1. Start Presentation © Prof. Dr. François E. Cellier Internal Seminar: Groups Arbenz/Cellier/Gander/Hinterberger May 29, 2007 Making Predictions in an Uncertain World ?

2. Start Presentation © Prof. Dr. François E. Cellier Internal Seminar: Groups Arbenz/Cellier/Gander/Hinterberger May 29, 2007 Table of Contents Inductive ModelingInductive Modeling Inductive ReasoningInductive Reasoning Fuzzy Inductive ReasoningFuzzy Inductive Reasoning Modeling the Error of a PredictionModeling the Error of a Prediction Economic ModelingEconomic Modeling Conclusions

3. Start Presentation © Prof. Dr. François E. Cellier Internal Seminar: Groups Arbenz/Cellier/Gander/Hinterberger May 29, 2007 Inductive Modeling In this presentation, we shall study general techniques for identifying complex non-linear models from observations of input/output behavior. These techniques make an attempt at mimicking human capabilities of vicarious learning, i.e., of learning from observation. These techniques should be perfectly general, i.e., the algorithms ought to be capable of capturing an arbitrary functional relationship for the purpose of reproducing it faithfully. The techniques will also be totally unintelligent, i.e., their capabilities of generalizing patterns from observations are almost non-existent.

4. Start Presentation © Prof. Dr. François E. Cellier Internal Seminar: Groups Arbenz/Cellier/Gander/Hinterberger May 29, 2007 Taxonomy of Modeling Methodologies Knowledge-Based Approaches Observation-Based Approaches Deep ModelsShallow Models Neural NetworksInductive Reasoners FIR SD

5. Start Presentation © Prof. Dr. François E. Cellier Internal Seminar: Groups Arbenz/Cellier/Gander/Hinterberger May 29, 2007 Observation-based Modeling and Complexity Observation-based modeling is very important, especially when dealing with unknown or only partially understood systems. Whenever we deal with new topics, we really have no choice, but to model them inductively, i.e., by using available observations. The less we know about a system, the more general a modeling technique we must embrace, in order to allow for all eventualities. If we know nothing, we must be prepared for anything. In order to model a totally unknown system, we must thus allow a model structure that can be arbitrarily complex.

6. Start Presentation © Prof. Dr. François E. Cellier Internal Seminar: Groups Arbenz/Cellier/Gander/Hinterberger May 29, 2007 Parametric vs. Non-parametric Models ANNs are parametric models. The observed knowledge about the system under study is mapped on the (potentially very large) set of parameters of the ANN. Once the ANN has been trained, the original knowledge is no longer used. Instead, the learnt behavior of the ANN is used to make predictions. This can be dangerous. If the testing data, i.e. the input patterns during the use of the already trained ANN differ significantly from the training data set, the ANN is likely to predict garbage, but since the original knowledge is no longer in use, is unlikely to be aware of this problem.

7. Start Presentation © Prof. Dr. François E. Cellier Internal Seminar: Groups Arbenz/Cellier/Gander/Hinterberger May 29, 2007 Parametric vs. Non-parametric Models II Non-parametric models, on the other hand, always refer back to the original training data, and therefore, can be made to reject testing data that are incompatible with the training data set. The Fuzzy Inductive Reasoning (FIR) engine that we shall discuss in this talk, is of the non-parametric type. During the training phase, FIR organizes the observed patterns, and places them in a data base. During the testing phase, FIR searches the data base for the five most similar training data patterns, the so-called five nearest neighbors, by comparing the new input pattern with those stored in the data base. FIR then predicts the new output as a weighted average of the outputs of the five nearest neighbors.

8. Start Presentation © Prof. Dr. François E. Cellier Internal Seminar: Groups Arbenz/Cellier/Gander/Hinterberger May 29, 2007 Quantitative vs. Qualitative Models Training a model (be it parametric or non-parametric) means solving an optimization problem. In the parametric case, we have to solve a parameter identification problem. In the non-parametric case, we need to classify the training data, and store them in an optimal fashion in the data base. Training such a model can be excruciatingly slow. Hence it may make sense to devise techniques that will help to speed up the training process.

9. Start Presentation © Prof. Dr. François E. Cellier Internal Seminar: Groups Arbenz/Cellier/Gander/Hinterberger May 29, 2007 Quantitative vs. Qualitative Models II How can the speed of the optimization be controlled? Somehow, the search space needs to be reduced. One way to accomplish this is to convert continuous variables to equivalent discrete variables prior to optimization. For example, if one of the variables to be looked at is the ambient temperature, we may consider to classify temperature values on a spectrum from very cold to extremely hot as one of the following discrete set: temperature = { freezing, cold, cool, moderate, warm, hot }

10. Start Presentation © Prof. Dr. François E. Cellier Internal Seminar: Groups Arbenz/Cellier/Gander/Hinterberger May 29, 2007 Inductive Reasoning: An Example Given the system: x = A · x + b · u y = C · x + d · u · = 0 0 1 · x + 0 · u -2 -3 -4 0 1 0 0 1 = 0 1 0 · x + 0 · u 0 0 1 1 0 0 0 0

11. Start Presentation © Prof. Dr. François E. Cellier Internal Seminar: Groups Arbenz/Cellier/Gander/Hinterberger May 29, 2007 Inductive Reasoning: An Example II We simulate using a binary random input stream:

12. Start Presentation © Prof. Dr. François E. Cellier Internal Seminar: Groups Arbenz/Cellier/Gander/Hinterberger May 29, 2007 Recoding We recode (discretize) the three output signals by sorting each trajectory (array) separately and choosing the landmarks between different classes such that each class contains the same number of members. Let us recode each quantitative output trajectory into a qualitative output episode consisting of three classes. Hence we end up with a single binary input episode and three ternary output episodes. Our goal is to predict future values of the output episodes given future binary input values from observations of past input/output behavior.

13. Start Presentation © Prof. Dr. François E. Cellier Internal Seminar: Groups Arbenz/Cellier/Gander/Hinterberger May 29, 2007 Qualitative Modeling Once the data have been recoded, we wish to determine, which among the possible sets of (past and current) variables best represent the observed (future) behavior. Of all possible combinations, we pick the one that gives us as deterministic an input/output relationship as possible, i.e., when the same input pattern is observed multiple times among the training data, we wish to obtain output patterns that are as consistent as possible. Each input pattern should be observed at least five times.

14. Start Presentation © Prof. Dr. François E. Cellier Internal Seminar: Groups Arbenz/Cellier/Gander/Hinterberger May 29, 2007 Qualitative Modeling II y 1 (t) = f ( y 3 (t  2  t), u 2 (t  t), y 1 (t  t), u 1 (t) ) Rule base

15. Start Presentation © Prof. Dr. François E. Cellier Internal Seminar: Groups Arbenz/Cellier/Gander/Hinterberger May 29, 2007 Qualitative Modeling III The qualitative model is the optimal mask, i.e., the set of input patterns that best predict a given output. Usually, the optimal mask is dynamic, i.e., the current output depends both on current and past values of inputs and outputs. The optimal mask can then be applied to the training data to obtain a set of rules that can be alphanumerically sorted. The rule base is our training data base.

16. Start Presentation © Prof. Dr. François E. Cellier Internal Seminar: Groups Arbenz/Cellier/Gander/Hinterberger May 29, 2007 Qualitative Simulation We create one optimal mask for each output variable. We can now shift the optimal masks beyond the measurement data, and predict the next output as the most likely output given the new input pattern. We repeat this for each optimal mask and then continue to shift the masks further down using the already predicted data as “measurement” data. In this way, we can predict future episodic behavior.

17. Start Presentation © Prof. Dr. François E. Cellier Internal Seminar: Groups Arbenz/Cellier/Gander/Hinterberger May 29, 2007 Qualitative Simulation II We got not a single prediction error.

18. Start Presentation © Prof. Dr. François E. Cellier Internal Seminar: Groups Arbenz/Cellier/Gander/Hinterberger May 29, 2007 Making Money Fast

19. Start Presentation © Prof. Dr. François E. Cellier Internal Seminar: Groups Arbenz/Cellier/Gander/Hinterberger May 29, 2007 Predicting Trajectory Behavior Being able to predict episodic behavior is not sufficient for technical applications. We need to be able to predict trajectory behavior. To this end, we require some kind of smoothing function that enables us to interpolate in between neighboring discrete class values. Fuzzy logic can provide us with a good tool for this purpose.

20. Start Presentation © Prof. Dr. François E. Cellier Internal Seminar: Groups Arbenz/Cellier/Gander/Hinterberger May 29, 2007 Fuzzy Variables Fuzzification proceeds as follows. A continuous variable is fuzzified, by decomposing it into a discrete class value and a fuzzy membership value. For the purpose of reasoning, only the class value is being considered. However, for the purpose of interpolation, the fuzzy membership value is also taken into account. Fuzzy variables are not discrete, but they are also referred to as qualitative.

21. Start Presentation © Prof. Dr. François E. Cellier Internal Seminar: Groups Arbenz/Cellier/Gander/Hinterberger May 29, 2007 Fuzzy Variables II { Class, membership } pairs of lower likelihood must be considered as well, because otherwise, the mapping would not be unique. Systolic blood pressure = 110  { normal, 0.78 }  { too low, 0.15 } Systolic blood pressure = 141  { normal, 0.78 }  { too high, 0.18 }

22. Start Presentation © Prof. Dr. François E. Cellier Internal Seminar: Groups Arbenz/Cellier/Gander/Hinterberger May 29, 2007 Fuzzy Variables in FIR FIR embraces a slightly different approach to solving the uniqueness problem. Rather than mapping into multiple fuzzy rules, FIR only maps into a single rule, that with the largest likelihood. However, to avoid the aforementioned ambiguity problem, FIR stores one more piece of information, the “side value.” It indicates, whether the data point is to the left or the right of the peak of the fuzzy membership value of the given class. Systolic blood pressure = 110  { normal, 0.78, left } left right Systolic blood pressure = 141  { normal, 0.78, right }

23. Start Presentation © Prof. Dr. François E. Cellier Internal Seminar: Groups Arbenz/Cellier/Gander/Hinterberger May 29, 2007 Fuzzy Inductive Reasoning (FIR) Discretization of quantitative information (Fuzzy Recoding) Reasoning about discrete categories (Qualitative Modeling) Inferring consequences about categories (Qualitative Simulation) Interpolation between neighboring categories using fuzzy logic (Fuzzy Regeneration)

24. Start Presentation © Prof. Dr. François E. Cellier Internal Seminar: Groups Arbenz/Cellier/Gander/Hinterberger May 29, 2007 Fuzzy Inductive Reasoning (FIR) II FIR Fuzzy Recoding Fuzzy Recoding Fuzzy Modeling Fuzzy Modeling Regene- ration Regene- ration Fuzzy Simulation Fuzzy Simulation Crisp data

25. Start Presentation © Prof. Dr. François E. Cellier Internal Seminar: Groups Arbenz/Cellier/Gander/Hinterberger May 29, 2007 Fuzzification in FIR

26. Start Presentation © Prof. Dr. François E. Cellier Internal Seminar: Groups Arbenz/Cellier/Gander/Hinterberger May 29, 2007 Qualitative Simulation in FIR Behavior Matrix Optimal Matrix Input Pattern Euclidean Distance Computation d j Output Computation f i =F(W*5-NN-out) Forecast Value 5-Nearest Neighbors Matched Input Pattern Class Side Membership 2 2 3 1 3 1 1 ? Raw Data Matrix

27. Start Presentation © Prof. Dr. François E. Cellier Internal Seminar: Groups Arbenz/Cellier/Gander/Hinterberger May 29, 2007 Time-series Prediction in FIR Water demand for the city of Barcelona, January 85 – July 86

28. Start Presentation © Prof. Dr. François E. Cellier Internal Seminar: Groups Arbenz/Cellier/Gander/Hinterberger May 29, 2007 Simulation Results Prediction Real

29. Start Presentation © Prof. Dr. François E. Cellier Internal Seminar: Groups Arbenz/Cellier/Gander/Hinterberger May 29, 2007 Quantitative vs. Qualitative Modeling Deductive Modeling Techniques * have a large degree of validity in many different and even previously unknown applications * are often quite imprecise in their predictions due to inherent model inaccuracies Inductive Modeling Techniques * have a limited degree of validity and can only be applied to predicting behavior of systems that are essentially known * are often amazingly precise in their predictions if applied carefully

30. Start Presentation © Prof. Dr. François E. Cellier Internal Seminar: Groups Arbenz/Cellier/Gander/Hinterberger May 29, 2007 Mixed Quantitative & Qualitative Modeling It is possible to combine qualitative and quantitative modeling techniques. Quantitative Subsystem Recode FIR Model Regenerate Quantitative Subsystem Recode FIR Model Regenerate

31. Start Presentation © Prof. Dr. François E. Cellier Internal Seminar: Groups Arbenz/Cellier/Gander/Hinterberger May 29, 2007 Sources of Uncertainties in Predictions

32. Start Presentation © Prof. Dr. François E. Cellier Internal Seminar: Groups Arbenz/Cellier/Gander/Hinterberger May 29, 2007 The Cardiovascular System Heart Rate Controller Myocardiac Contractility Controller Peripheric Resistance Controller Venous Tone Controller Coronary Resistance Controller Central Nervous System Control (Qualitative Model) Regenerate Heart Circulatory Flow Dynamics Carotid Sinus Blood Pressure Recode Hemodynamical System (Quantitative Model) TH B2 Q4 D2 Q6 PAC Pressure of the arteries in the brain.

33. Start Presentation © Prof. Dr. François E. Cellier Internal Seminar: Groups Arbenz/Cellier/Gander/Hinterberger May 29, 2007 Simulation Results II

34. Start Presentation © Prof. Dr. François E. Cellier Internal Seminar: Groups Arbenz/Cellier/Gander/Hinterberger May 29, 2007 Discussion The top graph shows the peripheric resistance controller, Q4, during a Valsalva maneuver. The true data are superposed with the simulated data. The simulation results are generally very good. However, in the center part of the graph, the errors are a little larger. Below are two graphs showing the estimate of the probability of correctness of the prediction made. It can be seen that FIR is aware that the simulation results in the center area are less likely to be of high quality.

35. Start Presentation © Prof. Dr. François E. Cellier Internal Seminar: Groups Arbenz/Cellier/Gander/Hinterberger May 29, 2007 Discussion II This can be exploited. Multiple predictions can be made in parallel together with estimates of the likelihood of correctness of these predictions. The predictions can then be kept that are accompanied by the highest confidence value. This is shown on the next graph. Two different models (sub-optimal masks) are compared against each other. The second mask performs better, and also the confidence values associated with these predictions are higher.

36. Start Presentation © Prof. Dr. François E. Cellier Internal Seminar: Groups Arbenz/Cellier/Gander/Hinterberger May 29, 2007 Simulation Results III

37. Start Presentation © Prof. Dr. François E. Cellier Internal Seminar: Groups Arbenz/Cellier/Gander/Hinterberger May 29, 2007 Economic Modeling We shall now deal with an application of Fuzzy Inductive Reasoning (FIR): making economic predictions. The presentation demonstrates how FIR can be used to improve the System Dynamics (SD) approach to soft- science modeling. It shows furthermore how hierarchical modeling can be used in the context of FIR, and demonstrates that by means of hierarchical modeling, the quality of economic predictions can dramatically be improved.

38. Start Presentation © Prof. Dr. François E. Cellier Internal Seminar: Groups Arbenz/Cellier/Gander/Hinterberger May 29, 2007 Using FIR for Identifying Laundry Lists One of the most daring (and dubious!) assumptions made by Forrester in his system dynamics approach to modeling soft-science systems was that a function of multiple variables can be written as a product of functions of a single variable each: This is obviously not generally true, and Forrester of course knew it. He made this assumption simply because he didn’t know how else to proceed. Birth_rate = Population · f (Pollution, Food, Crowding, Material_Standard_of_Living) Birth_rate = Population · f 1 (Pollution) · f 2 (Food) · f 3 (Crowding) · f 4 ( Material_Standard_of_Living) 

39. Start Presentation © Prof. Dr. François E. Cellier Internal Seminar: Groups Arbenz/Cellier/Gander/Hinterberger May 29, 2007 Using FIR for Identifying Laundry Lists II An alternative might be to make use of FIR instead of table look-up functions for identifying any one of these unknown relationships among variables forming a laundry list. This is what we shall attempt now. Since FIR models are by themselves usually dynamic (since the optimal mask usually spans several rows), the functional relationships of each laundry list may furthermore be dynamic rather than static.

40. Start Presentation © Prof. Dr. François E. Cellier Internal Seminar: Groups Arbenz/Cellier/Gander/Hinterberger May 29, 2007 Modeling in the Agricultural Sector

41. Start Presentation © Prof. Dr. François E. Cellier Internal Seminar: Groups Arbenz/Cellier/Gander/Hinterberger May 29, 2007 Modeling in the Agricultural Sector II In general, specific economic variables, such as food consumption patterns, depend on the general state of the economy. If the economy is doing well, Americans are more likely to consume steak, whereas otherwise, they may choose to buy hamburgers instead. The general state of the economy can in a first instance be viewed as depending primarily on two variables: availability of jobs, and availability of money. If people don’t have savings, they can’t buy much, and if they don’t have jobs, they are less likely to spend money, even if they have some savings. The general state of the economy is heavily influenced by population dynamics. It takes people to produce goods, and it takes customers to buy them.

42. Start Presentation © Prof. Dr. François E. Cellier Internal Seminar: Groups Arbenz/Cellier/Gander/Hinterberger May 29, 2007 Modeling in the Agricultural Sector III Food consumption is modeled in a hierarchical fashion. Each of the rate variables has a local delay block associated with it. This delay block represents the fact that the rates are modeled using FIR, which allows to obtain dynamic models of laundry lists. Three distinct layers are identified. The specific economic layer depends on a generic economic layer, which in turn depends on a demographic layer. Specific economic layer Generic economic layer Demographic layer

43. Start Presentation © Prof. Dr. François E. Cellier Internal Seminar: Groups Arbenz/Cellier/Gander/Hinterberger May 29, 2007 Population Dynamics Contraceptives The Great Depression Population Dynamics Macro-economy Food Demand Food Supply

44. Start Presentation © Prof. Dr. François E. Cellier Internal Seminar: Groups Arbenz/Cellier/Gander/Hinterberger May 29, 2007 Prediction of Growth Functions One of the major difficulties with (and greatest strengths of) FIR modeling is its inability to extrapolate. Thus, if a variable is growing, such as the population, FIR has no means of predicting this directly. A simple trick solves this dilemma. Economists have known about this problem for a long time, since many other, mostly statistical, approaches to making predictions share FIR’s inability to extrapolate. When economists wish to make predictions about the value of a stock, x, they make use of the so-called daily return variable. Whereas x may be increasing or decreasing, the daily return is mostly stationary. daily return = x(end of day) – x(end of previous day) x(end of day)

45. Start Presentation © Prof. Dr. François E. Cellier Internal Seminar: Groups Arbenz/Cellier/Gander/Hinterberger May 29, 2007 Prediction of Growth Functions II  k(n+1) = FIR [ k(n), P(n), k(n-1), P(n-1), … ]   If P(t) is growing exponentially, k(t) is constant.

46. Start Presentation © Prof. Dr. François E. Cellier Internal Seminar: Groups Arbenz/Cellier/Gander/Hinterberger May 29, 2007 10 6 % Population Dynamics II Population Dynamics Macro-economy Food Demand Food Supply Predicting 1 year ahead.Average error when predicting 1 year ahead. Predicting 3 years ahead. Average error when predicting 3 years ahead.

47. Start Presentation © Prof. Dr. François E. Cellier Internal Seminar: Groups Arbenz/Cellier/Gander/Hinterberger May 29, 2007 Macro-economy Population Dynamics Macro-economy Food Demand Food Supply $ % Average error incurred when making use of own past in predictions only Average error found when making use of predicted population in economy predictions as well

48. Start Presentation © Prof. Dr. François E. Cellier Internal Seminar: Groups Arbenz/Cellier/Gander/Hinterberger May 29, 2007 Macro-economy II Population Dynamics Macro-economy Food Demand Food Supply % % The unemployment rate is a controlled variable. It is influenced by the lending rate. For many years, the feds tried to keep it around 6%. The variations around this value are difficult to predict accurately.

49. Start Presentation © Prof. Dr. François E. Cellier Internal Seminar: Groups Arbenz/Cellier/Gander/Hinterberger May 29, 2007 £ % Food Demand/Supply I Population Dynamics Macro-economy Food Demand Food Supply Average error incurred when making use of own past in predictions only Average error found when making use of population dynamics and economy forecasts as well

50. Start Presentation © Prof. Dr. François E. Cellier Internal Seminar: Groups Arbenz/Cellier/Gander/Hinterberger May 29, 2007 Discussion III The models have shown that making use of predictions already made for the more generic layers of the architecture helps in improving the predictions of variables associated with the more specific layers. In most cases, the prediction errors are reduced by approximately a factor of three in this fashion. Notice that the best prediction techniques available were used in all cases. In particular, the confidence measure has been heavily exploited by always making several predictions in parallel, preserving in every step the one with the highest confidence value.

51. Start Presentation © Prof. Dr. François E. Cellier Internal Seminar: Groups Arbenz/Cellier/Gander/Hinterberger May 29, 2007 Refined Model Group Food Quantity Population Age Groups Demographics Wage rateUnempl.RatePer Capita Income Consumer Price Index Producer Price Index Food Prices Food Quantities Dollars Spent on Food

52. Start Presentation © Prof. Dr. François E. Cellier Internal Seminar: Groups Arbenz/Cellier/Gander/Hinterberger May 29, 2007 Age Groups Population Dynamics Macro-economy Food Demand Food Supply Total Population 0 - 4 5 - 14 15 - 24 25 - 34 35 - 44 45 - 54 65+ 55 - 64

53. Start Presentation © Prof. Dr. François E. Cellier Internal Seminar: Groups Arbenz/Cellier/Gander/Hinterberger May 29, 2007 Macro-economy III Jobs Money PPI CPI Lending RateUnemployment Inflation Population Dynamics Macro-economy Food Demand Food Supply

54. Start Presentation © Prof. Dr. François E. Cellier Internal Seminar: Groups Arbenz/Cellier/Gander/Hinterberger May 29, 2007 Food Demand/Supply II Population Dynamics Macro-economy Food Demand Food Supply Prices Food Income Inflation Climate Unemploy. Population

55. Start Presentation © Prof. Dr. François E. Cellier Internal Seminar: Groups Arbenz/Cellier/Gander/Hinterberger May 29, 2007 Population Dynamics III Population Dynamics Macro-economy Food Demand Food Supply

56. Start Presentation © Prof. Dr. François E. Cellier Internal Seminar: Groups Arbenz/Cellier/Gander/Hinterberger May 29, 2007 Population Dynamics IV Population Dynamics Macro-economy Food Demand Food Supply Average error found with original model Average error found using model with age groups and demographics

57. Start Presentation © Prof. Dr. François E. Cellier Internal Seminar: Groups Arbenz/Cellier/Gander/Hinterberger May 29, 2007 Macro-economy IV Population Dynamics Macro-economy Food Demand Food Supply

58. Start Presentation © Prof. Dr. François E. Cellier Internal Seminar: Groups Arbenz/Cellier/Gander/Hinterberger May 29, 2007 Food Demand Population Dynamics Macro-economy Food Demand Food Supply

59. Start Presentation © Prof. Dr. François E. Cellier Internal Seminar: Groups Arbenz/Cellier/Gander/Hinterberger May 29, 2007 Food Supply Population Dynamics Macro-economy Food Demand Food Supply

60. Start Presentation © Prof. Dr. François E. Cellier Internal Seminar: Groups Arbenz/Cellier/Gander/Hinterberger May 29, 2007 Discussion IV Making use of the more generic layers of the multi-layer architecture in making predictions has consistently helped in reducing the average prediction error. The same architecture can be applied to any segment of the economy, i.e., if the application changes, only the application layer needs to be re-identified. The more generic layers of the architecture are invariant to the application at hand.

61. Start Presentation © Prof. Dr. François E. Cellier Internal Seminar: Groups Arbenz/Cellier/Gander/Hinterberger May 29, 2007 Conclusions Fuzzy Inductive Reasoning offers an exciting alternative to neural networks for modeling systems from observations of behavior. Fuzzy Inductive Reasoning is highly robust when used correctly. Fuzzy Inductive Reasoning features a model synthesis capability rather than a model learning approach. It is therefore quite fast in setting up the model. Fuzzy Inductive Reasoning offers a self-assessment feature, which is easily the most important characteristic of the methodology. Fuzzy Inductive Reasoning is a practical tool with many industrial applications. Contrary to most other qualitative modeling techniques, FIR does not suffer in major ways from scale-up problems.

62. Start Presentation © Prof. Dr. François E. Cellier Internal Seminar: Groups Arbenz/Cellier/Gander/Hinterberger May 29, 2007 Industrial Applications Cardiovascular System Modeling for Classification of Anomalies. Anaesthesiology Model for Control of Depth of Anaesthesia During Surgery. Shrimp Growth Model for El Remolino Shrimp Farm in Northern Mexico. Prediction of Water Demand in Barcelona and Rotterdam. Design of Fuzzy Controller for Tanker Ship Steering. Fault Diagnosis of Nuclear Power Plants. Prediction of Technology Changes in Telecommunication Industry.

63. Start Presentation © Prof. Dr. François E. Cellier Internal Seminar: Groups Arbenz/Cellier/Gander/Hinterberger May 29, 2007 Recent Ph.D. Dissertations Nebot, A. (1994), Qualitative Modeling and Simulation of Biomedical Systems Using Fuzzy Inductive Reasoning.Qualitative Modeling and Simulation of Biomedical Systems Using Fuzzy Inductive Reasoning Mugica, F. (1995), Diseño Sistemático de Controladores Difusos Usando Razonamiento Inductivo.Diseño Sistemático de Controladores Difusos Usando Razonamiento Inductivo de Albornoz, A. (1996), Inductive Reasoning and Reconstruction Analysis: Two Complementary Tools for Qualitative Fault Monitoring of Large-Scale Systems.Inductive Reasoning and Reconstruction Analysis: Two Complementary Tools for Qualitative Fault Monitoring of Large-Scale Systems López, J. (1999), Qualitative Modeling and Simulation of Time Series Using Fuzzy Inductive Reasoning.Qualitative Modeling and Simulation of Time Series Using Fuzzy Inductive Reasoning Mirats, J.M. (2001), Large-Scale System Modeling Using Fuzzy Inductive Reasoning.Large-Scale System Modeling Using Fuzzy Inductive Reasoning