Classical
Definition
Assembly: Novacta.Analytics (in Novacta.Analytics.dll) Version: 2.1.0+428f3840cfab98dda567bb0ed350b302533e273a
public class ClassicalMultidimensionalScalingPublic Class ClassicalMultidimensionalScalingpublic ref class ClassicalMultidimensionalScalingtype ClassicalMultidimensionalScaling = class end- Inheritance
- Object ClassicalMultidimensionalScaling
Remarks
Instantiation
New instances of class ClassicalMultidimensionalScaling can be initialized by calling method Analyze(DoubleMatrix, NullableInt32), which implements the classical multidimensional scaling algorithm.
Given a matrix of proximities, say 
, let
be the matrix of its squared elements, and
define the transformed matrix of proximities as
where 
is the centering matrix.
Let 
be the number of positive eigenvalues
of matrix 
.
If the configuration dimension is not specified to method
Analyze(DoubleMatrix, NullableInt32),
it is set equal to 
if 
;
otherwise, an exception is thrown.
If the configuration dimension is specified to method
Analyze(DoubleMatrix, NullableInt32), then it must be less than or equal to
; otherwise, an exception is thrown.
Results
Let 
be the
eigenvalues of 
in decreasing order, and let
be the corresponding eigenvectors.
If the dimension is equal to 
, then
the configuration is given by matrix:
which can be inspected through property Configuration.
Goodness of fit
The goodness of fit of such configuration is given by the formula
and is accessible through property GoodnessOfFit.
Example
In the following example, a classical multidimensional scaling analysis is performed on a matrix of proximities.
using System;
namespace Novacta.Analytics.CodeExamples
{
public class ClassicalMultidimensionalScalingExample0
{
public void Main()
{
// Create a matrix of proximities among cities.
int numberOfCities = 10;
string[] cities = [
"Atlanta",
"Chicago",
"Denver",
"Houston",
"Los Angeles",
"Miami",
"New York",
"San Francisco",
"Seattle",
"Washington DC"];
var proximities = DoubleMatrix.Dense(
numberOfRows: numberOfCities,
numberOfColumns: numberOfCities,
data: [
0, 587, 1212, 701, 1936, 604, 748, 2139, 2182, 543,
587, 0, 920, 940, 1745, 1188, 713, 1858, 1737, 597,
1212, 920, 0, 879, 831, 1726, 1631, 949, 1021, 1494,
701, 940, 879, 0, 1374, 968, 1420, 1645, 1891, 1220,
1936, 1745, 831, 1374, 0, 2339, 2451, 347, 959, 2300,
604, 1188, 1726, 968, 2339, 0, 1092, 2594, 2734, 923,
748, 713, 1631, 1420, 2451, 1092, 0, 2571, 2408, 205,
2139, 1858, 949, 1645, 347, 2594, 2571, 0, 678, 2442,
2182, 1737, 1021, 1891, 959, 2734, 2408, 678, 0, 2329,
543, 597, 1494, 1220, 2300, 923, 205, 2442, 2329, 0],
storageOrder: StorageOrder.RowMajor);
// Add city names to the matrix of proximities.
for (int i = 0; i < numberOfCities; i++)
{
var city = cities[i];
proximities.SetRowName(i, city);
proximities.SetColumnName(i, city);
}
// Define the dimension of the configuration of points in the target space.
// Passing null, the dimension of the configuration is automatically selected.
int? configurationDimension = null;
// Execute a classical MDS analysis.
var metricMds =
ClassicalMultidimensionalScaling.Analyze(
proximities,
configurationDimension);
// Display the configuration.
Console.WriteLine("City coordinates:");
Console.WriteLine(metricMds.Configuration);
// Display the goodness of fit.
Console.WriteLine("Goodness of fit:");
Console.WriteLine(metricMds.GoodnessOfFit);
}
}
}
// Executing method Main() produces the following output:
//
// City coordinates:
// [Atlanta] -718.759381 142.994269 35.102499 -1.22496264 7.40947762 -1.5046461 6.58357404e-06
// [Chicago] -382.055766 -340.839623 29.6022277 -8.23788477 12.0242975 2.33830156 6.58357404e-06
// [Denver] 481.602336 -25.2850406 53.3938021 1.33927902 -15.6658897 0.952696317 6.58357404e-06
// [Houston] -161.466258 572.769911 1.45257147 -1.76231781 0.671865645 -2.70076208 6.58357404e-06
// [Los Angeles] 1203.73802 390.100291 -18.6350654 14.9748641 3.16920061 1.65614881 6.58357404e-06
// [Miami] -1133.52708 581.907309 -32.2688418 -2.37568536 -2.97185365 2.04718777 6.58357404e-06
// [New York] -1072.23569 -519.02423 -34.3418782 -14.2538571 -6.44732888 -0.270908782 6.58357404e-06
// [San Francisco] 1420.60332 112.589202 -7.75475501 -18.120276 0.805412307 -0.869519732 6.58357404e-06
// [Seattle] 1341.72248 -579.739278 -23.6507873 5.96145265 1.4286322 -0.614379418 6.58357404e-06
// [Washington DC] -979.621992 -335.47281 -2.89977264 23.6993879 -0.423813627 -1.03411834 6.58357404e-06
//
//
// Goodness of fit:
// 0.9963038236716906Properties
| Configuration | Gets the configuration of points in the target space. |
| GoodnessOfFit | Gets the goodness of fit at the Configuration. |
Methods
| Analyze | Executes a metric multidimensional scaling analysis. |
| Equals | Determines whether the specified object is equal to the current object. (Inherited from Object) |
| Finalize | Allows an object to try to free resources and perform other cleanup operations before it is reclaimed by garbage collection. (Inherited from Object) |
| GetHashCode | Serves as the default hash function. (Inherited from Object) |
| GetType | Gets the Type of the current instance. (Inherited from Object) |
| MemberwiseClone | Creates a shallow copy of the current Object. (Inherited from Object) |
| ToString | Returns a string that represents the current object. (Inherited from Object) |