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Flood forecasting using time series data mining
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by Chaitanya Damle.
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[Tampa, Fla.] :
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2005.
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Thesis (M.S.I.E.)University of South Florida, 2005.
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Includes bibliographical references.
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Text (Electronic thesis) in PDF format.
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ABSTRACT: Earthquakes, floods, rainfall represent a class of nonlinear systems termed chaotic, in which the relationships between variables in a system are dynamic and disproportionate, however completely deterministic. Classical linear time series models have proved inadequate in analysis and prediction of complex geophysical phenomena. Nonlinear approaches such as Artificial Neural Networks, Hidden Markov Models and Nonlinear Prediction are useful in forecasting of daily discharge values in a river. The focus of these methods is on forecasting magnitudes of future discharge values and not the prediction of floods. Chaos theory provides a structured explanation for irregular behavior and anomalies in systems that are not inherently stochastic. Time Series Data Mining methodology combines chaos theory and data mining to characterize and predict complex, nonperiodic and chaotic time series. Time Series Data Mining focuses on the prediction of events.
590
Adviser: Dr. Ali Yalcin.
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Phase space reconstruction.
Flood prediction.
Nonlinear time series prediction.
Clustering.
Genetic algorithm.
Event prediction.
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Dissertations, Academic
z USF
x Industrial Engineering
Masters.
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t USF Electronic Theses and Dissertations.
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u http://digital.lib.usf.edu/?e14.1038
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Flood Forecasting Using Time Series Data Mining b y Chaitanya Damle A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science in Industrial Engineering Department of Industrial and Managemen t Systems Engineering College of Engineering University of South Florida Major Professor: Ali Yalcin Ph.D. Bruce Lindsey, Ph.D. Jos L. Zayas Castro Ph.D. Date of Approval: April 1 2005 Keywords: phase space reconstruction, flood prediction, nonlinear time series prediction, clustering, genetic algorithm, event prediction Copyright 2005 Chaitanya Damle
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i Table of Contents L ist of Tables ................................ ................................ ................................ ........................ iii L ist of Figures ................................ ................................ ................................ ....................... iv A bstract ................................ ................................ ................................ ................................ vi C hapter 1. Introduction ................................ ................................ ................................ ........... 1 1.1 Effects of Floods ................................ ................................ ................................ ... 1 1.2 Nature of Geophysical Systems ................................ ................................ .............. 1 1.3 Existing Flood Forecasting Technique s ................................ ................................ ... 2 1.4 Time Series Data Mining ................................ ................................ ....................... 3 1.5 Flood Forecasting U sing Time Series Data Mining ................................ .................. 4 C hapter 2. Literature Review ................................ ................................ ................................ ... 6 2.1 P resence Chaos in Geophysical Systems ................................ ................................ 6 2.2 Existing Flood Forecasting Techniques ................................ ................................ ... 7 2.2.1 Stream and Rain Gauge Networks and Hydrograph Analyses .................... 8 2.2.2 Radar and Information Systems ................................ ............................... 8 2.2.3 Linear Statistical Models ................................ ................................ ........ 9 2.3 Nonlinear T ime S eries A nalysis and P rediction A pplied to F lood F orecasting ........... 9 2.3.1 Hidden Ma rkov Models ................................ ................................ .......... 9 2.3.2 Artificial Neural Networks ................................ ................................ .... 10 2.3.3 Non Linear Prediction Method ................................ .............................. 10 C hapter 3. Time Series Data Mining Methodology ................................ ................................ .. 17 3.1 Time Series Data Mining ................................ ................................ .................... 17 3.2 Training Stage ................................ ................................ ................................ .... 18 3.2.1 Step 1. Reconstruct the Phase Space ................................ ...................... 18 3.2.2 Step 2. a) Define the Event Character ization Function ............................. 19 Step 2. b) Define the Objective Function ................................ ................ 20 Step 2. c) Define Optimization Formulation ................................ ........... 23 3.2.3 Step 3. Create Augmente d Phase Space ................................ .................. 23 3.2.4 Step 4. Search for O ptimal Temporal Pattern Cluster .............................. 24 3.2.5 Step 5. Evaluate T raining R esults ................................ .......................... 25 3.3 Testing Stage ................................ ................................ ................................ ..... 25 3.3.1 Step 1. Embed the Testing T ime S eries into P hase S pace ........................ 25 3.3.2 Step 2. Use the O ptimal T emporal Pattern C luster to P redict E vents ........ 25 3.2.3 Step 3. Evaluate T esting R esults ................................ ............................ 26
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ii C hapter 4. Application of TSDM to Flood Forecasting and Results ................................ .......... 27 4.1 Detection of Chaos in River Daily Discharge Time Series ................................ ...... 27 4.2 Reconstru ction of Phase Space from Training Time Series ................................ ..... 2 8 4.3 Event Characterization Function, Objective Function and Optimization .................. 29 4.3.1 Event Characterization Function ................................ ............................ 29 4.3 .2 Objective Function ................................ ................................ ............... 30 4.3.3 Optimization Formulation ................................ ................................ ..... 33 4.4 Cluster Predicition Accuracy ................................ ................................ ................ 34 4.5 Problem Discription and Setup ................................ ................................ ............. 35 4.6 Results ................................ ................................ ................................ ............... 41 4.6.1 Prediction Accuracy with Objective Function I ................................ ....... 42 4.6.2 Earliness Prediction Accuracy ................................ ............................... 46 4.6.3 Prediction Accuracy w ith Objective Function II ................................ ..... 54 4.7 Summary of Results ................................ ................................ ............................ 57 C hapter 5. Conclusions and Future Work ................................ ................................ ............... 60 5 .1 Conclusions ................................ ................................ ................................ ........ 60 5 .2 Futu re Work ................................ ................................ ................................ ....... 62 R eferences ................................ ................................ ................................ ........................... 64
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iii List of Tables Table 3.1 Event Categorization ................................ ................................ .............................. 2 2 Table 4.1 Training Stage Results for Di fferent Values at St. Louis Gauging Station ............... 4 3 Table 4.2 Testing Stage Results for Different Values at St. Louis Gauging Station ................. 43 Table 4. 3 Training Stage Results for Different Values at Kansas City Station ........................ 44 Table 4.4 Testing Stage Results for Different Values at Kansas City Gauging Station ............ 4 5 T able 4.5 Training Stage Results for Different Values at Harrisburg Gauging Station ............. 4 5 Table 4.6 Testing Stage Results for Different Values at Harrisburg Gauging Station .............. 4 6 Table 4. 7 Training Earliness Prediction Accuracy for = 0.85 at St. Louis Gauging Station ...... 4 7 Table 4.8 Testing Earliness Prediction Accuracy for = 0.85 at St. Lou is Gauging Station ........ 4 8 Table 4.9 Training Earliness Prediction Accuracy for = 0. 2 5 at Kansas City Gauging Station 5 1 Table 4.10 Testing Earliness Predict ion Accuracy for = 0. 2 5 at Kansas City Gauging Station 5 2 Table 4.1 1 Training Earliness Prediction Accuracy for = 0. 7 5 at Harrisburg Gauging Station .. 5 3 Table 4.12 T e sting Earliness Prediction Accuracy for = 0. 7 5 at Harrisburg Gauging Station ... 5 4 Table 4. 13 Training Stage Results for Different Values Using Objective Function II .............. 56 Table 4. 14 Testing Stage Results for Different Va lues Using Objective Function II ................ 56
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iv List of Figures Figure 1.1 Block Diagram of TSDM Methodology [41] ................................ ............................ 3 Figure 1.2 Time S eries R epresenting D aily D ischarge of Suwanee River ................................ .... 4 Figure 1.3 Reconstructed P hase S pace with t = 1 in T wo D imensions ................................ ........ 5 Figure 2.1 Averag e Mutual Information Function for Suwanee River T ime S eries .................... 13 Figure 2.2 Plot S howing R elationship B etween m and P ercentage of F alse N earest N eighbors ... 15 Figure 3.1 Synthetic S eismic T ime S eries ( T raining) with Events I dentified [41] ...................... 19 Figure 3.2 Reconstructed P hase Space from S ynthetic S eismic T ime S eries [41] ....................... 19 Figure 3.3 Clusters I dentified in P hase S pace from Synthetic E arthquake T ime S eries [41] ....... 24 Figure 3.4 Synthetic Seismic P hase S pace with T emporal P attern Cluster I dentified [41] .......... 24 Figure 3.5 Synthetic S eismic T ime Series with Temporal P atterns H ighli ghted [41] .................. 25 Figure 3.6 Prediction R esults in a T esting T ime S eries [41] ................................ ..................... 26 Figure 4.1 2 D Embedding of Training Time Series from St. Louis Gauging Station ................. 30 Figure 4.2 Time Series Plot of Daily Discharge Values at St. Louis Gauging Station ................. 3 6 Figure 4.3 2 D Embedding of St. Louis Training Time Series ................................ .................. 3 7 Figure 4.4 Augmented Phase Space For Tr aining Series for St. Louis ................................ ....... 3 7 Figure 4.5 Daily Discharge Time Series at Kansas City Gauging Station, Missouri River ......... 3 8 Figure 4.6 2 D Embedding of Kansas City Training Time Series ................................ ............. 3 9 Figure 4. 7 Augmented Phase Space For Training Series for Kansas City ................................ .. 3 9 Figure 4.8 Daily Discharge Time Series at Harrisburg Gauging Station, Susquehanna River ...... 40 Figure 4.9 2 D Phase Embedding of Harrisburg Training Time Series ................................ ...... 4 1 Figure 4.10 Augmented Phase Space For Harrisburg Training Series ................................ ....... 41
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v Figure 4.11 Training Earliness Prediction Accuracy for = 0.85 at St. Louis Gauging Station ... 4 8 Figure 4.12 Testing Earliness Prediction Accuracy for = 0.85 at St. Louis Gauging Station ..... 4 9 Figure 4.13 Earliness Prediction Results for = 0 .85 at St. Louis Gauging Station .................... 50 Figure 4. 14 Training Earliness Prediction Accuracy for = 0. 2 5 at Kansas Ct Gauging Station 5 1 Figure 4. 15 Testing Earliness Prediction Accuracy for = 0. 2 5 at Kansas Ct Gauging Station ... 5 2 Figure 4.16 Training Earliness Prediction Accuracy at Harrisburg Gauging Station 5 3 Figure 4.17 Testing Earliness Prediction Accu racy at Harrisburg Gauging Station .................... 5 4 Figure 4.1 9 Comparison of Two Clusters in Phase Space ................................ ....................... 58 Figure 4. 2 0 Comparison of CPP for Objective Function I and II at St. Louis Gauging Station .... 58
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vi Flood Forecas ting using Time Series Data Mining C haitanya Damle ABSTRACT Earthquakes, floods, rainfall represent a class of nonlinear systems termed chaotic, in which the relationships between variables in a system are dynamic and disproportionate, however compl etely deterministic. Classical linear time series models have proved inadequate in analysis and prediction of complex geophysical phenomena. Nonlinear approaches such as Artificial Neural Networks, Hidden Markov Models and Nonlinear Prediction are useful i n forecasting of daily discharge values in a river. The focus of these methods is on forecasting magnitudes of future discharge values and not the prediction of floods. Chaos theory provides a structured explanation for irregular behavior and anomalies in systems that are not inherently stochastic. Time Series Data Mining methodology combines chaos theory and data mining to characterize and predict complex, nonperiodic and chaotic time series. Time Series Data Mining focuses on the prediction of events. Fl oods constitute the events in a river daily discharge time series. This research focuses on application of the Time Series Data Mining to prediction of floods.
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1 Chapter 1. Introduction This research is an application of Time Series Data Minin g methodology to prediction of floods. Chapter 1 is an introduction to effects of floods, nature of geophysical phenomena, existing flood forecasting techniques, the Time Series Data Mining approach and its application to flood forecasting. 1.1 Effects of Floods According to the United States Geological Survey (USGS) and National Weather Service (NWS), as much as 90 percent of the damage related to natural disasters (excluding droughts) is caused by floods and associated mud and debris flows. Over the last 10 years, floods have cost on average, $3.1 billion annually in damages. USGS and NWS estimate that more than 95 lives are lost, on average, per year and proper detection and prediction of these disasters can save countless lives and over 1 billion dollar s a year in damages [34]. 1.2 Nature of Geophysical Systems Traditionally, geophysical systems are viewed as systems that exhibit irregular behavior, essentially due to the large number of variables that govern and dominate the underlying systems [44]. Ge ophysical phenomena such as earthquakes, floods etc represent nonlinear systems whose occurrences are subject to high levels of uncertainty and unpredictability. The irregularity of these systems is not a transient phenomenon, but an intrinsic property [17 ]. A few deterministic approaches have been applied, but the stochastic approaches have proved better in the process of
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2 representing important statistical characteristics of the geophysical system and provide reasonably good predictions [44]. Nonlinear dyn amics or chaos addresses the set of systems that may be stochastic but also display correlations that are deterministic, and are known as deterministic chaotic systems. Deterministic chaos provides a structured explanation for irregular behavior and anom alies in systems which do not seem to be inherently stochastic [25]. Thus, chaotic systems are treated as "slightly predictable" and can be studied in the framework of nonlinear system dynamics. This fact has sparked a surge of interest in nonlinear models among researchers in applied sciences. 1.3 Existing Flood Forecasting Techniques Flood prediction is a complex process because of the numerous factors that affect river water levels such as the location, rainfall, soil types and size of catchments. The relationship between these factors has not been fully understood. Classical linear Gaussian time series (deterministic) models are inadequate in analysis and prediction of complex geophysical phenomena. Linear methods such as ARIMA approach are unable to identify complex characteristics due to the goal of characterizing all time series observations, the necessity of time series stationarity and, the requirement of normality and independence of residuals [41]. Nonlinear time series approaches such as Hidden Markov Models (HMM) [3], Artificial Neural Networks (ANN) [5] and Nonlinear Predict ion (NLP) [28], applied to discharge forecasting produce accurate predictions for short prediction periods of up to one day. Because the world of nonlinear models is so va st, much attention has been devoted to particular families of models, which have been found to perform well in a range of applications. Time delayed embedding is one such technique that has been applied to a variety of physical applications in the domains of physiology [53], economics [12, 13, 40], geophysics [9, 24, 36, 44,
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3 50, 51] and engineering applications [4, 42]. The Time Series Data Mining methodology [41] is based on a variant of time delayed embedding called the reconstruction of phase space 1.4 Time Series Data Mining The Time Series Data Mining (TSDM) methodology [41], proposed by Richard Povinelli follows the time delayed embedding process to predict future occurrences of important events. TSDM framework combines the methods of phase space re construction and data mining to reveal hidden patterns predictive of future events in nonlinear, nonstationary time series. TSDM has its theoretical justification in the theory of nonlinear dynamics, specifically the Takens Embedding Theorem and Sauers T heorem [41]. The TSDM methodology is explained in detail in Chapter 3. The general block diagram of TSDM methodology is shown in Figure 1.1. Define TSDM goal Reconstruct phase space Define criteria for cluster selection and cluster size Search phase space for optimal temporal cluster pattern cluster Evaluate training stage results TRAINING STAGE Observed time series Embed time series into phase spac e Predict events Testing time series TESTING STAGE Figure 1.1 Block Diagram of TSDM Methodology [41]
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4 TSDM methodology has been applie d to the areas of earthquake prediction based on seismic activity, stock market predictions and to manufacturing applications such as predicting release of welding droplets [41]. Phase space reconstruction provides a simplified, multidimensional representa tion of a nonlinear time series that facilitates further analysis. An example of nonlinear time series and its reconstructed phase space is shown in Figures 1.2 and 1.3 respectively. The details of phase space reconstruction and the calculation of embeddi ng parameters are explained later. Figure 1.2 Time Series Representing Daily Discharge of Suwanee River 1.5 Flood Forecasting Using Time Series Data Mining Motivation Nonlinear approaches such as the HMM, ANN and NLP have been applied to the area of f lood forecasting, with the prediction results decreasing in accuracy as the prediction periods exceed a day. Moreover, HMM, ANN and NLP are useful in the forecasting of all future values, and their accuracy is measured over all forecasted values and not fo r the accuracy of predicting
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5 events such as floods. The prediction of floods requires a technique that can predict events (floods) in particular. Figure 1.3 Reconstructed Phase Space with t = 1 in Two Dimensions This is where the event based data minin g approach of TSDM is useful because it focuses on the prediction of events (floods). The prediction accuracy can then be measured by how successful the methodology is in predicting an event (flood) rather than the whole series of forecasted values. The TS DM methodology will be applied in this research to flood prediction at the three gauging stations and, the accuracy in predicting floods over short and long prediction periods will be measured. Chapter 2 presents a literature review enlisting previous wor k done in discharge forecasting, flood prediction, time series predictions, and nonlinear prediction techniques. In Chapter 3 the TSDM methodology is reviewed. In Chapter 4 TSDM is applied to flood prediction problem. Three different gauging stations are considered and the results are presented. Chapter 5 includes the conclusion of this research and the directions for future research.
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6 Chapter 2. Literature Review This chapter includes an introduction to presence of chaos in geophysical systems, followe d by the classification and a brief review of existing flood prediction techniques along with their advantages and disadvantages. This is followed by a review of application of Non Linear Time Series Analysis to flood forecasting. 2.1 Presence of Chaos in Geophysical Systems It is necessary to confirm the presence of chaos in any system before applying techniques based on chaos theory. It has been observed that the application of chaos theory based methods to systems that are not chaotic may produce wrong results [25]. Additionally, if there is no proof of existence of chaos, other methods targeted towards deterministic or stochastic time series analysis can be applied with greater success. Sivakumar [44] presents a detailed literature review on applicati on of Chaos Theory in geophysics with applications to rainfall, river flow, rainfall runoff, sediment transport, temperature, pressure, wind velocity, wave amplitudes, sunshine duration and tree rings. M.N. Islam and B. Sivakumar [23] have performed a comp arative study of existing techniques to determine the presence of chaos in hydrological systems. Each technique has its own advantages and disadvantages and no single method guarantees a correct classification of a system as being chaotic or non chaotic. A lthough proof for existence of chaos in different river flow time series has been illustrated in [23, 25, 36, 50, 51], its presence in the river discharge time series needs to be confirmed for the three examples considered.
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7 2.2 Existing Flood Forecastin g Techniques In recent years, numerous studies from varied fields of hydrodynamics, civil engineering, statistics and data mining have contributed to the area of flood prediction. Some of the existing techniques used in flood prediction are: 1. Stream a nd Rain Gauge Networks and Hydrograph Analysis 2. Radar and Information Systems 3. Linear Statistical Models and 4. Nonlinear Time Series Analysis and Prediction. The first three techniques mentioned above use more than one input parameter (multivariate) for cha racterization and prediction of floods. For example one of the linear statistical models uses the flood discharge, weighted flood discharge, precipitation intensity, elevations, stream length, and main channel slope for flood prediction. Some of the nonli near time series approaches such as Hidden Markov Models [3] and Artificial Neural Networks [5] are also based on multiple time series. Nonlinear Prediction (NLP) [28] method developed by Farmer and Sidorowich [14] has been used in river discharge forecast ing by Porporato et al [36 38]. Researchers have experimented with the application of NLP to discharge forecasting based on both single variable time series [45 46] and, multiple variable time series [8, 37]. Laio et al [28] have performed a comparison of ANN and NLP approaches in daily discharge forecasting. The results have shown that the NLP method provides accurate forecasts over a shorter prediction period (1 6 hours), but over prediction periods exceeding 24 hours, the ANN approach is more accurate. However, for periods exceeding a day, the HMM, ANN and NLP methods lose their accuracy. Moreover, the HMM, ANN and NLP techniques are not adequate for event predictions for reasons explained in Section 1.5. These approaches are described in the following s ections.
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8 2.2.1 Stream and Rain Gauge Networks and Hydrograph Analysis River flows and precipitation volumes are measured and monitored by more than 7,300 gauge stations operated mainly by USGS out of which about 4,200 are telemetries by an earth satellite based communication system. There also exist more than 14,000 rain gauges operated by NWS [34]. Data from these two recording systems acts as an input to statistical hydraulic models that estimate the possible river stage and discharge that may result. Th e models often turn out to be inaccurate because they are built using historic data that hasnt been recorded for more than the past 25 years and changes in topography as a result of rapid urbanization. Since hydraulic models cannot predict exactly what wi ll happen to the river, rating curves or hydrographs that show the relationship between water flows and water levels are used simultaneously [34]. The disadvantage of using a hydrograph is that it does not consider the changes in river cross sections that result from changes in channel bed and as a result, the stage and discharge relationship is altered [34]. 2.2.2 Radar and Information Systems With the development of Advanced Hydrological Prediction Services (AHPS) and NEXRAD by the NWS, the Doppler Rada r and the Geographic Information System (GIS) is being used along with the traditional hydraulic models for improved flood forecasting [34]. This is complex system comprising of simulation programs and uses data from various sources such as telemetry, auto mated gauges to calculate runoff, infiltration and precipitation volumes using land use and elevation information. AHPS has been found to result in flood forecasts that are 20% more accurate than the stream and rain gauge analysis. NWS has implemented 478 AHPS forecast points by the year 2004 with a one time fee of $2.1 million and $300,000 annually for maintenance [34]. The major disadvantages of AHPS are the complexity and the high cost of implementation and maintenance.
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9 2.2.3 Linear Statistical Models L inear Statistical Models such as Autocorrelation functions Spectral Analysis Analysis of cross correlations Linear Regression and Autoregressive Integrated Moving Average (ARIMA) have been studied for the applicability to flood forecasting. Solomatine e t al. [46] have found in their study that the use of stationary (ARMA ) as well as non stationary (ARIMA) versions of linear prediction techniques does not provide accurate predictions. Application of other linear stochastic methods has also resulted in ina ccurate predictions, clearly indicating that linear statistical models do not accurately represent historical data and hence are not acceptable methods for a non linear application such as flood forecasting [46]. 2.3 Nonlinear Time Series Analysis and Pre diction Applied to Flood Forecasting 2.3.1 Hidden Markov Models The concept of the state of a system is powerful even for nondeterministic systems. A Markov chain consists of a continuous range of flow values and, given the transition probability of movin g from one state to another, will predict the most probable future state based on the current state. The objective of using Hidden Markov Models to predict floods is to provide a simpler, generalized data mining model that could be reused for various geogr aphical areas, in which independence of predictions could be obtained with minimal consideration of past events. Most of the applications of Hidden Markov Models in flood forecasting have used the spatio temporal approach, whereas the time series p rediction used in this research is a purely temporal approach [3, 29]. The drawbacks of this approach are that the initial structure of the Markov model may not be certain at the time of model construction and it is very difficult to change the transition probabilities as the model itself changes with time. It was also observed that the Hidden Markov Models have a higher error for longer prediction periods as well as for
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10 prediction of events with sudden occurrences such as flash floods, leading to the concl usion that Hidden Markov Models do not perform better than other data mining techniques [3]. 2.3.2 Artificial Neural Networks Artificial Neural Networks (ANN) are widely accepted as a potentially useful way of modeling complex nonlinear and dynamic syste ms. They are particularly useful in situations where the underlying physical process relationships are not fully understood or where the nature of the event being modeled (i.e. a flood) may display chaotic properties. Though neural networks do not remove t he need for knowledge or prior information about the systems of interest, they reduce the model's reliance on this prior information. This removes the need for an exact specification of the precise functional form of the relationship that the model seeks t o represent. Artificial Neural Networks represent input output connectionist models where different factors such as temperature, precipitation, flow rate, depth etc are provided as input to the model. This technique has been used by a number of researche rs in a variety of geophysical phenomena from predicting currents in sea to flood prediction [5, 10 11, 20, 22, 52 53]. There is however, no set algorithm that can be applied to ensure that the network will always yield an optimal solution as opposed to a local minimum value, and the nonlinear nature of the ANN often results in multiple predicted values [27]. 2.3.3 Non Linear Prediction Method Non Linear Prediction (NLP) method was developed by J. Doyne Farmer and John J. Sidorowich [14] and was subsequent ly used in flood prediction by Porporato et al [36 38]. Researchers have experimented with flood prediction based on single parameter (river flow) time series [45 46] and multiple variable time series as well [8, 37]. The first steps of NLP are same as tha t of Time Series Data Mining (TSDM) methodology used in this research, starting with the
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11 reconstruction of phase space from the measured time series of the variable to be forecast. The phase space reconstruction and Takens theorem which provides theoretica l justification for phase space reconstruction are explained next. Phase Space Reconstruction A ttractors are the states towards which a system evolves when starting from certain initial conditions Since the dynamic of the system is unknown, the original t heoretical attractor that gives rise to the observed time series cannot be constructed. Instead, a phase space is created where the attractor is reconstructed from the scalar observed data that preserves the invariant characteristics of the original unknow n attractor described by the time delay method to approximate the state space from a single time series data. The reconstructed phase space is a Q dimensional metric space into which a time series is embedded. It is a vector space for the system such that specifying a point in this space specifies the state of the system and vice versa. Time delayed embedding maps a set of Q time series observations taken from X onto x t where x t is a vector or point in phase space. A time series is represented by { x t (Q 1 )t ,, x t 2t x t t x t } where x t represents the current observation, and ( x t (Q 1)t ,, x t 2 ,, x t t ) are the past observations. If t is the current time index, then t t is a time index in the past, and t+t is a time index in the future. The emb edding delay ( t ) is the time difference in number of time units between adjacent components of delay vectors. The embedding dimension ( m ) is the number of dimensions of reconstructed phase space that are required to achieve an embedding. Any further analys is of deterministic properties of a nonlinear time series depends on the precondition of a successful reconstruction of a state space of the underlying process [17]. There are many theorems based on time delayed embedding used in reconstruction of phase sp ace such as the Takens Embedding Theorem [44] and Whitneys embedding Theorem [46]. Other methodologies that are not based on the method of time delays are Filtered Embedding [40] that comprise of Principal Components [17] and Derivatives and Legendre coor dinates [17].
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12 Takens Embedding Theorem Takens theorem gives the theoretical justification for the reconstruction of phase space. It states that when a system has a state space M that is Q dimensional, M M : j be a map that describes the dy namics of the system, and R M y : its twice differentiable function (function that is differentiable and has a differentiable derivative), which represents the observation of a single state variable. Then the map ? (f, y): M ?R 2Q+1 defined by ( ) ( ) ( ) ( ) ( ) ( ) x y x y x y x Q y 2 ) ( ,..., ) ( j j j = F ( 2. 1) is an embedding which is a mapping that retains the structure of the original attractor from one topological space to another. Takens Theorem guarantees that the reconst ructed dynamics are topologically identical to the true dynamics of the system. According to the Takens Embedding Theorem, the selection of any value for the delay (t) will result in an embedding, given the fact that the data is infinitesimally accurate an d does not contain any noise [17, 23]. The data collected from naturally occurring dynamical systems hardly matches these specifications. In regards to the embedding dimension ( m ), an accurate value of m is only required when we want to exploit determinis m with minimal computational effort. If an m dimensional embedding yields a faithful representation of state space, every m ` dimensional reconstruction with m`>m does so as well [23]. Selecting a large value of m for chaotic data adds redundancy, degrades the performance of prediction algorithms and increases the computation time [23]. Many algorithms have been proposed that provide an estimation of optimal embedding dimension and time delay. Some of the methods for estimation of embedding dimension are t he method of False Nearest Neighbors [24], The Fillfactor algorithm [6, 7], and The Integral Local Deformation algorithm [6, 7]. Selection of method for estimation of m and t is largely dependant on the application and the kind of analysis that is to be performed.
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13 Methods of Estimating t (Time Delay) A variety of methods have been proposed for approximation of ideal time delay to ensure an embedding. Some of these techniques are autocorrelation, power spectrum functions average mutual information (AMI) function degree of separation function and Lyapunov exponents [1, 17, 25]. In this research, the AMI [16] and the autocorrelation function [25] have been used to approximate the time delay. For the AMI method, first, a histogram is created from the existing time series for determining the probability distribution of the data. p i is the probability that the signal assumes a value inside the i th bin of the histogram, and p ij (t) is the probability that s(t) is in bin I and s(t + t) is in bin j The AMI function is based on Shannons entropy [16], which is a measure of uncertainty and is represented by the expression ( ) ( ) ( ) i i i ij j i ij p p p p I ln 2 ln = t t t ( 2. 2) The first minimum o f I (t) marks the delay where a (t=t) adds maximum information to the knowledge we have from s (t) i.e. the redundancy is minimum. However, the AMI function is only applicable to two dimensional embeddings [25]. The AMI function generated from the Suwane e daily water level recording time series is shown in Figure 2.1. Average Mutual Information Function 0.0000 0.2000 0.4000 0.6000 0.8000 1.0000 1.2000 1.4000 1.6000 1.8000 1 159 317 475 633 791 949 1107 1265 1423 1581 1739 1897 Number of Observations AMI Fuction Value Figure 2.1 Average Mutual Information Function for Suwanee River Time Series
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14 Autocorrelation function is a method used to find delay dimension for higher dimensional embeddings. Given a ti me series x t t = 0, 1, ( N 1), its correlation can be measured by checking whether the product of x t and x t+t for a given delay t differs significantly from zero on average over the whole time series [17]. The autocorrelation function is given by ( ) t t t t + = = t N t t x x N A 1 0 1 1 (2. 3) In principle the autocorrelation function should be zero at all lags equal or lar ger than t, however se lecting a delay higher than t lead s to completely uncorrelated elements in the phase space, which is undesirable. Therefore, a reasonable choice of delay is the time index where the autocorrelation function decays to zero or first min imum [25]. The accurate calculation of time delay is important for the NLP technique since it tries to exploit the predictive structure of the phase space. Method for Estimating the Embedding Dimension (m) False nearest neighbors [26] The Fillfactor Algo rithm [6] Integral Local Deformation Algorithm [6] are some of the methods used for estimating the phase space dimension, with the most popular being the method of false nearest neighbors. The basic concept behind the method of false nearest neighbors is that if a reconstructed phase space does not have enough dimensions, it cannot be described as an embedding. Suppose a phase space has been reconstructed with dimension m If this m is less than the optimal value, the points which in true state space are s eparated by large distances will be mapped as neighbors in the reconstructed phase space and hence are called false nearest neighbors. The False Nearest Neighbor algorithm by Kennel et al. [26] identifies the points that move the farthest in terms of Eucli dean distance when plotted in a dimension higher than the present one and gives the percentage of false nearest neighbors. As we move into higher dimensions, the attractor is unfolded and, a true embedding is achieved when all
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15 the false nearest neighbors a re removed. An example showing the reduction in fraction of false nearest neighbors with the increase in embedding dimension is shown in Figure 2.2. Results of False Nearest Neighbor Algorithm 0.00E+00 1.00E01 2.00E01 3.00E01 4.00E01 5.00E01 6.00E01 1 2 3 4 5 Number of Dimensions (m) Fraction of False nearest Neighbors Figure 2.2 Plot Showing Relationship Between m and Percentage of False Nearest Neighbors Once an embedd ing is achieved, the next step for prediction is to estimate a suitable function that is able to project the present state of the system, x (t) into the forecasted value x (t + t) at future time t+ t. The function is presented in Equation 2. 4. x F (t + t) = F (x (t)) ( 2. 4) The next step is to identify the sets into which the constructed domain can be subdivided. The number of mutually overlapping sets is taken to be the equal to the total number of points to be forecaste d; each set containing number n of vectors whose Euclidian distance from x (t) is the minimum. n is the number of neighbors and when a suitable number of neighbors are chosen the model can be completely identified [28]. When a linear relationship is chosen for the function F the model assumes the form as given in Equation 2. 5 x F (t + t) = 0 1 ) ( ) ( k t a k i i m i L tot + = ( 2. 5) where a i (t) are the components of x(t), k 0 is the bias and (k L ) i are weights to be found using information from the n nearest neighbors of x(t) The weights are different for each forecasted
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16 point because each po int lies in a different neighborhood in which weights are calibrated, thus ensuring the nonlinearity of the predicted times series. F. Laio et al. [28] have performed a comparison of ANN and NLP approaches in flood prediction. The results have shown that the NLP method provides more accurate forecasts over a shorter prediction period (1 6 hours), however for prediction periods exceeding 24 hours, the ANN approach is more accurate. Chapter 3 reviews the TSDM methodology.
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17 Chapter 3. Time Series Dat a Mining Methodology This chapter reviews the Time Series Data Mining methodology proposed by Richard Povinelli [41]. 3.1 Time Series Data Mining A primary difference between ANN, HMM and NLP and Richard Povinellis Time Series Data Mining (TSDM) is th at the focus of TSDM is to identify and predict the occurrences of events, which in our case are floods whereas, the methods mentioned in the sections above focus on forecasting all future values of a time series. The TSDM methodology in this research is b ased on a univariate river flow time series. The steps in TSDM methodology are: 1. Training Stage 1. Reconstruct the phase space from the observed time series by using the method of delays. 2. Frame TSDM goal in terms of event characterization function, objective function and optimization formulation. 1. Define the event characterization function g 2. Define the objective function f. 3. Define the optimization formulation, the constraints on the objective function. 4. Associate eventness with each time index represented by th e event characterization function. Create the augmented phase space.
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18 5. Search for optimal temporal pattern cluster in the augmented phase space that best characterizes the events. 6. Evaluate training stage results and repeat training stage as necessary. 2. Testin g Stage 1. Embed the testing time series into phase space. 2. Use optimal temporal pattern cluster for predicting events. 3. Evaluate testing results. These steps are explained in detail in the following sections. The steps in training stage are elaborated in Sect ion 3.2. Testing stage steps are explained in Section 3.3. 3.2 Training Stage 3.2.1 Step 1. Reconstruct the Phase Space Reconstruct the attractor in a two dimensional phase space using time delay of one. The accurate calculation of time delay and number of embedding dimensions is not a requirement in TSDM. The reason is that unlike the NLP, TSDM does not try to exploit the predictive structure of the reconstructed phase space. The only purpose of a phase space reconstruction is to provide a simplified r epresentation of the nonlinear time series. The identification of predictive patterns is accomplished by the data mining part of the TSDM methodology. Hence, the selection of time delay and embedding dimension is more a matter of choice. As an example, a synthetic seismic time series and its reconstructed phase space is shown in Figure 3.1 and 3.2 respectively.
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19 Figure 3.1 Synthetic Seismic Time Series (Training) with Events Identified [41] Figure 3.2 Reconstructed Phase Space f rom Synthetic Seismic T ime Series [41] 3.2.2 Step 2. a) Define the Event Characterization Function After an embedding has been achieved, the next step is to define a function that helps identify the events of interest and projects them in the phase space so as to identify the t emporal patterns and temporal pattern clusters. The event characterization function is an application dependent function, defined in such a way that value that represents the equation at time t correlates to the value of that event in the future. These typ es of event characterization functions are classified as causal and are useful in prediction type problems. The other type of event characterization
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20 function is the non causal type and is useful for system identification and classification problems. For ex ample, an event characterization function g (t) = x t+1 is useful in predicting an event one step before it actually occurs. Another example is ( ) t t t x x x t g = + 1 ( 3. 1) g (t) representing the percentage change of values for time t to time t +1 The g value is calculated for each phase space point and is called eventness The choice of event characterization function has a significant effect on the prediction results. 3.2.2 Step 2. b) Define the Objective Function The objective function is us ed to determine which temporal pattern cluster is efficient in its ability to characterize events and is consistent with the TSDM goal. The objective function calculates a value for temporal pattern cluster P which provides an ordering to temporal pattern clusters according to their ability to characterize events. The number of temporal pattern clusters could be one or more. Before finding the objective function, some basic definitions about average eventness variances are necessary. The index set ? is a set of all time indices t of phase space points described by ( ) { } 1,...,N t 1 Q t:t ? + = = ( 3. 2) where ( Q 1) t is the largest embedding time delay, and N is the number of observations in the time series. The time index set M is the set of al l time indices t when x t is within the temporal pattern cluster ( P ), i.e. { } L = t P x t M t : The average value of g also called average eventness of the phase space points within the temporal cluster P is given by ( ) ( ) = M t M t g M c 1 m ( 3. 3 )
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21 where ( ) M c is the cardinality of M Cardinality of M represents the number of points lying inside the temporal cluster. The average eventness of the phase space points of phase space points not in P is ( ) ( ) = M t M t g M c ~ ~ ~ 1 m ( 3. 4 ) where ( ) M c ~ is the number of points outside the temporal cluster. The average eventness of all the phase space points is given by ( ) ( ) L L = t x t g c 1 m ( 3. 5) The corresponding variances for respective average eventness functions a re given in Equations 3. 6, 3. 7 and 3. 8 ( ) ( ) ( ) 2 2 1 = M t M M t g M c m s ( 3. 6) ( ) ( ) ( ) 2 ~ ~ 2 ~ ~ 1 = M t M M t g M c m s ( 3. 7) ( ) ( ) ( ) 2 2 1 L L = t x x t g c m s ( 3. 8) From these definitions, many different objective functions can be created for the purpose of selecting one tempora l pattern cluster over the other. One example is the t test for the difference between two means of points inside and outside the cluster, shown in Equation 3. 9. ( ) ( ) ( ) M c M c P f M M M M ~ 2 ~ 2 ~ s s m m + = ( 3. 9) where P is a temporal pattern cluster. This function can be used for identifying a single statistically significant cluster that has high average eventness. For example, in Figure 3.3, cluster
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22 P 1 is the best collection for identifying events, whereas P 2 is not. The objective function as in Equation 3. 9 will map the collection of temporal pattern clusters such that f (C 1 ) > f (C 2 ). For identifying a single temporal pattern cluster where the purpose is to minimize the false predictions, an objective function of the type fp tp tp P f + = ) ( ( 3. 10) is useful, where values of tp tn fp and fn are shown in Table 3.1. However this type of objective function is only applicable to problems where the time series observations can be classified as tp tn fp and fn Table 3.1 Event Categoriz ation Actually an event Actually a nonevent Categorized as an event True positive, tp False positive, fp Categorized as a nonevent False negative, fn True negative, tn When the accuracy of prediction is of primary importance, the efficacy of a collect ion of temporal pattern clusters in total prediction accuracy is given by fn fp tn tp tn tp C f + + + + = ) ( ( 3. 11) where C is the collection of temporal pattern clusters. The category of TSDM problems, where the exact count of events and their magnitude are kno wn, an event characterization function of the type If c(M)/c(?) = ( ) ( ) ( ) + L = 0 0 ) ( g c M c g P f M M b m m otherwise ( 3. 12)
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23 where is the minimum proportion of points inside the cluster. And g 0 is the minimum eventness of the phase space. This objective function o rders temporal pattern clusters on the basis of time series observations with high eventness and characterizes at least a minimum number of events. The selection of objective function depends on the goal of TSDM. 3.2.2 Step 2. c) Define Optimization Formu lation It is possible that different temporal pattern clusters within the augmented phase space can contain the same set of phase space points. The optimization formulation becomes important in this case to determine the optimal size of cluster by maximizi ng the objective function f (P). Three types of biases, minimize, maximize or moderate can be possibly placed on d, the radius of the temporal pattern cluster hypersphere Formulation by minimizing the d subject to f (P) remaining constant can be used to minimize the false positive prediction errors, i.e. the error of classifying a non event as an event. This ensures that the temporal pattern cluster has as small coverage as possible, keeping the value of objective function constant. 3.2.3 Step 3. Create Augmented Phase Space The augmented phase space is a Q+1 dimensional phase space that is created by adding one more dimension to the existing phase space. The additional dimension is represented by the event characterization function g (.) in a way that e very phase space point is a vector, as shown in Figure 3.3. The augmented phase space is expressed in Equation 3. 13 given below ( ) 1 + Q t R t g x ( 3. 13) The next step is to be able to identify the optimal temporal cluster in the augmented phase sp ace and this requires the formulation of an objective function.
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24 Figure 3.3 Clus ters Identified in Phase Space f rom Synthetic Earthquake Time Series [41] 3.2.4 Step 4. Search for Optimal Temporal Patter Cluster The search for optimal temporal pattern clu ster is performed using the Genetic Algorithm [18]. Figure 3.4 shows the temporal pattern cluster identified by the Genetic Algorithm. Figure 3.4 Synthetic Seismic Phase Space with Temporal Pattern Cluster Identified [41]
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25 3.2.5 St ep 5. Evaluate Training Results From Figure 3.5 it is clear that all temporal patterns occurring before the events have been identified and thus training stage is successful. Figure 3.5 Synthetic Seismic Time Series with Temporal Patterns Highlighted [41] 3.3 Testing Stage 3 .3.1 Step 1. Embed The Testing Time Series Into Phase Space Embed the testing time series in phase space using the same embedding parameters as the training time series. 3.3.2 Step 2. Use The Optimal Temporal Pattern Cluster to Predict Events Whenever a point in the testing time series phase space falls inside the cluster identified in the training stage, an event is predicted.
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26 3.3.3 Step 3. Evaluate Testing Results Testing results are evaluated by measuring the number of events correctly identifie d and predicted. Figure 3.6 shows a testing time series and the events predicted. Figure 3.6 Prediction Results in a Testing Time Series [41] In Chapter 4, the TSDM is applied to flood prediction at the three gauging stations and results are presented.
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27 Chapter 4. Application of TSDM to Flood Forecasting and Results This chapter describes the application of TSDM to flood prediction at three gauging stations. Section 4.1 describes the Surrogate Data method for detection of chaos in the ri ver discharge time series. In Section 4. 2 the parameters for phase space reconstruction are explained. In Section 4. 3 the event characterization function, objective function and the optimization formulation are explained. Section 4. 4 describes the measures for cluster prediction accuracy This is followed by description of problem and setup in Section 4.5 Section 4.6 presents the results using two objective functions and the Earliness Prediction Accuracy. A summary of results is presented in Section 4.7. 4.1 Detection of Chaos in the River Daily Discharge Time Series Before the TSDM methodology is applied to flood forecasting, the presence of nonlinearity in the river discharge time series at the gauging stations needs to be confirmed. For this, the Surrog ate data method [25], a technique for detecting the presence of chaos is used. Surrogate data method consists of computing a nonlinear statistic from the data being analyzed and from an ensemble of realizations of a linear stochastic process, which mimics linear properties of the data under study. If the computed statistic for the original data is significantly different from the values obtained for the surrogate data set, it can be inferred that the data was not generated by a linear process; otherwise the null hypothesis, that a linear model fully explains the data, cannot be rejected and the data can be analyzed, characterized and predicted using well developed linear methods [31]. The TISEAN 2.1 software for Nonlinear Time Series Analysis [25] is used he re for
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28 detecting chaos in river daily discharge time series. The null hypothes is and alternate hypothesis are: H o : Daily discharge time Series is linear and stationary Root Mean Square Error for daily discharge time series = Root Mean Square Prediction Err or for its surrogate data H a : Daily Discharge Time Series is non linear and non stationary Root Mean Square Prediction Error for daily discharge time series > Root Mean Square Prediction Error for its surrogate data Level of Significance : a = 0.05 Test St atistic : Difference between Root Mean Square Prediction Error for Daily Discharge Time Series and its Surrogate Data All three discharge time series were tested for presence of chaos. Using the surrogate data testing, the null hypothesis of a stationary, l inear Gaussian random process was rejected at the 95% level of significance, since the prediction error of the river discharge data is found to be significantly higher than that of the surrogates. This proves that the daily discharge time series observed a t all three gauging stations are chaotic and justify the use of TSDM in their analyses. 4.2 Reconstruction of Phase Space f rom Training Time Series As explained in Section 3.2, the TSDM methodology does not require accurate estimates of time delay and embeddi ng dimension. The choice of time delay and number of dimensions does not have any effect on the prediction results of TSDM methodology. The phase space is reconstructed in two dimensions with a time delay of one for all three examples considered.
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29 4.3 E vent Characterization Function Objective Function and Optimization 4.3.1 Event Characterization Function Event characterization function represents the value of future eventness for the present time index. Different event characterization functions can be used to model the prediction of flood s The event characterization functions considered are: 1. ( ) t t t t x x x x g = + 1 : percentage change in discharge values between two consecutive days 2. ( ) 1 + = t t x x g :captures the goal of characterizing a flood one time step in future 3. ( ) i t t x x g + = :captures the goal of characterizing a flood i time steps in future It is found that the percentage change function could not highlight the events of interest in the augmented phase space. During the evaluati on of training stage results, the percentage change function was able to identify only 4 out of 15 events in the training time series from St. Louis gauging station. This indicates that a high rise in the discharge values over two successive days does not necessarily result in a flood. For example, if a heavy rainfall follows a period of draught, the discharge will increase rapidly over a period of days. These rapid increases in discharge will be highlighted in the augmented phase space although they may no t lead to a flood. On the other hand, if the discharge increases gradually and results in a flood, it would not be highlighted by the percentage change function since the changes are not drastic. The event characterization function is chosen depending o n how many days early the flood is to be predicted. The one step ahead event characterization function is useful for prediction of floods one day early. Different step ahead event characterization functions are applied and the effect of earliness of predic tion on the prediction accuracy is measured.
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30 4.3.2 Objective Function The differe nce of means objective function, ( ) ( ) ( ) M c M c P f M M M M ~ 2 ~ 2 ~ s s m m + = explained in Chapter 3 is not applicable to the flood prediction problem because of the linearly increasing natur e of the phase space as shown in Figure 4.1. The cluster, in order to maximize the difference between the g values of points inside and outside the cluster, includes all the points in the phase space, resulting in a large number of false alarms in the test ing time series. There is no upper limit on the size of the cluster. The difference of means objective function is only applicable to phase spaces that are disjoint or have multiple clusters, where the objective function selects one cluster over the other based on the ) ( P f value. Figure 4.1 2 D Embedding of Training Time Series from St. Louis Gauging Station
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31 Two objective functions, both of which are variations of the objective function that selects at least a minimum proportion of p oints in the phase space, are considered The objective function that selects a minimum proportion of points in phase space is If c(M)/c(?) = ( ) ( ) ( ) + L = 0 0 ) ( g c M c g P f M M b m m otherwise ( 4. 1) where c(M) is the cardinality of the cluster and c(? ) is the total number of points in the phase space. specifies the minimum proportion of points that the cluster can include. The objective function value is M m which is the mean of g values of points inside the cluster, if the cluster includes the minimum proportion of points Otherwise the objective function value is ( ) ( ) ( ) 0 0 g c M c g M + L b m which is lesser than M m This objec tive function tries to maximize the mean of g values of points inside the cluster. As in case of the difference of means function, t his event characterization function cannot be applied to the flood prediction problem because of the linearly increasing nat ure of phase space In order to maximize the mean of g values of points inside the cluster, the GA selects a cluster with a single point having the highest g value in the phase space. Every other point added to the cluster reduces the cluster mean and hen ce the GA terminates with only one point in the cluster and a radius of zero. There is no lower limit on the size of the cluster. In order to address this problem, two variations of the objective function for minimum proportion of points a re proposed. Obje ctive Function I Objective Function I is applicable to flood prediction problems where the history of floods is known. The minimum discharge value that results in a flood can be identified and, based on this value, each event can be classified as a true po sitive or a false positive according to the following definitions:
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32 1. tp (True Positives): If the event identified by the cluster as a flood is actually a flood it is called a true positive. 2. fp (False Positive): If an event identified by the cluster is not a flood it constitutes a false positive or a false alarm. Objective Function I is formally stated as T he Objective Function I uses the knowledge learnt from historical data. Instead o f specifying the minimum proportion of points to be included in the cluster as in Equation 4. 1, this objective function specifies the maximum proportion of false positives allowed in the cluster. Another difference is that the objective function tries to m aximize the product of sum of g values and the number of tp s. For every tp included, the ) ( P f function is rewarded by multiplying the summation of g values of points inside the cluster by the number of tp s inside the cluster. As a result, the cluster tries to include all true positives along with other phase space points with high g values, however, its size is restricted by the maximum proportion of fp s ( ). Objective Function II Objective Function II is applicable to the flood forecasting problems where no information is available regarding previous floods. The lack of information means that there is no way to classify an event as a tp or a fp In this ca se, another criteria which is based on flood zoning is used to train the cluster. Areas susceptible to floods are classified into 1 0 year, 50 year, 100 year flood zones Suppose an area has been classified as 100 year flood zone It means that there is a 1% probability of a flood occurring in that area in a ny given year. From this probability, the expected If b + p p p t f f = = = M i M M i M g g tp P f 1 1 ) ( otherwise ( 4. 2)
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33 number of floods for the period under study can be calculated. The Objective Function II uses the ratio of expected number of days of flooding to the to tal number of observations in the training time series for limit ing the cluster size. The Objective Function II is presented below If c(M)/c(?) = ? = = = M i M M i M g g P f 1 1 ) ( otherwise where, ? is the maximum proportion of phase space points inside the cl uster. If the proportion of phase space points inside the cluster is less than ? the objective function value is the positive summation of g values inside the cluster. If the proportion of phase space points inside the cluster exceeds ? the objective fu nction value is negative summation of g values inside the cluster. The goal of this objective function is to select the points with high g values to be a part of cluster in order to maximize the summation of g values. The objective function is maximum when all the phase space points lie inside the cluster, however the cluster size is restricted by ?, which the proportion of points based on the flood zone and the length (in number of days or observations) of the training time series. 4.3 .3 Optimization Form ulation The selection of optimal cluster size is accomplished with the use of optimization formulation. The o ptimization formulation is modeled as a multiobjective formulation with the two objectives as: 1. Maximize the value of objective function and 2. Minimiz e the radius of the cluster. An unsupervised clustering technique, the Genetic Algorithm [18] is used in the search process for optimal temporal pattern cluster. The Genetic Algorithm (GA) searches for a global maxima and, identifies the optimal cluster. A s explained in Section 3.2.2, priorities are assigned
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34 to the two objectives, with the maximization of objective function being the first priority and minimization of radius is the secondary priority. Thus, the GA will search for a temporal pattern cluster that maximizes the objective function value and if there are multiple clusters with equally high objective function value, it selects the cluster with minimum radius. The minimization objective is required to select the crispest cluster in order to minimiz e the number of false positives in the testing phase. The Genetic Algorithm Toolbox in Matlab, version 7.0.1 (Release 14) is used for the search and the output is the cluster center and its radius. 4.4 Cluster Prediction Accuracy In order to determine t he prediction accuracy of the cluster and measure its ability to identify and characterize the floods in the training and testing phases, the following set of parameters is defined. 1. tp (True Positives): If the event identified by the cluster as a flood is actually a flood it is called a true positive. 2. fp (False Positive): If an event identified by the cluster is not a flood it constitutes a false positive or a false alarm. 3. a (Type I Error): In hypothesis testing, a is defined as the probability of rejectin g a null hypothesis when its true. Analogically, applied to cluster prediction accuracy, a is the probability of missing a true positive. 4. (Type II Error): is defined as the probability of failing to reject the null hypothesis even though its false. Ap plied to cluster prediction accuracy, is the probability of selecting a false positive. 5. Positive P rediction A ccuracy : Positive Prediction A ccuracy (PPA) [41] is the p ercentage of true positives in the cluster. Since the events are classified either as t rue positives or
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35 false positives, the positive prediction accuracy of a cluster is calculated as 100 + fp tp tp PPA can also be calculated as ( ) 100 1 b since is fp tp fp + ( ) b 1 is the probability of selecting a true positive. 6. Correct Prediction Percentage : Correct Prediction Percentage (CPP) is defined as the percentage of true positives predicted. Correct Prediction Percentage = ) ( ) ( actual tp predicted tp The cluster having a high PPA as well as high CPP is the crispest cluster with maximum prediction accuracy. 7. Number of Start s Missed : Since the goal is to predict occurrences of floods, it is more important to predict the first instance when the discharge exceeded the flood threshold, causing the rive r to overflow, than to predict all events ( tp s) when the discharge exceeds the threshold. Hence, along with CPP measured with respect to number of tp s predicted, the number of starts of floods missed are also measured. 4.5 Problem Description and Setu p TSDM is applied to flood forecasting at three gauging stations. The Kansas City gauging station time series contains 3 floods, the St. Louis gauging station time series contains 6 floods and the Harrisburg gauging station time series contains 15 floods d uring the periods under study These three gauging stations are selected as examples because they represent low, medium and high number of flood occurrences. St. Louis Gauging Station, Mississippi River The daily discharge time series for St. Louis gaugin g station is obtained from the United States Geological Survey website [21], covering the time period from April 1933 to September 2003, consisting of 25750 data points. The daily discharge time series is shown in Figure 4.2
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36 Historical records show that f loods have occurred during the years 1943, 1944, 1947, 1973, 1993 and 1995. The events of interest are the peak dis charge values during the flooding periods Figure 4. 2 Time Series Plot of Daily Discharge Values at St. Louis Gauging Station The time se ries is split into two parts, the first 15000 data points making up the training time series and, the rest will be used in testing. TSDM is applied to the training data set and, the prediction accuracy is calculated by its ability to predict actual floods occurring in the testing time series. The minimum value of discharge, during the reported flooding periods is 780,000 cubic feet per second. This metric has been chosen as an identifier for floods, i.e. a value of discharge exceeding 780,000 cu. feet/sec results in a flood and hence constitutes an e vent. The 2 D phase space and the augmented phase space from the training time series are shown in F igures 4 .3 and 4.4 below. The temporal pattern clusters are identified from this time series and consequently u sed in the prediction of floods in the testing part.
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37 Figure 4.3 2 D Embedding of St. Louis Training Time Series Fi gure 4.4 Augmented Phase Space For Training Series for St. Louis
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38 Kansas City Gauging Station Missouri River The Missouri River that passe s through Kansas City, MO has flooded three times during the recorded history. These floods occurred during the years of 1951, 1952 and 1993. The daily discharge time series consists of 27393 observations [21], recorded between 10/1/1928 and 9/30/2003 as s hown in Figure 4. 5 The minimum discharge observed during the floods is 380,000 cubic feet per second and is chosen as the event identifier. The time series is split into the training and testing time series. The training time series has 13500 data points, contains two floods (1951 and 1952) and six events when the discharge exceeded 380,000 cubic feet per second The testing time series is comprised of 13893 data points, representing the time period between 16/9/1965 and 9/30/2003. One flood occurred duri ng this time period, during the year of 1993. The testing time series contains six events. Figure 4.5 Daily Discharge Time Series at Kansas City Gauging Station, Missouri River The 2 D phase space from the training time series and the augmented phase s pace are presented in Figures 4.6 and 4.7 below.
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39 Figure 4.6 2 D Embedding of Kansas City Training Time Series Figure 4.7 Augmented Phase Space For Training Series for Kansas City
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40 Harrisburg Gauging Station Susquehanna River The daily discharge time s eries at Harrisburg gauging station contains observations recorded between 10/01/1890 and 09/30/2003 (41272 data points) [21] Harrisburg is one of the highest flood hit areas of United States. Floods have occurred during the years of 1894, 1901, 1902, 192 0, 1936, 1940, 1943, 1950, 1964, 1972, 1975, 1980, 1985, 1994 and 1997. The time series is bisected into training and testing time series, each comprising of 20636 data points. The minimum value of discharge during flooding is 390,000 cubic feet per second This value is chosen as the identifier for flood events. The training time series covers time between 10/01/1890 to 04/01/1947, contains seven floods and 17 tp s. The testing time series contains eight floods and 14 tp s. The goal is to predict the eight floods in the testing time series accurately and as early as possible. The daily discharge time series is presented in Figure 4. 8 Figure 4.8 Daily Discharge Time Series at Harrisburg Gauging Station, Susquehanna River The 2 D phase space and the augmen ted phase space for the Harrisburg g auging s tation training time series are shown in Figure 4.9 and 4.10.
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41 Figure 4.9 2 D Phase Embedding of Harrisburg Training Time Series Figure 4.10 Augmented Phase Space for Harrisburg Training Series
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42 4.6 Results T his section presents the results from application of TSDM to flood forecasting. The goal is to predict floods accurately and as early as possible. In Section 4. 6 .1, the prediction accuracy is calculated for all three gauging stations using Objective Functi on I for different values of In Section 4. 6 .2, the Earliness Prediction Accuracy is evaluated for all three gauging stations using different step ahead event c haracterization f unctions. This is followed by application of Objective Function II to flood f orecasting at St. Louis gauging station. 4. 6 .1 Prediction Accuracy with Objective Function I This section presents the prediction accuracy for the three gauging stati ons using Objective Function I for different values. The following set of parameters is used : Event Characterization Function : One step ahead function ( ) 1 + = t t x x g Objective Function : Objective Function I Optimization Formulation : 1. Maximize the value of O bjective Function (Priority I) 2. Minimize the radius of cluster (Priority II) First, the training and testing results for St. Louis gauging station are presented followed by the results for Kansas City and Harrisburg gauging stations. Training an d Testing Results for St. Louis Gauging Station Training stage results for St. Louis gauging station are presented in Table 4.1. Objective Function I rewards the GA for including tp s in the cluster. Since the tp s are rewarded, the GA tries to include max imum possible tp s in the cluster. If b + p p p t f f = = = M i M M i M g g tp P f 1 1 ) ( otherwise
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43 Table 4. 1 Training Stage Results for Different Values at St. Louis Gauging Station Points PPA CPP Specified in Events ( tp /tp+fp) tp(predicted) Cluster tp fp Missed a or (1 ) tp(actual) 0.95 295 15 280 0 0 5.08 100 0.85 98 15 83 0 0 15.31 100 0.75 60 15 45 0 0 25.00 100 0. 65 42 15 27 0 0 35.71 100 0.55 33 15 18 0 0 45.45 100 0.45 27 15 12 0 0 55.56 100 0.35 23 15 8 0 0 65.22 100 0.25 20 15 5 0 0 75.00 100 0.15 17 15 2 0 0 88.24 100 0.05 No Cluster Identified For value until 0.15, the GA is able to identify clusters which include all tp s and hence have a 100% CPP and the a value is zero. No cluster was identified for value of 0.05 and below. The clusters identified in the training stage are used to predict floods in the testing stage. Clusters f or differen t values of from training stage are applied testing phase space and the resulting a , PPA and CPP values are shown in Table 4.2 Table 4.2 Testing St age Results for Different V alues at St. Louis Gauging Station Number of Floods in Testing Time Series= 2, Number of Events in Testing Time Series= 33 Points PPA CPP Number Spec. In in Events Resultant tp /tp+fp tp(pred.) of Starts Training Cluster tp fp Missed a or (1 ) tp(actual) Missed 0.95 272 12 260 21 0.64 0.96 4.41 36.36 0 0.85 90 20 70 13 0.39 0.78 22.22 60.61 0 0.75 53 17 36 16 0.48 0.68 32.08 51.52 0 0.65 26 12 14 21 0.64 0.54 46.15 36.36 0 0.55 18 14 4 19 0.58 0.22 77.78 42.42 0 0.45 18 15 3 18 0.55 0.17 83.33 45.45 0 0.35 22 20 2 13 0.39 0.09 90.91 60.61 0 0.25 11 10 1 23 0.70 0.09 90.91 30.30 0 0.15 11 10 1 23 0.70 0.09 90.91 30.30 0 0.05 No Cluster Identified Although the a value increases with a decrease in indicating that more events are being missed, none of the clusters miss the starts of floods. The Correct Prediction Percentage is highest
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44 for values of 0.85 and 0.35. From the testing time series results, it can b e seen that there is no clear relationship between decrease in value and the CPP. This happens because of the non uniform density of points in the phase space. If the density of points in phase space would have been uniform, a decrease in the value wou ld have meant a smaller cluster with less number of fp s and a lesser number of tp s as well. However, this is not the case as prediction a ccuracy also depends on the location of the cluster in the phase space as much as it does on the cluster size (contro lled by ). Training and Testing Results for Kansas City Gauging Station Using the same set of parameters as in St. Louis gauging station, the training stage results for Kansas City gauging station are presented in Table 4.3. Table 4.3 Training Time Series Result s for Different Values at Kansas City Gauging Station The testing time ser ies res ults are presented in Table 4.4 Points PPA CPP Specified in Events ( tp /tp+fp) tp(predicted) Cluster tp fp Missed a or (1 ) tp(actual) 0.95 97 5 92 1 0.17 5.15 83.33 0.85 24 5 19 1 0.17 20.83 83.33 0.75 11 5 6 1 0. 17 45.45 83.33 0.65 8 5 3 1 0.17 62.50 83.33 0.55 7 5 2 1 0.17 71.43 83.33 0.45 7 5 2 1 0.17 71.43 83.33 0.35 7 5 2 1 0.17 71.43 83.33 0.25 6 5 1 1 0.17 83.33 83.33 0.15 4 4 0 2 0.33 100.00 66.67 0.05 4 4 0 2 0.33 100.00 66.67
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45 Table 4. 4 Testing T ime Series Results for Different Values at Kansas City Gauging Station As in training time series there is no clear relationship between the and CP P be cause of the non uniform density of phase space points. The highest prediction accuracy is 100 % for = 0. 9 5 0.35 and 0.25 The clusters do not miss the starts of floods for any value of Training and Testing Results for Harrisburg Gauging Station The training stage results for Harrisburg gauging station are presented in Table 4.5. Table 4.5 Training Time Series Results for Different Values at Harrisburg Gauging Station The testing stage results are presented in Table 4.6 Number of Floods in Testing Time Series= 1, Number of Events in Testing Time Series= 6 Points PPA CPP Number Spec. In in Events Resultant tp /tp+fp tp(pred.) of Starts Training Cluster tp fp Missed a or (1 ) tp(actual) Missed 0.95 60 6 54 0 0.00 0.90 10.00 100.00 0 0.85 15 4 11 2 0.33 0.73 26.67 66.67 0 0.75 8 4 4 2 0.33 0.50 50.00 66.67 0 0.65 8 4 4 2 0.33 0.50 50.00 66.67 0 0.55 7 4 3 2 0.33 0.43 57.14 66.67 0 0.45 7 4 3 2 0.33 0.4 3 57.14 66.67 0 0.35 8 6 2 0 0.00 0.25 75.00 100.00 0 0.25 7 6 1 0 0.00 0.14 85.71 100.00 0 0.15 3 3 0 3 0.50 0.00 100.00 50.00 0 0.05 4 3 1 3 0.50 0.25 75.00 50.00 0 Points PPA CPP Specified in Events ( tp /tp+fp) tp(predicted) Cluster tp fp Missed a or (1 ) tp(actual) 0.95 229 13 216 4 0.24 5.68 76.47 0.85 60 1 3 47 4 0.24 21.67 76.47 0.75 45 13 32 4 0.24 28.89 76.47 0.65 24 12 12 5 0.29 50.00 70.59 0.55 21 10 11 7 0.41 47.62 58.82 0.45 18 11 7 6 0.35 61.11 64.71 0.35 13 9 4 8 0.47 69.23 52.94 0.25 14 11 3 6 0.35 78.57 64.71 0.15 9 8 1 9 0.53 88.89 47.06 0.05 6 6 0 11 0.65 100.00 35.29
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46 Table 4.6 Testing Time Series Results for Different Values at Harrisburg Gauging Station Number of Floods in Testing Time Series= 8 Number of Events in Testing Time Series= 14 Points PPA CPP Number Spec. In in Events Resultant tp /tp+fp tp(pred.) of Starts Training Cluster tp fp Missed a or (1 ) tp(actual) Missed 0.95 226 10 216 4 0.29 0.96 4.42 71.43 0 0.85 59 10 49 4 0.29 0.83 16 .95 71.43 0 0.75 51 10 41 4 0.29 0.80 19.61 71.43 0 0.65 19 8 11 6 0.43 0.58 42.11 57.14 2 0.55 18 9 9 5 0.36 0.50 50.00 64.29 1 0.45 17 8 9 6 0.43 0.53 47.06 57.14 2 0.35 14 9 5 5 0.36 0.36 64.29 64.29 1 0.25 12 8 4 6 0.43 0.33 66.67 57.14 1 0.15 8 7 1 7 0.50 0.13 87.50 50.00 3 0.05 7 7 0 7 0.50 0.00 100.00 50.00 3 Starting at = 0.65, the clusters determined from the training stage start missing floods. Again, as in the St. Louis and Kansas City examples, t he re is no clear relationship between and CPP in both the training and testing time series For a of 0.15 and below, t he clusters start missing out 3 floods out of the total 8 floods that occurred in the testing time series. 4.6.2 Earliness Prediction Accuracy An event is predicted when a phase space point from the testing time series falls inside the cluster from the tr aining time series. Since the goal is to predict the floods accurately and as early as possible, one performance measure is the E arliness Prediction A ccuracy of TSDM using different step ahead functions. For this, is kept constant and the prediction accuracy is measured in terms of tp, fp, a, PPA and CPP for different step ahead e vent c haracterization f unctions. The with highest CPP value in the testing stage using the Objective Function I is selected The o bjective function and optimization formulation are described on the next page
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47 Objective Function : Objective Function Type I Optimization Formulation : 1. Maximize the value of Object ive Function (Priority I) 2. Minimize the radius of cluster (Priority II) Earliness Prediction Accuracy for St. Louis Gauging Station The CPP in the testing time series is maximum for = 0.85 (Table 4.2) Using this value and the o bjective f unction and optimization f ormulation mentioned above, different event characterization f unctions are applied starting from the one step ahead function. The training stage results for St. Louis gau ging station are presented in Table 4.7 and F igure 4.11 From Table 4.7 and Figure 4.11 it is clear that the both the PPA and the CPP decrease as the prediction time horizon increases. Other indication of degradation of prediction accuracy with increasing prediction time horizon is the increase in a values. Table 4. 7 Training Earliness Prediction Accuracy for = 0.85 at St. Louis Gauging Station Event Points PPA CPP Char. in Events ( tp /tp+fp) tp(predicted) Func. Cluster tp fp Missed a or (1 ) tp(actual) x t+1 98 15 83 0 0 .00 0.85 15.31 100.00 x t+2 90 14 76 1 0.07 0.84 15.56 93.33 x t+3 81 14 67 1 0.07 0.83 17.28 93.33 x t+4 66 10 56 5 0.33 0.85 15.15 66.67 x t+5 43 7 36 8 0.53 0.84 16.28 46.67 x t+6 30 5 25 10 0.67 0.83 16.67 33.33 x t+7 13 3 10 12 0.80 0.77 23.08 20.00 x t+8 No Cluster Identified If b + p p p t f f = = = M i M M i M g g tp P f 1 1 ) ( otherwise
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48 Training Prediction Results 0.00 20.00 40.00 60.00 80.00 100.00 120.00 xt+1 xt+2 xt+3 xt+4 xt+5 xt+6 xt+7 xt+8 Event Characterization Function Percentage Alpha Beta PPA CPP Figure 4. 11 Training Earliness Prediction Accuracy for = 0.85 at St. Louis Gauging Station The clusters found in the training phase are used to predict the floods in the testing phase. The prediction results from testing phase are displayed in Table 4. 8 and Figure 4. 12. As in the training phase, the predicti on accuracy decreas es with the increase in prediction horizon. For prediction horizon of seven days and more, the cluster misses the starts of floods. Table 4. 8 Testing Earliness Prediction Accuracy for = 0.85 at St. Louis Gauging Station Number of Flood s in Testing Time Series=2, Number of Events in Testing Time Series=33 Event Points PPA CPP Number Char. in Events Resultant tp /tp+fp tp(pred.) of Starts Func. Cluster tp fp Missed a or (1 ) tp(actual) Missed x t+1 90 20 70 13 0.3 9 0.78 22.22 60.61 0 x t+2 75 10 65 23 0.70 0.87 13.33 30.30 0 x t+3 73 10 63 23 0.70 0.86 13.70 30.30 0 x t+4 43 7 36 26 0.79 0.84 16.28 21.21 0 x t+5 36 4 32 29 0.88 0.89 11.11 12.12 0 x t+6 24 3 21 30 0.91 0.88 12.50 9.09 0 x t+7 16 1 15 32 0.97 0.94 6. 25 3.03 1 x t+8 10 0 10 33 1.00 1.00 0.00 0.00 2
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49 Testing Prediction Results 0.00 20.00 40.00 60.00 80.00 100.00 120.00 xt+1 xt+2 xt+3 xt+4 xt+5 xt+6 xt+7 xt+8 Event Characterization Function Percentage Alpha Beta PPA CPP Figure 4. 12 Testing Earliness Prediction Accuracy for = 0.85 at St. Louis Gauging Station Figure 4. 13 displays the events predicted in the testing time series using some examples of step ahead functions. The first time series represents the testing time series and, the following time series display the pre dicted events for different step ahead event characterization functions It can be seen from Figure 4. 13 that as the prediction horizon increases from one day to seven days, the number of false alarms as well as the number of events predicted goes on decre asing. For a seven day prediction horizon, the cluster misses the start of one flood out of two
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50 Figure 4.13 Earliness Prediction Results for = 0.85 at St. Louis Gauging Station Starts of Floo ds False Alarms Start of Flood Missed Start of Flood Starts of Floods Start of Floods
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51 Earliness Prediction Accuracy for Kansas City Gauging Station The results for earliness prediction accuracy are presented in Table 4. 9 and Figure 4.1 4 value of 0. 2 5 is used for calculating the earliness prediction accuracy using different step ahead event characterization functions. Table 4. 9 Training Earliness Prediction Accuracy for = 0. 2 5 at Kansas Ct Gauging Station Training Prediction Results 0.00 20.00 40.00 60.00 80.00 100.00 120.00 xt+1 xt+2 xt+3 xt+4 Event Characterization Function Percentage Alpha Beta PPA CPP Figure 4. 1 4 Training Earliness Prediction Accuracy for = 0. 2 5 at K ansas Ct Gauging Station Testing time series results are presented in Table 4.10 and Figure 4.1 5 Event Points PPA CPP Char. in Events ( tp /tp+fp) tp(predicted) Func Clust er tp fp Missed a or (1 ) tp(actual) x t+1 6 5 1 1 0.17 0.17 83.33 83.33 x t+2 5 4 1 2 0.33 0.20 80.00 66.67 x t+3 2 2 0 4 0.67 0.00 100.00 33.33 x t+4 0 0 0 6 1.00 0.00 0.00 0.00
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52 Table 4.10 Testin g Earliness Prediction Accuracy for = 0. 2 5 at Kansas Ct Gauging Station Number of Floods in Testing Time Series= 1, Number of Events in Testing Time Series= 6 Event Points PPA CPP Number Char in Events Resultant tp /tp+fp tp(pred.) of Starts Func Cluster tp fp Missed a or (1 ) tp(actual) Missed x t+1 7 6 1 0 0.00 0.14 85.71 100.00 0 x t+2 3 2 1 4 0.67 0.33 66.67 3 3.33 0 x t+3 2 2 0 4 0.67 0.00 100.00 33.33 1 Testing Prediction Results 0.00 20.00 40.00 60.00 80.00 100.00 120.00 xt+1 xt+2 xt+3 Event Characterization Function Percentage Alpha Beta PPA CPP Figure 4. 1 5 Testin g Earliness Prediction Accuracy for = 0. 2 5 at Kansas Ct Gauging Station The CPP decreases as the prediction horizon is increased. The highest CPP is with the use of one step ahead Event C haracterization F unction and it decreases drastically as the prediction horizon is increased to two days and more. The clusters start missing the flood for a prediction horizon of 3 days and more.
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53 Earliness Prediction Accuracy for Harrisburg Gaugin g Station The CPP is maximum at of 0.75 Using this value the earliness prediction accuracy is calculated using different step ahead functions. The training stage results for E arliness P rediction A ccuracy at Harrisburg gauging station are presen ted in Table 4.11 and Figure 4.1 6 Tabl e 4.1 1 Trainin g Earliness Prediction Accuracy for = 0.75 at Harrisburg Gauging Station Event Points PPA CPP Char. in Events ( tp /tp+fp) tp(predicted) Func. Cluster tp fp Missed a or (1 ) tp(actual) x t+1 45 13 32 4 0.31 0.71 28.8 9 76.47 x t+2 25 8 17 9 0.53 0.68 32.00 47.06 x t+3 3 2 1 15 0.88 0.33 66.67 11.76 x t+4 No Cluster Identified Training Prediction Results 0.00 20.00 40.00 60.00 80.00 100.00 xt+1 xt+2 xt+3 Event Characterization Function Percentage Alpha Beta PPA CPP Figure 4. 1 6 Trainin g Earliness Prediction Accuracy at Harrisburg Gauging Station The results show that the CPP degrades for longer prediction horizon and for a prediction horizon of 4 days, no cluster can be identified. The results from testing phase are presented in Table 4.1 2 and Figure 4.1 7
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54 Table 4.1 2 Testin g Earliness Prediction Accuracy for = 0.75 at Harrisburg Gauging Station Number of Floods in Testing Time Series=8, Number of Events in Testing Time Series=14 Event Points PPA CPP Number Char. in Events Resultant tp /tp+fp tp(pred.) of Starts Func. Cluster tp fp Missed a or (1 ) tp(actual) Missed x t+1 51 10 41 4 0.29 0.80 0.20 0.71 0 x t+2 25 3 22 11 0.79 0.88 0.12 0.21 6 x t+3 1 0 1 14 1.00 1.00 0.00 0.00 8 x t+4 No Cluster Identified Testing Prediction Results 0.00 20.00 40.00 60.00 80.00 100.00 120.00 xt+1 xt+2 xt+3 Event Characterization Function Percentage Alpha Beta PPA CPP Figure 4. 17 T estin g Earliness Prediction Accuracy at Harrisburg G auging Station For prediction periods longer than a day, the cluster misses the starts of floods. Out of the 8 floods in the testing time series, 6 floods are missed for a prediction horizon of 2 days and for a 3 day prediction horizon all 8 floods are mis sed. 4.6.3 Prediction Accuracy with Objective Function II The results presented in this section are for different values of using Objective Function II applied to the S t Louis gauging station The following set of parameters is used : Event Characterization Function : One step ahead function ( ) 1 + = t t x x g
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55 Objective Function : Objective Function I If c(M)/c(?)= ? = = = M i M M i M g g P f 1 1 ) ( otherwise Optimization Formulation : 1. Maximize the value of Objective Function (Priority I) 2. Minimize the radius of cluster (Priority II) The difference between Objective Function II and Objective Function I used i n Section 4.6.1 is that the summation of g values inside the cluster is not multiplied by the number of true positives inside the cluster. This means that the GA algorithm will not be rewarded for selecting a cluster with high number of tp s. Objective Fun ction II is useful when the history of floods is not known. T he GA will select the points with high g values to be a part of cluster in order to maximize the summation of g values. Since the p hase space points representing floods are points with high g val ues the GA would try to include them in the cluster to maximize the objective function The cluster size is restricted by the ? value which is determined from the flood zone specification. The information regarding flood zone at St. Louis gauging station was not available. Therefore, for the purpose of demonstration, instead of the ? value, the cluster size is restricted using values as in Section 4.6.1. The objective function is modified as If b + p p p t f f = = = M i M M i M g g P f 1 1 ) ( otherwise The training and testing stage results are presented next. Table 4. 1 3 displays the training stage results for different values of
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56 Table 4. 1 3 Training Stage Results for Different Values Using Objective Function II Points PPA CPP Specified in Events ( tp /tp+fp) tp(predicted) Cluster tp fp Missed a or (1 ) tp(actual) 0.95 295 15 280 0 0.00 5 100 0.85 98 15 83 0 0.00 15 100 0.75 60 15 45 0 0.00 25 100 0.65 42 15 27 0 0.00 36 100 0.55 33 15 18 0 0.00 45 100 0.45 27 15 12 0 0.00 56 1 00 0.35 17 14 3 1 0.07 82 93 0.25 13 10 3 5 0.33 77 67 0.15 7 6 1 9 0.60 86 40 0.05 6 6 0 9 0.60 100 40 As shown in Table 4. 1 3, the Prediction Accuracy is 100% for value of 0.45 and after that, the cluster starts missing tp s which decreases the p rediction accuracy. Testing stage results using O bjective F unction I I are presented in Table 4. 1 4. The cluster starts missing starts of floods at of 0.05. The a and Positive Prediction Accuracy increase with decrease in and Prediction Accuracy. This ha ppens because a reduction in reduces the number of fp s, making the cluster crisper and increasing its PPA However, it also starts to miss more tp s thereby reducing CPP Table 4. 1 4 Testing Stage Results for Different Values Using Objective Function I I Number of Floods in Testing Time Series= 2, Number of Events in Testing Time Series= 33 Points PPA CPP Number Spec. In in Events Resultant tp /tp+fp tp(pred.) of Starts Training Cluster tp fp Missed a or (1 ) tp(actual) Misse d 0.95 272 12 260 21 0.64 0.96 4 36 0 0.85 90 20 70 13 0.39 0.78 22 61 0 0.75 53 17 36 16 0.48 0.68 32 52 0 0.65 26 12 14 21 0.64 0.54 46 36 0 0.55 18 14 4 19 0.58 0.22 78 42 0 0.45 18 15 3 18 0.55 0.17 83 45 0 0.35 10 9 1 24 0.73 0.10 90 27 0 0.25 6 5 1 28 0.85 0.17 83 15 0 0.15 7 7 0 26 0.79 0.00 100 21 0 0.05 1 1 0 32 0.97 0.00 100 3 1
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57 4.7 Summary of Results 1. From the training and testing results for all three gauging stations, it can be seen that no clear relationship exists between the specif ied for a cluster and its Correct Prediction Percentage. As explained earlier, the non uniform density of phase space is responsible for th is If the phase space would have been uniform in density, a higher value of would have led to a cluster with lar ger radius since more fp s are allowed. The CPP would also be high considering the fact that a cluster with a larger radius would include a high number of tp s. However, the density of phase space points is non uniform. Consider two clusters in the Harrisb urg gauging station phase space as shown in Figure 4.19. Specifying a smaller value for (allowable proportion of fp s in the cluster) may lead to Cluster 1 being identified as the optimal cluster. The density of points in Cluster 1 is sparse which allows a larger radius for the cluster. Since it includes more number of tp s, the CPP for Cl uster 1 is high. On the other hand, specifying a large may lead to Cluster 2 with a smaller radius (since points are dense) and less CPP because it misses some of the tp s. Thus the CPP not only depends on the specified, but also on the location of th e cluster in the non uniform phase space. 2. An inverse relationship exists between the earliness of prediction and the CPP. It can be observed from Tables 4.8, 4.10 and 4.13 that as the prediction horizon increases, the Correct Prediction Accuracy decreases and, the number of starts of floods missed also increases. Ideally, one would like to predict a flood as early as possible, however, associated with the earliness of prediction is the probability of missing the start of floods. 3. Using Objective Function II the CPP was found to be either to equal or lower than that using Objective Function I, for different values of This result was expected because the Objective Function I uses the historical floods and threshold discharge values to train the GA in the s earch for optimal temporal pattern clusters. Figure 4.20 presents a graph comparing the CPP values for the St. Louis gauging station example using Objective Function I and II.
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58 The ? value in Objective Function II was replaced by since the information on flood zoning was not available. However, using the ?, the CPP would be even lower than the CPP using because the flood zoning values are calculated using linear regression models which are inadequate in the analysis of nonlinear time series. Figure 4. 19 Comparison of Two Clusters in Phase Space Comparison of CPP for Objective Function I and II 0 10 20 30 40 50 60 70 0.95 0.85 0.75 0.65 0.55 0.45 0.35 0.25 0.15 Value CPP(Percentage) Objective Function I Objective Function II Figure 4.20 Comparison of CPP for Objective Function I and II at St. Louis Gauging Station
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59 In this Chapter the results were presented for flood prediction problem applied to three gauging stations. Chapter 5 p resents the conclusion and the directions for future research.
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60 Chapter 5. Conclusions and Future Work This c hapter present s the conclusion s and directions for further r esearch. Section 5.1 summarizes this research and presents it s application as a decision making tool. Section 5.2 enlists the directions for future research in this area. 5.1 Conclusion s Time Series Data Mining is applied to the area of flood forecasting with the goal of predicting floods accurately and as early as possible. Three examples of gauging stations, representing high, medium and low flood occurrences are considered. The prediction accuracy is evaluated in terms of a , Positive Prediction Accuracy (PPA) and Correct Prediction Accuracy (CPP). T wo variations of objective function are presented Objective Function I can be used in flood forecasting problems where the information about history of floods is available. Where th is information is not available, t he flood zoning information can be used along with Objective Function II The effect of earliness of prediction on the prediction accuracy is also presented. Earlier approaches have dealt with forecasting magnitud es of future discharge values. This research focuses on the early prediction of floods (events). It is the first application of an event based data mining technique to flood forecasting. The predictions are specific to the location of the gauging station and its catchment area. Variations in factors such as river depth, cross section, rainfall runoff snowmelt affect the flood characteristics. For example, for the St. Louis gauging station, a discharge of 780,000 cubic feet per second causes the river to overflow. On the other hand, for the Kansas City gauging station, a
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61 discharge of 380,000 cubic feet per second is enough to make the river overflow. Hence, the predictions are not generic and for a flood prediction problem at another location, the GA needs to be trained on the discharge time series from that location. The prediction results from TSDM can be used as a decision making tool by city planners. The decision variables in this approach are (Proportion of False Positives) and, the Ev ent Characterization Function (Step ahead function). Depending on the location of the gauging station and its catchment area, the impact of flood on surrounding human population and economy, the city planners can decide the values for the above mentioned v ariables that would provide them enough time to plan for flood mitigation and evacuation procedures. For example, the flooding on Hillsborough River passing through the City of Tampa will affect a huge population and have large scale economic repercussions Whereas, a flood on the Suwanee River that passes through large unpopulated areas in Florida would have a lesser impact. Thus, for a n impact area such as the City of Tampa, the planners would not want to miss predicting a flood and would also accept a ce rtain number of false alarms as long as the no flood is missed. This criterion determines the value of (proportion of false positives) for the cluster. From the results it can be seen that a higher leads to high number of false alarms in the testing time series. However, associated with there is an a (proportion of points missed). Hence, when choosin g the value of the decision must also consider the a value for that cluster. There is no identifiable relationship between a and of a cluster, due to the non uniform nature of the phase space. In most cases, the decision makers would want to select parameters such that both a and are minimized and PPA and CPP are maximized Another factor that the city planners have to account for is the earliness of prediction. This factor affects the choice of event characterization function. The planners would w ant to predict the flood as early as possible; however, a tradeoff exists between the earliness of prediction and
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62 the Correct Prediction Percentage From the results it can be seen that the earlier the TSDM methodology tries to predict the flood, less is t he and Correct Prediction Percentage Co ncluding, this research leads to the development of a decision making tool for planning flood mitigation and evacuation procedures for use by planning authorities. Based on location and the possible impact the pla nners can make a choice in selecting the parameters for TSDM. The tradeoffs involved in selecting different values for these parameters are also presented. This is a general approach and can be applied to any gauging station and its catchment area for floo d prediction. 5.2 Future Work 1. Decision Support Software Development A Decision Support Software can be developed for a general purpose use by city planners and emergency operations agencies. The proposed software would have a graphical user interface with input screens to allow users to input the training time series for any gauging station. Depending on whether the history of floods is available the software will train the GA using one of the two objective functions. In cases where the history of floods is available, the users are allowed to choose the value for As mentioned in Section 5.1, this selection would depend on the impact the flood can have on the affected area. For cases where the historical information on floods is not available, the GA can be trained using the Objective Function II. Generally, all residential areas that have a probability of being affected by a flood are classified into flood zones for insurance and mortgage calculations by government o r private agencies. The Objective Function II makes use of this zoning information to search for o ptimal clusters indicative of floods. The software would also allow the decision makers to choose how many days early they want the prediction to be.
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63 The possible customers for this software would be insurance agencies and city planning authorities. 2. Use of Multiple Parameters In this research the daily discharge time series is used for flood prediction. A future study could use other data sets such as rainfall runoff, the height of river (water level), snowmelt. The use of combination of multiple paramet ers such as a discharge and rainfall runoff time series may lead to higher prediction accuracy. Another approach would be to modify the Event Characterization Function itself to include multiple variables. 3. Changing Embedding Time Delay It was observed th at the efficiency of the GA in finding a cluster depends on the spread of points in the phase space. For St. Louis gauging station, the spread of phase space points is minimum and the points are concentrated around the diagonal. The spread of points can b e controlled by specifying different values for delay. A higher delay would possibly lead to lower spread in the phase space points, however it also results in loss of information. An interesting direction for future research would be to experiment with di fferent time delays to observe the effect on the prediction accuracy of the GA. 4. TSDM in Conjunction with NLP TSDM and NLP can be used in conjunction to predict both the time and magnitude of floods. This will help in real time alerting and evacuation plan ning. If the time of floods and magnitude is known the area of impact can be estimated more accurately. For example, discharge over the threshold would cause the river to overflow, however the magnitude of discharge will determine what area is actually af fected by the flood.
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