Outline Predictions Are Important Case Study: . . . Enter Delay and Chaos Predicting Volcanic How to Detect Delay . . . Eruptions: Case Study of Let’s Apply This to . . . What We Expected . . . Rare Events in Chaotic Discussion This Is Bad News and . . . Systems with Delay Home Page Title Page Justin Parra 1 , Olac Fuentes 1 , Elizabeth Anthony 2 , and Vladik Kreinovich 1 ◭◭ ◮◮ ◭ ◮ Departments of 1 Computer Science and 2 Geological Sciences University of Texas at El Paso, El Paso, TX 79968, USA, Page 1 of 19 jrparra2@miners.utep.edu, ofuentes@utep.edu, eanthony@utep.edu, vladik@utep.edu Go Back Full Screen Close Quit
Outline Predictions Are Important 1. Outline Case Study: . . . • Volcanic eruptions can be disastrous. Enter Delay and Chaos How to Detect Delay . . . • It is therefore important to be able to predict them as Let’s Apply This to . . . accurately as possible. What We Expected . . . • Theoretically, we can use the general machine learning Discussion techniques for such predictions. This Is Bad News and . . . Home Page • However, in general, such methods require an unreal- istic amount of computation time. Title Page • It is therefore desirable to look for additional informa- ◭◭ ◮◮ tion that would enable us to speed up computations. ◭ ◮ • In this talk, we provide an empirical evidence that the Page 2 of 19 volcanic system exhibit chaotic and delayed character. Go Back • We also show how this can speed up computations. Full Screen Close Quit
Outline Predictions Are Important 2. Predictions Are Important Case Study: . . . • Often, we want to predict future values y ( t f ) ( t f > t 0 ) Enter Delay and Chaos of different quantities y . How to Detect Delay . . . Let’s Apply This to . . . • To predict a future value, we can use the values of the What We Expected . . . related quantities x 1 ( t ), . . . , x n ( t ) for t ≤ t 0 . Discussion • For that, we need to know the dependence of y ( t f ) on This Is Bad News and . . . the values x ( t ) = ( x 1 ( t ) , . . . , x n ( t )). Home Page • In some practical situations, we know the desired de- Title Page pendence. ◭◭ ◮◮ • For example, we know Newton’s equations that de- ◭ ◮ scribe the orbit of an asteroid. Page 3 of 19 • Thus, we can use these known equation to make the corresponding predictions. Go Back Full Screen • In other cases, however, we do not know the desired dependence. Close Quit
Outline Predictions Are Important 3. Need for Machine Learning Case Study: . . . • We can use the general techniques for determining the Enter Delay and Chaos desired dependence from data. How to Detect Delay . . . Let’s Apply This to . . . • Such techniques are known machine learning. What We Expected . . . • Examples: neural networks (in particular, deep learn- Discussion ing networks), support vector machines, etc. This Is Bad News and . . . Home Page • To predict m steps into the future, we use patterns ( x, y ( t f )), where: Title Page • y ( t f ) is the observed value y at moment t f and ◭◭ ◮◮ • x is a collection of all the x -tuples x ( t ) observed at ◭ ◮ moments t ≤ t f − m . Page 4 of 19 Go Back Full Screen Close Quit
Outline Predictions Are Important 4. We Face a Practical Challenge Case Study: . . . • The machine learning computation time grows fast Enter Delay and Chaos with the number of unknowns. How to Detect Delay . . . Let’s Apply This to . . . • As possible inputs, we have each of n values x i mea- What We Expected . . . sured at each of N t moments of time. Discussion • So, we need the dependence on N t · n unknowns. This Is Bad News and . . . Home Page • When N t is large, the number of unknowns is large. Title Page • Thus, the corresponding computation require too much computation time. ◭◭ ◮◮ • And indeed, successful predictions – e.g., using deep ◭ ◮ learning – require high-performance computers. Page 5 of 19 • To overcome this challenge, we need to limit moments Go Back of time used for training. Full Screen Close Quit
Outline Predictions Are Important 5. Such a Limitation Is Indeed Possible Case Study: . . . • Suppose that we want to predict the weather in the Enter Delay and Chaos next hour. How to Detect Delay . . . Let’s Apply This to . . . • The weather usually does not change during an hour. What We Expected . . . • Thus, the most informative are current values x ( t 0 ). Discussion • Knowing last year’s weather will not help. This Is Bad News and . . . Home Page • To get predictions for the next day, it may be a good Title Page idea to also look for yesterday’s weather. ◭◭ ◮◮ • We will see if there is a tendency for the temperature to increase or to decrease. ◭ ◮ • If we are currently in the Fall, then, to get predictions Page 6 of 19 for the next summer: Go Back – today’s data is probably useless, Full Screen – it is much more useful to get data from last summer. Close Quit
Outline Predictions Are Important 6. Case Study: Predicting Volcanic Eruptions Case Study: . . . • An unexpected eruption can be a big disaster. Enter Delay and Chaos How to Detect Delay . . . • The ancient city of Pompei was destroyed by a nearby Let’s Apply This to . . . volcano. What We Expected . . . • The Cretan civilization was destroyed by a tsunami Discussion caused by a volcanic eruption. This Is Bad News and . . . Home Page • Nowadays, millions of people live in the close vicinity of active volcanos: Naples, Mexico City. Title Page • This makes the task of predicting volcanic eruptions ◭◭ ◮◮ even more critical. ◭ ◮ Page 7 of 19 Go Back Full Screen Close Quit
Outline Predictions Are Important 7. Specific Volcanoes Case Study: . . . • We used the Aleutian chain of volcanoes that reaches Enter Delay and Chaos from Alaska to Russia. How to Detect Delay . . . Let’s Apply This to . . . • Because of their location, their eruption affect major What We Expected . . . flight paths in the Pacific. Discussion • As a result, they are heavily monitored, with seismic This Is Bad News and . . . sensors near almost all of them. Home Page • Of course, volcanos near Naples and Mexico City are Title Page heavily monitored too. ◭◭ ◮◮ • However, these are solo volcanos, while there are about ◭ ◮ 30 Aleutian volcanos. Page 8 of 19 • Hence, Aleutian eruptions are more frequent, and we Go Back have more data to study. Full Screen Close Quit
Outline Predictions Are Important 8. What Information We Can Use to Predict Vol- Case Study: . . . canic Eruptions Enter Delay and Chaos • When magma ascends to the surface, this massive How to Detect Delay . . . movement causes some seismic activity. Let’s Apply This to . . . What We Expected . . . • This causes ground deformation. Discussion • Also, volcanic gases come out. This Is Bad News and . . . • Detecting deformations and gases requires complex on- Home Page site equipment, and all we get is a few numbers. Title Page • In contrast, seismic waves can detected far way, and ◭◭ ◮◮ carry a lot of information. ◭ ◮ • So, volcanic prediction techniques are based mostly on Page 9 of 19 seismic activities. Go Back • There exist techniques for predicting eruptions. Full Screen • However, these techniques are not perfect, more effi- cient and more accurate methods are needed. Close Quit
Outline Predictions Are Important 9. Enter Delay and Chaos Case Study: . . . • Delay means that inputs x i affect y only after some Enter Delay and Chaos time T d (example: incubation period). How to Detect Delay . . . Let’s Apply This to . . . • Thus, to predict y ( t f ), we only need to consider x i ( t ) What We Expected . . . for t ≤ t f − T d . Discussion • Chaos means even if we know the current state, we This Is Bad News and . . . cannot predict the distant future. Home Page • A small change in the initial conditions can lead to a Title Page drastic changes of the future. ◭◭ ◮◮ • This is known as the butterfly effect . ◭ ◮ • In precise terms, chaos means that the effect of x i dis- Page 10 of 19 appears after some time T c . Go Back • So, to predict y ( t f ), we only need to consider x i ( t ) for t ≥ t f − T c . Full Screen • Thus, we only need values x i ( t ) for t ∈ [ t f − T c , t f − T d ]. Close Quit
Outline Predictions Are Important 10. How to Detect Delay and Chaos Based on Case Study: . . . Data Enter Delay and Chaos • Delay and chaos means that: How to Detect Delay . . . Let’s Apply This to . . . – for some m 0 , y ( t f ) is not effected by x i ( t f − m 0 ), What We Expected . . . – for other m 0 , y ( t f ) strongly depends on x i ( t f − m 0 ). Discussion • So, to detect T d and T c , we need to find values m 0 for This Is Bad News and . . . which y ( t ) strongly depends on q = x i ( t f − m 0 ). Home Page • Each q is uniquely determined by properties q < q 0 and Title Page q ≥ q 0 for different q 0 . ◭◭ ◮◮ • So, in effect, we must find m 0 and q 0 for which y ( t f ) ◭ ◮ most depends on whether x i ( t f − m 0 ) < q 0 . Page 11 of 19 • For a discrete event like an eruption, we can build a Go Back decision three based on whether x i ( t f − m 0 ) < q 0 . Full Screen • Inequalities close to the top of the tree correspond to important inputs. Close Quit
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