Bodhisattva Sen

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• Bodhisattva Sen

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Topics

• Fractile Graphical Analysis (FGA) definition
  motivation Prof. Mahalanobis Method
  Statistical Inference Using FGA ours ideas.
• Two notions of Multivariate Quantiles
  Geometric Quantiles and PCMs Fractile.
• Extension of FGA to the Multiple Covariate Setup
  Using the above two notions of Quantiles.
• Evaluation of the performance of the methods
  using synthetic data and real life data.

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Want to compare the regression functions for two
populations with single covariate.

• We could look at

• Instead we look at

and
and
where F, F are the distribution functions of X
and X.
When X and X are not in comparable scales the
above does the necessary standardization.
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Mahalanobis Idea for Comparing Fractile
Graphs(Econometrica,1960)

• Divide the dataset into fractile groups
  according to the x-variable. Plot the y-averages
  for each fractile group and join the consecutive
  points (y-averages) by straight lines.
• Compute the area between the two fractile graphs
  for the two different samples.This is called the
  separation area.
• Using two sub samples from the same population
  the significance of the observed separation area
  is tested.

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Limitations of Mahalanobis Idea

• The exact distribution of the test statistic
  separation area is not known.Only some
  approximations were tried by Mahalanobis and his
  co-workers.
• He divided the original dataset into two parts
  and used the area between the two sub-samples
  (called error area) to test the significance of
  the observed separation area. The error area
  calculated in this way has different variance
  than the separation area due to the decrease in
  the number of sample points.
• The accuracy of the approximation was poor and
  it was known to Mahalanobis.

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Statistical Inference in FGAOur Ideas

• We discuss two methods to test the equality of
  the regression functions (single covariate).
• Swap the Response, (Method I)
• Resampling from the Joint Density of (X,Y)
  (Method II).

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Swap the Response

• Transform the covariates into the corresponding
  quantiles, i,e, the ith ranked x-value is
  transformed to i/(n1) (where n data points).
• Draw the fractile graphs by smoothing the
  y-values using usual kernel regression estimates
  and compute the separation area between them.
• To test the significance of the observed
  separation area we resample from the two
  samples in the following way

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We form two resampled datasets and compute the
separation area between them. Case (I)
Suppose n1n2 (i.,e., both the samples are of the
same size). For the ith data point (i.,e., with
quantile value i/(n1)) we interchange or swap
the y-value for the two datasets with probability
0.5 and keep the original y-values with
probability 0.5. Case (II) The sample sizes
are different. We interpolate the dataset from
the 2nd population to find the y-value at
i/(n11)th quantile and then repeat the same idea
as above. Similarly we get the other resampled
dataset. (Modified Swap Method)
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Resample from the Joint Density of (X,Y)

• We use usual kernel density estimates with very
  small bandwidth (bandwidth proportional to the
  inverse of sample size) to generate a pair of
  resampled datasets from each population. It is
  like resampling from a smoothed version of the
  empirical distribution of (X,Y).
• We compute the separation area for each pair
  of resampled datasets from the same population.
  It gives us the distribution of the separation
  area under Ho.(Note that we have 2 separation
  areas corresponding to the 2 populations).

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Plots of datasets along with the Fractile
Graphs. The models are Y 1.0 X e and Y
1.2 X e, where eN(0,0.09),XN(0,1) where N
data points 100.
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Swap the Response Method an illustration
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Resample from Joint Distribution of (X,Y) an
illustration.
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How do we define Quantiles in the Multivariate
setup?
We discuss two such notions of
Multivariate Quantiles

• 1.Geometric Quantiles and 2.PCMs Fractile

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Geometric Quantiles
Chaudhuri (JASA 1996), Koltchinskii (Annals of
Stat. 1997)

• If XP (X in Rd) then we define the Geometric
  Quantile QP(u) for u in B(d) u u lt 1 as
  

where
and the norm is the usual Euclidean norm
and lt , gt denotes the usual inner product.
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Properties of Geometric Quantiles ?

• The solution QP(u) always exist for any u and it
  is unique if P is not supported on a straight
  line.
• QP(u) characterizes the associated distribution,
  i.e, QP1(u) QP2(u) implies P1P2 .
• Computation of the sample geometric quantile for
  the data data set X1,X2,Xn ,via

is straightforward.

• The Geometric Quantiles have an asymptotic
• Multivariate normal distribution and the
• convergence rate is n1/2.

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Drawbacks of Geometric Quantile

• No simple distributional interpretation as was
  the case with the univariate quantiles.
• Though the Geometric Quantile is equivariant with
  respect to change in shift,any orthogonal
  transformation and homogeneous scale
  transformation, it is not equivariant under
  heterogeneous scale transformation.
• (Affine equivariant versions of Geometric
  Quantiles are defined using Transformation-Retrans
  formation methods.) (see On Afiine Equivariant
  Multivariate Quantiles, Biman Chakraborty, Ann.
  Inst. Stat. Math, Vol. 53, No.2, 380-403 (2001))
  . However we do not intend to consider it here.

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Another notion of Multivariate Quantile PCMs
Fractile - Mahalanobis(1970)
For a d-dimensional random vector X
(X1,X2,,Xd) P, we define the PCMs
distribution function HP from Rd to 0,1d as
where
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Why should we consider PCMs fractile ?

• If P and Q are two distributions with continuous
  densities of Rd and HP(x) HQ(x) for all x in Rd
  then
• P Q.
• Define Z(Z1,Z2,,Zd) Q s.t Zi fi (xi) for
  all i1,,d where each fi is a strictly monotone
  function on R. Then HP(x) HQ(f (x)) where f(x)
  (f1(x1), f2(x2),,fd(xd)). This shows that the
  PCMs fractile is equivariant under co-ordinate
  wise monotonic transformations.
• The PCMs fractile has simple probabilistic
• interpretations.

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Drawbacks of PCMs Fractile

• Requires the computation of the conditional
  distributions which is estimated by kernel
  density estimation.This estimation is very
  unstable in high dimensions due to the lack of
  data points. It is also computationally very
  intensive and time consuming.
• PCMs Fractile map is not n1/2 consistent as it
  uses density estimates which are nv consistent
  (vlt1/2). v may be much smaller than ½ when the
  covariate dimension is high(Curse of
  Dimensionality in Non Parametric Function
  Estimation).
• PCMs Fractile depends on the ordering of the
  co-ordinate random variables we have to fix an
  order and then work with it.

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Extension of Fractile Graphical Analysis in the
Multiple Covariate Case

• We use two different notions of Multivariate
  Quantiles
• (Geometric Quantiles and PCMs Fractile).
• We extend the method of Resampling from the
  joint density of (X,Y) using both the above
  notions of Multivariate Quantiles.

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Methodology using PCMs Fractile

• Transform the datasets to

and
where H is the PCMs distribution function.

• Smooth the two transformed datasets by using the
  usual multivariate kernel regression estimates to
  get the two fractile surfaces.

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3. The difference between the two fractile
graphs gives the separation volume between the
two populations. 4. To find the significance
level for the observed volume we resample a
pair of data sets from the same population using
very small bandwidth(from a fitted joint
density of X and Y) and recalculate the
volumebetween the two graphs for the same
population.In this way we try to estimate the
distribution of the separation volume under Ho.
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Remarks

• We used Least Square Cross Validation methods in
  choosing the bandwidth parameters in kernel
  regression and kernel density estimates (note
  that we use LSCV optimal bandwidths for KDE for
  the transformation x?H(x)).
• The method described depends highly on the
  fitted joint distribution of the covariate
  (i.,e., X) as we are computing the Quantiles
  from this kernel density estimate.

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Methodology using Geometric Quantiles

• We transform the covariates by using the
  tranformation x ? FP(x) where FP is the
  M-distribution function. The empirical
  M-distribution function is 1/n Si (x-Xi)
  (x-Xi)-1.
• We use the kernel regression estimates to smooth
  the transformed data sets using LSCV (Least
  Squares Cross Validation) optimal bandwidths.

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• We compute the fractile volume between the two
  surfaces (for the two populations).
• We resample a pair of datasets from each
  population using kernel density estimates with
  very small bandwidths and compute the
  resampled fractile volume.In this way we
  simulate the distribution of the fractile volume
  under both the populations separately.

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Remarks

• For choosing the kernel regression bandwidths
  for X we assume h1h2hd where hi is the
  bandwidth for Xi (note that X (X1,Xd)).This
  makes the computation relatively simple and
  computationally feasible.
• This method has the advantage that we do not
  need any LSCV optimal kernel density estimates as
  was the case with PCMs Fractile.

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Evaluation of the Methods using Synthetic Data

• Models considered
• Single covariate
• Y a bX e ,where e N(0,s2), XN(0,1).
• Y 1/(1 X2k) e, where k0.5,1.0,1.5,2.0,2.5.
  
• Y cs X2s eX2 e , where s1,2,3,4 cs
  normalizing constant
• Multiple covariates (2 covariates)
• Y a b1X1 b2X2 e ,where e N(0,s2),
  X1N(0,1), X2N(0,1).

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Remarks

• The Swap Method has a natural tendency to lower
  the resampled fractile area. Thus it exhibits
  high power but has high observed level also.
• The Swap Method essentially works on the number
  of crossings of the two datasets.
• The Swap Method works better when the error
  variance is large compared to the variance of X.

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• As the difference in the variance of the error
  terms in the two models increase, the two
  P-values for the Resampling from Joint Density
  Method show significant difference.
• The Resampling from Joint Density Method gives
  good results in all the models considered by us
  whereas the Swap the Response Method behaves
  very unsatisfactorily in the two nonlinear models
  considered.
• We recommend the Resampling from Joint Density
  method.

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Case Studies

• State wise data from scheduled commercial banks -
  RBI BSR data
• Amount Outstanding (as a fraction of Credit
  Limit) VS Credit Limit.
• Credit Deposit Ratio VS Total Deposit.
• Credit deposit Ratio VS Distribution of Employees
  (officers,clerks and sub-ordinates) (2
  covariates and 3 covariates).
• Credit Deposit Ratio VS accounts and offices.
• RBI data on private corporate firms
• Gross Company Profits VS Sales and Paid Up
  Capital.

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Amount Outstanding (as a ratio of Credit Limit)
VS Credit Limit
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Credit Deposit Ratio VS Total Deposit
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Grey Scale Image of the P-values for the two
methods with varying kernel regression
bandwidths. (Amount Outstanding (as ratio of
Credit Limit) VS Credit Limit for the year 1996
and 1998).
Using 1996
Using 1998
Resample from the Joint density of (X,Y)
Swap the Response Method
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Grey Scale Image of the P-values for the two
methods with varying kernel regression
bandwidths. (Amount Outstanding (as ratio of
Credit Limit) VS Credit Limit for the year 1997
and 2000).
Using 1997
Using 2000
Resample from the Joint density of (X,Y)
Swap the Response Method
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Remarks (for Single Covariate)

• The fractile graph of the year 2002 is very
  different from the other years.
• Over the years the ratio Amount Ouststanding to
  Credit Limit has decreased.
• For the pair wise comparison tests between
  Credit Deposit Ratio and Total Deposit most of
  the P-values for the are very high.The the
  fractile graph for the year 2002 is somewhat
  different from the rest.

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Remarks (for Multiple Covariates)

• With the inclusion of the subordinates the
  regressions functions change (as indicated by the
  change in P-values).
• The subordinates have decreased steadily over
  the years from 1996 to 2002.
• The P-values for the pair wise comparisons for
  Credit Deposit Ratio VS offices and accounts
  are very high.

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Thank you