Markov Transition Research Articles

Partial observability and management of ecological systems.

The actual state of ecological systems is rarely known with certainty, but management actions must often be taken regardless of imperfect measurement (partial observability). Because of the difficulties in accounting for partial observability, it is usually treated in an ad hoc fashion, or simply ignored altogether. Yet incorporating partial observability into decision processes lends a realism that has the potential to improve ecological outcomes significantly. We review frameworks for dealing with partial observability, focusing specifically on dynamic ecological systems with Markovian transitions, i.e., transitions among system states that are influenced by the current system state and management action over time. Fully observable states are represented in an observable Markov decision process (MDP), whereas obscure or hidden states are represented in a partially observable process (POMDP). POMDPs can be seen as a natural extension of observable MDPs. Management under partial observability generalizes the situation for complete observability, by recognizing uncertainty about the system's state and incorporating sequential observations associated with, but not the same as, the states themselves. Decisions that otherwise would depend on the actual state must be based instead on state probability distributions (“belief states”). Partial observability requires adaptation of the entire decision process, including the use of belief states and Bayesian updates, valuation that includes expectations over observations, and optimal strategy that identifies actions for belief states over a continuous belief space. We compare MDPs and POMDPs and highlight POMDP applications to some common ecological problems. We clarify the structure and operations, approaches for finding solutions, and analytic challenges of POMDPs for practicing ecologists. Both observable and partially observable MDPs can use an inductive approach to identify optimal strategies and values, with a considerable increase in mathematical complexity with POMDPs. Better understanding of POMDPs can help decision makers manage imperfectly measured ecological systems more effectively.

Rolling bearing fault diagnosis by Markov transition field and multi-dimension convolutional neural network

Safe and reliable operation of mechanical equipment depends on timely and accurate fault diagnosis. When the actual working conditions are complex and variable and the available sample data set is small, recognition accuracy of the rolling bearing fault diagnosis model is low. To solve this problem, a novel method based on Markov transition field (MTF) and multi-dimension convolutional neural network (MDCNN) is proposed in this paper. Firstly, the original vibration signals are converted into two-dimensional images containing temporal correlation by MTF. Then, a neural network model is constructed by using multi-dimension attention and E-rectified linear units (E-Relu) activation function to fully extract fault feature information. Finally, the MTF images are input into the model and the data is normalized using the group normalization method. The MDCNN model is validated on two different data sets, and the results show that compared with other intelligent fault diagnosis methods, the MDCNN has higher fault diagnosis accuracy and stronger robustness under both variable working conditions and small sample data sets conditions.

Reliability and Availability Optimization of Smart Microgrid Using Specific Configuration of Renewable Resources and Considering Subcomponent Faults

In this paper, renewable resources, namely photovoltaic panels (PV), are placed in a specific configuration to obtain the maximum reliability and availability of a microgrid and study the subcomponent-level reliability and availability. The reliability of components can be increased by trying different configurations of the components. We identify the preferred configuration used for the PV panels as bridged linked. The overall reliability of the microgrid is increased when component-wise reliability is considered. Even components are further divided into subcomponents, and the multiple faults of each component are considered. The method used for the reliability evaluation and availability study is Markov state transition modeling. The microgrid’s reliability and availability are plotted concerning time using Matlab. The optimization of reliability and availability is conducted through optimization techniques such as the genetic algorithm (GA) and artificial neural networks (ANN). The results are compared and validated for the optimal values of mean time to failure (MTTF) and mean time to repair (MTTR). Using a genetic algorithm, there is a 96% of improvement in the reliability, and after applying the neural networks, a significant improvement of 97% along with quick results is achieved.

An Advanced Multi-Agent Reinforcement Learning Framework of Bridge Maintenance Policy Formulation

In its long service life, bridge structure will inevitably deteriorate due to coupling effects; thus, bridge maintenance has become a research hotspot. The existing algorithms are mostly based on linear programming and dynamic programming, which have low efficiency and high economic cost and cannot meet the actual needs of maintenance. In this paper, a multi-agent reinforcement learning framework was proposed to predict the deterioration process reasonably and achieve the optimal maintenance policy. Using the regression-based optimization method, the Markov transition matrix can better describe the uncertain transition process of bridge components in the maintenance year and the real-time updating of the matrix can be realized by monitoring and evaluating the performance deterioration of components. Aiming at bridges with a large number of components, the maintenance decision-making framework of multi-agent reinforcement learning can adjust the maintenance policy according to the updated Markov matrix in time, which can better adapt to the dynamic change of bridge performance in service life. Finally, the effectiveness of the framework was verified by taking the simulation data of a simply supported beam bridge and a cable-stayed bridge as examples.

Smoothing effect and derivative formulas for Ornstein–Uhlenbeck processes driven by subordinated cylindrical Brownian noises

We investigate the concept of cylindrical Wiener process subordinated to a strictly α–stable Lévy process, with α∈(0,1), in an infinite–dimensional, separable Hilbert space, and consider the related stochastic convolution. We then introduce the corresponding Ornstein–Uhlenbeck process, focusing on the regularizing properties of the Markov transition semigroup defined by it. In particular, we provide an explicit, original formula –which is not of Bismut–Elworthy–Li's type– for the Gateaux derivatives of the functions generated by the operators of the semigroup, as well as an upper bound for the norm of their gradients. In the case α∈(12,1), this estimate represents the starting point for studying the Kolmogorov equation in its mild formulation.

Multi-robot adversarial patrolling strategies via lattice paths

In full-knowledge multi-robot adversarial patrolling, a group of robots has to detect an adversary who knows the robots' strategy. The adversary can easily take advantage of any deterministic patrolling strategy, which necessitates the employment of a randomised strategy. While the Markov decision process has been the dominant methodology in computing the penetration detection probabilities on polylines, we apply enumerative combinatorics to characterise the penetration detection probabilities for four penetration configurations. It allows us to provide the closed formulae of these probabilities and facilitates characterising optimal random defence strategies. Comparing to iteratively updating the Markov transition matrices, we empirically show that our method reduces the runtime by up to several hours. This allows us extensive simulations on the two dominant robot movement types for patrolling a perimeter showing that a movement with direction is up to 0.4 more likely to detect an adversary. Therefore, our approach greatly benefits the theoretical and empirical analysis of optimal patrolling strategies with extendability to more complicated attacks and other structured environments.

Adaptive Hybrid Robust Filter for Multi-Sensor Relative Navigation System

This paper provides an adaptive hybrid robust filter (AHRF) for multi-sensor relative navigation systems that can be used to support cooperative intelligent transport systems. It is known that Huber’s M-estimation based robust filter and the fault detection and exclusion (FDE) based RAIM filter each has its own drawbacks, depending on the nature of the observation error biases. Based on the interactive multiple model (IMM) framework, our proposed AHRF in this paper can take advantage of both filters in a complementary sense. A new adaptive IMM (AIMM) algorithm with Markov transition probability prediction is proposed to allow AHRF to switch efficiently between the two filters. We consider the relative navigation system with Global Navigation Satellite System (GNSS) and ultra-wideband (UWB) as observations to verify AHRF in three cases of possible failure modes and multipath-induced errors. Our results show that AHRF outperforms both the FDE and robust filter in all cases. AHRF framework can be further adapted to include many other fault-tolerant filters to improve the robustness of multi-sensor relative navigation system even further.

Markov Transition Field Combined with Convolutional Neural Network Improved the Predictive Performance of Near-Infrared Spectroscopy Models for Determination of Aflatoxin B1 in Maize

This work provides a novel approach to monitor the aflatoxin B1 (AFB1) content in maize by near-infrared (NIR) spectra-based deep learning models that integrates Markov transition field (MTF) image coding and a convolutional neural network (CNN) strategy. According to the data structure characteristics of near-infrared spectra, new structures of one-dimensional CNN (1D-CNN) and two-dimensional MTF-CNN (2D-MTF-CNN) were designed to construct a deep learning model for the monitoring of AFB1 in maize. The results obtained showed that compared with the 1D-CNN model, the performance of the 2D-MTF-CNN model had been significantly improved, and its root mean square error of prediction, coefficient of predictive determination, and relative percent deviation were 1.3591 μg·kg−1, 0.9955, and 14.9386, respectively. The results indicate that the MTF is an effective data encoding technique for converting one-dimensional spectra into two-dimensional images. It more intuitively reflects the intrinsic characteristics of the NIR spectra from a new perspective and provides richer spectral information for the construction of deep learning models, which can ensure the detection accuracy and generalization performance of deep learning quantitative detection models. This study provides a new analytical perspective for the chemometrics analysis of the NIR spectroscopy.

Muscle Fatigue Analysis by Visualization of Dynamic Surface EMG Signals Using Markov Transition Field.

Muscle fatigue analysis is important in the diagnosis of neuromuscular diseases. Analysis of surface electromyography (sEMG) signals by non-linear probabilistic approach is useful in studying their transitions and thus the neuromuscular system. In this study, a method to visualize sEMG signals using Markov transition field (MTF) under fatigue conditions is proposed. sEMG signals are acquired from 45 healthy participants during biceps curl exercise. They are filtered and divided into ten equal segments. Markov transition matrix is constructed and corresponding MTF image is generated. The average self-transition probability is extracted and compared for both non-fatigue and fatigue segments. It is observed that the extracted feature shows high statistical significance with p value less than 0.001. The increase in average self-transition probability under fatigue condition correlates with the reduction in the degree of signal complexity. Thus, encoding of sEMG signals to images is helpful in analyzing the complexity of the neuromuscular system. Clinical Relevance- This approach may be helpful in analyzing muscle fatigue related with various myoneural conditions.

Multistate models for the natural history of cancer progression.

Multistate models can be effectively used to characterise the natural history of cancer. Inference from such models has previously been useful for setting screening policies. We introduce the basic elements of multistate models and the challenges of applying these models to cancer data. Through simulation studies, we examine (1) the impact of assuming time-homogeneous Markov transition intensities when the intensities depend on the time since entry to the current state (i.e., the process is time-inhomogenous semi-Markov) and (2) the effect on precancer risk estimation when observation times depend on an unmodelled intermediate disease state. In the settings we examined, we found that misspecifying a time-inhomogenous semi-Markov process as a time-homogeneous Markov process resulted in biased estimates of the mean sojourn times. When screen-detection of the intermediate disease leads to more frequent future screening assessments, there was minimal bias induced compared to when screen-detection of the intermediate disease leads to less frequent screening. Multistate models are useful for estimating parameters governing the process dynamics in cancer such as transition rates, sojourn time distributions, and absolute and relative risks. As with most statistical models, to avoid incorrect inference, care should be given to use the appropriate specifications and assumptions.

Encoding Cardiopulmonary Exercise Testing Time Series as Images for Classification using Convolutional Neural Network.

Exercise testing has been available for more than a half-century and is a remarkably versatile tool for diagnostic and prognostic information of patients for a range of diseases, especially cardiovascular and pulmonary. With rapid advancements in technology, wearables, and learning algorithm in the last decade, its scope has evolved. Specifically, Cardiopulmonary exercise testing (CPX) is one of the most commonly used laboratory tests for objective evaluation of exercise capacity and performance levels in patients. CPX provides a non-invasive, integrative assessment of the pulmonary, cardiovascular, and skeletal muscle systems involving the measurement of gas exchanges. However, its assessment is challenging, requiring the individual to process multiple time series data points, leading to simplification to peak values and slopes. But this simplification can discard the valuable trend information present in these time series. In this work, we encode the time series as images using the Gramian Angular Field and Markov Transition Field and use it with a convolutional neural network and attention pooling approach for the classification of heart failure and metabolic syndrome patients. Using GradCAMs, we highlight the discriminative features identified by the model. Clinical relevance- The proposed framework can process multivariate exercise testing time-series data and accurately predict cardiovascular diseases. Interpretable Grad-CAMs can be obtained to explain the prediction.

HRSG early tube leak detection with a transfer learning neural network and Gramian Angular Difference Field

This paper proposes a novel heat recovery steam generator (HRSG) early tube leak detection model which leverages a convolution neural network classifier by utilising transfer learning with ResNet50 architecture. The design goal of this model was to achieve high classification accuracy with a minimal amount of leakage data. The model is also intended to be user-friendly and require minimal hyperparameter tuning. The proposed neural network was trained on the drum-specific conductivity time series data of HRSGs encoded in the Gramian Angular Difference Field (GADF). The model yielded a validation accuracy of 96.64%, true-positive rate of 93.28% and precision of 100% in regard to the validation set. The study included experiments on the influence of different encoding algorithms, Markov Transition Field (MTF) and Recurrence Plot (RP), and architectures on the performance of the model. This paper further discusses the viability of adapting the design to other time series classification problems.

Nonparametric Power-Law Surrogates

Power-law distributions are essential in computational and statistical investigations of extreme events and complex systems. The usual technique to generate power-law distributed data is to first infer the scale exponent $\alpha$ using the observed data of interest and then sample from the associated distribution. This approach has important limitations because it relies on a fixed $\alpha$ (e.g., it has limited applicability in testing the {\it family} of power-law distributions) and on the hypothesis of independent observations (e.g., it ignores temporal correlations and other constraints typically present in complex systems data). Here we propose a constrained surrogate method that overcomes these limitations by choosing uniformly at random from a set of sequences exactly as likely to be observed under a discrete power-law as the original sequence (i.e., regardless of $\alpha$) and by showing how additional constraints can be imposed in the sequence (e.g., the Markov transition probability between states). This non-parametric approach involves redistributing observed prime factors to randomize values in accordance with a power-law model but without restricting ourselves to independent observations or to a particular $\alpha$. We test our results in simulated and real data, ranging from the intensity of earthquakes to the number of fatalities in disasters.

Learning-Based Spectrum Sensing and Access in Cognitive Radios via Approximate POMDPs

A novel LEarning-based Spectrum Sensing and Access (LESSA) framework is proposed, wherein a cognitive radio (CR) learns a time-frequency correlation model underlying spectrum occupancy of licensed users (LUs) in a radio ecosystem; concurrently, it devises an approximately optimal spectrum sensing and access policy under sensing constraints. A Baum-Welch algorithm is proposed to learn a parametric Markov transition model of LUs’ spectrum occupancy based on noisy spectrum measurements. Spectrum sensing and access are cast as a Partially-Observable Markov Decision Process, approximately optimized via randomized point-based value iteration. Fragmentation, Hamming-distance state filters and Monte-Carlo methods are proposed to alleviate the inherent computational complexity, and a weighted reward metric to regulate the trade-off between CR’s throughput and interference to the LUs. Numerical evaluations demonstrate that LESSA performs within 5% of a genie-aided upper bound with foreknowledge of LUs’ spectrum occupancy, and outperforms state-of-the-art algorithms across the entire trade-off region: 71% over correlation-based clustering, 26% over Neyman-Pearson-based spectrum sensing, 6% over the Viterbi algorithm, and 9% over adaptive Deep Q-Network. LESSA is then extended to a distributed Multi-Agent setting (MA-LESSA), by proposing novel neighbor discovery and channel access rank allocation. MA-LESSA improves CRs’ throughputs by 43% over cooperative TD-SARSA, 84% over cooperative greedy distributed learning, and 3× over non-cooperative learning via g-statistics and ACKs. Finally, MA-LESSA is implemented on the DARPA SC2 platform, manifesting superior performance over competitors in a real-world TDWR-UNII WLAN emulation; its implementation feasibility is further validated on an ad-hoc distributed wireless testbed of ESP32 radios, exhibiting 96% success probability.

Markov Transition Field Enhanced Deep Domain Adaptation Network for Milling Tool Condition Monitoring

Tool condition monitoring (TCM) is of great importance for improving the manufacturing efficiency and surface quality of workpieces. Data-driven machine learning methods are widely used in TCM and have achieved many good results. However, in actual industrial scenes, labeled data are not available in time in the target domain that significantly affect the performance of data-driven methods. To overcome this problem, a new TCM method combining the Markov transition field (MTF) and the deep domain adaptation network (DDAN) is proposed. A few vibration signals collected in the TCM experiments were represented in 2D images through MTF to enrich the features of the raw signals. The transferred ResNet50 was used to extract deep features of these 2D images. DDAN was employed to extract deep domain-invariant features between the source and target domains, in which the maximum mean discrepancy (MMD) is applied to measure the distance between two different distributions. TCM experiments show that the proposed method significantly outperforms the other three benchmark methods and is more robust under varying working conditions.

A New Fault Diagnosis of Rolling Bearing Based on Markov Transition Field and CNN.

The rolling bearing is a crucial component of the rotating machine, and it is particularly vital to ensure its normal operation. In addition, the selection of different category features will add uncertainty and bias to the classification results. In order to decrease the interference of these factors to fault diagnosis, a new method that automatically learns the features of the data combined with Markov transition field (MTF) and convolutional neural network (CNN) is proposed in this paper, namely MTF-CNN. The MTF contributes to convert the original time series into corresponding figures, and the CNN is used to extract the deep feature information in the figure to complete the fault diagnosis. The effectiveness of the proposed method is verified by two public data sets. The experimental results show that MTF-CNN can classify different types of faults, and the highest accuracy rate can reach 100%. Likewise, the classification accuracy of this method is higher than some existing methods.

Cost-effectiveness of Low-Dose Computed Tomography With a Plasma-Based Biomarker for Lung Cancer Screening in China

China, which has one-third of the worldwide smoking population, has a substantial cancer burden, with lung cancer being the leading cause of cancer-related death. The effectiveness of lung cancer screening for mortality reduction has been confirmed, but the cost-effectiveness of diverse screening modalities remains unclear. To compare the cost-effectiveness of low-dose computed tomography (LDCT) with a biomarker (micro-RNA signature classifier [MSC]) with that of LDCT alone by screening interval and cumulative smoking exposure. In this economic evaluation, a comparative cost-effectiveness analysis used Markov state transition models that simulated the 1947 to 1971 China birth cohort. Simulated individuals in 8 cohorts of 10 000 entered the study between ages 50 and 74 years and were followed up until death or age 79 years, corresponding to a study period from January 1, 2021, to December 31, 2050. The model was run with a cycle length of 1 year. All the transition probabilities were validated, and health utility values were extracted from published literature. Cost parameters were derived from the databases of local medical insurance bureaus. Primary outcomes included life-years, quality-adjusted life-years (QALYs), and incremental cost-effectiveness ratios (ICERs) with future costs and outcomes discounted by 5%. Screening strategies with a mean ICER less than Chinese yuan (CNY) 212 676 per QALY gained were deemed to be cost-effective. The cost-effectiveness of 7 alternative screening strategies with a screening starting age of 50 years, minimum cumulative smoking exposure of 20 vs 30 pack-years, and screening interval of annual vs 1 time was estimated, including the 2021 China guideline-recommended strategy (LDCT, annual, 30 pack-years) and the 2018 China guideline-recommended strategy (LDCT, annual, 20 pack-years). In a simulated population of 80 000 individuals, the conjunctive LDCT and MSC screening strategy was estimated to obtain an ICER of CNY -793 995.17 to 254 417.46 (minimum cumulative smoking exposure, 20-30 pack-years) per QALY gained compared with LDCT screening alone. China's 2021 guideline-recommended strategy was not cost-effective compared with the 2018 guideline-recommended strategy, with higher costs and fewer QALYs gained; the QALY loss ranged from 0.02 to 0.15 per person and the increase in cost ranged from CNY 945.89 to CNY 5131.29 per person. LDCT and MSC screening beginning at age 70 to 74 years in individuals with a 20 pack-year smoking history was the most cost-effective strategy, with an ICER of CNY -793 995.17 per QALY gained. Lowering the minimum cumulative smoking exposure for screening from 30 to 20 pack-years and maintaining annual screening were associated with greater cost savings regardless of the screening tool. This economic evaluation found that China's 2018 recommendation for lung cancer screening was more cost-effective than the 2021 recommendation. Moreover, the cost-effectiveness of lung cancer screening was improved when MSC was included with LDCT. These findings may be useful for the modification of guidelines for lung cancer screening.

Comparison of CT, MRI, and F-18 FDG PET/CT for initial N-staging of oral squamous cell carcinoma: a cost-effectiveness analysis

Background and purposeTreatment of oral squamous cell carcinoma (OSCC) is based on clinical exam, biopsy, and a precise imaging-based TNM-evaluation. A high sensitivity and specificity for magnetic resonance imaging (MRI) and F-18 FDG PET/CT are reported for N-staging. Nevertheless, staging of oral squamous cell carcinoma is most often based on computed tomography (CT) scans. This study aims to evaluate cost-effectiveness of MRI and PET/CT compared to standard of care imaging in initial staging of OSCC within the US Healthcare System.MethodsA decision model was constructed using quality-adjusted life years (QALYs) and overall costs of different imaging strategies including a CT of the head, neck, and the thorax, MRI of the neck with CT of the thorax, and whole body F-18 FDG PET/CT using Markov transition simulations for different disease states. Input parameters were derived from literature and willingness to pay (WTP) was set to US $100,000/QALY. Deterministic sensitivity analysis of diagnostic parameters and costs was performed. Monte Carlo modeling was used for probabilistic sensitivity analysis.ResultsIn the base-case scenario, total costs were at US $239,628 for CT, US $240,001 for MRI, and US $239,131 for F-18 FDG PET/CT whereas the model yielded an effectiveness of 5.29 QALYs for CT, 5.30 QALYs for MRI, and 5.32 QALYs for F-18 FDG PET/CT respectively. F-18 FDG PET/CT was the most cost-effective strategy over MRI as well as CT, and MRI was the cost-effective strategy over CT. Deterministic and probabilistic sensitivity analysis showed high robustness of the model with incremental cost effectiveness ratio remaining below US $100,000/QALY for a wide range of variability of input parameters.ConclusionF-18 FDG PET/CT is the most cost-effective strategy in the initial N-staging of OSCC when compared to MRI and CT. Despite less routine use, both whole body PET/CT and MRI are cost-effective modalities in the N-staging of OSCC. Based on these findings, the implementation of PET/CT for initial staging could be suggested to help reduce costs while increasing effectiveness in OSCC.

Rolling Bearing Fault Diagnosis Based on Markov Transition Field and Residual Network.

Data-driven rolling-bearing fault diagnosis methods are mostly based on deep-learning models, and their multilayer nonlinear mapping capability can improve the accuracy of intelligent fault diagnosis. However, problems such as gradient disappearance occur as the number of network layers increases. Moreover, directly taking the raw vibration signals of rolling bearings as the network input results in incomplete feature extraction. In order to efficiently represent the state characteristics of vibration signals in image form and improve the feature learning capability of the network, this paper proposes fault diagnosis model MTF-ResNet based on a Markov transition field and deep residual network. First, the data of raw vibration signals are augmented by using a sliding window. Then, vibration signal samples are converted into two-dimensional images by MTF, which retains the time dependence and frequency structure of time-series signals, and a deep residual neural network is established to perform feature extraction, and identify the severity and location of the bearing faults through image classification. Lastly, experiments were conducted on a bearing dataset to verify the effectiveness and superiority of the MTF-ResNet model. Features learned by the model are visualized by t-SNE, and experimental results indicate that MTF-ResNet showed better average accuracy compared with several widely used diagnostic methods.

Analysis of the Gross Ecosystem Product—Gross Domestic Product Synergistic States, Evolutionary Process, and Their Regional Contribution to the Chinese Mainland

The trade-off and synergy relationship between ecosystem services (ESs) and human well-being (HWb) in the land-use process has become a research hotspot. The evolutionary process and regional contribution of the accurate quantification of ESs and HWb can provide a reference for government departments to formulate macroeconomic policies. Therefore, this study first constructed an analysis framework to identify the synergistic states/evolutionary stages of the gross ecosystem product (GEP) per capita (PGEP) and gross domestic product (GDP) per capita (PGDP) and the regional contribution of the GEP–GDP synergy of 362 municipal units in mainland China from 2000 to 2015. We did this by employing the Markov transition probability matrix, land use data, and economic data based on satellite remote sensing images. The findings of this study show that (1) the PGEP of the Chinese mainland has a remarkable spatial divergence featuring a higher value in the northwestern Chinese mainland and a lower value in the southeastern Chinese mainland on both sides of the Hu Line during the investigation period; despite the eastern Chinese mainland having a higher PGDP, the PGDP at the national level is distributed in dispersion on both sides of the Hu Line; (2) during the first half of the investigation period, the GEP–GDP synergy in the Chinese mainland was generally in the pseudo-synergy stage or the transition stage from pseudo-synergy to primary/intermediate synergy, while in the second half of the period, the GEP–GDP synergy in the Chinese mainland continued to improve; (3) the GEP–GDP synergy was relatively stable on the Chinese mainland during the investigation period, with 24.28% of the units shifting to a lower or higher level synergy, and the GEP–GDP synergy was more stable in the western and central Chinese mainland, while it was more likely to shift to a higher state in the northeastern and eastern Chinese mainland. This study suggests that the GEP–GDP relationship varies with spatial scales; a hierarchical, multiscale approach is necessary to study and improve both of these relationships, as simply extrapolating policies across single administrative levels may lead to unintended outcomes.