be more fully understood from the following description of preferred embodiments, when read together with the accompanying drawings.

[0026] The accompanying drawings, which are incorporated into and form a part of the instant specification, illustrate several aspects and embodiments of the present invention and, together with the description herein, serve to explain the principles of the invention. The drawings are provided only for the purpose of illustrating select embodiments of the invention and are not to be construed as limiting the invention.

[0027] Figure 1 schematically illustrates an example computing environment that may implement the systems and methods at hand.

[0028] Figure 2 is Hourly heatmap of probability and surprisal for a sample inpatient ward (floor). We show the relationship between probability and surprisal. a In the probabilityheatmap, lighter colors mean higher probabilities. Each row sums to 1. Higher probabilities are found during business hours. The band at 5:00 P. M. shows the time that routine, repeat AM lab orders are released so that they may be drawn and collected at 5:00 A. M. in time for morning hospital rounds, b In the surprisal heat map, lighter colors imply higher surprisal -the log inverse of probability. The highest surprisal is noted overnight. CUIMC Columbia University Irving Medical Center, PTT Partial Thromboplastin Time, PaO2 Partial Pressure of Oxygen.

[0029] Figure 3 shows surprisal heatmaps by hospital and unit. Surprisal for two laboratory tests across different units at three hospitals. Lighter colors indicate higher surprisal. Surprisal for hemoglobin (a) and PaO2 (b) across units at UF. Surprisal for hemoglobin (c) and PaO2 (d) across units at UVA. At CUIMC (e, f), MICU A at is staffed by advanced practice providers, MICU B is staffed by resident physicians; both are under the direction of attending physicians. This illustrates different surprisal patterns even in units that appear nominally similar. UF University of Florida, UVA University of Virginia, CUIMC Columbia University Irving Medical Center, ED emergency department, STICU Surgical Trauma Intensive Care Unit, NEURO Neurologic Intensive Care Unit, IMC Intermediate Medical Care Unit, MICU Medical Intensive Care Unit, NICU Neonatal Intensive Care Unit.

[0030] Figure 4 shows Jensen-Shannon plots over time and space. We highlight a single bed in each unit (bed NICUB09 and bed 4103 A for the NICU and floor, respectively) to show that the divergence between the bed and unit (bottom lines) is negligible while the difference between units is substantial (top line). IS Jensen-Shannon.

[0031] Figure 5 shows entropy declines on average over a patient stay. Surprisal heatmaps based on day of admission with histogram demonstrating overall values, a Low surprisal, as reflected in the darker colors, is shown within two hours of admission, as patients can arrive in the ED or unit at any time, though transfers to ICU or Floor happen later in the day. b There is increased surprisal of lab orders outside routine times, as reflected by the lighter colors distributed evenly throughout the day. c This pattern continues on the second day of admission, d Total entropy declines based on the day of admission.

[0032] Figure 6 shows Effects of COVID-19 on entropy measures. The Shannon Entropy (left yaxis),as plotted against time (x-axis) and COVID-19 hospitalizations (right y-axis), reveals changes in unit entropy patterns during the pandemic, a UF was minimally impacted by the pandemic, as demonstrated by the low number of COVID-19 hospitalizations with stable fluctuations in entropy, b This was also true for UVA. c At CUIMC, all units exceptthe ED showed a drop in entropy following the spike in COVID hospitalizations, with only MICU A and B appearing to return to baseline.

[0033] DETAILED DESCRIPTION OF EXEMPLARY EMBODIMENTS

[0034] Figure 1 is a block diagram illustrating an example of a machine upon which one or more aspects of embodiments of the present invention can be implemented.

[0035] Referring to Figure 1, an aspect of an embodiment of the present invention includes, but not limited thereto, a system, method, and computer readable medium implemented as a machine that can include logic, one or more components, circuits (e.g., modules), or mechanisms. Circuits are tangible entities configured to perform certain operations. In an example, circuits can be arranged (e.g., internally or with respect to external entities such as other circuits) in a specified manner. In an example, one or more computer systems (e.g., a standalone, client or server computer system) or one or more hardware processors (processors) can be configured by software (e.g., instructions, an application portion, or an application) as a circuit that operates to perform certain operations as described herein. In an example, the software can reside (1) on a non-transitory machine readable medium or (2) in a transmission signal. In an example, the software, when executed by the underlying hardware of the circuit, causes the circuit to perform the certain operations.

[0036] In an example, a circuit can be implemented mechanically or electronically. For example, a circuit can comprise dedicated circuitry or logic that is specifically configured to perform one or more techniques such as discussed above, such as including a special-purpose processor, a field programmable gate array (FPGA) or an application-specific integrated circuit (ASIC). In an example, a circuit can comprise programmable logic (e.g., circuitry, as encompassed within a general-purpose processor or other programmable processor) that can be temporarily configured (e.g., by software) to perform the certain operations. It will be appreciated that the decision to implement a circuit mechanically (e.g., in dedicated and permanently configured circuitry), or in temporarily configured circuitry (e.g., configured by software) can be driven by cost and time considerations.

[0037] Accordingly, the term “circuit” is understood to encompass a tangible entity, be that an entity that is physically constructed, permanently configured (e.g., hardwired), or temporarily (e.g., transitorily) configured (e.g., programmed) to operate in a specified manner or to perform specified operations. In an example, given a plurality of temporarily configured circuits, each of the circuits need not be configured or instantiated at any one instance in time. For example, where the circuits comprise a general -purpose processorconfigured via software, the general-purpose processor can be configured as respective different circuits at different times. Software can accordingly configure a processor, for example, to constitute a particular circuit at one instance of time and to constitute a different circuit at a different instance of time.

[0038] In an example, circuits can provide information to, and receive information from, other circuits. In this example, the circuits can be regarded as being communicatively coupled to one or more other circuits. Where multiple of such circuits exist contemporaneously, communications can be achieved through signal transmission (e.g., over appropriate circuits and buses) that connect the circuits. In embodiments in which multiple circuits are configured or instantiated at different times, communications between such circuits can be achieved, for example, through the storage and retrieval of information in memory structures to which the multiple circuits have access. For example, one circuit can perform an operation and store the output of that operation in a memory device to which it is communicatively coupled. A further circuit can then, at a later time, access the memory device to retrieve and process the stored output. In an example, circuits can be configured to initiate or receive communications with input or output devices and can operate on a resource (e.g., a collection of information).

[0039] The various operations of method examples described herein can be performed, at least partially, by one or more processors that are temporarily configured (e.g., by software) or permanently configured to perform the relevant operations. Whether temporarily or permanently configured, such processors can constitute processor-implemented circuits that operate to perform one or more operations or functions. In an example, the circuits referred to herein can comprise processor-implemented circuits.

[0040] Similarly, the methods described herein can be at least partially processor- implemented. For example, at least some of the operations of a method can be performed by one or processors or processor-implemented circuits. The performance of certain of the operations can be distributed among the one or more processors, not only residing within a single machine, but deployed across a number of machines. In an example, the processor or processors can be located in a single location (e.g., within a home environment, an office environment or as a server farm), while in other examples the processors can be distributed across a number of locations.

[0041] The one or more processors can also operate to support performance of the relevant operations in a "cloud computing" environment or as a "software as a service” (SaaS). For example, at least some of the operations can be performed by a group of computers (asexamples of machines including processors), with these operations being accessible via a network (e.g., the Internet) and via one or more appropriate interfaces (e.g., Application Program Interfaces (APIs)).

[0042] Example embodiments (e.g., apparatus, systems, or methods) can be implemented in digital electronic circuitry, in computer hardware, in firmware, in software, or in any combination thereof. Example embodiments can be implemented using a computer program product (e.g., a computer program, tangibly embodied in an information carrier or in a machine readable medium, for execution by, or to control the operation of, data processing apparatus such as a programmable processor, a computer, or multiple computers).

[0043] A computer program can be written in any form of programming language, including compiled or interpreted languages, and it can be deployed in any form, including as a stand¬ alone program or as a software module, subroutine, or other unit suitable for use in a computing environment. A computer program can be deployed to be executed on one computer or on multiple computers at one site or distributed across multiple sites and interconnected by a communication network.

[0044] In an example, operations can be performed by one or more programmable processors executing a computer program to perform functions by operating on input data and generating output. Examples of method operations can also be performed by, and example apparatus can be implemented as, special purpose logic circuitry (e.g., FPGA or ASIC).

[0045] The computing system can include clients and servers. A client and server are generally remote from each other and generally interact through a communication network. The relationship of client and server arises by virtue of computer programs running on the respective computers and having a client-server relationship to each other. In embodiments deploying a programmable computing system, it will be appreciated that both hardware and software architectures require consideration. Specifically, it will be appreciated that the choice of whether to implement certain functionality in permanently configured hardware (e.g., an ASIC), in temporarily configured hardware (e.g., a combination of software and a programmable processor), or a combination of permanently and temporarily configured hardware can be a design choice. Below are set out hardware (e.g., machine 400) and software architectures that can be deployed in example embodiments.

[0046] In an example, the machine 400 can operate as a standalone device or the machine 400 can be connected (e.g., networked) to other machines.

[0047] In a networked deployment, the machine 400 can operate in the capacity of either a server or a client machine in server-client network environments. In an example, machine400 can act as a peer machine in peer-to-peer (or other distributed) network environments. The machine 400 can be a personal computer (PC), a tablet PC, a set-top box (STB), a Personal Digital Assistant (PDA), a mobile telephone, a web appliance, a network router, switch or bridge, or any machine capable of executing instructions (sequential or otherwise) specifying actions to be taken (e.g., performed) by the machine 400. Further, while only a single machine 400 is illustrated, the term “machine” shall also be taken to include any collection of machines that individually or jointly execute a set (or multiple sets) of instructions to perform any one or more of the methodologies discussed herein.

[0048] Example machine (e.g., computer system) 400 can include a processor 402 (e.g., a central processing unit (CPU), a graphics processing unit (GPU) or both), a main memory 404 and a static memory 406, some or all of which can communicate with each other via a bus 408. The machine 400 can further include a display unit 410, an alphanumeric input device 412 (e.g., a keyboard), and a user interface (UI) navigation device 411 (e.g., a mouse). In an example, the display unit 410, input device 412 and UI navigation device 414 can be a touch screen display. The machine 400 can additionally include a storage device (e.g., drive unit) 416, a signal generation device 418 (e.g., a speaker), a network interface device 420, and one or more sensors 421, such as a global positioning system (GPS) sensor, compass, accelerometer, or other sensor.

[0049] The storage device 416 can include a machine readable medium 422 on which is stored one or more sets of data structures or instructions 424 (e.g., software) embodying or utilized by any one or more of the methodologies or functions described herein. The instructions 424 can also reside, completely or at least partially, within the main memory 404, within static memory 406, or within the processor 402 during execution thereof by the machine 400. In an example, one or any combination of the processor 402, the main memory 404, the static memory 406, or the storage device 416 can constitute machine readable media.

[0050] While the machine readable medium 422 is illustrated as a single medium, the term "machine readable medium" can include a single medium or multiple media (e.g., a centralized or distributed database, and / or associated caches and servers) that configured to store the one or more instructions 424. The term “machine readable medium” can also be taken to include any tangible medium that is capable of storing, encoding, or carrying instructions for execution by the machine and that cause the machine to perform any one or more of the methodologies of the present disclosure or that is capable of storing, encoding or carrying data structures utilized by or associated with such instructions. The term “machine readable medium” can accordingly be taken to include, but not be limited to, solid-statememories, and optical and magnetic media. Specific examples of machine readable media can include non-volatile memory, including, by way of example, semiconductor memory devices (e.g., Electrically Programmable Read-Only Memory (EPROM), Electrically Erasable Programmable Read-Only Memory (EEPROM)) and flash memory devices; magnetic disks such as internal hard disks and removable disks; magneto-optical disks; and CD-ROM and DVD-ROM disks.

[0051] The instructions 424 can further be transmitted or received over a communications network 426 using a transmission medium via the network interface device 420 utilizing any one of a number of transfer protocols (e.g., frame relay, IP, TCP, UDP, HTTP, etc.).

[0052] Example communication networks can include a local area network (LAN), a wide area network (WAN), a packet data network (e.g., the Internet), mobile telephone networks (e.g., cellular networks), Plain Old Telephone (POTS) networks, and wireless data networks (e.g., IEEE 802.11 standards family known as Wi-Fi®, IEEE 802.16 standards family known as WiMax®), peer-to-peer (P2P) networks, among others. The term “transmission medium” shall be taken to include any intangible medium that is capable of storing, encoding or carrying instructions for execution by the machine, and includes digital or analog communications signals or other intangible medium to facilitate communication of such software.

[0053] Any of the specified time periods, durations, frequencies, intervals, time of day, and percentages disclosed herein may be varied to meet the operational requirements of the embodiments of the present invention and, of course, be employed within the context of the embodiments of the present invention.

[0054] An aspect of an embodiment of the disclosure provides a method and system for monitoring and detecting changes in the operational dynamics of enterprises by leveraging Shannon entropy and ergodicity theory. The method identifies deviations from expected entropy distributions or ergodic properties within the system. The method quantifies the regularity and predictability of transactional data and identifies deviations indicative of systemic shifts, inefficiencies, or anomalies. It is applicable to healthcare systems, financial institutions, logistics, and other enterprises where timestamped transactions are recorded, enabling enhanced operational monitoring, early warning systems, and adaptive management An aspect of an embodiment of the disclosure extends the application of high dimensional Shannon entropy and information-theoretic measures into a dynamic, actionable framework for system monitoring and process control. By combining probabilistic modeling, divergence metrics, and statistical analysis, an aspect of an embodiment of the disclosure addressescritical gaps in existing methodologies and provides a comprehensive tool for managing complex systems.

[0055] An aspect of an embodiment of the present disclosure provides a novel approach for characterizing and detecting changes in transactional systems by:

[0056] Quantifying high-dimensional Shannon entropy: Entropy is calculated for the probability distributions of event times for systems (e.g., hospitals, corporations) and their individual components (e.g., hospital beds, departments). The results are used to compare the dynamical characteristics of operational units.

[0057] Evaluating ergodicity. Assessing the similarity between localized entropy values and those of the entire system to determine whether the system remains ergodic. Deviations indicate non-ergodic behavior, suggesting systemic or localized change. Comparing to reference distributions representing system-wide behavior. Reference distributions include historical or theoretical multidimensional probability distributions. Differences in the observed and expected probability distributions are used to monitor the system behavior.

[0058] Detecting change: Identifying operational changes based on deviations in entropy and ergodicity metrics, enabling early intervention to optimize workflows, improve patient care, or enhance enterprise efficiency.

[0059] An aspect of an embodiment of this disclosure introduces a method and system for monitoring enterprise dynamics by leveraging concepts of Shannon entropy and ergodicity Shannon entropy quantifies the average surprisal of timestamped events, providing a measure of the regularity and predictability of operational processes. Ergodicity theory is applied to assess whether the time-averaged behavior of a subset of transactions aligns with the space-averaged behavior of the entire system, enabling detection of systemic shifts.

[0060] The method of analyzing entropy and ergodicity has broad applications across enterprises where timestamped transactions occur, providing powerful tools for monitoring, optimization, and adaptation. By examining the temporal patterns and systemic behaviors of these transactions, organizations can gain actionable insights, improve efficiency, and adapt to shifting dynamics in diverse operational domains.

[0061] In healthcare, this method is particularly valuable for monitoring lab test orders, patient admissions, and clinician actions-all of which are timestamped events. By tracking entropy and ergodicity, hospitals can detect shifts in clinician ordering practices, which may signal data drift or label leakage in predictive models.These techniques also help identify changes in operational dynamics caused by external factors, such as pandemics, allowing for timely interventions and improved resource allocation.

[0062] In financial trading, entropy analysis can uncover anomalies in market dynamics by identifying shifts in the timing and frequency of trades or orders. Monitoring ergodicity helps ensure that the behavior of individual trading strategies aligns with overall market trends, leading to more robust and adaptive trading algorithms.

[0063] Additionally, irregular patterns deviations from expected behaviors can be flagged as potential indicators of fraud or market manipulation. For logistics and supply chain management, entropy analysis aids in detecting inefficiencies, bottlenecks, or disruptions in order placements, shipments, and deliveries. Ergodicity helps evaluate whether historical patterns of order timing and fulfillment are reliable predictors of future demand, improving inventory and resource management. By identifying variability in logistics operations, organizations can dynamically adjust routing, warehousing, or personnel deployment to optimize throughput and maintain efficiency.

[0064] In manufacturing, entropy can be used to monitor process stability by analyzing timestamped events like machine operations, maintenance schedules, and product inspections. Changes in entropy or ergodicity may signal emerging inefficiencies or faults, enabling predictive maintenance to reduce downtime and improve reliability. Similarly, deviations in time-stamped quality control events can highlight process drift or inconsistent performance, supporting more effective quality assurance measures.

[0065] In the retail and e-commerce sector, entropy analysis provides insights into customer behavior by tracking the timing of online purchases, browsing activity, and cart abandonments. Shifts in transaction entropy can reveal changing customer preferences, while ergodic analysis of purchasing patterns ensures inventory models align with demand, minimizing overstock or stockouts. Anomalous patterns, such as rapid changes in purchase frequency, may also indicate fraud or misuse, allowing for timely detection and mitigation.

[0066] In transportation and urban planning, entropy analysis helps optimize traffic flow by identifying irregularities in time stamped data, such as vehicle counts at intersections or public transit usage. Ergodic analysis assesses whether individual components of transportation systems, like parking facilities or bus routes, operate in harmony with overall demand, supporting more efficient infrastructure utilization and planning.

[0067] By leveraging the principles of entropy and ergodicity, organizations across these industries can deepen their understanding of operational dynamics, identify inefficiencies oranomalies, and proactively adapt to changing circumstances. This approach enables more informed decision-making, better resource optimization, and enhanced long-term strategic planning, making it an invaluable tool for modern enterprise management.

[0068] An aspect of an embodiment of the disclosure relates to a method and system for detecting changes in enterprise operations by analyzing the entropy and ergodic properties of time-stamped transactional data. The method leverages Shannon entropy to quantify the randomness or predictability of transaction timings and uses ergodic analysis to assess the consistency of operational patterns over time and across organizational units.

[0069] In a hospital setting, lab test orders are analyzed for their entropy and ergodicity. Normal operations exhibit stable entropy and ergodic properties. Deviations from these metrics, such as increased entropy or loss of ergodicity, may indicate changes in patient demographics, clinician behavior, or resource allocation, warranting further investigation.

[0070] For banking or financial systems, transactional data such as payment times and volumes are monitored for entropy fluctuations. A loss of ergodicity might indicate a market shift, fraud, or operational inefficiency.

[0071] The system integrates data collection, real-time analysis, and alert generation into a single framework, enabling enterprises to detect and respond to operational changes promptly. The method is applicable to any enterprise where transaction data is time-stamped and can be analyzed for entropy and ergodic properties.

[0072] Entropy, a concept rooted in thermodynamics and later adapted to information theory, offers a framework for quantifying surprise and uncertainty in complex systems. Originally introduced by Clausius in the 19th century to describe energy transformations, entropy has evolved into a measure of probabilistic uncertainty, as formalized by Shannon in 1948.

[0073] Shannon entropy quantifies the average surprise of observing a set of events, expressed mathematically as:

[0074] H = -Σ p(xi) log p(xi),

[0075] where p(xi) represents the probability of event xi. This principle has since been adapted to characterize dynamical systems, including hospital operations, wards, and individual patient care.

[0076] An intuitive explanation is that we wish to have a measure of the surprise that we feel when we see the next thing happen, like the next point in a time series, xl. One measure is the inverse of the probability p(x). This is 1 / p(x). Rare events are surprising. Very rare events are very surprising.Now consider the surprise of not one but two events - multiplying the two probabilities seems extreme. Rather, it seems one should be adding instead. Thus, this disclosure uses the log p(x), or, in this case, (-)log p(x) for the inverse.

[0077] We can then estimate the surprise of the entire time series as the sum of all the log p(x).

[0078] And to estimate the average, we can take the expectation, or

[0079] H(X) = -E [log p(xi)] = -Σ p(xi) log p(xi).

[0080] Thus, Shannon entropy is the average surprisal of an event.

[0081] In a hospital context, the entropy of clinical processes, such as lab test ordering or medication administration, can reveal underlying patterns and anomalies. Each action's surprisal, defined as S = -In (p) measures its unexpectedness based on historical probabilities. By using a summation of surprisal values over a set of actions, a total surprisal score quantifies the aggregate unpredictability of a process. The temporal distribution of these scores, and their comparison across units or wards, provides insights into systemic behaviors.

[0082] Introduction: Ergodicity

[0083] Ergodic theory is one property of the embodiments herein. A system is ergodic if its time-averaged properties for a single unit converge to space-averaged properties across all units. In this framework, entropy serves as an invariant characteristic of well-behaved dynamical systems as conceptualized by Kolmogorov and Sinai. As a practical example, this means that the entropy distribution for a single bed in a ward should align with that of the entire ward or hospital, enabling focused monitoring of specific units as proxies for broader system behavior.

[0084] An intuitive explanation is that a bee visits as many sites in a lifetime (time average) as a hive visits in a day (space average). An aspect of an embodiment of the disclosure introduces a method to test for ergodicity in real-world systems as well as advanced metrics for monitoring the dynamical state of the system to measure variability and differences in entropy distributions. These measures include but are not limited to such as sample entropy and Kullback-Leibler divergence.

[0085] Sample entropy assesses the regularity of time series data, with smaller values indicating more predictable patterns. Kullback-Leibler divergence quantifies differences between observed and idealized distributions, providing a robust method for identifying anomalies and deviations in clinical processes. For example, it is used to detect variations in lab test ordering patterns, distinguishing between expected external influences, such as day-night cycles or seasonal trends, and systemic inefficiencies or rogue behavior.An embodiment includes describes clustering techniques to group operational units by their entropy profiles, offering a method for system-wide characterization and anomaly detection. These insights can be used to inform operational decisions, optimize resource allocation, and maintain consistency across operational units.

[0086] Ergodicity and stationarity

[0087] Ergodicity and stationarity are foundational concepts in the analysis of dynamical systems and stochastic processes. Ergodic systems and stationary time series provide fundamental frameworks for understanding and analyzing systems in physics, finance, climate science, and other disciplines. Both concepts describe stability and predictability in their respective domains, allowing researchers to derive meaningful insights and apply practical models. In the context of an aspect of an embodiment of the disclosure, a key¬ concept is that ergodic systems and stationary time series.

[0088] Ergodicity pertains to dynamical systems and describes a scenario where, over a long enough time period, a system's trajectory covers all accessible states in its phase space. This means that the time average of a function along the system’s trajectory equals the space average over the entire phase space. Ergodic systems exhibit behavior that is representative of the entire system when observed over a sufficient period.

[0089] For example, in a hospital setting, the time-averaged entropy of a single bed might reflect the space-averaged entropy across an entire ward. This property simplifies analysis by ensuring that data collected from a single trajectory (e.g., a single hospital bed or process) is meaningful for the broader system.

[0090] Stationarity, on the other hand, applies to stochastic processes and describes systems whose statistical properties remain constant over time. In a stationary time series, metrics such as mean, variance, and autocorrelation do not change as time progresses. Strict stationarity requires that all statistical properties are invariant under time shifts, while weak stationarity focuses on the invariance of the first two moments (mean and variance).

[0091] Stationary processes are particularly advantageous for modeling because their consistent statistical properties allow for straightforward application of predictive and analytical tools.

[0092] Ergodicity often implies a form of stationarity, as time averages in an ergodic system converge to ensemble averages, suggesting stable statistical behavior. However, not all stationary processes are ergodic; stationarity describes consistency in statistical properties over time, while ergodicity emphasizes equivalence of time and space averages.

[0093] Both ergodicity and stationarity contribute to the classification of “well-behaved” systems. Such systems are stable, predictable, and amenable to analysis, which is crucial fordeveloping robust predictive models. In an ergodic system, long-term predictions can be made based on single trajectories, eliminating the need to account for multiple initial conditions or realizations. Similarly, a stationary time series provides a consistent framework for modeling, forecasting, and hypothesis testing, as its statistical structure does not change over time.

[0094] The implications of these properties are significant for the an aspect of an embodiment of the disclosure at hand. If a system is found to be ergodic or stationary, existing predictive models remain valid, as the system’s behavior is stable and consistent. However if a system departs from ergodicity or stationarity, it signals a fundamental shift in its behavior, rendering prior models unreliable and necessitating the development of new ones. For example, if the statistical properties of hospital lab test ordering patterns change-perhaps due to a systemic disruption such as a pandemic-new models would be required to account for the altered dynamics.

[0095] In summary, both ergodicity and stationarity offer powerful tools for understanding and monitoring complex systems. Their stability and predictability simplify analysis and ensure meaningful inferences, making them invaluable for designing and validating predictive models. The ability to identify and respond to departures from these properties further enhances their utility, ensuring that systems can adapt effectively to changing conditions.

[0096] An example of an ergodic system can be found in statistical mechanics, specifically in the behavior of an ideal gas. In such a system, a large number of particles move randomly within a container, exploring all accessible microstates (positions and velocities) over time, as constrained by the system's total energy. This ergodic property means that time averages, such as the average kinetic energy of the particles, can accurately determine macroscopic properties like temperature. Similarly, in a simple harmonic oscillator, such as a mass oscillating on a spring, slight damping ensures that the phase space trajectory predictably covers the available state space. This predictable motion allows precise calculations of the system’s behavior, exemplifying the utility of ergodic properties in deterministic systems.

[0097] In contrast, stationary time series are characterized by consistent statistical properties over time. For example, while stock prices are not stationary, daily stock returns often exhibit stationarity, maintaining a consistent mean and variance. This property enables financial analysts to apply predictive models like AR1MA (Auto-Regressive Integrated Moving Average) to forecast future returns. Another example is monthly temperature anomalies, which represent deviations from average temperatures. Unlike absolute temperatures, theseanomalies often exhibit stationarity, allowing climate scientists to study long-term trends and detect anomalies within the climate system.

[0098] The practical benefits of these properties are far-reaching. Ergodic systems underpin many applications in thermodynamics, where statistical mechanics relies on the ergodic hypothesis to derive macroscopic properties from microscopic behaviors. They also contribute to chaos theory, where long-term statistical behavior can be understood and predicted even when short-term behavior appears unpredictable.

[0099] Stationary time series, on the other hand, are crucial in fields like econometrics, where models for predicting economic indicators depend on consistent statistical properties. In signal processing, stationarity simplifies the analysis of signals and noise, leading to improved design of filters and communication systems.

[0100] To visualize these concepts, a phase space plot of an ergodic system, such as the trajectory of a particle in an ideal gas, could illustrate how the system uniformly fills the phase space over time. Similarly, a time series plot of a stationary process, such as stock returns, would reveal a consistent mean and variance, highlighting the stability required for predictive modeling.

[0101] Together, ergodic systems and stationary time series offer powerful frameworks for analyzing systems across diverse fields. Their properties provide the stability and predictability necessary for meaningful analysis, allowing researchers and practitioners to develop models that drive decision-making, optimize systems, and predict future behavior.

[0102] An example of an ergodic system can be found in statistical mechanics, specifically in the behavior of an ideal gas. In such a system, a large number of particles move randomly within a container, exploring all accessible microstates (positions and velocities) over time, as constrained by the system's total energy. This ergodic property means that time averages, such as the average kinetic energy of the particles, can accurately determine macroscopic properties like temperature. Similarly, in a simple harmonic oscillator, such as a mass oscillating on a spring, slight damping ensures that the phase space trajectory predictably covers the available state space. This predictable motion allows precise calculations of the system’s behavior, exemplifying the utility of ergodic properties in deterministic systems.

[0103] In contrast, stationary time series are characterized by consistent statistical properties over time. For example, while stock prices are not stationary, daily stock returns often exhibit stationarity, maintaining a consistent mean and variance. This property enables financial analysts to apply predictive models like ARIMA (AutoRegressive Integrated Moving Average) to forecast future returns. Another example is monthly temperature anomalies,which represent deviations from average temperatures. Unlike absolute temperatures, these anomalies often exhibit stationarity, allowing climate scientists to study long-term trends and detect anomalies within the climate system.

[0104] The practical benefits of these properties are far reaching. Ergodic systems underpin many applications in thermodynamics, where statistical mechanics relies on the ergodic hypothesis to derive macroscopic properties from microscopic behaviors. They also contribute to chaos theory, where long-term statistical behavior can be understood and predicted even when short-term behavior appears unpredictable. Stationary time series, on the other hand, are crucial in fields like econometrics, where models for predicting economic indicators depend on consistent statistical properties. In signal processing, stationarity simplifies the analysis of signals and noise, leading to improved design of filters and communication systems.

[0105] To visualize these concepts, a phase space plot of an ergodic system, such as the trajectory of a particle in an ideal gas, could illustrate how the system uniformly fills the phase space over time. Similarly, a time series plot of a stationary process, such as stock returns, would reveal a consistent mean and variance, highlighting the stability required for predictive modeling.

[0106] Together, ergodic systems and stationary time series offer powerful frameworks for analyzing systems across diverse fields. Their properties provide the stability and predictability necessary for meaningful analysis, allowing researchers and practitioners to develop models that drive decision-making, optimize systems, and predict future behavior.

[0107] The system evolves over time according to deterministic rules, such as the rate of lab test ordering, patient flow, and staff scheduling protocols. These rules can be expressed as mathematical equations-differential equations or discrete time-step models-depending on the granularity and nature of the data.

[0108] The system's evolution is best visualized in its phase space, representing the entire range of possible states of the hospital floor. As the system progresses, its behavior can be understood as trajectories through this phase space, revealing patterns and transitions in hospital operations.

[0109] The modeling process involves three key steps. First, state variables must be clearly defined and informed by comprehensive historical data. For instance, patient arrival times, lab test orders, discharge times, and staff schedules serve as essential variables in a hospital setting. Once these are established, evolution equations are developed to describe how each variable changes over time, incorporating interactions such as how patient arrivals influencelab test demand or how staffing levels affect. The workflow includes simulating the system's dynamics using numerical methods, which allows for testing the model against real-world data to refine its accuracy.

[0110] This modeling approach offers significant predictive power, enabling hospital administrators forecast future states, such as peak lab test times or bottlenecks in patient care. With these insights, administrators can optimize resource allocation, staff scheduling, and workflows, leading to reduced wait times and improved patient outcomes. Furthermore, the model provides a deeper understanding of the underlying dynamics, revealing how variables interact and identifying critical intervention points to enhance efficiency.

[0111] Operational efficiency is another major benefit. By simulating how variables evolve over time, hospitals can better manage resources such as staff and equipment, minimizing waste and lowering costs. For example, lab test ordering could be optimized by adjusting scheduling protocols to align with predictable fluctuations in demand.

[0112] An example of this approach applied to lab test ordering illustrates its practicality. State variables in this scenario include the number of lab orders, patient health states, and staff availability. Evolution equations might model lab order rates as functions of patient needs and staff dynamics as functions of shift schedules and workload. A simulation could start with historical data to establish initial conditions and then explore how lab test demand fluctuates over time, providing actionable strategies for improved management.

[0113] Despite its promise, this approach faces challenges. Complexity is a significant obstacle, as hospital operations involve many interconnected processes that are difficult to fully capture. Accurate modeling also requires detailed and comprehensive data, which may not always be available. Additionally, nonlinear interactions among variables and external factors, such as policy changes or pandemics, can complicate predictions.

[0114] Modeling a hospital floor as a deterministic dynamical system offers a powerful tool for understanding and optimizing operations. While it requires overcoming challenges of complexity and data requirements, the insights gained from this approach have the potential to transform hospital management. By enabling predictive analytics, improving resource allocation, and enhancing patient care, this framework could play a pivotal role in modernizing healthcare delivery.

[0115] Monitoring ergodicity and stationarity

[0116] The ability to monitor ergodicity and stationarity in complex systems, such as hospital operations, is critical for maintaining predictive accuracy, optimizing resources, and ensuring consistent quality of care. These properties offer stability and predictability, but their losscan signal significant changes in the system’s dynamics, requiring adjustments to models and operational strategies. When a previously ergodic system loses this property, time averages no longer represent space averages, and single trajectories fail to reflect the behavior of the entire system. Such a loss may occur due to external influences, parameter changes, or new dynamics disrupting the system's equilibrium. This breakdown has profound implications: predictive models based on ergodic assumptions become unreliable, more extensive sampling is needed to understand the system, and decomposable subunits may need reanalysis to uncover new organizational patterns. The need for new models and expanded data collection adds analytical complexity and requires significant resource investment. Similarly, the loss of stationarity in a time series-evident when statistical properties such as mean, variance, and autocorrelation begin to shift affects the reliability of models built on these properties.

[0117] Trends, seasonal effects, or data drift can disrupt the constancy that stationarity assumes, invalidating statistical models like ARIMA or linear regression. These shifts necessitate recalibration of models, adoption of adaptive forecasting techniques, or development of entirely new methodologies. Non-stationary data often requires preprocessing through methods like de-trending or differencing to restore a stable structure for analysis.

[0118] Continuous monitoring is essential to detect early signs of ergodicity or stationarity loss, allowing for timely interventions. Techniques such as the Kullback-Leibler divergence and Jensen-Shannon divergence can track changes in distributions, signaling shifts in ergodicity. For stationarity, tests like the Augmented Dickey-Fuller test can identify evolving trends in time series data. These tools enable organizations to recognize emerging disruptions and respond proactively.

[0119] The practical benefits of monitoring these properties in a hospital setting are far-reaching. Early detection of changes in patient demographics, clinician behavior, or resource utilization allows hospitals to adjust operations dynamically, maintaining efficiency and improving outcomes. For instance, identifying shifts in test-ordering patterns can inform staffing adjustments or resource reallocations, while monitoring entropy and divergence measures ensures predictive models remain accurate and relevant.

[0120] Automated monitoring systems and regular data analysis are vital for operational excellence. Dashboards displaying real-time metrics, combined with periodic model validation and scenario testing, enhance the hospital's ability to adapt to evolving conditions. Interdisciplinary collaboration ensures that data scientists, clinicians, and administrators collectively interpret findings and develop effective response strategies.Importantly, when ergodicity or stationarity remains intact, it provides assurance that the system’s fundamental structure has not changed. This continuity eliminates the need for frequent model updates, reducing the costs and risks associated with constant recalibration of predictive tools. Stable systems allow organizations to rely on established models, focusing instead on optimizing performance within the existing framework.

[0121] The loss of ergodicity or stationarity in a system has significant implications, making previously reliable models and predictions less accurate and necessitating more complex analytical approaches. Detecting these changes early and adapting models accordingly is crucial for maintaining accurate analysis and effective management of the system. When a previously ergodic or stationary system loses that property, it can have significant implications for the analysis, predictability, and management of the system. This can happen due to external influences, changes in the system’s parameters such as data drift, or the introduction of new dynamics that disrupt the system's previous state.

[0122] The implications include the need for new predictive models. Moreover, since single trajectories may no longer adequately represent the entire system, more extensive sampling and new data collection to understand the system’s state. Loss of ergodicity may reflect new organizations of subsystems, requiring reanalysis of decomposable ergodic subunits to understand the system's behavior. Monitoring ergodicity can be informed by monitoring stationarity of time series data relating to that system. When a system loses stationarity, the time series statistical properties such as mean, variance, and autocorrelation start to change over time. As a result, statistical models assume stationarity (e.g., ARIMA, linear regression) may become invalid or less accurate. The implications include a need for models to be recalibrated or new models to be developed to account for changing properties.

[0123] Alternatively, adaptive forecasting methods or models that can handle non-stationarity (e.g., state-space models) may be required. Continuous monitoring to detect early signs of ergodicity or stationarity loss will be useful. Methods include monitoring the Kullback-Liebler distance or the Jensen-Shannon divergence for ergodicity of distributions and the Augmented Dickey-Fuller test for stationarity of time series. Time series methods such as detrending and differencing may help filter noisy data and enhance the signal within.

[0124] Monitoring for ergodicity and stationarity is an important exercise for a complex enterprise like a hospital. It allows for the early detection of changes, maintains the accuracy of predictive models, optimizes resource allocation, ensures consistent quality of care, and supports informed decision-making. Implementing regular and automated monitoring systems can significantly enhance the hospital's ability to respond to dynamic changes inpatient demographics, clinician behavior, and resource utilization, ultimately improving operational efficiency and patient outcomes.

[0125] Importantly, the finding of continued ergodicity or stationarity argues against the need for model updating. Constant re-education of predictive monitoring is expensive and allows room for error. If the fundamental elements of the data structure are not changing, then there is less need for new models.

[0126] In summary, monitoring ergodicity and stationarity in a complex enterprise like a hospital ensures robust predictive accuracy, resource optimization, and quality care. Early detection of changes prevents inefficiencies and supports informed decision-making.

[0127] Conversely, confirming these properties over time avoids unnecessary model recalibration, preserving resources while maintaining operational stability. Together, these efforts contribute to a resilient, adaptive healthcare system capable of meeting evolving demands effectively.

[0128] Loss of ergodicity or stationarity

[0129] When a previously ergodic or stationary system loses those properties due to changes such as data drift or external disruptions, the implications for analysis, predictability, and management can be profound. Understanding these implications for both types of systems is essential for maintaining reliable models and adapting to evolving dynamics.

[0130] In the case of ergodic systems, losing ergodicity means that the time averages derived from a single trajectory no longer match the space averages across the system. This breakdown occurs when external influences, parameter shifts, or new dynamics disrupt the system's prior state. As a result, predictions based on long-term observations of individual trajectories become unreliable, as they fail to represent the entire system. The loss of ergodicity necessitates more extensive sampling and data collection to capture the system's true behavior, increasing the complexity of analysis. Models that assume ergodicity must be revised or replaced to accommodate the new dynamics, as the system's behavior can no longer be simplified to a representative single trajectory.

[0131] For stationary systems, the loss of stationarity occurs when statistical properties like mean, variance, and autocorrelation begin to change over time. Common causes include trends, seasonal effects, or structural breaks that introduce variability into the system. When stationarity is lost, models relying on constant statistical properties, such as ARIMA or linear regression, become invalid or less accurate. Forecasting accuracy declines as the underlying assumptions of consistency are violated, requiring recalibration of existing models or the adoption of adaptive methods capable of handling non-stationarity. Additionally, dataanalysis becomes more complex, often necessitating methods such as detrending or differencing to make the data suitable for analysis.

[0132] To address the loss of ergodicity or stationarity, organizations must employ strategies for detection, adaptation, and transformation. Continuous monitoring is critical for detecting early signs of these changes, using tools such as the Augmented Dickey-Fuller test for stationarity or metrics for ergodicity. Once changes are identified, adaptive models that adjust to evolving system dynamics or robust statistical methods designed to handle non-stationarity must be implemented. Data transformations, such as de-trending or differencing, can restore stationarity, while periodic reevaluation of assumptions ensures models aligned with the system's current state.

[0133] The loss of these properties underscores the importance of flexibility and vigilance in system management. Without early detection and adaptation, previously reliable models and predictions become obsolete, leading to ineffective decision-making and operational inefficiencies. Conversely, proactive strategies enable organizations to maintain accurate analysis and respond effectively to changing dynamics.

[0134] In summary, while the loss of ergodicity or stationarity introduces significant challenges, it also highlights need for continuous reassessment and innovation in modeling approaches. By detecting changes early, updating models, and employing adaptive methods, systems can remain robust and responsive, preserving their utility in an evolving landscape.

[0135] Total surprisal

[0136] In monitoring complex systems, calculating the surprisal for individual elements or agents within the system provides a quantitative measure of unexpectedness, enabling deeper insights into dynamics and anomalies. Surprisal, a concept rooted in information theory, is defined mathematically for a single event as S = -ln(p) where here p represents the probability of the event occurring. For a series of events associated with an individual agent, such as a patient in a hospital, the total surprisal is calculated as the sum of the individual surprisals. This can be expressed as

[0137] Stotal = Σi=1 to N -ln (pi), where p is the probability of each event i, and N is the total number of events observed. By aggregating the surprisal values of individual events, this metric provides a comprehensive measure of the agent's interaction with the system reflecting the degree of deviation from expected patterns and enabling the identification of anomalies or trends within the system. In the context of monitoring complex systems, the calculation of surprisal for individual elements or agents within the system provides a powerful tool forunderstanding and predicting system behavior. Surprisal, derived from information theory, quantifies the unexpectedness of an event based on its probability.

[0138] By applying these principles, the calculation of surprisal for individual agents enables effective monitoring of complex systems. This approach has wide-ranging applications, including identifying anomalies, optimizing resource allocation, and predicting future states. In healthcare, for instance, total surprisal for a patient can indicate deviations from typical clinical pathways, alerting providers to potential issues. More broadly, it offers a scalable method to monitor and manage systems with high complexity and interdependence, such as transportation networks, financial systems, and industrial processes.

[0139] At least one innovation lies in the application of surprisal as a metric for individual elements within complex systems, allowing for efficient monitoring and actionable insights into system dynamics.

[0140] Generating the idealized probability distribution

[0141] An initial step in implementing an aspect of an embodiment of the disclosure is to determine a baseline multidimensional multivariable probability distribution for comparison to new data. The baseline model may be created using historical data gathered from the system, from knowledge of the principles or rules of the underlying system, or by a deterministic mathematical model. As an example, an initial step in implementing an aspect of an embodiment of the disclosure in the specific context of lab test order times is to develop a historical baseline multi-dimensional probability matrix model using aggregated lab test data. Future sources of heuristic data include the Internet of Things and generative AI models.

[0142] As another example, in another system the initial step might be rules governing the behavior of agents. When rules are insufficient to provide probability estimates for all possible states, the minimum entropy principle can be applied. An example is given below in the context of managing restaurant personnel.

[0143] As another example, in another system the realization of an aspect of an embodiment of the disclosure is to use a multi-dimensional probability matrix that is based on theory. There are two general approaches. The first is to use a stochastic model, of which several are listed below. Particular attention is paid to comparing and contrasting agent-based models. Another is to use a deterministic model, which explicitly follows a system of equations. In some circumstances, stochastic models can have quasi-detemiinistic properties. These considerations are important in the context of using entropy-based analyses in the context of ergodic dynamical systems.The integration of Internet of Things (IoT) devices with advanced modeling techniques offers a novel approach to monitoring the ergodicity of complex enterprises. By¬ leveraging loT’s capacity for real-time data acquisition, it becomes possible to assess the ergodic properties of a system dynamically through the continuous analysis of time series data. Specifically, this method uses IO devices to collect detailed, timestamped data on key operational variables, which are then used to calculate the Kullback-Leibler (KL) distances from an idealized high-dimensional probability matrix representing the system's expected ergodic state.

[0144] In very high dimensional systems, it may not be possible to construct mathematical models. In these cases, the baseline multidimensional probability matrix model can be constructed from observed historical data. In this framework, the IoT devices serve as a distributed network of sensors, capturing data from multiple facets of the enterprise, such as resource utilization, operational workflows, and event timing. This data populates a multidimensional probability matrix that represents the distribution of system states over time.

[0145] By applying methods of this disclosure, an enterprise gains the ability to dynamically track its operational stability. For example, in a hospital setting, IoT-enabled monitoring of lab test ordering patterns, patient flow, and resource allocation can reveal whether the system maintains consistent probabilistic behaviors over time. If a significant drift is detected, suggesting a loss of ergodicity, decision-makers can investigate the root causes-such as shifts in clinician practices, policy changes, or unforeseen disruptions-and implement corrective measures. Conversely, when the system retains low KL distances over time, it confirms that predictive models and operational strategies remain valid, obviating the need for costly model recalibration or redesign.

[0146] The use of IoT devices to monitor ergodicity via KL distances from an idealized probability matrix represents a significant advancement in enterprise management. This approach enables real-time, data-driven decision-making, ensuring that complex systems operate efficiently while maintaining their predictive reliability. By embedding this capability into the fabric of enterprise operations, organizations can achieve a level of adaptability and resilience that was previously unattainable.

[0147] Example 1

[0148] The information theory concepts of surprisal, entropy, and ergodicity help reveal these dynamics. Departures from the standard narrative are considered surprises and offer quantitative insight into clinical decision making. A dynamic system is deemed ergodic when the statistical properties observed over a single long-term trajectory (the time average)are the same as those observed across all accessible states at a single moment (the space average). Ergodicity simplifies analysis of complex systems, lends confidence to long-term forecasting, and adapts well to fields as diverse as biology, physics, and economics. This disclosure follows convention in defining surprisal as the logarithm of the inverse of probability, a term synonymous with “information content”. The average surprisal of events is the eponymous Shannon entropy, which was later extended to dynamical systems by Kolmogorov and Sinai. In the context of this disclosure, higher entropy arises from a more uniform distribution of lab test orders, while lower entropy connotes a more regular pattern, as seen when lab test orders cluster around scheduled rounding times.

[0149] Any type of mathematical modeling that aims to predict the future from the past relies on some form of temporal consistency - either the future will be like the past, or the trends in the future will resemble trends in the past. However, model performance is limited by data drift-where input data and their relationship with the target outcome change, such as with changes in practice patterns or documentation habits within and between hospitals. Machine learning operations (MLOps) is a discipline tasked with continuous monitoring for data drift over time and space. Yet the field is still nascent, lacks consensus definitions, and its standard tools of mean and variance do not adequately capture higher-dimensional processes.

[0150] Demonstrating ergodicity and tracking the entropy of ergodic systems can indicate when the assumption of stationarity is being violated. Moreover, this can be observed at any level of a complex system, allowing for rigorous analysis of granular changes in dynamic systems with temporal data.

[0151] Results

[0152] Surprisal analysis demonstrates latent information in ordering practice. Figure 1 depicts the relative frequency of lab orders, treated as empirical estimates of probability, for each hour of the day based on pre-COVID data (2018-2020) data from an inpatient ward at Columbia University Irving Medical Center (CUIMC). Derivations of these numbers and explanations of their definitions can be found in the Methods section. In each matrix, rows are lab tests, columns are hours of the day, and cells are color-coded to represent the probability p or the surprisal (-In p) of the lab being ordered at that time. Individual probability values range from 0 to 0.27, with surprisal ranging from 1.31 to 5.2 over 24 h. We selected five different labs to represent repeat and ad hoc ordering practices. For the former, we chose serum sodium (Na) concentrations to represent serum chemistries and hemoglobin level (Hgb) for blood counts, which are typically drawn daily on admitted patients. For the latter, we selected partial pressure of oxygen in arterial blood (PaO2) from arterial blood gasanalysis, which requires more invasive arterial access, and fibrinogen, a measure of coagulopathy that is typically a one-time order. Partial Thromboplastin Time (PTT), an additional measure of coagulopathy, may be routine or ad hoc.

[0153] We find that surprisal follows a diurnal pattern with a decreased probability of overnight orders for draws. This pattern was most evident in the repeat labs (Na, Hgb) and least evident in the ad hoc labs (PTT), which demonstrated a more uniform distribution throughout the day as care providers place one-time orders.

[0154] In addition to demonstrating the relationship between probability and surprisal, Fig. 5 also highlights the digital behavior of EHRs as noted by the low surprisal (dark) bands. When a care provider places an order, the EHR records three distinct times: order placement, collection, and result. This analysis uses the first definition, but it is complicated by the priority of the lab order. “STAT” orders are always released immediately.

[0155] However, when routine labs are to be repeated at future intervals, they are stored in batches and released at different times of day, depending on the institution’s practices. At CUIMC, the lab system releases repeat lab orders 36 h before they are due. For example, at 5:00 P. M., the laboratory system releases recurring lab orders due to be collected at 5:00 A. M. two days later. Since repeat labs are typically drawn at 5:00 A. M. for morning rounds on floor, the surprisal for repeat floor labs like sodium and hemoglobin is lowest at 5:00 P. M.

[0156] Extending this analysis across units and hospitals reveals that each has a unique surprise thumbprint. Figure 5 shows the addition of individual departments and two other hospitals, along with the measured surprisal for Na (left column) and PaO2 (right column). At each institution, surprisal was most evenly distributed throughout the day in the Emergency Departments (EDs), reflecting ad hoc ordering as patients appeared at random intervals.

[0157] Other patterns are explained by the behavior of each institution’s EHR. At the University of Florida (UF), same-day tests ordered as “routine” are released immediately if they are set to be drawn before midnight and are otherwise released at midnight. Therefore, UF showed very low surprisal only at midnight (Fig. 2a, b) when repeat labs are released, with more even distribution throughout the day for both Na and PaO2. PaO2 behaved more as a repeat lab in the Neuro and 'Trauma Surgery Intensive Care Units (NEURO and STICU), perhaps because of the large number of mechanically ventilated patients.

[0158] At the University of Virginia (UVA), repeat labs are released four hours before they are scheduled to be collected. Therefore, UVA’s heat maps showed bands of low surprisal at 3:00 A. M. and every 4 h thereafter (Fig. 2b, e). There is high surprisal between these draws, suggesting STAT orders, which approach maximum on the floor and Neonatal Intensive CareUnit (NICU). PaO2 has a more even distribution at UVA and does not behave as a repeat lab in the STICU.

[0159] CUIMC showed a more diurnal dispersal of surprisal (Fig. 2c, f) across units, though each had its own pattern, and there was less difference between Na and PaO2 order times. Medical Intensive Care Unit (MICU) A cares for all ECMO (ExtracorporealMembrane Oxygenation) patients and is staffed by advanced care practitioners. In contrast, MICU B cares for a variety of non-ECMO, medically complex conditions and is staffed by residents. Both units are under the direction of faculty physicians. For ECMO patients in MICU A, labs are typically ordered STAT as labs are placed at the start of shifts and taper off throughout the day, with high surprisal noted overnight for both Na and PaO2. For MICU B, alternatively, repeat morning labs are released at 3:00 P. M. (for 3:00 A. M. draws), with another low surprisal band at 3:00 A. M. for twice-daily lab draws. MICU B also places STAT orders at the start of the shift (6:00 A. M.) and has higher surprisal overnight. The Neuro-ICU has two bands of low surprisal at 12:00 A. M. and 12:00 P. M. for their repeat labs.

[0160] The result is shown in the two lower lines of the figure. This disclosure also compared the month-by -month distributions of the two units. The result is shown in the upper line. The average divergence between the individual beds and their respective units was small for both the NICU and inpatient ward (floor), at 0.013 and 0.007, respectively. Our interpretation is that the time average of lab orders for a single bed over a long period is similar to the time and space average of the unit. These distributions of probabilities, with similar mean, variance, and correlation, are consistent with ergodicity. On the other hand, the average Jensen-Shannon divergence between the neonatal ICU and cardiac in-patient ward was more than tenfold larger at 0.192. Thus, hospital units behave as ergodic systems. On the other hand, the whole hospital does not; rather, we find that the distributions of lab test order times vary from unit to unit. As expected for ergodic systems, the time series of the divergences, shown in Fig. 3, were stationary (Augmented Dickey-Fuller tests, p < 0.001).

[0161] Entropy falls after the first day of hospital admission and is independent of the number of lab test orders Entropy is the weighted average of surprisal. As patients are admitted at any time of day or night, we expected many admission lab orders to have higher entropy than orders later in admission. To test this, we generated surprisal heatmaps filtered by hours and days following admission. Within two hours of admission (Fig. 4a), the distributions of surprisal across the day are relatively even at UF, with the notable exception of the Intermediate Care Unit (IMC), where patients are typically transferred from an ICU bed. Over the following days (Fig. 4b, c), the probability of routine orders released atmidnight is high, suggesting that regular patterns emerge on the inpatient ward, with higher-surprisal for ad hoc labs. Accordingly, the entropy on the

[0162] first day of admission is highest, as labs are ordered at unscheduled times. After admission, entropy falls as most labs are clustered to one or two time points for scheduled lab draws (Fig. 4d).

[0163] To ensure that entropy was not simply a function of the number of test orders, we plotted lab test order times as a function of the number of orders (Supplementary Fig. 1). Each data point is the entropy calculated for a 28-day window. We initially observed that the Emergency Departments of all hospitals had the highest number of orders and the highest entropy; the UVA neonatal ICU had the fewest laboratory test orders and the low entropy. However, inspection of the plots does not show consistent correlations.

[0164] Specifically, in some cases, entropy is unchanged in a unit over a wide range of lab test orders. For example, the Neurologic ICU unit at UF has a mean entropy of 1.57 over an extensive range of test numbers (218-1693 orders), and the inpatient ward (floor) has an entropy of 1.12-1.47 over an even more extensive range of test numbers (4270-8268 orders). Moreover, some hospitals had a wide range of entropy in different units with similar numbers of lab test orders. In another example, entropy in two UVA ICUs and one in-patient ward ranged from 1.23 to 2.41 for comparable numbers of orders. Likewise, at CUIMC, the entropies for one in-patient ward and the MICU B ranged from 1.63 to 3.01 over a similar number of orders. Thus, entropy contains information about a hospital unit independent of the number of labs ordered on that unit.

[0165] Entropy as a hospital metric: impact of the Coronavirus- 19 pandemic Figure 6 shows the time course of entropy over three years that include the COVID-19 pandemic. We also show the number of COVID-19 hospitalizations (plotted in gray) for calibration. For this analysis, the entropy lines show the difference between each 28-day window. Before CO VID- 19, monthly entropy generally oscillated around a characteristic mean in each of the wards and each of the hospitals. Again, we show that entropy was highest and near maximum for the ED in all hospitals.

[0166] Entropy changed little during the pandemic in the UF and UVA hospitals, where the proportion of COVID-19 patients was low. In contrast, at CUIMC, where the effect of the pandemic was most severe, there were marked changes correlating with the steep rise inCOVID cases. Entropy fell across all units other than the ED, where it remained near the maximum. The change in entropy persisted in some units and appeared to reverse in others, with the MICUs returning to pre-pandemic levels and the floor appearing to remain lower.We thus see how entropy measures can visualize shifts in patient population and the impact of these changes on clinician ordering behavior.

[0167] Surprisal measures inform predictive models. To test the possibility that surprisal may signal suspicion of an imminent clinical event, we examined its ability to predict hemorrhage in a previous dataset using a cardiorespiratory monitoring data set of 3688 consecutively admitted patients to the University of Virginia Medical ICU, 141 of whom had 155 hemorrhagic events, defined as three units of packed red blood cell transfusion within 24 h having had no transfusions for a day prior. The features included the means, variances, pairwise correlation coefficients of heart rate, respiratory rate, and oxygen saturation along with dynamical measures such as detrended fluctuation analysis and coefficient of sample entropy. We used regularized logistic regression adjusted for repeated measures. We found that the surprisal of the reporting time of a hemoglobin level was a statistically significant predictor when all other continuous cardiorespiratory monitoring parameters were considered (p < 0.001). While the most important predictor was oxygen saturation variability, hemoglobin surprisal was the next most informative and had the same impact on the model as blood pressure.

[0168] In an embodiment, a computer implemented system for assessing stability of an operational environment, modeled by at least one dynamic variable during operations includes a computer comprising a processor connected to computer memory storing software that when executed, causes the following steps to be performed. First the computer is continuously storing, in the computer memory, data retrieved from the operational environment and corresponding to the dynamic variables of the operations. Using the software and the data retrieved from the operational environment, the computer calculates entropy distribution and ergodic properties witliin the operations; monitoring the entropy distribution and the ergodic properties with the computer; identifying entropy deviations of the entropy distribution or ergodic deviations of the ergodic properties during a period of time; and determining if the entropy deviations or the ergodic deviations are within a threshold to quantify a stability profile for the operations.

[0169] The computer implemented system includes aspects wherein the stability profile of the operations is accessible by the computer to identify shifts in an order of the operations, inefficiencies in the operations, disruptions in the operations, or anomalies in the operations.The computer implemented system includes aspects further comprising retrieving data with sensors connected to the computer over a network connected to the operational environment.

[0170] The computer implemented system includes aspects of retrieving the data from graphical user interfaces connected to the computer over a network connected to the operational environment, wherein the graphical user interfaces provide human machine interfaces to the computer for data entry.

[0171] The computer implemented system includes aspects of calculating the ergodic properties of the operations with a space average of the data over a discrete epoch of time for a single active component within the operational environment and a time average over a longer period of time, greater than the discrete epoch of time, for a plurality of active units within the operational environments.

[0172] The computer implemented system includes aspects of using the software to assess distributional stationarity of the dynamic variables within the operational environment as a function of a variation between the space average of the operational environment and the time average of the operational environment.

[0173] The computer implemented system includes aspects of calculating the entropy distribution of the data retrieved from the operational environment as a function of probability distributions for events corresponding to the dynamic variables.

[0174] The computer implemented system includes aspects of calculating using the software to compare the relative probability of the events at specified times.

[0175] The computer implemented system includes aspects of calculating a Shannon entropy value for the data on a continuous basis and identifying differences in observed and expected Shannon entropy.

[0176] The computer implemented system includes aspects of using the Shannon entropy to quantify average surprisal of time stamped events within the operations.

[0177] The computer implemented system includes aspects of assessing deviations in the entropy as indications of non-ergodic behavior of the operations.The computer implemented system includes aspects of representing the dynamic variables of the operations in a state space comprising dimensions that correspond to respective dynamic variables.

[0178] The computer implemented system includes aspects of using the state space for the dynamic variables to identify equations to describe how respective dynamic variables change over time within the operations.

[0179] The computer implemented system includes aspects of encoding operational restraints on the equations and applying a maximum entropy principle to predict operations upon presence of a new constraint on the dynamic variables.

[0180] The computer implemented system includes aspects of using the entropy to quantify surprisal of time stamped events within the operations, and factoring surprisal calculations into the equations to account for data drift in the data collected from the operational environment.

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[0231] Although example embodiments of the present disclosure are explained in some instances in detail herein, it is to be understood that other embodiments are contemplated. Accordingly, it is not intended that the present disclosure be limited in its scope to the details of construction and arrangement of components set forth in the following description or illustrated in the drawings. The present disclosure is capable of other embodiments and of being practiced or carried out in various ways.

[0232] It should be appreciated that any element, part, section, subsection, or component described with reference to any specific embodiment above may be incorporated with, integrated into, or otherwise adapted for use with any other embodiment described herein unless specifically noted otherwise or if it should render the embodiment device nonfunctional. Likewise, any step described with reference to a particular method or process may be integrated, incorporated, or otherwise combined with other methods or processes described herein unless specifically stated otherwise or if it should render the embodiment method nonfunctional. Furthermore, multiple embodiment devices or embodiment methods may be combined, incorporated, or otherwise integrated into one another to construct or develop further embodiments of the invention described herein.

[0233] It should be appreciated that any of the components or modules referred to with regards to any of the present invention embodiments discussed herein, may be integrally or separately formed with one another. Further, redundant functions or structures of the components or modules may be implemented. Moreover, the various components may be communicated locally and / or remotely with any user / clinician / patient or machine / system / computer / processor. Moreover, the various components may be in communication via wireless and / or hardwire or other desirable and available communication means, systems and hardware. Moreover, various components and modules may be substituted with other modules or components that provide similar functions.

[0234] It should be appreciated that the device and related components discussed herein may take on all shapes along the entire continual geometric spectrum of manipulation of x, y and z planes to provide and meet the anatomical, environmental, and structural demands and operational requirements. Moreover, locations and alignments of the various components may vary as desired or required.It should be appreciated that various sizes, dimensions, contours, rigidity, shapes, flexibility and materials of any of the components or portions of components in the various embodiments discussed throughout may be varied and utilized as desired or required.

[0235] It should be appreciated that while some dimensions are provided on the aforementioned figures, the device may constitute various sizes, dimensions, contours, rigidity, shapes, flexibility and materials as it pertains to the components or portions of components of the device, and therefore may be varied and utilized as desired or required.

[0236] It must also be noted that, as used in the specification and the appended claims, the singular forms “a,” “an” and “the” include plural referents unless the context clearly dictates otherwise. Ranges may be expressed herein as from “about” or “approximately” one particular value and / or to “about” or “approximately” another particular value. When such a range is expressed, other exemplary embodiments include from the one particular value and / or to the other particular value.

[0237] By “comprising” or “containing” or “including” is meant that at least the named compound, element, particle, or method step is present in the composition or article or method, but does not exclude the presence of other compounds, materials, particles, or method steps, even if the other such compounds, material, particles, or method steps have the same function as what is named.

[0238] In describing example embodiments, terminology will be resorted to for the sake of clarity. It is intended that each term contemplates its broadest meaning as understood by those skilled in the art and includes all technical equivalents that operate in a similar manner to accomplish a similar purpose. It is also to be understood that the mention of one or more steps of a method does not preclude the presence of additional method steps or intervening method steps between those steps expressly identified. Steps of a method may be performed in a different order than those described herein without departing from the scope of the present disclosure. Similarly, it is also to be understood that the mention of one or more components in a device or system does not preclude the presence of additional components or intervening components between those components expressly identified.

[0239] Some references, which may include various patents, patent applications, and publications, are cited in a reference list and discussed in the disclosure provided herein. The citation and / or discussion of such references is provided merely to clarify the description of the present disclosure and is not an admission that any such reference is “prior art” to any aspects of the present disclosure described herein. In terms of notation, “[n]” corresponds to the nth reference in the list. All references cited and discussed in this specification areincorporated herein by reference in their entireties and to the same extent as if each reference was individually incorporated by reference.

[0240] It should be appreciated that as discussed herein, a subject may be a human or any animal. It should be appreciated that an animal may be a variety of any applicable type, including, but not limited thereto, mammal, veterinarian animal, livestock animal or pet type animal, etc. As an example, the animal may be a laboratory animal specifically selected to have certain characteristics similar to human (e.g. rat, dog, pig, monkey), etc. It should be appreciated that the subject may be any applicable human patient, for example.

[0241] The term “about,” as used herein, means approximately, in the region of, roughly, or around. When the term “about” is used in conjunction with a numerical range, it modifies that range by extending the boundaries above and below the numerical values set forth. In general, the term “about” is used herein to modify a numerical value above and below the stated value by a variance of 10%. In one aspect, the term “about” means plus or minus 10% of the numerical value of the number with which it is being used. Therefore, about 509o means in the range of 45%-55%. Numerical ranges recited herein by endpoints include all numbers and fractions subsumed within that range (e.g. 1 to 5 includes 1, 1.5, 2, 2.75, 3, 3.90, 4, 4.24, and 5). Similarly, numerical ranges recited herein by endpoints include subranges subsumed within that range (e.g. 1 to 5 includes 1-1.5, 1.5-2, 2-2.75, 2.75-3, 3-3.90, 3.90-4, 4-4.24, 4.24-5, 2-5, 3-5, 1 -4, and 2-4). It is also to be understood that all numbers and fractions thereof are presumed to be modified by the term “about.”

Claims

CLAIMSWhat is claimed is:

1. A computer implemented system for assessing stability of an operational environment modeled by at least one dynamic variable during operations, the system comprising:a computer comprising a processor connected to computer memory storing software that when executed, causes the following steps to be performed:continuously storing, in the computer memory, data retrieved from the operational environment and corresponding to the dynamic variables of the operations;using the software and the data retrieved from the operational environment to calculate entropy distribution and ergodic properties within the operations:monitoring the entropy distribution and the ergodic properties with the computer; identifying entropy deviations of the entropy distribution or ergodic deviations of the ergodic properties during a period of time;determining if the entropy deviations or the ergodic deviations are within a threshold to quantify a stability profile for the operations.

2. The computer implemented system of Claim 1, wherein the stability profile of the operations is accessible by the computer to identify shifts in an order of the operations, inefficiencies in the operations, disruptions in the operations, or anomalies in the operations.

3. The computer implemented system of Claim 1, further comprising retrieving data with sensors connected to the computer over a network connected to the operational environment.

4. The computer implemented system of Claim 1, further comprising retrieving the data from graphical user interfaces connected to the computer over a network connected to the operational environment, wherein the graphical user interfaces provide human machine interfaces to the computer for data entry.

5. The computer implemented system of Claim 1, further comprising calculating the ergodic properties of the operations with a space average of the data over a discrete epoch of time for a single active component within the operational environment and a time average over a longer period of time, greater than the discrete epoch of time, for a plurality of active units within the operational environments.

6. The computer implemented system of Claim 1, further comprising using the software to assess distributional stationarity of the dynamic variables within the operational environment as a function of a variation between the space average of the operational environment and the time average of the operational environment.

7. The computer implemented system of Claim 1, further comprising calculating the entropy distribution of the data retrieved from the operational environment as a function of probability distributions for events corresponding to the dynamic variables.

8. The computer implemented system of Claim 7, further comprising using the software to compare the relative probability of the events at specified times.

9. The computer implemented system of Claim 8, further comprising calculating a Shannon entropy value for the data on a continuous basis and identifying differences in observed and expected Shannon entropy.

10. The computer implemented system of Claim 9, further comprising using the Shannon entropy to quantify average surprisal of time stamped events within the operations.

11. The computer implemented system of Claim 9, further comprising assessing deviations in the entropy as indications of non-ergodic behavior of the operations.

12. The computer implemented system of Claim 1, further comprising representing the dynamic variables of the operations in a state space comprising dimensions that correspond to respective dynamic variables.

13. The computer implemented system of Claim 12, further comprising using the state space for the dynamic variables to identify equations to describe how respective dynamic variables change over time within the operations.

14. The computer implemented system of Claim 13, further comprising encoding operational restraints on the equations and applying a maximum entropy principle to predict operations upon presence of a new constraint on the dynamic variables.

15. The computer implemented system of Claim 14, further comprising using the entropy to quantify surprisal of time stamped events within the operations, and factoring surprisal calculations into the equations to account for data drift in the data collected from the operational environment.

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