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plCndNormal

Definition 17   Given $\Omega=\{X_1,X_2,...,X_n\}$ and one or both of the variables sets $\Omega_{\mu}=\{\mu_1,\mu_2,...,\mu_n\}$, $\Omega_{\sigma}=\{\omega_{\sigma(1,1)},\omega_{\sigma(1,2)}, ...\omega_{\sigma(n,n)}\}$ a plCndNormal is a represents a collection of plNormal kernels either with variable means ($Omega_{\mu}$), variable variance-covariance ( $\Omega_{\sigma}$) or both.

Example 11   In this example we show the result of compute and instantiate in a plCnNormal kernel.

Program 9: plCndNormal.

     1  /*=============================================================================
     2   * File           : cndNormal.cpp
     3   *=============================================================================
     4   *
     5   *------------------------- Description ---------------------------------------
     6   * This program shows the main functionalities of a plCndNormalKernel.
     7   *
     8   *-----------------------------------------------------------------------------
     9  */
    10
    11
    12  #include <pl.h>
    13
    14
    15  main()
    16  {
    17    /**********************************************************************
    18       Defining the variable type, set and values
    19    ***********************************************************************/
    20    plRealType Tx(0.0,2.10);
    21    plRealType Ty(0.0,120);
    22    plRealType Std_Type(0.0,20);
    23
    24    plSymbol X("x",Tx);
    25    plSymbol Y("y",Ty);
    26
    27    plSymbol X_mean("mx",Tx);
    28    plSymbol Y_mean("my",Ty);
    29    plVariable Mean(X_mean^Y_mean);
    30    plArray Sigma("sigma",Std_Type,2,2,2);
    31
    32    plValues values(X^Y^Mean^Sigma);
    33
    34    /**********************************************************************
    35       Constructing the conditional kernel P(X Y | Mean Sigma)
    36    ***********************************************************************/
    37
    38    plCndNormal my_cnd_kernel(X^Y,Mean,Sigma);
    39    cout<<"The conditional kernel is :"<<my_cnd_kernel<<"\n\n";
    40
    41    /**********************************************************************
    42       Capturing the values of the mean and variance-covariance matrix
    43    ***********************************************************************/
    44
    45    plFloat v;
    46
    47    cout<<"Give me the mean on x: "; cin>>v;
    48    values[X_mean] = v;
    49    cout<<"Give me the mean on y: "; cin>>v;
    50    values[Y_mean] = v;
    51    cout<<"Give me the (X,X) variance: "; cin>>v;
    52    values[Sigma(0,0)] = v;
    53    cout<<"Give me the (X,Y) variance: "; cin>>v;
    54    values[Sigma(0,1)] = v;
    55    // The variance-covariance matrix is simetric so
    56    values[Sigma(1,0)] = v;
    57    cout<<"Give me the (Y,Y) variance: "; cin>>v;
    58    values[Sigma(1,1)] = v;
    59
    60    /**********************************************************************
    61       Illustrating the main functionalities: instantiate and compute
    62    ***********************************************************************/
    63
    64    plKernel new_kernel;
    65
    66    // Select a kernel from the family of kernels with the previous values
    67    my_cnd_kernel.instantiate(new_kernel,values);
    68
    69    // Print the resulting kernel
    70    cout<<"\nThe new kernel is: "<<new_kernel<<"\n\n";
    71
    72    // Capture the values of X and Y
    73    cout<<"Give me the value of X: "; cin>>v;
    74    values[X] = v;
    75    cout<<"Give me the value of Y: "; cin>>v;
    76    values[Y] = v;
    77
    78    // Execute a compute with the new kernel
    79    cout<<"\nCompute with the new kernel ";
    80    cout<<new_kernel.compute(values)<<endl;
    81
    82    // Execute a compute with the conditional kernel
    83    cout<<"Compute with the conditional kernel: ";
    84    cout<<my_cnd_kernel.compute(values)<<endl;
    85  }

The plVariable Mean at line 29 contains two variables mx and my. They represent the means of $x$ and $y$ respectively. In line 30 the plArray Sigma represents the variance-covariance variable matrix. The values of the matrix are given by rows, that is sigma(0,0) corresponds to $\sigma_{xx}$, sigma(0,1) to $\sigma_{xy}$, sigma(1,0) to $\sigma_{yx}$ and sigma(1,1) to $\sigma{yy}$. The conditional normal distribution is created at line 38. Note that, both the mean and variance are variables. However, ProBT provides other constructors allowing the creation of conditional normal distributions with only one variable parameter. Lines 45 to 58 captures a particular instance of the mean and variance-covariance variables. This values are then used at lines 60 to 84 to show the functionalities of the functions $compute$ and $instantiate$. At line 67 a new kernel is obtained from the conditional kernel by instantiating it with the captured values. Lines 80 and 84 executes the $compute$ function for both the conditional kernel my_cnd_kernel and the kernel new_kernel. The difference between the two $compute$ functions is that in my_cnd_kernel it is defined over $\{x,y,mx,my,sigma(0,0),sigma(0,1),sigma(1,0),sigma(1,1)\}$ and in new_kernel it is defined over $\{x,y\}$. my_cnd_kernel represents then a family of normal kernels with variable mean an variance-covariance matrix. The function $instantiate$ permits to select one of these normal kernels.

One output of the program is the following:

The conditional kernel is :P(x y | mx my sigma(0,0) sigma(0,1)
sigma(1,0) sigma(1,1)) = plCndNormal(x y, mx my sigma(0,0) 
sigma(0,1) sigma(1,0) sigma(1,1))


Give me the mean on x: 1.10
Give me the mean on y: 50
Give me the (X,X) variance: 0.5
Give me the (X,Y) variance: 25.6
Give me the (Y,Y) variance: 10.5

The new kernel is: P(x y) = plNormal(x y,1.1,50)

Give me the value of X: 1.98
Give me the value of Y: 60.5

Compute with the new kernel 0.238076
Compute with the conditional kernel: 0.238076