Multidimensional Exploratory Analysis of a Structural Model using a general costructure criterion: THEME (THematic Equation Model Explorator) X. Bry I3M, Univ. Montpellier II T. Verron ITG - SEITA, Centre de recherche P. Redont I3M, Univ. Montpellier II
THEME - Bry, Redont, Verron ; COMPSTAT 2010 Introducing the Data and Problem: 52 Variables: 15 var. 5 var. 8 var. 3 var. 3 var. 9 var. 9 var. Data: Data: Tobacco Blend Paper Tobacco Blend Filtration Hoffmann Filter Hoffmann Chemistry Combustion Combustion / ISO behaviour smoke contents smoke contents smoking / ISO /Intense smoking /ISO smoking smoking 19 CIGARETTE SMOKE Observations: Cigarettes Problem: Regulations → Hoffmann Compounds control ⇒ HC modeling
THEME - Bry, Redont, Verron ; COMPSTAT 2010 Introducing the Data and Problem: 52 Variables: 15 var. 5 var. 8 var. 3 var. 3 var. 9 var. 9 var. Data: Data: Tobacco Blend Paper Tobacco Blend Filtration Hoffmann Filter Hoffmann Chemistry Combustion Combustion / ISO behaviour smoke contents smoke contents smoking / ISO /Intense smoking /ISO smoking smoking 19 Observations: Cigarettes Problem: Regulations → Hoffmann Compounds control ⇒ HC modeling 1) The thematic partitioning of variables must be kept (to separate roles , and keep explanatory ) ⇒ Dimension reduction in groups 2) Many (redundant) variables ⇒ Look for dimensions: reflecting their group's structure & interpretable with respect to their theme
THEME - Bry, Redont, Verron ; COMPSTAT 2010 Introducing the Data and Problem: Dependency network of Data: Dependency network of Data: Thematic (conceptual) model Model design motivations : Equation 1 : X 1: Tob Ch X 5: Fil Iso Hoffmann compounds are generated / transferred to X 6: Hoff Iso smoke through combustion. Filter only plays a X 2: Cb Pap retention role (pores blocked in intense mode) X 7: Hoff Int X 3: Cb Blend Equation 2 : Equation 2 Final output of Hoffmann compounds is conditioned X 4: Cb Fil by other filter properties, as ventilation/dilution. Equation 1
THEME - Bry, Redont, Verron ; COMPSTAT 2010 Introducing the Data and Problem: Dependency network of Data: Dependency network of Data: Thematic (conceptual) model Model design motivations : Equation 1 : X 1: Tob Ch X 5: Fil Iso Hoffmann compounds are generated / transferred to X 6: Hoff Iso smoke through combustion. Filter only plays a X 2: Cb Pap retention role (pores blocked in intense mode) X 7: Hoff Int X 3: Cb Blend Equation 2 : Equation 2 Final output of Hoffmann compounds is conditioned X 4: Cb Fil by other filter properties, as ventilation/dilution. Equation 1 ⇒ Structural dimensions should be informative with respect to the model too 1) How many dimensions do play a proper role? 2) Which ?
THEME - Bry, Redont, Verron ; COMPSTAT 2010 Path modeling methods optimizing a criterion: ● Likelihood → LISREL (Jöreskog 1975-2002) ● Residual Sum of Squares → Multiblock Multiway Components and Covariates Regression Models (Smilde, Westerhuis, Bocqué 2000) Generalized structured component analysis (Hwang, Takane, 2004). RSS(group models) RSS = < Y > + RSS(component-based model) (minimized via A lternated L east S quares) < X 1 > < X 2 > ➔ Model residuals need weighting: How? ➔ Convergence problems in case of collinearity (small samples) ➔ The Methods do not extend PLS Regression to K Predictor Groups. based on a covariance criterion...
THEME - Bry, Redont, Verron ; COMPSTAT 2010 Extending covariance ● Multiple Covariance (Bry, Verron, Cazes 2009) y being linearly modeled as a function of x 1 ,..., x S , Multiple Covariance of y on x 1 ,..., x S is: S = [ V y ∏ S ] s R 1 S 1 , ... , x 2 2 y ∣ x 1 , ... ,x MC y ∣ x V x s = 1 Product of Linear Model all variances Fit ● Use for single « equation » structural model estimation: SEER (Bry, Verron, Cazes 2009) ➢ One component per group: 2 Yv ∣ X 1 u 1 , ... , X R u R max MC v , u 1 , ... , u R g | f 1 , … , f R X 1 X R ∥ v ∥ 2 = 1 2 = 1 ∀ r, ∥ u r ∥ f R f 1 ... g Y
THEME - Bry, Redont, Verron ; COMPSTAT 2010 Extending covariance ● Multiple Covariance (Bry, Verron, Cazes 2009) y being linearly modeled as a function of x 1 ,..., x S , Multiple Covariance of y on x 1 ,..., x S is: S = [ V y ∏ S ] s R 1 S 1 , ... , x 2 2 y ∣ x 1 , ... ,x MC y ∣ x V x s = 1 Product of Linear Model all variances Fit ● Use for single « equation » structural model estimation: SEER (Bry, Verron, Cazes 2009) ➢ One component per group: 2 Yv ∣ X 1 u 1 , ... , X R u R max MC v , u 1 , ... , u R g | f 1 , … , f R X 1 X R ∥ v ∥ 2 = 1 2 = 1 ∀ r, ∥ u r ∥ f R f 1 ... → The weighting of Groups is naturally balanced 2 = 0 ∇ log MC ⇔ relative variations compensate → The Method extends PLS Regression to K g Y Predictor Groups
THEME - Bry, Redont, Verron ; COMPSTAT 2010 Extending covariance ● Multiple Covariance (Bry, Verron, Cazes 2009) y being linearly modeled as a function of x 1 ,..., x S , Multiple Covariance of y on x 1 ,..., x S is: S = [ V y ∏ S ] s R 1 S 1 , ... , x 2 2 y ∣ x 1 , ... ,x MC y ∣ x V x s = 1 Product of Linear Model all variances Fit ● Use for single « equation » structural model estimation: SEER (Bry, Verron, Cazes 2009) ➢ One component per group: 2 Yv ∣ X 1 u 1 , ... , X R u R max MC v , u 1 , ... , u R g | f 1 , … , f R X 1 X R ∥ v ∥ 2 = 1 2 = 1 ∀ r, ∥ u r ∥ 1 ⊥ … f R K f R f 1 ... → The weighting of Groups is naturally balanced 2 = 0 ∇ log MC ⇔ relative variations compensate → The Method extends PLS Regression to K 1 ⊥ … g L g Y Predictor Groups ➢ Several components per group: → Model Local Nesting Principle: Xr 's components f r 1 , f r 2 ... are mutually ⊥ and calculated sequentially in one batch, controlling for all components in the other groups
THEME - Bry, Redont, Verron ; COMPSTAT 2010 Extending covariance ● Multiple Covariance (Bry, Verron, Cazes 2009) y being linearly modeled as a function of x 1 ,..., x S , Multiple Covariance of y on x 1 ,..., x S is: S = [ V y ∏ S ] s R 1 S 1 , ... , x 2 2 y ∣ x 1 , ... ,x MC y ∣ x V x s = 1 Product of Linear Model all variances Fit ● Use for single « equation » structural model estimation: SEER (Bry, Verron, Cazes 2009) ➢ One component per group: 2 Yv ∣ X 1 u 1 , ... , X R u R max MC v , u 1 , ... , u R g | f 1 , … , f R X 1 X R ∥ v ∥ 2 = 1 2 = 1 ∀ r, ∥ u r ∥ 1 ⊥ … f R K 1 ⊥ f 1 2 f R f 1 ... → The weighting of Groups is naturally balanced 2 = 0 ∇ log MC ⇔ relative variations compensate → The Method extends PLS Regression to K 1 ⊥ … g L g Y Predictor Groups ➢ Several components per group: → Model Local Nesting Principle: Xr 's components f r 1 , f r 2 ... are mutually ⊥ and calculated sequentially in one batch, controlling for all components in the other groups
THEME - Bry, Redont, Verron ; COMPSTAT 2010 Extending covariance ● Multiple Covariance (Bry, Verron, Cazes 2009) y being linearly modeled as a function of x 1 ,..., x S , Multiple Covariance of y on x 1 ,..., x S is: S = [ V y ∏ S ] s R 1 S 1 , ... , x 2 2 y ∣ x 1 , ... ,x MC y ∣ x V x s = 1 Product of Linear Model all variances Fit ● Use for single « equation » structural model estimation: SEER (Bry, Verron, Cazes 2009) ➢ One component per group: 2 Yv ∣ X 1 u 1 , ... , X R u R max MC v , u 1 , ... , u R g | f 1 , … , f R X 1 X R ∥ v ∥ 2 = 1 2 = 1 ∀ r, ∥ u r ∥ 1 ⊥ … f R K 1 ⊥ f 1 2 ⊥ ... f R f 1 ... → The weighting of Groups is naturally balanced 2 = 0 ∇ log MC ⇔ relative variations compensate → The Method extends PLS Regression to K 1 ⊥ … g L g Y Predictor Groups ➢ Several components per group: → Model Local Nesting Principle: Xr 's components f r 1 , f r 2 ... are mutually ⊥ and calculated sequentially in one batch, controlling for all components in the other groups
THEME - Bry, Redont, Verron ; COMPSTAT 2010 Extending covariance Predictor space < X > ● Beyond Covariance: Costructure Bundle A ➢ Broadened approach to structural strength Bundle B
THEME - Bry, Redont, Verron ; COMPSTAT 2010 Extending covariance Predictor space < X > ● Beyond Covariance: Costructure PC2 Bundle A ➢ Broadened approach to structural strength PC1 Bundle B
THEME - Bry, Redont, Verron ; COMPSTAT 2010 Extending covariance Predictor space < X > ● Beyond Covariance: Costructure PC2 Bundle A ➢ Broadened approach to structural strength PC1 OLS predictor Bundle B
THEME - Bry, Redont, Verron ; COMPSTAT 2010 Extending covariance Predictor space < X > ● Beyond Covariance: Costructure PC2 Bundle A ➢ Broadened approach to structural strength PC1 original THEME predictor OLS predictor Bundle B
THEME - Bry, Redont, Verron ; COMPSTAT 2010 Extending covariance Predictor space < X > ● Beyond Covariance: Costructure PC2 Bundle A ➢ Broadened approach to structural strength PC1 original THEME predictor OLS predictor ➢ General Costructure Criterion Bundle B ∀ component f r = X r u r , V ( f r ) = u r 'X r' PX r u r is replaced by: S u r = ∑ a u r ' A h u r h = 1, H a = bundle focus parameter
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