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VI.7.  The Johnson relative weight
VI.7.  The Johnson relative weight
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VI.7.  The Johnson relative weight

This section introduces indices whose purpose is mainly to obtain good estimators of the Shapley's values defined in [metho]. The underlying assumption is to state that the model can be considered linear so that the results can be considered as proper estimation of the Shapley indices (with or without correlation between the input variables).

VI.7.1. General overview

In Uranie, computing sensitivity indices with Johnson's relative weights method is dealt with the eponymous class TJohnsonRW which inherits from the TSensitivity class.

This class handles all the steps to compute the indices depending on provided informations:

  • one can only provide a correlation matrix without data. The indices would be estimated as such.
  • one can provide a sample containing input and outputs variables. From there the class will compute the correlation matrix and estimate the indeces from them.
  • If one provides either a code, a function, or a runner (see Section VIII.4) then, the class can
    • generate the deterministic sample if no data are found;
    • run the code, the analytic function or the evaluator if the output is not provided as well;
    • computing the indices.

VI.7.1.1. Example: simple computation of Johnson's relative wieght using a function

The example script below uses the TJohnsonRW class to compute and display the relative weights: 

"""
Example of Johnson relative weight method applied to flowrate
"""
from rootlogon import ROOT, DataServer, Sensitivity

ROOT.gROOT.LoadMacro("UserFunctions.C")

# Define the DataServer
tds = DataServer.TDataServer("tdsflowrate", "DataBase flowrate")
tds.addAttribute(DataServer.TUniformDistribution("rw", 0.05, 0.15))
tds.addAttribute(DataServer.TUniformDistribution("r", 100.0, 50000.0))
tds.addAttribute(DataServer.TUniformDistribution("tu", 63070.0, 115600.0))
tds.addAttribute(DataServer.TUniformDistribution("tl", 63.1, 116.0))
tds.addAttribute(DataServer.TUniformDistribution("hu", 990.0, 1110.0))
tds.addAttribute(DataServer.TUniformDistribution("hl", 700.0, 820.0))
tds.addAttribute(DataServer.TUniformDistribution("l", 1120.0, 1680.0))
tds.addAttribute(DataServer.TUniformDistribution("kw", 9855.0, 12045.0))

# param Size of a sampling.
nS = 1000
FuncName = "flowrateModel"

tjrw = Sensitivity.TJohnsonRW(tds, FuncName, nS,
                              "rw:r:tu:tl:hu:hl:l:kw", FuncName)
tjrw.computeIndexes()

# Get the results on screen
tjrw.getResultTuple().Scan("Out:Inp:Method:Value", "Order==\"First\"")

# Get the results as plots
cc = ROOT.TCanvas("canhist", "histgramme")
tjrw.drawIndexes("Flowrate", "", "nonewcanv,hist,first")
cc.Print("appliUranieFlowrateJohnsonRW1000Histogram_py.png")

ccc = ROOT.TCanvas("canpie", "TPie")
tjrw.drawIndexes("Flowrate", "", "nonewcanv,pie")
ccc.Print("appliUranieFlowrateJohnsonRW1000Pie_py.png")

The figures resulting from this script are shown below and display the sensitivity indices in a histogram and in a pie chart. The structure of both plots and tables, as they are common with the rest of the sensitivity class, provides the results in the ntuple that can be reached with getResultTuple(). This explains why the results are stored in the Sobol column.

Figure VI.13. Histogram of JohnsonRW's indices

Histogram of JohnsonRW's indices

Figure VI.14. Pie chart of JohnsonRW's indices

Pie chart of JohnsonRW's indices

VI.7.1.2. Specifying the input parameters

First, define the uncertain parameters and add them to a TDataServer object:

# Define the DataServer
tds = DataServer.TDataServer("tdsflowreate", "DataBase flowreate")
tds.addAttribute(DataServer.TUniformDistribution("rw", 0.05, 0.15))
tds.addAttribute(DataServer.TUniformDistribution("r", 100.0, 50000.0))
tds.addAttribute(DataServer.TUniformDistribution("tu", 63070.0, 115600.0))
tds.addAttribute(DataServer.TUniformDistribution("tl", 63.1, 116.0))
tds.addAttribute(DataServer.TUniformDistribution("hu", 990.0, 1110.0))
tds.addAttribute(DataServer.TUniformDistribution("hl", 700.0, 820.0))
tds.addAttribute(DataServer.TUniformDistribution("l", 1120.0, 1680.0))
tds.addAttribute(DataServer.TUniformDistribution("kw", 9855.0, 12045.0))

VI.7.1.3. TJohnsonRW constructor

There are four different kinds of constructors to build a TJohnsonRW object, each corresponding to a different problem:

  • the model is either provided data or just a correlation matrix;
  • the model is an analytic function run by Uranie,
  • the model is a code run by Uranie.
  • the model is either a function or a code and the problem is specified through a Relauncher architecture.

VI.7.1.3.1. TJohnsonRW constructor with provided data

The constructor prototype used when data are provide (or one will not used data but just a correlation matrix): 

# Create a TJohnsonRW object data
TJohnsonRW(tds, inp,  out , Option = "")

This constructor takes up to four arguments:

  • a pointer to a TDataServer object,
  • a string to specify the names of the input factors separated by ':' (ex. "rw:r:tu:tl:hu:hl:l:kw"), no default is accepted,
  • a string to specify the names of the output variables of the model, no default is accepted.
  • a string to specify the options, by default empty "";

Here is an example of how to use the constructor with provided data: 

tjrw = Sensitivity.TJohnsonRW(tds, "rw:r:tu:tl:hu:hl:l:kw", "flowrateModel")

The way to inject the correlation matrix is discussed below.

VI.7.1.3.2. TJohnsonRW constructor for an analytic function

The constructor prototype used with an analytic function is: 

# Create a TJohnsonRW object with an analytic function
TJohnsonRW(tds, fcn, inp, out, ns)  # fcn is a function pointer
TJohnsonRW(tds, fcn, ns, inp = "", out = "")  # fcn is string

This constructor takes five arguments:

  • a pointer to a TDataServer object,
  • a pointer to an analytic function (either a void or a const char that represents the function's name when it has been loaded in ROOT's memory),
  • an integer to specify the number of simulations, its default value is 100,
  • a string to specify the names of the input factors separated by ':' (ex. "rw:r:tu:tl:hu:hl:l:kw"), its default value might be the empty string "",
  • a string to specify the names of the output variables of the model, its default value might be the empty string "".

Here is an example of how to use the constructor with an analytic function: 

nS = 50
tjrw_func = Sensitivity.TJohnsonRW(tds, "flowrateModel", nS, "rw:r:tu:tl:hu:hl:l:kw", "flowrateModel")
VI.7.1.3.3. TJohnsonRW constructor for a code

The constructor prototype used with a code is: 

# Create a TJohnsonRW object with a code
TJohnsonRW(tds, fcode, ns, option = "")

This constructor takes three arguments:

  • a pointer to a TDataServer object,
  • a pointer to a TCode,
  • an integer to specify the number of simulations.

Here is an example of the use of this constructor on the flowrate case: 

# The reference input file
sJDDReference = "flowrate_input_with_keys.in"

# Set the reference input file and the key for each input attributes
tds.getAttribute("rw").setFileKey(sJDDReference, "Rw")
tds.getAttribute("r").setFileKey(sJDDReference, "R")
tds.getAttribute("tu").setFileKey(sJDDReference, "Tu")
tds.getAttribute("tl").setFileKey(sJDDReference, "Tl")
tds.getAttribute("hu").setFileKey(sJDDReference, "Hu")
tds.getAttribute("hl").setFileKey(sJDDReference, "Hl")
tds.getAttribute("l").setFileKey(sJDDReference, "L")
tds.getAttribute("kw").setFileKey(sJDDReference, "Kw")

# The output file of the code
fout = Launcher.TOutputFileRow("_output_flowrate_withRow_.dat")
# The attribute in the output file
fout.addAttribute(DataServer.TAttribute("yhat"))

# Instanciation de mon code
mycode = Launcher.TCode(tds, "flowrate -s -k")
# Adding the outputfile to the tcode object
mycode.addOutputFile(fout)

# Size of a sampling.
nS2 = 50
tjrw2 = Sensitivity.TJohnsonRW(tds, mycode, nS2)
VI.7.1.3.4. TJohnsonRW constructor for a runner

The constructor prototype used with a runner is: 

# Create a TJohnsonRW object with a runner
TJohnsonRW(tds, frun, ns, option = "")

This constructor takes three arguments:

  • a pointer to a TDataServer object,
  • a pointer to a TRun,
  • an integer to specify the number of simulations.

Here is an example of the use of this constructor on the flowrate code in sequential mode: 

# The input file
infile = Relauncher.TKeyScript("flowrate_input_with_keys.in")
# provide the input and their key
infile.addInput(tds.getAttribute("rw"), "Rw")
infile.addInput(tds.getAttribute("r"), "R")
infile.addInput(tds.getAttribute("tu"), "Tu")
infile.addInput(tds.getAttribute("tl"), "Tl")
infile.addInput(tds.getAttribute("hu"), "Hu")
infile.addInput(tds.getAttribute("hl"), "Hl")
infile.addInput(tds.getAttribute("l"), "L")
infile.addInput(tds.getAttribute("kw"), "Kw")

yhat = DataServer.TAttribute("yhat")
# The output file of the code
outfile = Relauncher.TKeyResult("_output_flowrate_withKey_.dat")
# The attribute in the output file
outfile.addOutput(yhat, "yhat")

# Instanciation de mon code
code = Relauncher.TCodeEval("flowrate -s -k")
# Adding the intput/output file to the code
code.addInputFile(infile)
code.addOutputFile(outfile)

run = Relauncher.TSequentialRun(code)
run.startSlave()
if run.onMaster():

    # Size of a sampling.
    nS2 = 50
    tjrwR = Sensitivity.TJohnsonRW(tds, run, nS2)
    #...

VI.7.1.4. Not using a sample but only a correlation matrix

With the Johnson relative weight method, it is possible to use either data loaded from an existing file or no data at all but only corration matrix to estimate the weights and the . This former is done, as usual, by calling the fileDataRead of the TDataserver object. This is not discussed in details as these as this aspect if fairly classical. The later on the other hand, is new and specific to the TJohnsonRW class. This possibility is very specific and can be used through the method setCorrelationMatrix(TMatrixD Corr);

Warning


It is important to pay attention not to mix up setInputCorrelationMatrix, setCorrelationMatrix. The former is useful to generate a correlated sample before submitting the computation with function or code (see Section VI.7.1.5 for instance) while the latter is a correlation matrix that does not only contains the input variables but also the output ones.

Here is an exemple of code that shows how to use this possibility. It starts with the input attribute definition (using StochasticAttributes is not compulsory anymore has no design-of-experiments will be drawn), along with the output ones. Once done, the full correlation matrix (inputs and outputs) can be provided through the setCorrelationMatrix() (for more details on the GenCorr method, see Section XIV.6.18.2). The rest is pretty much straightforward. 

# Load the GenCorr function
ROOT.gROOT.LoadMacro("UserFunctions.C")

# Define the DataServer
tds = DataServer.TDataServer("tdsflowrate", "Ex. Flowrate")
tds.addAttribute("rw")
tds.addAttribute("r")
tds.addAttribute("tu")
tds.addAttribute("tl")
tds.addAttribute("hu")
tds.addAttribute("hl")
tds.addAttribute("l")
tds.addAttribute("kw")
# outputs
tds.addAttribute("flowrateModel")
tds.getAttribute("flowrateModel").setOutput()

# Get the full correlation matrix
inCorr = ROOT.TMatrixD(8,8)
ROOT.GenCorr(inCorr, True, True)
# inCorr is now 9 by 9 as linearOutput has been added in GenCorr

tjrw = Sensitivity.TJohnsonRW(tds, "rw:r:tu:tl:hu:hl:l:kw", "flowrateModel")
tjrw.setCorrelationMatrix(inCorr)
tjrw.computeIndexes()

VI.7.1.5. Generating the sample

To generate the sample that can be used to get the relative weights, one can use the generateSample method defined in TSensitivity but the purpose of the relative weight method is focusing on correlated inputs issue. In order to do this, one should specify the correlation structure of the inputs through the method setInputCorrelationMatrix(TMatrixD Corr) where the only argument is a correlation matrix that has to be a symmetrical positive definite matrix whose coefficients have to be in while the diagonal ones must be set to 1. With the usual variable definition, this could look like this: 


import numpy as np
inCorr = ROOT.TMatrixD(8, 8)
corrValue = np.array([1, 0.184641, -0.613412, -0.214481, 0.373538, -0.0926293, 0.656586, 0.194991,
                      0.184641, 1, 0.134385, 0.0704593, -0.284958, 0.105629, -0.536972, 0.663751,
                      -0.613412, 0.134385, 1, 0.200747, -0.174041, 0.0817216, -0.547402, 0.0202687,
                      -0.214481, 0.0704593, 0.200747, 1, -0.128116, -0.248905, -0.263717, -0.0613039,
                      0.373538, -0.284958, -0.174041, -0.128116, 1, -0.0832623, 0.397725, -0.675141,
                      -0.0926293, 0.105629, 0.0817216, -0.248905, -0.0832623, 1, 0.111711, 0.100442,
                      0.656586, -0.536972, -0.547402, -0.263717, 0.397725, 0.111711, 1, -0.302142,
                      0.194991, 0.663751, 0.0202687, -0.0613039, -0.675141, 0.100442, -0.302142, 1])
inCorr.Use(8, 8, corrValue)

# Create the jrw object
tjrw = Sensitivity.TJohnsonRW(tds, "flowrateModel", nS, "rw:r:tu:tl:hu:hl:l:kw", "flowrateModel")
tjrw.setInputCorrelationMatrix(inCorr)
tjrw.generateSample()

Then, the sample generated can be exported in a file and used outside of Uranie to compute the simulations associated.

VI.7.1.6. Computing the indices

To compute the indices, run the method computeIndexes: 

tjrw.computeIndexes()

Note that this method is all inclusive: it constructs the sample, launches the simulations and computes the indices.

VI.7.1.7. Displaying the indices

To display graphically the coefficients, use the method drawIndexes: 

drawIndexes(sTitre, select, option)

The method needs:

  • a TString containing the title of the figure,
  • a string containing a selection (empty if no selection),
  • a string containing the options of the graphics separated by commas.

Some of the options available are:

  • "nonewcanv": to not create a new canvas,
  • "pie": to display a pie chart,
  • "hist": to display a histogram,
  • "first": to display the indices of the first order (with "hist" only),

In our example the use of this method is: 

cc = ROOT.TCanvas("canhist", "histogramme")
tjrw.drawIndexes("Flowrate", "", "nonewcanv,hist,first")
cc.Print("appliUranieFlowrateJohnsonRW500Histogram.png")

ccc = ROOT.TCanvas("canpie", "TPie", 5, 64, 1270, 560)
tjrw.drawIndexes("Flowrate", "", "nonewcanv,pie")
ccc.Print("appliUranieFlowrateJohnsonRW500Pie.png")

VI.7.1.8. Extracting the coefficients

The coefficients, once computed, are stored in a TTree. To get this TTree, use the method TSensitivity::getResultTuple(): 

results = tjrw.getResultTuple()

Several methods exist in ROOT to extract data from a TTree, it is advised to look for them into the ROOT documentation. We propose two ways of extracting the value of each coefficient from the TTree.

VI.7.1.8.1. Method of extraction

The method use the method getValue of the TJohnsonRW object specifying the order of the extract value ("First"), the related input and possibly more selected options. 

hl_first_index = tjrw.getValue("First","hl");
VI.7.1.8.2. Second method of extraction

The second method uses 3 steps to extract an index:

  • scan the TTree for the chosen input variable (with a selection) in order to obtain its row number. In our example, if we chose the variable "hl", we'll use the command: 
    results.Scan("*", "Inp==\"hl\"")
    and in the resulting figure below we see that the first order index of "hl" is in the row 10: 
    *************************************************************************
*    Row   *       Out *  Inp *  Order * Method *      Algo *     Value *
*************************************************************************
*       10 * flowrateM *   hl *  First * Johnso * --first-- * 0.0435594 *
*       11 * flowrateM *   hl *  Total * Johnso * --total-- * 0.0435594 *
*************************************************************************
  • set the entry of the TTree on this row with the method GetEntry;
  • get the value of the index with GetValue method on the "Value" leaf of the TTree.

Below is an example of extraction of the index for "hl" in our flowrate case: 

results.Scan("*", "Inp==\"hl\"")
results.GetEntry(100)
S_Rw_Indexe = results.GetLeaf("Value").GetValue()
VI.7.1.8.3. Third method of extraction

The third method uses 2 steps to extract an index:

  • use the Draw method with a selection to select the index, for example the selection for the first order index of "rw" is "Inp==\"rw\" && Algo==\"--first--\"";
  • get the pointer on the value of the index with GetV1 method on the TTree.

Below is another example of extraction of the first order index for "rw" in our flowrate case: 

results.Draw("Value", "Inp==\"rw\" && Algo==\"--first--\"","goff")
S_Rw_IndexeS = results.GetV1()[0]

Summary: TJohnsonRW constructors

  • TJohnsonRW(TDataServer *tds, const char *inp, const char *out, Option_t *option="")

    • tds: an empty dataserver containing the input variables or a dataserver filled with an already existing design-of-experiments (input and output data available).
    • out: attribute names which represent the responses of the problem. it must refer to existing attributes in the "tds".
    • inp: input attribute names separated by colons. They represent all the inputs of the problem. They must refer to existing attributes in the "tds".
    • option: usual option field.

  • TJohnsonRW(TDataServer *tds, void *fcn(double *,double *), const char *inp, const char *out, Int_t ns=100)

    • tds: an empty dataserver containing the input variables.
    • fcn: the function to analyse (a pointer).
    • ns: an integer to specify the number of simulations.
    • out: attribute names which represent the responses of the problem. it must refer to existing attributes in the "tds".
    • inp: input attribute names separated by colons. They represent all the inputs of the problem. They must refer to existing attributes in the "tds".

  • TJohnsonRW(TDataServer *tds, const char *fcn, Int_t ns=100, const char *inp="", const char *out="")

    • tds: an empty dataserver containing the input variables.
    • fcn: the function to analyse (the function's name).
    • ns: an integer to specify the number of simulations.
    • out: attribute names which represent the responses of the problem. it must refer to existing attributes in the "tds".
    • inp: input attribute names separated by colons. They represent all the inputs of the problem. They must refer to existing attributes in the "tds".

  • TJohnsonRW(TDataServer *tds, TCode *fcode, Int_t ns)

    • tds: an empty dataserver containing the input variables.
    • fcode: the code to analyse .
    • ns: an integer to specify the number of simulations.

  • TJohnsonRW(TDataServer *tds, TRun *frun, Int_t ns)

    • tds: an empty dataserver containing the input variables.
    • frun: the runner to use code/function to analyse .
    • ns: an integer to specify the number of simulations.

Summary: TJohnsonRW methods

  • setInputCorrelationMatrix(TMatrixD inCorr): define the correlation matrix that is used to intricate the input variables one to another.

    • inCorr: a symmetrical positive definite matrix whose coefficients should all be in but the diagonal ones which should be set to 1.

  • setCorrelationMatrix(TMatrixD Corr): define the global correlation matrix that is used to intricate the input and output variables one to another.

    • Corr: a symmetrical positive definite matrix whose coefficients should all be in but the diagonal ones which should be set to 1.

  • generateSample(Option_t *option=""): generate the sample used for computing the sensitivity indices.

    • option: no option has been implemented

  • computeIndexes(Option_t *option=""): compute the Sobol sensitivity indices.

    • option: string specifying the drawIndexes option used .

  • void drawIndexes(TString sTitre, const char* select="", Option_t *option=""): draw the coefficients.

    • sTitre: a string containing the title of the figure.
    • select: a string containing a solution.
    • option: a string containing the options of the graphics separated by commas.
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