SOFT COMPUTING METHODOLOGY APPLIED TO TIME SERIES ANALYSIS

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SOFT COMPUTING METHODOLOGY APPLIED TO TIME SERIES
ANALYSIS

• Barbara Diaz
• Visiting Scholar, BISC Program, UC. Berkeley
• Socio-economical Analysis Methods (MAS fuzzy)
• University of Malaga, Spain

BISC Seminar, October 12
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Outline

• Objective, data and methodology
• Results in a classical approach
• Neuro-fuzzy approach
• Feature selection
• Input space partition
• Fuzzy inference system
• Parameter optimization
• Results
• Conclusions and further research activities

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FIS applied to wage employment in Spain. Study
and prediction.

• OBJECTIVE
• Evaluate the capacity of fuzzy inference systems
  and classic methodologies from two points of
  view
• Prediction
• Conceptual interpretation of economic theory
• DATA
• Data is quarterly based (22 years data)
• The information sources are
• Active Population Survey (EPA)
• Industry and Services Salary Survey (ESIS)
• Quarterly National Accounting (CNT)

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METHODOLOGY

• Classic time series model. AR(I)MA
• Fuzzy Inference System with Takagi-Sugeno Kang
  first order rules. ANFIS.
• The predictions are compared over the last five
  quarterly periods

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PREDICTION OF WAGE EMPLOYMENT IN SPAIN

• Labor demand Modelled by a single equation
  considering wages as an exogenous variable,
  focusing on demand factors
• The labor demand is a derivate demand -- an
  increase in the demand of goods and services
  implies an increase in labor demand
• Change in the relative costs for the labor factor
  is studied through the relationship between
  wages and productivity

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WAGES AND PRODUCTIVITY

• ?Productivity ??employment?
• ?Productivity ? ? salaries ? ? demand ? ?
  production ? ? employment?
• Salaries
• Substitution effect
• ?salaries (production unchanged) ? ? employment
• Product effect (supply)
• ?salaries ? ? employment ? ? production ? ?
  employment
• Product effect (demand)
• ? salaries ? ? demand ? ? production ? ?
  employment

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VARIABLES

• Private, non-agricultural wage earners (asan)
• Private, non-agricultural gross added value
  (VABnap)
• Wages (sal)
• Productivity In place of marginal productivity,
  mean productivity is calculated as output divided
  by employed people. To improve the approximation
  of productivity for full employment, the maximum
  historical occupation productivity is used

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PRODUCTIVITY (pr)
VARIABLES
WAGE EARNERS (asan)
GROSS ADDED VALUE(VABnap)
Source Encuesta de Población Activa. INE
Source Contabilidad Nacional Trimestral. INE
WAGES (sal)
Source Contabilidad Nacional Trimestral,
Encuesta de Población Activa.INE
Source Encuesta de salarios de la industria y
los servicios. INE
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TRANSFORMATIONS

• For variables asan and VABnap the basic growth
  series is used

• The unitary costs are

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Wage earners growth rate ASANGross added value
growth rate VABNAPUnitary labor costs SUP
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VARIABLES RELATIONSHIP
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REGRESSION (I)

• Vt was generated by an ARMA(p,q) model
• Lagged variables werent significant for SUP
• VABNAPt-1 is significant, but we lost
  significance in the other variables, hence there
  was no improvement
• A variable t was introduced but was not found to
  be statistically significant

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REGRESSION ADJUSTMENT (II)

• Breusch-Godfrey test, and the correlograms
  observation gave as a result a model for the
  residuals with AR(1) MA(4)
• This improved the adjustment, but it didnt reach
  the predictions results in FIS
• The Theil inequality coefficient is 0.130063.
  Different kind of errors BIAS, 78, Variance,
  0.58, covariance (random), 20

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FUZZY INFERENCE SYSTEM (Methodology)
Input Variable Selection
RC/SRC
Input Space Partition
Subtractive Clustering
Fuzzy Inference System (F.I.S.)
Parameters Optimization
ANFIS
Application to Prediction
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INPUT SELECTION. RC method.Sugeno Yasukawa
(1993)
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INPUT SELECTION.SRC methodTanka, Sano Watanabe
(1995)
2n-1 n(n1)/2
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INPUT VARIABLE SELECTION
Y ASAN x1 VABNAP x2 SUP x3 t
SRC algorithm
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INPUT SPACE PARTITION (Subtractive clustering)
Chiu (1994)
ra
c1
rb
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INPUT SPACE PARTITION
Subtractive clustering
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Initial membership functions
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FUZZY INFERENCE SYSTEM (F.I.S.) Takagi-Sugeno-Kang
if x is A and y is B then zpxqy
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PARAMETER OPTIMIZATION (ANFIS) Jang (1993)
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PARAMETER OPTIMIZATION (ANFIS) Jang (1993).
Hybrid learning algorithm
Least squares
Error back propagation
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Optimized fuzzy membership functions
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FUZZY RULES
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VARIABLE RELATIONSHIPS (F.I.S.)
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F.I.S. RESULTS
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COMPARISON TO REGRESSION
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COMPARISON BETWEEN PREDICTIONS
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COMPARISON BETWEEN PREDICTIONS (II)

• Theil inequality coefficient in FIS is 0.046426,
  Different kind of errors BIAS, 9.3, Variance,
  19.8, covariance (random), 70.8
• Theil inequality coefficient in regression is
  0.130063. Different kind of errors BIAS, 78,
  Variance, 0.58, covariance (random), 20

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COMPARISON BETWEEN SYSTEM ADJUSTMENTS
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Conclusions and further research activities

• Good approximation and prediction results
• Good to model non linear relationships
• Risk of overfitting
• Possible Improvements
• feature selection
• Prediction intervals
• Other approaches
• Mamdani fuzzy rules and supervised competitive
  learning techniques. (confidence intervals
  defuzzification methods. (Dusan Marcek, 2004)
• Qualitative description of time series based on
  moving approximations.(Batyrshin et al, 2004).
• Fuzzy clustering of short time series (2003).
• Support vector regression applied to time series
  analysis..