In this paper we study asymptotic properties of random forests within the framework of nonlinear time series modeling. While random forests have been successfully applied in various fields, the theoretical justification has not been considered for their use in a time series setting. Under mild conditions, we prove a uniform concentration inequality for regression trees built on nonlinear autoregressive processes and, subsequently, we use this result to prove consistency for a large class of random forests. The results are supported by various simulations.
The random forest algorithm, proposed by L. Breiman in 2001, has been extremely successful as a general-purpose classification and regression method. The approach, which combines several randomized decision trees and aggregates their predictions by averaging, has shown excellent performance in settings where the number of variables is much larger than the number of observations. Moreover, it is versatile enough to be applied to large-scale problems, is easily adapted to various ad-hoc learning tasks, and returns measures of variable importance. The present article reviews the most recent theor…
Random forest (RF) is one of the most popular methods for estimating regression functions. The local nature of the RF algorithm, based on intra-node means and variances, is ideal when errors are i.i.d. For dependent error processes like time series and spatial settings where data in all the nodes will be correlated, operating locally ignores this dependence. Also, RF will involve resampling of correlated data, violating the principles of bootstrap. Theoretically, consistency of RF has been established for i.i.d. errors, but little is known about the case of dependent errors. We propose RF-GLS,…
Random forest and deep neural network are two schools of effective classification methods in machine learning. While the random forest is robust irrespective of the data domain, the deep neural network has advantages in handling high dimensional data. In view that a differentiable neural decision forest can be added to the neural network to fully exploit the benefits of both models, in our work, we further combine convolutional autoencoder with neural decision forest, where autoencoder has its advantages in finding the hidden representations of the input data. We develop a gradient boost modul…
Shapelet is a discriminative subsequence of time series. An advanced shapelet-based method is to embed shapelet into accurate and fast random forest. However, it shows several limitations. First, random shapelet forest requires a large training cost for split threshold searching. Second, a single shapelet provides limited information for only one branch of the decision tree, resulting in insufficient accuracy and interpretability. Third, randomized ensemble causes interpretability declining. For that, this paper presents Random Pairwise Shapelets Forest (RPSF). RPSF combines a pair of shapelet…
Dept. of Mathematics \& Statistics Utah State University Logan, Utah, USA；Dept. of Computer Science Brigham Young University Provo, UT, USA；Dept. of Statistics Brigham Young University Provo, UT, USA；Dept. of Mathematics \& Statistics Utah State University Logan, UT, USA
We introduce random spatial forests, a method of bagging regression trees allowing for spatial correlation. Our main contribution is the development of a computationally efficient tree building algorithm which selects each split of the tree adjusting for spatial correlation. We evaluate two different approaches for estimation of random spatial forests, a pseudo-likelihood approach combining random forests with kriging and a non-parametric version for a general class of spatial smoothers. We show improved prediction accuracy of our method compared to existing two-step approaches combining rando…
This paper presents a new ensemble learning method for classification problems called projection pursuit random forest (PPF). PPF uses the PPtree algorithm introduced in Lee et al. (2013). In PPF, trees are constructed by splitting on linear combinations of randomly chosen variables. Projection pursuit is used to choose a projection of the variables that best separates the classes. Utilizing linear combinations of variables to separate classes takes the correlation between variables into account which allows PPF to outperform a traditional random forest when separations between groups occurs i…
University of Liège（列日大学）；Aristotle University of Thessaloniki（塞萨洛尼基亚里士多德大学）；Katholieke Universiteit Leuven（鲁汶天主教大学）
Within machine learning, the supervised learning field aims at modeling the input-output relationship of a system, from past observations of its behavior. Decision trees characterize the input-output relationship through a series of nested $if-then-else$ questions, the testing nodes, leading to a set of predictions, the leaf nodes. Several of such trees are often combined together for state-of-the-art performance: random forest ensembles average the predictions of randomized decision trees trained independently in parallel, while tree boosting ensembles train decision trees sequentially to ref…
Decision forests (Forests), in particular random forests and gradient boosting trees, have demonstrated state-of-the-art accuracy compared to other methods in many supervised learning scenarios. In particular, Forests dominate other methods in tabular data, that is, when the feature space is unstructured, so that the signal is invariant to a permutation of the feature indices. However, in structured data lying on a manifold (such as images, text, and speech) deep networks (Networks), specifically convolutional deep networks (ConvNets), tend to outperform Forests. We conjecture that at least pa…
An increasing array of biomedical and computer vision applications requires the predictive modeling of complex data, for example images and shapes. The main challenge when predicting such objects lies in the fact that they do not comply to the assumptions of Euclidean geometry. Rather, they occupy non-linear spaces, a.k.a. manifolds, where it is difficult to define concepts such as coordinates, vectors and expected values. In this work, we construct a non-parametric predictive methodology for manifold-valued objects, based on a distance modification of the Random Forest algorithm. Our method i…
Survival random forest is a popular machine learning tool for modeling censored survival data. However, there is currently no statistically valid and computationally feasible approach for estimating its confidence band. This paper proposes an unbiased confidence band estimation by extending recent developments in infinite-order incomplete U-statistics. The idea is to estimate the variance-covariance matrix of the cumulative hazard function prediction on a grid of time points. We then generate the confidence band by viewing the cumulative hazard function estimation as a Gaussian process whose d…
Vencera Consulting（Vencera咨询公司）；Massachusetts Institute of Technology（麻省理工学院）；Southern University of Science and Technology（南方科技大学）
Conventionally, random forests are built from "greedy" decision trees which each consider only one split at a time during their construction. The sub-optimality of greedy implementation has been well-known, yet mainstream adoption of more sophisticated tree building algorithms has been lacking. We examine under what circumstances an implementation of less greedy decision trees actually yields outperformance. To this end, a "stepwise lookahead" variation of the random forest algorithm is presented for its ability to better uncover binary feature interdependencies. In contrast to the greedy appr…
Semantic segmentation of land cover classes is fundamental for agricultural and economic development work, from sustainable forestry to urban planning, yet existing training datasets have significant limitations. To generate an open and comprehensive training library of high resolution Earth imagery and high quality land cover classifications, public Sentinel-2 data at 10 m spatial resolution was matched with accurate GlobeLand30 labels from 2010, which were filtered by agreement with an intermediary Sentinel-2 classification at 20 m produced during atmospheric correction. Scene-level classifi…
Random Forests [Breiman:2001] (RF) are a fully non-parametric statistical method requiring no distributional assumptions on covariate relation to the response. RF are a robust, nonlinear technique that optimizes predictive accuracy by fitting an ensemble of trees to stabilize model estimates. The randomForestSRC package (http://cran.r-project.org/package=randomForestSRC) is a unified treatment of Breiman's random forests for survival, regression and classification problems. Predictive accuracy make RF an attractive alternative to parametric models, though complexity and interpretability of the…
We discuss an application of Generalized Random Forests (GRF) proposed by Athey et al.(2019) to quantile regression for time series data. We extracted the theoretical results of the GRF consistency for i.i.d. data to time series data. In particular, in the main theorem, based only on the general assumptions for time series data in Davis and Nielsen (2020), and trees in Athey et al.(2019), we show that the tsQRF (time series Quantile Regression Forests) estimator is consistent. Davis and Nielsen (2020) also discussed the estimation problem using Random Forests (RF) for time series data, but the…
University of Pittsburgh（匹兹堡大学）；Cornell University（康奈尔大学）
In 2001, Leo Breiman wrote of a divide between "data modeling" and "algorithmic modeling" cultures. Twenty years later this division feels far more ephemeral, both in terms of assigning individuals to camps, and in terms of intellectual boundaries. We argue that this is largely due to the "data modelers" incorporating algorithmic methods into their toolbox, particularly driven by recent developments in the statistical understanding of Breiman's own Random Forest methods. While this can be simplistically described as "Breiman won", these same developments also expose the limitations of the pred…
This work presents a new classifier that is specifically designed to be fully interpretable. This technique determines the probability of a class outcome, based directly on probability assignments measured from the training data. The accuracy of the predicted probability can be improved by measuring more probability estimates from the training data to create a series expansion that refines the predicted probability. We use this work to classify four standard datasets and achieve accuracies comparable to that of Random Forests. Because this technique is interpretable by design, it is capable of…
Research into the classification of time series has made enormous progress in the last decade. The UCR time series archive has played a significant role in challenging and guiding the development of new learners for time series classification. The largest dataset in the UCR archive holds 10 thousand time series only; which may explain why the primary research focus has been in creating algorithms that have high accuracy on relatively small datasets. This paper introduces Proximity Forest, an algorithm that learns accurate models from datasets with millions of time series, and classifies a time…
Time Series Classification (TSC) has seen enormous progress over the last two decades. HIVE-COTE (Hierarchical Vote Collective of Transformation-based Ensembles) is the current state of the art in terms of classification accuracy. HIVE-COTE recognizes that time series data are a specific data type for which the traditional attribute-value representation, used predominantly in machine learning, fails to provide a relevant representation. HIVE-COTE combines multiple types of classifiers: each extracting information about a specific aspect of a time series, be it in the time domain, frequency dom…
We propose a method for causal inference using satellite image time series, in order to determine the treatment effects of interventions which impact climate change, such as deforestation. Simply put, the aim is to quantify the 'before versus after' effect of climate related human driven interventions, such as urbanization; as well as natural disasters, such as hurricanes and forest fires. As a concrete example, we focus on quantifying forest tree cover change/ deforestation due to human led causes. The proposed method involves the following steps. First, we uae computer vision and machine lea…
Label-Guided Imputation via Forest-Based Proximities for Improved Time Series Classification
1 Department of Statistics, Brigham Young University, Provo, UT, USA Emails；2 Department of Computer Science, Brigham Young University, Provo, UT, USA Email；3 Department of Public Health, Brigham Young University, Provo, UT, USA Email；4 Department of Neurology, Vagelos College of Physicians \& Surgeons, Columbia University, New York, NY, USA Email；5 Department of Neurology, Miller School of Medicine, University of Miami, Miami, FL, USA Email；6 Elmhurst Hospital Center, NYC Health + Hospitals, New York, NY, USA Email；7 Department of Mathematics \& Statistics, Utah State University, Logan, UT, USA Email
Threshold Autoregressive (TAR) models have been widely used by statisticians for non-linear time series forecasting during the past few decades, due to their simplicity and mathematical properties. On the other hand, in the forecasting community, general-purpose tree-based regression algorithms (forests, gradient-boosting) have become popular recently due to their ease of use and accuracy. In this paper, we explore the close connections between TAR models and regression trees. These enable us to use the rich methodology from the literature on TAR models to define a hierarchical TAR model as a …
Unsupervised Feature Based Algorithms for Time Series Extrinsic Regression
Department of Computer Science and Numerical Analysis, University of Cordoba（科塔达大学计算机科学与数值分析系）；School of Computing Sciences, University of East Anglia（东安格利亚大学计算科学学院）
Stochastic forest transition model dynamics and parameter estimation via deep learning
Joint Graduate School of Mathematics for Innovation, Kyushu University（创新数学联合研究生院，九州大学）；Graduate School of Bioresource and Bioenvironmental Sciences, Kyushu University（生物资源与生物环境科学研究生院，九州大学）；Center for Promotion of International Education and Research, Faculty of Agriculture, Kyushu University（国际教育与研究促进中心，农业学部，九州大学）
Analysis of water and environmental data is an important aspect of many intelligent water and environmental system applications where inference from such analysis plays a significant role in decision making. Quite often these data that are collected through sensible sensors can be anomalous due to different reasons such as systems breakdown, malfunctioning of sensor detectors, and more. Regardless of their root causes, such data severely affect the results of the subsequent analysis. This paper demonstrates data cleaning and preparation for time-series data and further proposes cost-sensitive …
University of Massachusetts Dartmouth（马萨诸塞大学达特茅斯分校）；Dali University（大理大学）
K-nearest neighbor (kNN) search has wide applications in many areas, including data mining, machine learning, statistics and many applied domains. Inspired by the success of ensemble methods and the flexibility of tree-based methodology, we propose random projection forests (rpForests), for kNN search. rpForests finds kNNs by aggregating results from an ensemble of random projection trees with each constructed recursively through a series of carefully chosen random projections. rpForests achieves a remarkable accuracy in terms of fast decay in the missing rate of kNNs and that of discrepancy i…
Institute for Development and Research in Banking Technology（银行技术发展与研究所）；University of Hyderabad（海得拉巴大学）
This paper proposes to model chaos in the ATM cash withdrawal time series of a big Indian bank and forecast the withdrawals using deep learning methods. It also considers the importance of day-of-the-week and includes it as a dummy exogenous variable. We first modelled the chaos present in the withdrawal time series by reconstructing the state space of each series using the lag, and embedding dimension found using an auto-correlation function and Cao's method. This process converts the uni-variate time series into multi variate time series. The "day-of-the-week" is converted into seven feature…
