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- /* manual_MDS.cpp
- *
- * Copyright (C) 1993-2014, 2015 David Weenink
- *
- * This code is free software; you can redistribute it and/or modify
- * it under the terms of the GNU General Public License as published by
- * the Free Software Foundation; either version 2 of the License, or (at
- * your option) any later version.
- *
- * This code is distributed in the hope that it will be useful, but
- * WITHOUT ANY WARRANTY; without even the implied warranty of
- * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
- * General Public License for more details.
- *
- * You should have received a copy of the GNU General Public License
- * along with this work. If not, see <http://www.gnu.org/licenses/>.
- */
- /*
- djmw 20020422 GPL + removed "C syntax" part of manpage
- djmw 20030317 Latest modification.
- djmw 20030825 Spelling corrections.
- djmw 20040118 Procrustes modifications.
- djmw 20040214 Removed last link to @praat program@.
- djmw 20040407 Change Arguments -> Settings.
- djmw 20040513 Spelling.
- djmw 20070108 Latest modification.
- */
- #include "ManPagesM.h"
- #include "MDS.h"
- static void drawLetterRConfigurationExample (Graphics g)
- {
- autoConfiguration me = Configuration_createLetterRExample (1);
- Graphics_setWindow (g, -6, 4, -7, 5);
- Configuration_draw (me.get(), g, 1, 2, -6, 4, -7, 5, 0, 1, U"", 1);
- }
- static void drawLetterRConfigurationExample2 (Graphics g)
- {
- autoConfiguration me = Configuration_createLetterRExample (2);
- Configuration_draw (me.get(), g, 1, 2, -0.8, 1.2, -0.8, 0.7, 0, 1, U"", 1);
- }
- static void drawLetterRShepard (Graphics g)
- {
- autoDissimilarity d = Dissimilarity_createLetterRExample (32.5);
- autoConfiguration c = Configuration_createLetterRExample (2);
- Dissimilarity_Configuration_drawShepardDiagram (d.get(), c.get(), g, 0, 200, 0, 2.2, 1, U"+", 1);
- }
- static void drawLetterRRegression (Graphics g)
- {
- autoDissimilarity d = Dissimilarity_createLetterRExample (32.5);
- autoConfiguration c = Configuration_createLetterRExample (2);
- Dissimilarity_Configuration_drawMonotoneRegression (d.get(), c.get(), g, MDS_PRIMARY_APPROACH, 0, 200, 0, 2.2, 1, U"+", 1);
- }
- static void drawCarrollWishConfigurationExample (Graphics g)
- {
- autoConfiguration me = Configuration_createCarrollWishExample ();
- Graphics_setWindow (g, -2, 2, -2, 2);
- Configuration_draw (me.get(), g, 1, 2, -2, 2, -2, 2, 0, 1, U"", 1);
- }
- static void drawCarrollWishSalienceExample (Graphics g)
- {
- autoSalience me = Salience_createCarrollWishExample ();
- Salience_draw (me.get(), g, 1, 2, 1);
- }
- static void drawMsplineExample (Graphics g)
- {
- drawSplines (g, 0, 1, 0, 10, 1, 3, U"0.3 0.5 0.6", 1);
- }
- static void drawIsplineExample (Graphics g)
- {
- drawSplines (g, 0, 1, 0, 1.5, 2, 3, U"0.3 0.5 0.6", 1);
- }
- void manual_MDS_init (ManPages me);
- void manual_MDS_init (ManPages me)
- {
- MAN_BEGIN (U"CANDECOMP", U"djmw", 19971201)
- ENTRY (U"An algorithm to solve the INDSCAL problem.")
- NORMAL (U"In the analysis of the INDSCAL three-way data matrix (%numberOfPoints "
- "\\xx %numberOfDimensions \\xx %numberOfSources) we seek to minimize the "
- "function: ")
- FORMULA (U"%f(%X, %W__1_,..., %W__%numberOfSources_) = "
- "\\su__%i=1..%numberOfSources_ | %S__%i_ \\-- %X%W__%i_%X\\'p |^2")
- NORMAL (U"where %S__%i_ is a known symmetric %numberOfPoints \\xx %numberOfPoints "
- "matrix with scalar products of distances for source %i, %X is the unknown configuration "
- "%numberOfPoints \\xx %numberOfDimensions matrix, %X\\'p its transpose, and, %W__%i_ is "
- "the diagonal %numberOfDimensions \\xx %numberOfDimensions weight matrix for source %i. The function "
- "above has no analytical solution for %X and the %W__%i_. It can be solved, however, "
- "by an iterative procedure which Carroll & Chang have christened CANDECOMP "
- "(CANonical DECOMPosition). This method minimizes, instead of the function "
- "given above, the following function:")
- LIST_ITEM (U"%g(%X, %Y, %W__1_,..., %W__%numberOfSources_) = \\su__%i=1..%numberOfSources_ "
- "| %S__%i_ \\-- %X%W__%i_%Y\\'p |^2")
- NORMAL (U"where %X and %Y are both %numberOfPoints \\xx %numberOfDimensions configuration matrices.")
- NORMAL (U"The algorithm proceeds as follows:")
- NORMAL (U"1. Initialize the $W matrices and the configuration matrix %X. This can for example be "
- "done according to a procedure given in @@Young, Takane & Lewyckyj (1978)@.")
- NORMAL (U"2. An alternating least squares minimization process is started as described that "
- "sequentially updates %Y, %X an %W (@@Carroll & Chang (1970)@):")
- LIST_ITEM (U"2.1. Solve for a new %Y given %X and the %W__%i_")
- LIST_ITEM (U"2.2. Solve for a new %X given the %W__%i_ and the new %Y.")
- LIST_ITEM (U"2.3. Solve for the %W__%i_ given the new %X and %Y.")
- NORMAL (U"Evaluate the goodness-of-fit criterion and either repeat the minimization sequence "
- "(2.1\\--2.3) or continue.")
- NORMAL (U"3. Done: make %Y equal to %X and solve a last time for the %W__%i_.")
- NORMAL (U"Note: during the minimization the following constraints are effective:")
- LIST_ITEM (U"The configuration should be centered.")
- LIST_ITEM (U"The sum of squared coordinates in the configuration space is one for "
- "each dimension, i.e., the configuration always has unit variance in each dimension.")
- MAN_END
- MAN_BEGIN (U"Configuration", U"djmw", 20101102)
- INTRO (U"One of the @@types of objects@ in P\\s{RAAT}.")
- NORMAL (U"An object of type Configuration represents the positions of a number "
- "of labelled points in a multidimensional space.")
- ENTRY (U"How to create a Configuration")
- NORMAL (U"From the New menu:")
- LIST_ITEM (U" \\bu @@Create Configuration...")
- NORMAL (U"By multidimensional scaling:")
- LIST_ITEM (U" \\bu @@Dissimilarity: To Configuration (monotone mds)...")
- LIST_ITEM (U" \\bu @@Dissimilarity: To Configuration (i-spline mds)...")
- LIST_ITEM (U" \\bu @@Dissimilarity: To Configuration (interval mds)...")
- LIST_ITEM (U" \\bu @@Dissimilarity: To Configuration (ratio mds)...")
- LIST_ITEM (U" \\bu @@Dissimilarity: To Configuration (absolute mds)...")
- NORMAL (U"By multidimensional scaling with weights (@@Dissimilarity & Weight: To Configuration...@):")
- LIST_ITEM (U" \\bu ##Dissimilarity & Weight: To Configuration (monotone mds)...")
- LIST_ITEM (U" \\bu ##Dissimilarity & Weight: To Configuration (i-spline mds)...")
- LIST_ITEM (U" \\bu ##Dissimilarity & Weight: To Configuration (interval mds)...")
- LIST_ITEM (U" \\bu ##Dissimilarity & Weight: To Configuration (ratio mds)...")
- LIST_ITEM (U" \\bu ##Dissimilarity & Weight: To Configuration (absolute mds)...")
- NORMAL (U"By multidimensional scaling with a start Configuration:")
- LIST_ITEM (U" \\bu @@Dissimilarity & Configuration: To Configuration (monotone mds)...")
- LIST_ITEM (U" \\bu @@Dissimilarity & Configuration: To Configuration (i-spline mds)...")
- LIST_ITEM (U" \\bu @@Dissimilarity & Configuration: To Configuration (interval mds)...")
- LIST_ITEM (U" \\bu @@Dissimilarity & Configuration: To Configuration (ratio mds)...")
- LIST_ITEM (U" \\bu @@Dissimilarity & Configuration: To Configuration (absolute mds)...")
- NORMAL (U"By transforming an existing Configuration:")
- LIST_ITEM (U" \\bu @@Configuration: To Configuration (varimax)...")
- LIST_ITEM (U" \\bu @@Configuration & AffineTransform: To Configuration")
- LIST_ITEM (U" \\bu @@Configuration & Procrustes: To Configuration")
- NORMAL (U"From @@Principal component analysis@:")
- LIST_ITEM (U" \\bu @@TableOfReal: To Configuration (pca)...")
- LIST_ITEM (U" \\bu @@PCA & TableOfReal: To Configuration...")
- NORMAL (U"From @@Discriminant analysis@:")
- LIST_ITEM (U"@@TableOfReal: To Configuration (lda)...")
- LIST_ITEM (U" \\bu @@Discriminant & TableOfReal: To Configuration...")
- ENTRY (U"How to draw a Configuration")
- LIST_ITEM (U"\\bu @@Configuration: Draw...")
- LIST_ITEM (U"\\bu ##Configuration: Draw as numbers...")
- LIST_ITEM (U"\\bu ##Configuration: Draw as squares...")
- ENTRY (U"How to modify a Configuration")
- LIST_ITEM (U"\\bu @@Configuration: Randomize")
- LIST_ITEM (U"\\bu @@Configuration: Rotate (pc)@ (to principal directions)")
- LIST_ITEM (U"\\bu @@Configuration: Rotate...@ (in a plane around the origin)")
- LIST_ITEM (U"\\bu @@Configuration: Invert dimension...")
- LIST_ITEM (U"\\bu @@Configuration: Normalize...")
- ENTRY (U"Inside a Configuration")
- NORMAL (U"With @Inspect you will see the following attributes:")
- TAG (U"%numberOfRows")
- DEFINITION (U"the number of points (%numberOfPoints\\>_1).")
- TAG (U"%numberOfColumns")
- DEFINITION (U"the dimension of the space (%numberOfDimensions\\>_1).")
- TAG (U"%rowLabels")
- DEFINITION (U"the names associated with the points.")
- TAG (U"%columnLabels")
- DEFINITION (U"the names for the dimensions.")
- TAG (U"%data [1..%numberOfPoints][1..%numberOfDimensions]")
- DEFINITION (U"the coordinates of the points.")
- TAG (U"%metric")
- DEFINITION (U"determines the way distances between points are measured. In general "
- "the distance between points #x__%i_ and #x__%j_ is:")
- FORMULA (U"%d__%ij_ = "
- "(\\su__%k=1..%numberOfDimensions_ %w__%k_ |%x__%ik_ \\-- "
- "%x__%jk_|^^%metric^)^^1/%metric^")
- DEFINITION (U"For Euclidean distances %metric is 2.")
- TAG (U"%w [1..%numberOfDimensions]")
- DEFINITION (U"weight given to each dimension in the distance calculation.")
- MAN_END
- MAN_BEGIN (U"Configuration: Centralize", U"djmw", 19980413)
- INTRO (U"Makes the centre of the selected @Configuration equal to the origin.")
- NORMAL (U"")
- MAN_END
- MAN_BEGIN (U"Configuration: Draw...", U"djmw", 20040407)
- INTRO (U"Draws a projection of the selected @Configuration on a coordinate plane.")
- ENTRY (U"Settings")
- TAG (U"##X-coordinate#, ##Y-coordinate#")
- DEFINITION (U"control the dimensions that will show in the plot.")
- TAG (U"##xmin#, ##xmax#; ##ymin#, ##ymax#")
- DEFINITION (U"range for horizontal and vertical axes, respectively.")
- TAG (U"##Garnish")
- DEFINITION (U"when on, draws a bounding box with decoration.")
- MAN_END
- MAN_BEGIN (U"Configuration: Invert dimension...", U"djmw", 20040407)
- INTRO (U"Inverts one dimension of a @Configuration.")
- NORMAL (U"Setting")
- TAG (U"##Dimension")
- DEFINITION (U"the dimensions that has to be inverted.")
- ENTRY (U"Behaviour")
- NORMAL (U"For all points %i=1..%numberOfPoints: if %j == %dimension then %x__%ij_ "
- "= \\--%x__%ij_.")
- MAN_END
- MAN_BEGIN (U"Configuration: Normalize...", U"djmw", 20040407)
- INTRO (U"Normalizes the selected @Configuration.")
- ENTRY (U"Settings")
- TAG (U"##Sum of squares# (standard value: 0.0)")
- DEFINITION (U"The desired value for the variance.")
- TAG (U"##Each dimension separately")
- DEFINITION (U"When on, the sum of squares in each dimension (column) will be scaled to %sumOfSquares "
- "When off, the sum of squares of all the matrix elements will equal %sumOfSquares.")
- NORMAL (U"With the default value (0.0) for %sumOfSquares, and %eachDimensionSeparately chosen, "
- "an INDSCAL-like normalization is applied: the sum of squares for each column is scaled to "
- "equal 1.0. When %eachDimensionSeparately is off, a Kruskal-like normalization is applied: "
- "the sum of squares of the whole matrix is scaled equal to %numberOfRows.")
- ENTRY (U"Behaviour")
- NORMAL (U"Before the normalization will be applied, however, we first translate the centre of the "
- "configuration to the origin by subtracting the mean for each dimension. "
- "The sum of squares than equals variance.")
- MAN_END
- MAN_BEGIN (U"Configuration: Randomize", U"djmw", 19971201)
- INTRO (U"Changes all coordinates of the points #%x__%i_ in the @Configuration according to:")
- LIST_ITEM (U"%x__%ij_ = randomUniform (-1, 1)")
- MAN_END
- MAN_BEGIN (U"Configuration: Rotate...", U"djmw", 20100303)
- INTRO (U"Rotates the @Configuration in a plane around the origin.")
- NORMAL (U"Settings")
- TAG (U"##Dimension 1#, ##Dimension 2#")
- DEFINITION (U"the dimensions that span the plane. The order of dimension 1 and dimension 2 is not important: "
- "the lowest number always determines the first dimension.")
- TAG (U"##Angle")
- DEFINITION (U"the counter-clockwise rotation angle in degrees.")
- MAN_END
- MAN_BEGIN (U"Configuration: Rotate (pc)", U"djmw", 19971201)
- INTRO (U"Rotates the @Configuration to principal directions. The principal directions "
- "correspond to the principal components.")
- MAN_END
- MAN_BEGIN (U"Configuration: To Configuration (procrustes)", U"djmw", 19971219)
- INTRO (U"A command that transforms the second selected @Configuration object "
- "to match the first selected Configuration object as closely as possible."
- "This problem of fitting one configuration (testee) to another (target) "
- "as closely as possible is called the Procrustes problem. We use a "
- "special @@Procrustes transform@ algorithm that does not "
- "mutilate or distort the testee configuration.")
- NORMAL (U"Both Configuration objects must have the same dimensions.")
- MAN_END
- MAN_BEGIN (U"Configuration: To Configuration (varimax)...", U"djmw", 20040407)
- INTRO (U"A command that rotates the selected @Configuration object to a new "
- "Configuration object whose coordinates have maximum %squared variance. ")
- ENTRY (U"Settings")
- TAG (U"##Normalize rows")
- DEFINITION (U"when selected, the distances of all points to the origin will "
- "be made equal before iteration starts. We remember these scale factors "
- "and restore the original distances after the iteration process has "
- "stopped.")
- TAG (U"##Quartimax")
- DEFINITION (U"when selected, the sum of fourth powers, normalized or raw, "
- "will be maximized. ")
- TAG (U"##Maximum number of iterations")
- DEFINITION (U"sets a limit to the number of iterations. One iteration consists "
- "of %numberOfDimensions\\.c (%numberOfDimensions\\--1)/2 planar rotations "
- "of all pairs of dimensions.")
- TAG (U"##Tolerance")
- DEFINITION (U"also determines when the iteration stops. This happens if "
- "|%v__%i_\\--%v__%i+1_| < %tolerance \\.c %v__%i_, where %v__%i_ is the "
- "squared variance for the %i^^th^ iteration.")
- NORMAL (U"The iteration process stops when either the %%maximum number of "
- "iterations% is reached or the %tolerance criterion is met, which ever "
- "one is first.")
- ENTRY (U"Algorithm")
- NORMAL (U"The Varimax rotation procedure was first proposed by @@Kaiser "
- "(1958)@. Given a %numberOfPoints \\xx %numberOfDimensions configuration "
- "#A, the procedure tries to find an orthonormal rotation matrix #T such "
- "that the sum of variances of the columns of #B*#B is a maximum, where #B "
- "= #A#T and * is the element wise (Hadamard) product of matrices. A direct "
- "solution for the optimal #T is not available, except for the case when "
- "%numberOfDimensions equals two. Kaiser suggested an iterative "
- "algorithm based on planar rotations, i.e., alternate rotations of all "
- "pairs of columns of #A.")
- NORMAL (U"However, this procedure is not without problems: the varimax function "
- "may have stationary points that are not even local maxima. We have "
- "incorporated an algorithm of @@Ten Berge (1995)@ "
- "that prevents this unpleasant situation from happening.")
- MAN_END
- MAN_BEGIN (U"Configuration: To Distance", U"djmw", 19971207)
- INTRO (U"A command that computes a @Distance object for each selected "
- "@Configuration.")
- ENTRY (U"Algorithm")
- NORMAL (U"The distance %d__%ij_ between objects %i and %j is calculated as:")
- FORMULA (U"%d__%ij_ = %d__%ji_ = (\\su__%k=1..%numberOfDimensions_ |%x__%ik_ "
- "\\-- %x__%jk_|^2)^^1/2^")
- MAN_END
- MAN_BEGIN (U"Configuration: To Similarity (cc)", U"djmw", 19980130)
- INTRO (U"A command that create one @Similarity object from the selected "
- "@Configuration objects.")
- NORMAL (U"In the Similarity object entry %s__%ij_ equals the @@congruence "
- "coefficient@ for the %i-th and %j-th selected Configuration object.")
- NORMAL (U"All Configuration objects must have the same number of points and "
- "the same dimensions.")
- MAN_END
- MAN_BEGIN (U"Configuration & AffineTransform: To Configuration", U"djmw", 20011008)
- INTRO (U"A command that transforms the selected @Configuration to a new "
- "Configuration object according to the specifications in the selected "
- "@AffineTransform object.")
- MAN_END
- MAN_BEGIN (U"Configuration & Procrustes: To Configuration", U"djmw", 20011008)
- INTRO (U"A command that transforms the selected @Configuration to a new "
- "Configuration object according to the specifications in the selected "
- "@Procrustes object.")
- MAN_END
- MAN_BEGIN (U"Configurations: To AffineTransform (congruence)...", U"djmw", 20040407)
- INTRO (U"A command that creates an @AffineTransform object from two selected "
- "@Configuration objects.")
- NORMAL (U"We calculate the affine transform that transforms the second "
- "selected Configuration object to match the first selected Configuration "
- "object as closely as possible. "
- "The degree of proportionality is the congruence between corresponding "
- "dimensions.")
- ENTRY (U"Settings")
- TAG (U"##Maximum number of iterations")
- DEFINITION (U"sets a limit to the number of iterations.")
- TAG (U"##Tolerance")
- DEFINITION (U"also determines when the iteration stops. This happens if "
- "|%f(#T__%i_)\\--%f(#T__%i+1_)| < %tolerance \\.c %f(#T__%i_), where "
- "%f(#T__%i_) is the sum of the congruences for the %i^^th^ "
- "iteration (see below).")
- NORMAL (U"The iteration process stops when either the %%maximum number of "
- "iterations% is reached or the %tolerance criterion is met, which ever "
- "one is first.")
- ENTRY (U"Algorithm")
- NORMAL (U"Sometimes the criterion used in a @@Procrustes transform@ is too "
- "restrictive for comparing two configurations. This criterion is only "
- "zero when the positions in the rotated configuration (#A#T) equal the "
- "positions in the other configuration (#B). @@Brokken (1983)@ proposed an "
- "algorithm to maximize instead the sum of congruences between "
- "corresponding dimensions of #AT and #B. "
- "Specifically he proposed to maximize")
- FORMULA (U"%f(#T) = \\su__%i=1..%numberOfDimensions_ #t\\'p__%i_#A\\'p#b__%i_ /"
- " ((#t\\'p__%i_#A\\'p#A#t__%i_)^^1/2^(#b\\'p__%i_#b__%i_)^^1/2^),")
- NORMAL (U"where #t\\'p__%i_ and #b\\'p__%i_ are the %i^^th^ column of #T and "
- "#B, respectively. A direct solution for #T is not available, it can only "
- "be obtained by an iterative procedure. The implemented algorithm is from "
- "@@Kiers & Groenen (1996)@ and shows excellent convergence properties.")
- MAN_END
- MAN_BEGIN (U"Configuration & Configuration: To Procrustes...", U"djmw", 20011008)
- INTRO (U"A command that creates a @Procrustes object from two selected "
- "@Configuration objects.")
- ENTRY (U"Setting")
- TAG (U"##Orthogonal transform")
- DEFINITION (U"determines whether or not a translation and a scaling are allowed in the transform.")
- NORMAL (U"We calculate the @@Procrustes transform@ that transforms the second "
- "selected Configuration object to match the first selected Configuration "
- "object as closely as possible.")
- MAN_END
- MAN_BEGIN (U"Confusion: To Dissimilarity...", U"djmw", 20040407)
- INTRO (U"A command that creates a @Dissimilarity from every selected "
- "@Confusion.")
- ENTRY (U"Settings")
- TAG (U"##Normalize")
- DEFINITION (U"when on, normalize rows by dividing each row element by the row "
- "sum. In this way you correct for unequal stimulus numbers.")
- TAG (U"##No symmetrization#, #Average, #Houtgast")
- DEFINITION (U"determine the symmetrization procedure. See "
- "@@Confusion: To Similarity...")
- TAG (U"##Maximum dissimilarity")
- DEFINITION (U"determines the maximum dissimilarity possible. When the default "
- "value, 0.0, is chosen, %maximumDissimilarity "
- "is calculated as the maximum element in the Similarity object.")
- ENTRY (U"Algorithm")
- NORMAL (U"We first transform the Confusion to a Similarity. See "
- "@@Confusion: To Similarity...")
- NORMAL (U"To obtain dissimilarities from similarities we \"reverse\" the "
- "latter:")
- FORMULA (U"%%dissimilarity__%ij_ = %maximumDissimilarity \\-- %similarity__%ij_")
- MAN_END
- MAN_BEGIN (U"Confusion: To Dissimilarity (pdf)...", U"djmw", 20040407)
- INTRO (U"A command that creates a @Dissimilarity from every selected "
- "@Confusion.")
- ENTRY (U"Settings")
- TAG (U"##Symmetrize first")
- DEFINITION (U"when on, the confusion matrix is symmetrized before we calculate "
- "dissimilarities.")
- TAG (U"##Maximum dissimilarity (units of sigma)")
- DEFINITION (U"specifies the dissimilarity from confusion matrix elements that "
- "are zero.")
- ENTRY (U"Algorithm")
- TAG (U"1. Normalize rows by dividing each row element by the row sum (optional).")
- TAG (U"2. Symmetrize the matrix by averaging %f__%ij_ and %f__%ji_.")
- TAG (U"3. Transformation of the confusion measure which is a sort of "
- "%similarity measure to the %dissimilarity measure.")
- NORMAL (U"Similarity and dissimilarity have an inverse relationship: the "
- "greater the similarity, the smaller the dissimilarity and vice versa. "
- "Both have a monotonic relationship with distance. "
- "The most simple way to transform the similarities %f__%ij_ into "
- "dissimilarities is:")
- FORMULA (U"%dissimilarity__%ij_ = %maximumSimilarity \\-- %similarity__%ij_")
- NORMAL (U"For ordinal analyses like Kruskal this transformation is fine because "
- "only order relations are important in this analysis. However, for "
- "metrical analyses like INDSCAL this is not optimal. "
- "In INDSCAL, distance is a linear function of dissimilarity. This means "
- "that, with the transformation "
- "above, you ultimately fit an INDSCAL model in which the distance "
- "between object %i and %j will be linearly related to the confusion "
- "between %i and %j.")
- NORMAL (U"For the relation between confusion and dissimilarity, the model "
- "implemented here, makes the assumption that the amount of confusion "
- "between objects %i and %j is related to the amount that their "
- "probability density functions, pdf's, overlap. Because we do not know "
- "these pdf's we make the assumption that both are normal, have equal "
- "%sigma and are one-dimensional. The parameter to be determined is the "
- "distance between the centres of both pdf's. "
- "According to formula 26.2.23 in @@Abramowitz & Stegun (1970)@, for each "
- "fraction %f__%ij_, we have to find an %x that solves:")
- FORMULA (U"%f__%ij_ = 1 / \\Vr(2%\\pi) \\in__%x_^^\\oo^ e^^-%t\\.c%t/2^ %dt")
- NORMAL (U"This %x will be used as the dissimilarity between %i and %j. The "
- "relation between %x and %f__%ij_ is monotonic. This means that the "
- "results for a Kruskal analysis will not change much. For INDSCAL, in "
- "general, you will note a significantly better fit.")
- MAN_END
- MAN_BEGIN (U"Confusion: To Similarity...", U"djmw", 20040407)
- INTRO (U"A command that creates a @Similarity from every selected @Confusion.")
- ENTRY (U"Settings")
- TAG (U"##Normalize")
- DEFINITION (U"when on, normalize rows by dividing each row element by the row "
- "sum. In this way you correct for unequal stimulus numbers.")
- TAG (U"##No symmetrization#, #Average, #Houtgast")
- DEFINITION (U"determine the symmetrization procedure.")
- ENTRY (U"Algorithm")
- NORMAL (U"The %Average procedure averages:")
- FORMULA (U"%similarity__%ij_= %similarity__%ji_ = (%confusion__%ij_ + "
- "%confusion__%ji_) / 2")
- NORMAL (U"The %Houtgast procedure as described in the paper by @@Klein, Plomp "
- "& Pols (1970)@, expresses similarity between stimuli %i and %j by the "
- "number of times that stimulus %i and %j have "
- "resulted in the same response, summated over all response categories.")
- NORMAL (U"We use the following formula to calculate the %Houtgast "
- "dissimilarities:")
- FORMULA (U"%similarity__%ij_ = \\su__%k=1..%numberOfColumns_ min "
- "(%confusion__%ik_, %confusion__%jk_)")
- NORMAL (U"which is equivalent to the formula in the Klein et al. paper:")
- FORMULA (U"%similarity__%ij_ = \\su__%k=1..%numberOfColumns_ (%confusion__%ik_ "
- "+ %confusion__%jk_ \\-- |%confusion__%ik_ \\-- %confusion__%jk_|)")
- MAN_END
- MAN_BEGIN (U"congruence coefficient", U"djmw", 20040407)
- INTRO (U"The %%congruence coefficient% is a measure of similarity between two "
- "@@Configuration@s.")
- NORMAL (U"The congruence coefficient %c(#X, #Y) for the configurations #X and "
- "#Y is defined as:")
- FORMULA (U"%c(%X, %Y) = \\su__%i<%j_ %w__%ij_ %d__%ij_(#X) %d__%ij_(#Y) / "
- "([\\su__%i<%j_ %w__%ij_ %d^2__%ij_(#X)]^^1/2^ [\\su__%i<%j_ %w__%ij_ "
- "%d^2__%ij_(#Y)]^^1/2^),")
- NORMAL (U"where %d__%ij_(#X) is the distance between the points %i and %j in "
- "configuration #X and %w__%ij_ are nonnegative weights (default: %w__%ij_"
- " = 1).")
- NORMAL (U"Since distances are nonnegative, the congruence coefficient has a "
- "value between 0 and 1.")
- NORMAL (U"The %%congruence coefficient% is a better measure of the similarity "
- "between configurations than the %%correlation coefficient% of the "
- "distances. @@Borg & Groenen (1997)@ give a simple example where things "
- "go wrong with correlation coefficients: two configurations #X and #Y with three points each, have "
- "distances %d__12_(#X) = 1, %d__13_(#X) = 2, %d__23_(#X) = 3 and "
- "%d__12_(#Y) = 2, %d__13_(#Y) = 3, %d__23_(#Y) = 4. "
- "These distances have a correlation coefficient of 1. "
- "However, in #X the three points lie on a straight line and in #Y the "
- "points form a triangle. This unwanted situation occurs because "
- "in the calculation of the correlation coefficient the mean is subtracted "
- "from the distances and the resulting values are no longer distances "
- "(they may become negative). In calculating the correlation "
- "between the distances we should not subtract the mean. "
- "In fact, the congruence coefficient is exactly this correlation "
- "coefficient calculated with respect to the origin and "
- "not with respect to the centroid position (the \"mean\").")
- NORMAL (U"For further information on how well one number can assess the "
- "similarity between two configurations see @@Borg & Groenen (1997)@ "
- "section 19.7.")
- MAN_END
- MAN_BEGIN (U"ContingencyTable", U"djmw", 19971216)
- INTRO (U"One of the @@types of objects@ in P\\s{RAAT}.")
- NORMAL (U"In a two-way contingency table, cell %f__%ij_ contains the frequency "
- "with which row category %i co-occurs with column category %j. "
- "Necessarily, all %f__%ij_ \\>_ 0.")
- ENTRY (U"Commands")
- NORMAL (U"Creation")
- LIST_ITEM (U"\\bu ##TableOfReal: To ##ContingencyTable")
- NORMAL (U"Query")
- LIST_ITEM (U"\\bu ##ContingencyTable: Get chi squared probability")
- LIST_ITEM (U"\\bu ##ContingencyTable: Get Cramer's statistic")
- LIST_ITEM (U"\\bu ##ContingencyTable: Get contingency coefficient")
- NORMAL (U"Analysis")
- LIST_ITEM (U"\\bu @@ContingencyTable: To Configuration (ca)...")
- MAN_END
- MAN_BEGIN (U"ContingencyTable: To Configuration (ca)...", U"djmw", 20040407)
- INTRO (U"A command that creates a @Configuration object from the selected "
- "@ContingencyTable object by means of @@Correspondence analysis@.")
- ENTRY (U"Settings")
- TAG (U"##Number of dimensions")
- DEFINITION (U"The dimensionality of the Configuration.")
- TAG (U"##Scaling of the final configuration")
- DEFINITION (U"determines whether row points are in the centre of gravity of "
- "column points, or, column points are in the centre of gravity of row "
- "points, or, whether roes and columns are treated symmetrically.")
- ENTRY (U"Algorithm")
- NORMAL (U"1. We start with the following transformation of the entries "
- "%f__%ij_:")
- FORMULA (U"%h__%ij_ = %f__%ij_ / \\Vr (%f__%i+_%f__+%j_) - \\Vr "
- "(%f__%i+_%f__+%j_) / %N,")
- NORMAL (U"where %h__%ij_ is the entry for a cell in the matrix #H with "
- "transformed data, %f__%i+_ "
- "is the total count for row %i, %f__+%j_ is the total count for column %j "
- "and %N is the grand total. "
- "This can be written in matrix form as:")
- FORMULA (U"#H = #R^^\\--1/2^#F#C^^\\--1/2^ \\-- #R^^1/2^#uu\\'p#C^^1/2^ / %N,")
- NORMAL (U"where #R and #C are diagonal matrices with the row and column totals, "
- "respectively and #u a column vector with all elements equal to 1. ")
- NORMAL (U"2. Next the singular value decomposition of matrix #H is performed:")
- FORMULA (U"#H = #K #\\La #L\\'p,")
- NORMAL (U"where #K\\'p#K = #I, #L\\'p#L = #I, and #\\La is a diagonal matrix "
- "with singular values.")
- NORMAL (U"3. Now the row (#X) and column points (#Y) can be determined. "
- "Three normalizations are possible:")
- TAG (U"\\bu Scale row points in the centre of gravity of column points")
- DEFINITION (U"#X = \\Vr%N #R^^\\--1/2^#K#\\La")
- DEFINITION (U"#Y = \\Vr%N #C^^\\--1/2^#U")
- TAG (U"\\bu Scale column points in the centre of gravity of row points")
- DEFINITION (U"#X = \\Vr%N #R^^\\--1/2^#K")
- DEFINITION (U"#Y = \\Vr%N #C^^\\--1/2^#L#\\La")
- TAG (U"\\bu Treat row points and column points symmetrically")
- DEFINITION (U"#X = \\Vr%N #R^^\\--1/2^#K#\\La^^\\--1/2^")
- DEFINITION (U"#Y = \\Vr%N #C^^\\--1/2^#L\\La^^\\--1/2^")
- NORMAL (U"For more details see @@Gifi (1990)@, chapter 8.")
- MAN_END
- MAN_BEGIN (U"Correspondence analysis", U"djmw", 19971216)
- INTRO (U"Correspondence analysis provides a method for representing data in "
- "an Euclidean space so that the results can be visually examined for "
- "structure. For data in a typical two-way @ContingencyTable both the row "
- "variables and the column variables are represented in the same space. "
- "This means that one can examine relations not only among row "
- "or column variables but also between row and column variables.")
- NORMAL (U"In correspondence analysis the data matrix is first transformed by "
- "dividing each cell by the square root of the corresponding row and column "
- "totals. "
- "The transformed matrix is then decomposed with singular value "
- "decomposition resulting in the singular values (which in this case are "
- "canonical correlations) and a set of row vectors and column vectors. "
- "Next the row and column vectors are rescaled with the original total "
- "frequencies to obtain optimal scores. "
- "These optimal scores are weighted by the square root of the singular "
- "values and become the coordinates of the points in the @Configuration.")
- NORMAL (U"Examples can be found in the books by @@Weller & Romney (1990)@ and "
- "@@Gifi (1990)@.")
- MAN_END
- MAN_BEGIN (U"Create Configuration...", U"djmw", 19980413)
- INTRO (U"A command to create a @Configuration with the specified number of "
- "points and number of dimensions. The location of the points will be "
- "determined by the formula (see @@Formulas@ for more "
- "information about possible formulas).")
- MAN_END
- MAN_BEGIN (U"Create INDSCAL Carroll & Wish example...", U"djmw", 19971201)
- INTRO (U"Creates eight @Dissimilarity objects that bear names \"1\" ... \"8\".")
- NORMAL (U"These objects contain the interpoint distances for a twodimensional "
- "3\\xx3 @Configuration of points, labelled A, B, C, ... I. "
- "All Dissimilarity objects are based on the following underlying configuration.")
- PICTURE (4.0, 4.0, drawCarrollWishConfigurationExample)
- NORMAL (U"The eight sources weigh this configuration in the following manner:")
- PICTURE (4.0, 4.0, drawCarrollWishSalienceExample)
- NORMAL (U"For each source, the distances were subjected to the transformation: ")
- FORMULA (U"%dissimilarity__%ij_ = %distance__%ij_ + %noiseRange \\.c #u, ")
- NORMAL (U"where #u is a uniform random variable between 0 and 1.")
- NORMAL (U"Now you can do the following for example:")
- TAG (U"Select all the Dissimilarity objects and choose @@Dissimilarity: To Distance...|"
- "To Distance...@.")
- DEFINITION (U"Uncheck scale (add \"additive constant\").")
- TAG (U"Select all the Distance objects and choose @@Distance: To Configuration (indscal)...|"
- "To Configuration (indscal)...@.")
- DEFINITION (U"and an @@INDSCAL analysis@ will be performed. In order to reproduce the saliences, "
- "you have to uncheck the \"Normalize scalar products\" option.")
- NORMAL (U"This example was adapted from @@Carroll & Wish (1974)@.")
- MAN_END
- MAN_BEGIN (U"Create letter R example...", U"djmw", 19971201)
- INTRO (U"Creates a @Dissimilarity object that bears the name %R. The "
- "dissimilarities in this object were chosen to be a monotone "
- "transformation of the distances between the 32 two-dimensional points "
- "that make up the capital letter #R.")
- PICTURE (4.0, 4.0, drawLetterRConfigurationExample)
- NORMAL (U"All 32 \\.c (32-1)/2 interpoint distances were subjected to the "
- "transformation: ")
- FORMULA (U"%dissimilarity__%ij_^ = %distance__%ij_^2 + 5 + %noiseRange \\.c #u, ")
- NORMAL (U"where #u is a uniform random variable between 0 and 1.")
- NORMAL (U"This example was chosen from @@Green, Carmone & Smith (1989)@.")
- MAN_END
- MAN_BEGIN (U"disparities", U"djmw", 19980111)
- INTRO (U"The numbers %d\\'p__%ij_ that result from applying an admissible "
- "transformation %f on the dissimilarities %\\de__%ij_, i.e., %d\\'p__%ij_ "
- "= %f (%\\de__%ij_). Disparities have the same dimension as distances. "
- "Other names for disparities are %%pseudo distances% and %%target "
- "distances%.")
- MAN_END
- MAN_BEGIN (U"Dissimilarity", U"djmw", 20010327)
- INTRO (U"One of the @@types of objects@ in P\\s{RAAT}.")
- NORMAL (U"It represents a one-way table with "
- "dissimilarities between \"objects\".")
- ENTRY (U"Creating a Dissimilarity from data in a text file")
- NORMAL (U"Suppose you have three objects A, B and C. "
- "In one way or another, you have acquired the following (symmetric) "
- "dissimilarities: %\\de__%AB_ = 2 (= %\\de__%BA_) , %\\de__%AC_ = 1 "
- "(= %\\de__%CA_), and %\\de__%BC_ = 1.4 (= %\\de__CB_), where %\\de__%AB_"
- " represents the dissimilarity between object A and object B.")
- NORMAL (U"You can create a simple text file like the following:")
- CODE (U"\"ooTextFile\" ! The line by which Praat can recognize your file")
- CODE (U"\"Dissimilarity\" ! The line that tells Praat about the contents")
- CODE (U"3 \"A\" \"B\" \"C\" ! Number of columns, and column labels")
- CODE (U"3 ! Number of rows")
- CODE (U"\"A\" 0 2 1 ! Row label (A), A-B value, A-C value")
- CODE (U"\"B\" 2 0 1.4 ! Row label (B), B-A value, B-C value")
- CODE (U"\"C\" 1 1.4 0 ! Row label (C), C-A value, C-B value")
- NORMAL (U"Notice that:")
- LIST_ITEM (U"\\bu the row and column labels are identical.")
- LIST_ITEM (U"\\bu the matrix elements on the diagonal are zero.")
- LIST_ITEM (U"\\bu the matrix is symmetrical.")
- NORMAL (U"This text file can be read with the @@Read from file...@ command. "
- "Since a Dissimilarity object has the data structure of a square "
- "symmetrical TableOfReal, you could also start from an appropriate "
- "@TableOfReal object and cast it to a Dissimilarity object.")
- ENTRY (U"Commands")
- NORMAL (U"Creation")
- LIST_ITEM (U"\\bu @@Confusion: To Dissimilarity...")
- NORMAL (U"Drawing")
- LIST_ITEM (U"\\bu ##Draw as numbers...")
- LIST_ITEM (U"\\bu ##Draw as squares...")
- NORMAL (U"Query")
- LIST_ITEM (U"\\bu ##Get column mean (index)...")
- LIST_ITEM (U"\\bu ##Get column mean (label)...")
- LIST_ITEM (U"\\bu ##Get column stdev (index)...")
- LIST_ITEM (U"\\bu ##Get column stdev (label)...")
- LIST_ITEM (U"\\bu @@Dissimilarity: Get additive constant")
- NORMAL (U"Modification")
- LIST_ITEM (U"\\bu @@Formula...")
- LIST_ITEM (U"\\bu ##Set value...")
- LIST_ITEM (U"\\bu ##Remove column (index)...")
- LIST_ITEM (U"\\bu ##Insert column (index)...")
- LIST_ITEM (U"\\bu ##Set row label (index)...")
- LIST_ITEM (U"\\bu ##Set row label (label)...")
- LIST_ITEM (U"\\bu ##Set column label (index)...")
- LIST_ITEM (U"\\bu ##Set column label (label)...")
- NORMAL (U"Multidimensional scaling analysis")
- LIST_ITEM (U"\\bu @@Dissimilarity: To Configuration (monotone mds)...")
- LIST_ITEM (U"\\bu @@Dissimilarity: To Configuration (i-spline mds)...")
- LIST_ITEM (U"\\bu @@Dissimilarity: To Configuration (interval mds)...")
- LIST_ITEM (U"\\bu @@Dissimilarity: To Configuration (ratio mds)...")
- LIST_ITEM (U"\\bu @@Dissimilarity: To Configuration (absolute mds)...")
- LIST_ITEM (U"\\bu @@Dissimilarity: To Configuration (kruskal)...")
- LIST_ITEM (U"Transformations")
- LIST_ITEM (U"\\bu @@Dissimilarity: To Distance...")
- LIST_ITEM (U"\\bu @@Dissimilarity: To Weight")
- MAN_END
- MAN_BEGIN (U"Dissimilarity: Get additive constant", U"djmw", 19971201)
- INTRO (U"A command that calculates the \"additive constant\" from the selected @Dissimilarity.")
- NORMAL (U"Distances %d__%ij_ will be obtained from dissimilarities %\\de__%ij_ according to:")
- FORMULA (U" %distance__%ij_ = %dissimilarity__%ij_ + %additiveConstant")
- NORMAL (U"We use a procedure by @@Cailliez (1983)@ to solve the \"additive constant problem\", i.e. "
- "find the smallest %additiveConstant such that all %distance__%ij_ in the above equation "
- " have a Euclidean representation.")
- MAN_END
- MAN_BEGIN (U"Dissimilarity: To Configuration (absolute mds)...", U"djmw", 19980105)
- INTRO (U"A command that creates a @Configuration object from a @Dissimilarity object.")
- NORMAL (U"The @disparities %d\\'p__%ij_ will be obtained from dissimilarities %\\de__%ij_ according to:")
- FORMULA (U"%d\\'p__%ij_ = %\\de__%ij_")
- MAN_END
- MAN_BEGIN (U"Dissimilarity: To Configuration (interval mds)...", U"djmw", 19980105)
- INTRO (U"A command that creates a @Configuration object from a @Dissimilarity object.")
- NORMAL (U"The @disparities %d\\'p__%ij_ will be obtained from dissimilarities %\\de__%ij_ according to:")
- FORMULA (U"%d\\'p__%ij_ = %a + %b \\.c %\\de__%ij_")
- MAN_END
- MAN_BEGIN (U"Dissimilarity: To Configuration (i-spline mds)...", U"djmw", 20040407)
- INTRO (U"A command that creates a @Configuration object from a @Dissimilarity object.")
- NORMAL (U"Dissimilarities %\\de__%ij_ and @disparities %d\\'p__%ij_ will be related by a @spline function:")
- FORMULA (U"%d\\'p__%ij_ = \\su__%k=1..(%%numberOfInteriorKnots%+%order)_ spline__%k_ (%knots, %order, %\\de__%ij_),")
- NORMAL (U"where spline__%k_ (\\.c) is the value of the %k^^th^ I-spline of order %order and knot sequence "
- "%knot evaluated at %\\de__%ij_.")
- ENTRY (U"Settings")
- TAG (U"##Number of dimensions")
- DEFINITION (U"determines the dimensionality of the configuration.")
- TAG (U"##Number of interior knots")
- DEFINITION (U"determines the number of segment boundaries. Each interior knot "
- "is the boundary between two segments. The splines in each segment will "
- "be joined as continuously as possible.")
- TAG (U"##Order of I-spline")
- DEFINITION (U"The order of the polynomial basis of the I-spline.")
- NORMAL (U"Finding the optimal Configuration involves a minimization process:")
- TAG (U"##Tolerance")
- DEFINITION (U"When successive values for the stress differ by less than "
- "#Tolerance, the minimization process stops.")
- TAG (U"##Maximum number of iterations")
- DEFINITION (U"Minimization stops after this number of iterations has been "
- "reached.")
- TAG (U"##Number of repetitions")
- DEFINITION (U"If chosen larger than 1, the minimization process will be "
- "repeated, each time with another random start configuration. "
- "The configuration that results in minimum stress, will be saved.")
- ENTRY (U"Hints")
- NORMAL (U"If %numberOfInteriorKnots is zero, polynomial regression will be "
- "performed. Therefore , the combination %numberOfInteriorKnots = 0 and "
- "%order = 1 also gives interval "
- "scaling (in fact, it is the implementation in this program).")
- NORMAL (U"In the limit when %order = 0 and %numberOfInteriorKnots = "
- "%numberOfDissimilarities, monotone regression is performed.")
- MAN_END
- MAN_BEGIN (U"Dissimilarity: To Configuration (kruskal)...", U"djmw", 20040407)
- INTRO (U"A command that creates a @Configuration object from a @Dissimilarity "
- "object.")
- ENTRY (U"Settings")
- TAG (U"##Number of dimensions# (standard value: 2)")
- DEFINITION (U"The dimensionality of the Configuration.")
- TAG (U"##Distance metric% (standard value: 2, i.e. Euclidean)")
- DEFINITION (U"the general distance between points #x__%i_ and #x__%j_ (%i,%j "
- "= 1..%numberOfPoints) is:")
- DEFINITION (U"(\\su__%k=1..%numberOfDimensions_ |%x__%ik_ \\--%x__%jk_|"
- "^^%metric^)^^1/%metric^")
- TAG (U"##Sort distances")
- DEFINITION (U"determines the handling of ties in the data. When off, whenever "
- "two or more dissimilarities are equal we do not care whether the fitted "
- "distances are equal or not. "
- "Consequently, no constraints are imposed on the fitted distances. "
- "When on, however, we impose the constaint that the fitted distances be "
- "equal whenever the dissimilarities are equal.")
- NORMAL (U"For the calculation of stress:")
- TAG (U"##Formula1 (default)") // ??
- FORMULA (U"%stress = \\Vr(\\su(%distance__%k_ \\-- %fittedDistance__%k_)^2 / "
- "\\su %distance__%k_^2)")
- TAG (U"##Formula2")
- FORMULA (U"%stress = \\Vr(\\su(%distance__%k_ \\-- %fittedDistance__%k_)^2 / "
- "\\su (%distance__%k_ \\-- %averageDistance)^2)")
- DEFINITION (U"Note that values of stress 2 are generally more than double those "
- "of stress 1 for the same degree of fit.")
- NORMAL (U"Finding the optimal Configuration involves a minimization process:")
- TAG (U"##Tolerance")
- DEFINITION (U"When successive values for the stress differ less than %Tolerance "
- "the minimization process stops.")
- TAG (U"##Maximum number of iterations")
- DEFINITION (U"Minimization stops after this number of iterations has been "
- "reached.")
- TAG (U"##Number of repetitions")
- DEFINITION (U"When chosen larger than 1, the minimalization process will be "
- "repeated, each time with another random start configuration. "
- "The configuration that results in minimum stress will be saved.")
- ENTRY (U"Precautions")
- NORMAL (U"When there are few objects it is impossible to recover many "
- "dimensions. A rough rule of thumb is that there should be at least twice "
- "as many number of observations, i.e. the %numberOfPoints \\.c "
- "(%numberOfPoints - 1) / 2 (dis)similarities, than parameters "
- "to be estimated, i.e. the %numberOfPoints \\.c %numberOfDimensions "
- "position coordinates. A practical guide is:")
- LIST_ITEM (U"for %numberOfDimensions = 1 you need \\>_ 5 objects")
- LIST_ITEM (U"for %numberOfDimensions = 2 you need \\>_ 9 objects")
- LIST_ITEM (U"for %numberOfDimensions = 3 you need \\>_ 13 objects")
- NORMAL (U"There is no feasible way to be certain that you have found the "
- "true global minimum. However, by using a great number of different "
- "random starting configurations to scale the same data it is often "
- "possible to obtain practical certainty. "
- "Although the procedure for obtaining an initial configuration is based "
- "on a %linear relation between distance and (dis)similarity, it gives a "
- "very good approximation of the optimal #Configuration and "
- "the #Minimizer practically always finds the global minimum from it "
- "(I guess...). A way to find out is to try the %numberOfRepetitions "
- "parameter which gives you the possibility to fit many times and each "
- "time start with another random initial configuration.")
- ENTRY (U"Algorithm")
- LIST_ITEM (U"1. The Dissimilarity object is converted to a Distance object in "
- "the same way as in @@Dissimilarity: To Distance...@.)")
- LIST_ITEM (U"2. From the Distance object an initial Configuration is found by "
- "first transforming the Distance object to a matrix with scalar products "
- "of distances and subsequently solving for the first %numberOfDimensions "
- "eigenvectors of this matrix.")
- LIST_ITEM (U"3. A minimalization algorithm is started that tries to minimize a "
- "function. In this function:")
- LIST_ITEM (U"\\bu 3.1 We normalize the current Configuration from the minimizer")
- LIST_ITEM (U"\\bu 3.2 Calculate a new Distance object from the configuration")
- LIST_ITEM (U"\\bu 3.3 Do a monotone regression of this Distance on the "
- "Dissimilarity. This results in a new Distance object.")
- LIST_ITEM (U"\\bu 3.4 Calculate stress from this Distance and the Distance "
- "obtained from Dissimilarity.")
- NORMAL (U"The optimization process is ccontrolledby a conjugate gradient "
- "minimization algorithm that tries to minimize the %stress function. "
- "In @@Kruskal (1964)@, a steepest descent "
- "algorithm is used wwhichis less efficient. ")
- MAN_END
- MAN_BEGIN (U"Dissimilarity: To Configuration (monotone mds)...", U"djmw", 20040407)
- INTRO (U"A command that creates a @Configuration object from a @Dissimilarity "
- "object.")
- NORMAL (U"Dissimilarities %\\de__%ij_ and @disparities %d\\'p__%ij_ are "
- "related by:")
- FORMULA (U"%d\\'p__%ij_ \\<_ %d\\'p__%kl_ if %\\de__%ij_ \\<_ %\\de__%kl_")
- ENTRY (U"Settings")
- TAG (U"##Number of dimensions")
- DEFINITION (U"determines the number of dimensions of the configuration.")
- TAG (U"##Primary or secondary approach to ties")
- DEFINITION (U"When dissimilarities are equal, i.e., %\\de__%ij_ = %\\de__%kl_, "
- "the primary approach imposes no conditions on the corresponding "
- "@disparities %d\\'p__%ij_ and %d\\'p__%kl_, while the %secondary "
- "approach demands that also %d\\'p__%ij_ = %d\\'p__%kl_.")
- NORMAL (U"Finding the optimal Configuration involves a minimization process:")
- TAG (U"##Tolerance")
- DEFINITION (U"When successive values for the stress differ less than %Tolerance "
- "the minimization process stops.")
- TAG (U"##Maximum number of iterations")
- DEFINITION (U"Minimization stops after this number of iterations has been reached.")
- TAG (U"##Number of repetitions")
- DEFINITION (U"When chosen larger than 1, the minimalization process will be "
- "repeated, each time with another random start configuration. "
- "The configuration that results in minimum stress will be saved.")
- MAN_END
- MAN_BEGIN (U"Dissimilarity: To Configuration (ratio mds)...", U"djmw", 19980105)
- INTRO (U"A command that creates a @Configuration object from a @Dissimilarity "
- "object.")
- NORMAL (U"The @disparities %d\\'p__%ij_ will be obtained from dissimilarities "
- "%\\de__%ij_ according to:")
- FORMULA (U"%d\\'p__%ij_ = %b \\.c %\\de__%ij_")
- MAN_END
- MAN_BEGIN (U"Dissimilarity: To Distance...", U"djmw", 20040407)
- INTRO (U"A command that creates a @Distance object from a selected "
- "@Dissimilarity object.")
- ENTRY (U"Settings")
- TAG (U"##Scale")
- DEFINITION (U"when on, the @@Dissimilarity: Get additive constant|"
- "additiveConstant@ is determined, when off the %additiveConstant = 0.")
- NORMAL (U"dissimilarities are transformed to distances according to:")
- FORMULA (U" %distance__%ij_ = %dissimilarity__%ij_ + %additiveConstant.")
- MAN_END
- MAN_BEGIN (U"Dissimilarity: To Weight", U"djmw", 19980108)
- INTRO (U"Creates an object of type @Weight for each selected @Dissimilarity "
- "object.")
- NORMAL (U"The values in the weight matrix will be:")
- LIST_ITEM (U"%w__%ii_ = 0")
- LIST_ITEM (U"%w__%ij_ = 1 if %\\de__%ij_ > 0")
- MAN_END
- MAN_BEGIN (U"Dissimilarity & Configuration: Draw regression (absolute mds)...",
- U"djmw", 20040407)
- INTRO (U"Draws a scatterplot of the dissimilarities %\\de__%ij_ from the "
- "selected @Dissimilarity object versus @disparities %d\\'p__%ij_ obtained "
- "from the \"regression\" of distances %d__%ij_ "
- "from @Configuration on the dissimilarities %\\de__%ij_.")
- FORMULA (U"%d\\'p__%ij_ = %\\de__%ij_")
- ENTRY (U"Settings")
- TAG (U"##Minimum proximity#, ##Maximum proximity#")
- DEFINITION (U"minimum and maximum values for the proximities (horizontal axis).")
- TAG (U"##Minimum distance#, ##Maximum distance#")
- DEFINITION (U"minimum and maximum values for the distances (vertical axis).")
- TAG (U"##Mark size (mm)#, ##Mark string#")
- DEFINITION (U"size and kind of the marks in the plot.")
- TAG (U"##Garnish")
- DEFINITION (U"when on, draws a bounding box with decoration.")
- MAN_END
- MAN_BEGIN (U"Dissimilarity & Configuration: Draw regression (interval mds)...",
- U"djmw", 20040407)
- INTRO (U"Draws a scatterplot of the dissimilarities %\\de__%ij_ from the "
- "selected @Dissimilarity versus @disparities %d\\'p__%ij_ obtained "
- "from the regression of distances %d__%ij_ "
- "from @Configuration on the dissimilarities %\\de__%ij_.")
- FORMULA (U"%d\\'p__%ij_ = %a + %b \\.c %\\de__%ij_,")
- NORMAL (U"where the values of %a and %b are determined by regression.")
- ENTRY (U"Settings")
- TAG (U"##Minimum proximity#, ##Maximum proximity#")
- DEFINITION (U"minimum and maximum values for the proximities (horizontal axis).")
- TAG (U"##Minimum distance#, ##Maximum distance#")
- DEFINITION (U"minimum and maximum values for the distances (vertical axis).")
- TAG (U"##Mark size (mm)#, ##Mark string#")
- DEFINITION (U"size and kind of the marks in the plot.")
- TAG (U"##Garnish")
- DEFINITION (U"when on, draws a bounding box with decoration.")
- MAN_END
- MAN_BEGIN (U"Dissimilarity & Configuration: Draw regression (i-spline mds)...", U"djmw", 20040407)
- INTRO (U"Draws a scatterplot of the dissimilarities %\\de__%ij_ from the "
- "selected @Dissimilarity versus @disparities %d\\'p__%ij_ obtained "
- "from the regression of distances %d__%ij_ from @Configuration on the "
- "@spline transformed dissimilarities %\\de__%ij_.")
- ENTRY (U"Settings")
- TAG (U"##Number of interior knots")
- DEFINITION (U"determines the number of segments.")
- TAG (U"##Order of I-spline")
- DEFINITION (U"The order of the polynomial basis of the I-spline.")
- TAG (U"##Minimum proximity#, ##Maximum proximity#")
- DEFINITION (U"minimum and maximum values for the proximities (horizontal axis).")
- TAG (U"##Minimum distance#, ##Maximum distance#")
- DEFINITION (U"minimum and maximum values for the distances (vertical axis).")
- TAG (U"##Mark size (mm)#, ##Mark string#")
- DEFINITION (U"size and kind of the marks in the plot.")
- TAG (U"##Garnish")
- DEFINITION (U"when on, draws a bounding box with decoration.")
- MAN_END
- MAN_BEGIN (U"Dissimilarity & Configuration: Draw regression (monotone mds)...", U"djmw", 20040407)
- INTRO (U"Draws a scatterplot of the dissimilarities %\\de__%ij_ from the "
- "selected @Dissimilarity versus @disparities %d\\'p__%ij_ obtained "
- "from the monotone regression of distances %d__%ij_ "
- "from @Configuration on the dissimilarities %\\de__%ij_.")
- ENTRY (U"Settings")
- TAG (U"##Primary or secondary approach to ties")
- DEFINITION (U"When dissimilarities are equal, i.e., %\\de__%ij_ = %\\de__%kl_ "
- "the primary approach imposes no conditions on the corresponding distances "
- "%d__%ij_ and %d__%kl_, while the %secondary approach demands that also "
- "%d__%ij_ = %d__%kl_.")
- TAG (U"##Minimum proximity#, ##Maximum proximity#")
- DEFINITION (U"minimum and maximum values for the proximities (horizontal axis).")
- TAG (U"##Minimum distance#, ##Maximum distance#")
- DEFINITION (U"minimum and maximum values for the distances (vertical axis).")
- TAG (U"##Mark size (mm)#, ##Mark string#")
- DEFINITION (U"size and kind of the marks in the plot.")
- TAG (U"##Garnish")
- DEFINITION (U"when on, draws a bounding box with decoration.")
- MAN_END
- MAN_BEGIN (U"Dissimilarity & Configuration: Draw regression (ratio mds)...", U"djmw", 20040407)
- INTRO (U"Draws a scatterplot of the dissimilarities %\\de__%ij_ from the "
- "selected @Dissimilarity versus @disparities %d\\'p__%ij_ obtained "
- "from the \"regression\" of distances %d__%ij_ "
- "from @Configuration on the dissimilarities %\\de__%ij_.")
- FORMULA (U"%d\\'p__%ij_ = %b \\.c %\\de__%ij_,")
- NORMAL (U"where the value of %b is determined by regression.")
- ENTRY (U"Settings")
- TAG (U"##Minimum proximity#, ##Maximum proximity#")
- DEFINITION (U"minimum and maximum values for the proximities (horizontal axis).")
- TAG (U"##Minimum distance#, ##Maximum distance#")
- DEFINITION (U"minimum and maximum values for the distances (vertical axis).")
- TAG (U"##Mark size (mm)#, ##Mark string#")
- DEFINITION (U"size and kind of the marks in the plot.")
- TAG (U"##Garnish")
- DEFINITION (U"when on, draws a bounding box with decoration.")
- MAN_END
- MAN_BEGIN (U"Dissimilarity & Configuration: Draw Shepard diagram...", U"djmw", 20040407)
- INTRO (U"Draws the Shepard diagram. This is a scatterplot of the "
- "dissimilarities from the @Dissimilarity object versus distances (as "
- "calculated from the @Configuration).")
- ENTRY (U"Settings")
- TAG (U"##Minimum proximity#, ##Maximum proximity#")
- DEFINITION (U"minimum and maximum values for the proximities (horizontal axis).")
- TAG (U"##Minimum distance#, ##Maximum distance#")
- DEFINITION (U"minimum and maximum values for the distances (vertical axis).")
- TAG (U"##Mark size (mm)#, ##Mark string#")
- DEFINITION (U"size and kind of the marks in the plot.")
- TAG (U"##Garnish")
- DEFINITION (U"when on, draws a bounding box with decoration.")
- MAN_END
- MAN_BEGIN (U"Dissimilarity & Configuration: Get stress (absolute mds)...",
- U"djmw", 19980119)
- INTRO (U"A command to obtain the @stress value for the selected @Dissimilarity "
- "and @Configuration object.")
- ENTRY (U"Behaviour")
- NORMAL (U"Stress formula's are #dependent of the scale of the Configuration: "
- "you will get #another stress value if you had pre-multiplied the "
- "selected Configuration with any number greater than zero.")
- MAN_END
- MAN_BEGIN (U"Dissimilarity & Configuration: Get stress (interval mds)...",
- U"djmw", 19980119)
- INTRO (U"A command to obtain the @stress value for the selected @Dissimilarity "
- "and @Configuration object.")
- ENTRY (U"Behaviour")
- NORMAL (U"We use stress formula's that are independent of the scale of the "
- "Configuration: you would have got the same stress value if you had "
- "pre-multiplied the selected Configuration with any number greater "
- "than zero.")
- MAN_END
- MAN_BEGIN (U"Dissimilarity & Configuration: Get stress (i-spline mds)...",
- U"djmw", 19980119)
- INTRO (U"A command to obtain the @stress value for the selected @Dissimilarity "
- "and @Configuration object.")
- ENTRY (U"Behaviour")
- NORMAL (U"We use stress formula's that are independent of the scale "
- "of the Configuration: you would have got the same stress value if "
- "you had pre-multiplied the selected Configuration with any number "
- "greater than zero.")
- MAN_END
- MAN_BEGIN (U"Dissimilarity & Configuration: Get stress (monotone mds)...",
- U"djmw", 19980119)
- INTRO (U"A command to obtain the @stress value for the selected @Dissimilarity "
- "and @Configuration object.")
- ENTRY (U"Behaviour")
- NORMAL (U"We use stress formula's that are independent of the scale "
- "of the Configuration: you would have got the same stress value if "
- "you had pre-multiplied the selected Configuration with any number "
- "greater than zero.")
- MAN_END
- MAN_BEGIN (U"Dissimilarity & Configuration: Get stress (ratio mds)...",
- U"djmw", 19980119)
- INTRO (U"A command to obtain the @stress value for the selected @Dissimilarity "
- "and @Configuration object.")
- ENTRY (U"Behaviour")
- NORMAL (U"We use stress formula's that are independent of the scale "
- "of the Configuration: you would have got the same stress value if "
- "you had pre-multiplied the selected Configuration with any number "
- "greater than zero.")
- MAN_END
- MAN_BEGIN (U"Dissimilarity & Configuration: To Configuration (absolute mds)...",
- U"djmw", 19980119)
- INTRO (U"A command that creates a @Configuration object from a @Dissimilarity "
- "object. The selected Configuration object serves as a starting "
- "configuration for the minimization process.")
- MAN_END
- MAN_BEGIN (U"Dissimilarity & Configuration: To Configuration (interval mds)...",
- U"djmw", 19980119)
- INTRO (U"A command that creates a @Configuration object from a @Dissimilarity "
- "object. The selected Configuration object serves as a starting "
- "configuration for the minimization process.")
- MAN_END
- MAN_BEGIN (U"Dissimilarity & Configuration: To Configuration (i-spline mds)...",
- U"djmw", 19980119)
- INTRO (U"A command that creates a @Configuration object from a @Dissimilarity "
- "object. The selected Configuration object serves as a starting "
- "configuration for the minimization process.")
- MAN_END
- MAN_BEGIN (U"Dissimilarity & Configuration: To Configuration (kruskal)...",
- U"djmw", 19971201)
- INTRO (U"A command to fit an optimal @Configuration for the selected "
- "@Dissimilarity object. The selected @Configuration will be used as the "
- "starting configuration in the kruskal analysis.")
- MAN_END
- MAN_BEGIN (U"Dissimilarity & Configuration: To Configuration (monotone mds)...",
- U"djmw", 19980119)
- INTRO (U"A command that creates a @Configuration object from a @Dissimilarity "
- "object. The selected Configuration object serves as a starting "
- "configuration for the minimization process.")
- MAN_END
- MAN_BEGIN (U"Dissimilarity & Configuration: To Configuration (ratio mds)...",
- U"djmw", 19980119)
- INTRO (U"A command that creates a @Configuration object from a @Dissimilarity "
- "object. The selected Configuration object serves as a starting "
- "configuration for the minimization process.")
- MAN_END
- MAN_BEGIN (U"Dissimilarity & Configuration & Weight: Get stress...", U"djmw", 20040407)
- INTRO (U"A command that calculates the @stress between distances %d__%ij_ "
- "derived from the selected @Configuration object and @disparities "
- "%d\\'p__%ij_ derived from the selected @Dissimilarity object. "
- "With the selected @Weight object the evaluation of the influence "
- "of each dissimilarity on stress can be influenced.")
- ENTRY (U"Settings")
- LIST_ITEM (U"%%Normalized stress%, %%Kruskal's stress-1%, %%Kruskal's "
- "stress-2% or %Raw stress%")
- ENTRY (U"Behaviour")
- NORMAL (U"Except for %absolute %mds, we us stress formula's that are "
- "independent of the scale of the Configuration (see @stress): you would "
- "have got the same stress value if you had pre-multiplied the selected "
- "Configuration with any number greater than zero.")
- MAN_END
- MAN_BEGIN (U"Dissimilarity & Configuration & Weight: To Configuration...", U"djmw", 20040407)
- INTRO (U"A command that creates a @Configuration object from a @Dissimilarity "
- "object. With the selected @Weight object the influence of each "
- "dissimilarity on @stress can be influenced. The selected Configuration "
- "object serves as a starting configuration for the minimization process.")
- ENTRY (U"Settings")
- LIST_ITEM (U"\\bu @@Dissimilarity: To Configuration (monotone mds)...")
- LIST_ITEM (U"\\bu @@Dissimilarity: To Configuration (i-spline mds)...")
- LIST_ITEM (U"\\bu @@Dissimilarity: To Configuration (interval mds)...")
- LIST_ITEM (U"\\bu @@Dissimilarity: To Configuration (ratio mds)...")
- LIST_ITEM (U"\\bu @@Dissimilarity: To Configuration (absolute mds)...")
- MAN_END
- MAN_BEGIN (U"Dissimilarity & Weight: To Configuration...", U"djmw", 20040407)
- INTRO (U"A command that creates a @Configuration object from a @Dissimilarity "
- "object. With the selected @Weight object the influence of each "
- "dissimilarity on @stress can be influenced.")
- ENTRY (U"Settings")
- NORMAL (U"May be different and depend on the representation function, i.e. "
- "the scale of measurement.")
- LIST_ITEM (U"\\bu @@Dissimilarity: To Configuration (monotone mds)...")
- LIST_ITEM (U"\\bu @@Dissimilarity: To Configuration (i-spline mds)...")
- LIST_ITEM (U"\\bu @@Dissimilarity: To Configuration (interval mds)...")
- LIST_ITEM (U"\\bu @@Dissimilarity: To Configuration (ratio mds)...")
- LIST_ITEM (U"\\bu @@Dissimilarity: To Configuration (absolute mds)...")
- MAN_END
- MAN_BEGIN (U"Dissimilarity & Weight: To Configuration...", U"djmw", 20040407)
- INTRO (U"A command that creates a @Configuration object from a @Dissimilarity "
- "object. With the selected @Weight object the influence of each "
- "dissimilarity on @stress can be influenced.")
- ENTRY (U"Settings")
- NORMAL (U"May be different and depend on the representation function, i.e. the scale of measurement.")
- LIST_ITEM (U"\\bu @@Dissimilarity: To Configuration (monotone mds)...")
- LIST_ITEM (U"\\bu @@Dissimilarity: To Configuration (i-spline mds)...")
- LIST_ITEM (U"\\bu @@Dissimilarity: To Configuration (interval mds)...")
- LIST_ITEM (U"\\bu @@Dissimilarity: To Configuration (ratio mds)...")
- LIST_ITEM (U"\\bu @@Dissimilarity: To Configuration (absolute mds)...")
- MAN_END
- MAN_BEGIN (U"Distance", U"djmw", 19971124)
- INTRO (U"One of the @@types of objects@ in P\\s{RAAT}.")
- NORMAL (U"An object of type Distance represents distances between objects in a metrical space.")
- NORMAL (U"Creation")
- LIST_ITEM (U"\\bu @@Confusion: To Dissimilarity (pdf)...")
- LIST_ITEM (U"\\bu @@Dissimilarity: To Distance...")
- MAN_END
- MAN_BEGIN (U"Distance: To Configuration (indscal)...", U"djmw", 19971124)
- INTRO (U"Perform an @@INDSCAL analysis@ on the selected object(s) of type @Distance that "
- "results in a @Configuration and a @Salience object.")
- MAN_END
- MAN_BEGIN (U"Distance: To Configuration (ytl)...", U"djmw", 19971124)
- INTRO (U"A command that creates one @Configuration and one @Salience object "
- "from a collection of one or more @Distance objects.")
- NORMAL (U"This Configuration and Salience object normally serve as starting points "
- "for an @@individual difference scaling@ such as an @@INDSCAL analysis@.")
- NORMAL (U"The algorithm is ddescribedin @@Young, Takane & Lewyckyj (1978)@.")
- MAN_END
- MAN_BEGIN (U"Distance: To ScalarProduct...", U"djmw", 20040407)
- INTRO (U"A command that creates a @ScalarProduct for each selected @Distance.")
- ENTRY (U"Setting")
- TAG (U"##Make sum of squares equal 1.0")
- DEFINITION (U"when selected, the elements in the resulting matrix part will be scaled such that "
- "the sum of all the squared elements in the matrix equals 1.0.")
- ENTRY (U"Algorithm")
- NORMAL (U"ScalarProduct entries %b__%ij_ are created from distances %d__%ij_ bij double centering "
- "the matrix with elements \\--1/2 %d__%ij_^2, i.e.,")
- FORMULA (U"%b__%ij_= \\--1/2(%d__%ij_^2 \\-- %d__%\\.cj_^2 \\-- %d__%i\\.c_^2 + %d__%\\.c\\.c_^2),")
- NORMAL (U"where the dot (\\.c) means averaging over that dimension.")
- MAN_END
- MAN_BEGIN (U"Distance & Configuration: Draw scatter diagram...", U"djmw", 19971201)
- MAN_END
- MAN_BEGIN (U"Distance & Configuration: Get VAF...", U"djmw", 19971201)
- INTRO (U"Calculates the \"%%variance accounted for%\" from the selected collection of @Distance objects and the "
- "selected @Configuration. The optimal @Salience necessary for the calculation "
- "will be inferred from the selected Distance and Configuration objects.")
- MAN_END
- MAN_BEGIN (U"Distance & Configuration: To Configuration (indscal)...", U"djmw", 19971201)
- INTRO (U"Performs an @@INDSCAL analysis@ on the selected objects of type @Distance and calculates "
- "a @Configuration from them. Uses the selected Configuration object as the initial Configuration in "
- "the iteration process.")
- MAN_END
- MAN_BEGIN (U"Distance & Configuration & Salience: Get VAF...", U"djmw", 19971201)
- INTRO (U"Calculates the \"variance accounted for\" from the selected collection "
- "of @Distance objects, the "
- "selected @Configuration and the selected #Salience.")
- MAN_END
- MAN_BEGIN (U"Distance & Configuration & Salience: To Configuration "
- "(indscal)...", U"djmw", 19971201)
- INTRO (U"A command that creates a new @Configuration from the selected "
- "collection of @Distance objects, the selected @Configuration and the "
- "selected #Salience. The selected Configuration and Salience "
- "serve as start values for the @@INDSCAL analysis@.")
- MAN_END
- MAN_BEGIN (U"individual difference scaling", U"djmw", 19970502)
- INTRO (U"The purpose of individual difference scaling is to represent objects, "
- "whose dissimilarities are given, as points in a metrical space. "
- "The distances in the space should be in accordance with the "
- "dissimilarities as well as is possible. Besides the configuration a "
- "@Salience matrix is calculated.")
- NORMAL (U"The basic Euclidean model is:")
- LIST_ITEM (U"%\\de__%ijk_ \\~~ (\\su__%s=1..%r_ %w__%ks_(%x__%is_ \\-- "
- "%x__%js_)^2)^^1/2^")
- NORMAL (U"Here \\de__%ijk_ is the (known) dissimilarity between %objects %i and "
- "%j, as measured on %data %source %k. The %x's are the %coordinates of "
- "the objects in an %r-dimensional space and the %w's are weights or "
- "saliences. Because straight minimization of the expression above "
- "is difficult, one applies transformations on this expression. "
- "Squaring both sides gives the model:")
- LIST_ITEM (U"%\\de^2__%ijk_ \\~~ \\su__%s=1..%r_ %w__%ks_(%x__%is_ \\-- "
- "%x__%js_)^2")
- NORMAL (U"and the corresponding least squares loss function:")
- LIST_ITEM (U"\\su__%k=1..%numberOfSources_ \\su__%i=1..%numberOfPoints_ "
- "\\su__%j=1..%numberOfPoints_ (%\\de^2__%ijk_ \\-- %d^2__%ijk_)^2")
- NORMAL (U"This loss function is minimized in the (ratio scale option of the) "
- "#ALSCAL program of @@Takane, Young & de Leeuw (1976)@.")
- NORMAL (U"The transformation used by @@Carroll & Chang (1970)@ in the INDSCAL "
- "model, transforms the data from each source into scalar products "
- "of vectors. For the dissimilarities:")
- LIST_ITEM (U"%\\be__%ijk_ = \\--{ %\\de^2__%ijk_ \\-- %\\de^2__%i.%k_ \\-- "
- "%\\de^2__.%jk_ + %\\de^2__..%k_ } / 2,")
- NORMAL (U"where dots replacing indices indicate averaging over the range of "
- "that index. In the same way for the distances:")
- LIST_ITEM (U"%z__%ijk_ = \\--{ %d^2__%ijk_ \\-- %d^2__%i.%k_ \\-- %d"
- "^2__%.%jk_ + %d^2__%..%k_ } / 2.")
- LIST_ITEM (U"%\\be__%ijk_ \\~~ %z__%ijk_ = \\su__%s=1..%numberOfDimensions_ "
- "%w__%ks_ %x__%is_ %x__%js_")
- NORMAL (U"Translated into matrix algebra, the equation above translates to:")
- LIST_ITEM (U"%B__%k_ \\~~ %Z__%k_ = %X %W__%k_ %X\\'p,")
- NORMAL (U"where %X is a %numberOfPoints \\xx %numberOfDimensions configuration "
- "matrix, %W__%k_, a non-negative %numberOfDimensions \\xx "
- "%numberOfDimensions matrix with weights, and %B__%k_ "
- "the %k^^th^ slab of %\\be__%ijk_.")
- NORMAL (U"This translates to the following INDSCAL loss function:")
- FORMULA (U"%f(%X, %W__1_,..., %W__%numberOfSources_) = "
- "\\su__%k=1..%numberOfSources_ | %B__%k_ \\-- %X%W__%k_%X\\'p |^2")
- NORMAL (U"")
- MAN_END
- MAN_BEGIN (U"INDSCAL analysis", U"djmw", 20120306)
- INTRO (U"A method for @@individual difference scaling@ analysis in P\\s{RAAT}.")
- NORMAL (U"An INDSCAL analysis can be performed on objects of type Distance.")
- NORMAL (U"If you start with @Dissimilarity objects you first have to transform "
- "them to Distance objects.")
- LIST_ITEM (U"\\bu @@Dissimilarity: To Distance...@")
- NORMAL (U"If you start with a @Confusion you can use:")
- LIST_ITEM (U"\\bu @@Confusion: To Dissimilarity (pdf)...@")
- ENTRY (U"Examples")
- LIST_ITEM (U"\\bu @@Distance: To Configuration (indscal)...@")
- DEFINITION (U"Perform an INDSCAL analysis on one or more objects of type "
- "@Distance to calculate a @Configuration.")
- LIST_ITEM (U"\\bu @@Distance & Configuration: To Configuration (indscal)...@")
- DEFINITION (U"Perform an INDSCAL analysis on one or more objects of type "
- "@Distance to calculate a @Configuration. Use the selected Configuration "
- "object as the initial Configuration in the iteration process.")
- ENTRY (U"Algorithm")
- NORMAL (U"The function to be minimized in INDSCAL is the following:")
- FORMULA (U"%f(%X, %W__1_,..., %W__%numberOfSources_) = "
- "\\su__%i=1..%numberOfSources_ |%S__%i_ \\-- %XW__%i_%X\\'p|^2")
- NORMAL (U"where %X an unknown %numberOfPoints x %numberOfDimensions "
- "configuration matrix, the %W__%i_ are %numberOfSources unknown "
- "diagonal %numberOfDimensions x %numberOfDimensions matrices with weights, "
- "often called saliences, and the %S__%i_ are known symmetric "
- "matrices with scalar products of dimension %numberOfPoints x "
- "%numberOfPoints.")
- NORMAL (U"In the absence of an algorithm that minimizes %f, @@Carroll & "
- "Chang (1970)@ resorted to the @CANDECOMP algorithm, which instead of the "
- "function given above minimizes the following function:")
- FORMULA (U"%g(%X, %Y, %W__1_,..., %W__%numberOfSources_) = "
- "\\su__%i=1..%numberOfSources_ |%S__%i_ \\-- %XW__%i_%Y\\'p|^2.")
- NORMAL (U"Carroll & Chang claimed that for most practical circumstances %X "
- "and %Y converge to matrices that will be columnwise proportional. "
- "However, INDSCAL does not only require symmetry of the solution, "
- "but also non-negativity of the weights. Both these aspects cannot be "
- "guaranteed with the CANDECOMP algorithm.")
- NORMAL (U"@@Ten Berge, Kiers & Krijnen (1993)@ describe an algorithm that "
- "automatically satisfies symmetry because it solves %f directly, and, "
- "also, can guarantee non-negativity of the weights. "
- "This algorithm proceeds as follows:")
- NORMAL (U"Let #%x__%h_ be the %h-th column of %X. We then write the function %f above as:")
- FORMULA (U"%f(#%x__%h_, %w__1%h_, ..., %w__%numberOfSources %h_) = \\su__%i=1.."
- "%numberOfSources_ |%S__%ih_ \\-- #%x__%h_%w__%ih_#%x\\'p__%h_|^2,")
- NORMAL (U"with %S__%ih_ defined as:")
- FORMULA (U"%S__%ih_ = (%S__%i_ - \\su__%j\\=/%h, %j=1..%numberOfDimensions_ "
- "#%x__%j_%w__%ij_#%x\\'p__%j_).")
- NORMAL (U"Without loss of generality we may require that")
- FORMULA (U"#%x\\'p__%h_#%x__%h_ = 1")
- NORMAL (U"Minimizing %f over #%x__%h_ is equivalent to minimizing")
- FORMULA (U"\\su__%i=1..%numberOfSources_ |%S__%ih_|^2 \\-- 2tr \\su "
- "%S__%ih_#%x__%h_%w__%ih_#%x\\'p__%h_ + \\su %w^2__%ih_")
- NORMAL (U"This amounts to maximizing")
- FORMULA (U"%g(#%x__%h_) = #%x\\'p__%h_(\\su %w__%ih_%S__%ih_)#%x__%h_")
- NORMAL (U"subject to #%x\\'p__%h_#%x__%h_ = 1. The solution for #%x__%h_ is "
- "the dominant eigenvector of (\\su %w__%ih_%S__%ih_), "
- "which can be determined with the power method (see @@Golub & van Loan "
- "(1996)@). The optimal value "
- "for the %w__%ih_, given that all other parameters are fixed:")
- FORMULA (U"%w__%ih_ = #%x\\'p__%h_%S__%ih_#%x__%h_")
- NORMAL (U"In an alternating least squares procedure we may update columns of "
- "%X and the diagonals of the %W matrices in any sensible order.")
- MAN_END
- MAN_BEGIN (U"Kruskal analysis", U"djmw", 19971201)
- INTRO (U"One of the @@MDS models@ in P\\s{RAAT}.")
- NORMAL (U"You can perform a Kruskal-type multidimensional scaling only on "
- "objects of type @Dissimilarity. Objects of other types first have to "
- "be converted to objects of Dissimilarity type.")
- ENTRY (U"Example")
- NORMAL (U"Convert a @Dissimilarity object into a @Configuration object.")
- LIST_ITEM (U"\\bu @@Dissimilarity: To Configuration (monotone mds)...@")
- DEFINITION (U"choose appropriate parameters")
- LIST_ITEM (U"\\bu @@Dissimilarity & Configuration: Get stress (monotone mds)...@")
- DEFINITION (U"choose stress-1 to obtain the value for the @stress according "
- "to Kruskal.")
- ENTRY (U"How to get started")
- NORMAL (U"You can create an example @Dissimilarity object with the @@Create "
- "letter R example...@ button which you can find under the "
- "##Multidimensional scaling# option in the #New menu.")
- MAN_END
- MAN_BEGIN (U"MDS models", U"djmw", 20101109)
- INTRO (U"Multidimensional scaling (MDS) models are defined by specifying "
- "how given @Dissimilarity data, %\\de__%ij_, are "
- "mapped into distances of an %m-dimensional MDS @Configuration %#X. "
- "The mapping is specified by a %%representation function%, %f : "
- "%\\de__%ij_ \\-> %d__%ij_(#X), which specifies how "
- "dissimilarities should be related to the distances. The MDS analysis "
- "tries to find the configuration (in a given dimensionality) whose "
- "distances satisfy %f as closely as possible. "
- "This closeness is quantified by a badness-of-fit measure which is often "
- "called @stress.")
- ENTRY (U"Representation functions")
- NORMAL (U"In the application of MDS we try to find a configuration #X such that "
- "the following relations are satisfied as well as possible:")
- FORMULA (U"%f(%\\de__%ij_) \\~~ %d__%ij_(#X)")
- NORMAL (U"The numbers that result from applying %f on %\\de__%ij_ are sometimes "
- "called @disparities %d\\'p__%ij_. In most applications of MDS, besides "
- "the configuration #X, also the function %f is not "
- "completely specified, i.e., the exact parameters of %f are unknown and "
- "must also be estimated during the analysis. If no particular %f can be "
- "derived from a theoretical model, one often restricts %f to a particular "
- "class of functions on the basis of the scale level of the dissimilarity "
- "data. If the disparities are related to the proximities by a specific "
- "parametric function we speak of %metric MDS otherwise we speak of "
- "%ordinal or %%non-metric% MDS.")
- LIST_ITEM (U"\\bu %absolute mds: %d\\'p__%ij_ = \\de__%ij_")
- DEFINITION (U"No parameters need to be estimated.")
- LIST_ITEM (U"\\bu %ratio mds: %d\\'p__%ij_ = %b \\.c \\de__%ij_,")
- DEFINITION (U"where the value of %b can be estimated by a linear regression of "
- "%d__%ij_ on %\\de__%ij_.")
- LIST_ITEM (U"\\bu %interval mds: %d\\'p__%ij_ = %a + %b \\.c %\\de__%ij_,")
- DEFINITION (U"where the values of %a and %b can be estimated by a linear "
- "regression of %d__%ij_ on %\\de__%ij_.")
- LIST_ITEM (U"\\bu %i-spline mds: %d\\'p__%ij_ = %i-spline(%\\de__%ij_),")
- DEFINITION (U"where %i-spline(\\.c) is a smooth monotonically increasing "
- "@spline curve. The conceptual idea is that it is not possible to map "
- "all dissimilarities into disparities by one simple function.")
- LIST_ITEM (U"\\bu %monotone mds: %d\\'p__%ij_ = %monotone(\\de__%ij_),")
- DEFINITION (U"where %monotone(\\.c) is restricted to be a monotonic function "
- "that preserves the order of the dissimilarities:")
- FORMULA (U"if %\\de__%ij_ < %\\de__%kl_, then %d__%ij_(#X) < %d__%kl_(#X)")
- DEFINITION (U"If %\\de__%ij_ = %\\de__%kl_ and no particular constraint is involved for %d__%ij_(#X) "
- "and %d__%kl_(#X) this is referred to as the %%primary approach% to ties. The %%secondary "
- "approach% to ties requires that if %\\de__%ij_ = %\\de__%kl_, then also %d__%ij_(#X) = %d__%kl_(#X).")
- NORMAL (U"More information on all aspects of multidimensional scaling can be found in: "
- "@@Borg & Groenen (1997)@ and @@Ramsay (1988)@.")
- NORMAL (U"The most important object types used in Praat for MDS and the conversions between these types are "
- "shown in the following figure.")
- PICTURE (6, 6, drawMDSClassRelations)
- MAN_END
- MAN_BEGIN (U"Measurement levels", U"djmw", 20151014)
- INTRO (U"According to the measurement theory of @@Stevens (1951)@, there are four measurement levels, namely "
- "#Nominal, #Ordinal, #Interval and #Ratio. In the light of multidimensional scaling, the first "
- "two levels, Nominal and Ordinal, are often called %non-%metric. The last two are %metric.")
- MAN_END
- MAN_BEGIN (U"Multidimensional scaling", U"djmw", 20140117)
- INTRO (U"This tutorial describes how you can use P\\s{RAAT} to "
- "perform ##M#ulti ##D#imensional ##S#caling (MDS) analysis.")
- NORMAL (U"MDS helps us to represent %dissimilarities between objects as "
- "%distances in a %%Euclidean space%. In effect, the more dissimilar two "
- "objects are, the larger the distance between the objects in the Euclidean "
- "space should be. The data types in P\\s{RAAT} that "
- "incorporate these notions are @Dissimilarity, @Distance and "
- "@Configuration.")
- NORMAL (U"In essence, an MDS-analysis is performed when you select a "
- "Dissimilarity object and choose one of the ##To Configuration (xxx)...# "
- "commands to obtain a Configuration object. In the above, method (xxx) "
- "represents on of the possible @@MDS models|multidimensional scaling "
- "models@.")
- ENTRY (U"MDS-analysis")
- NORMAL (U"Let us first create a Dissimilarity object. You can for example "
- "@@Dissimilarity|create a Dissimilarity object from a file@. Here we "
- "will the use the Dissimilarity object from @@Create letter R example...|"
- "the letter R example@. We have chosen the default value (32.5) for the "
- "(uniform) noise range. Note that this may result in substantial "
- "distortions between the dissimilarities and the distances.")
- NORMAL (U"Now you can do the following, for example:")
- NORMAL (U"Select the Dissimilarity and choose @@Dissimilarity: To Configuration "
- "(monotone mds)...|To Configuration (monotone mds)...@, and you perform "
- "a @@Kruskal analysis|kruskal@-like multidimensional scaling which "
- "results in a new "
- "Configuration object. (This Configuration could subsequently be used as "
- "the starting Configuration for a new MDS-analysis!).")
- NORMAL (U"Select the Configuration and choose @@Configuration: Draw...|Draw...@ "
- "and the following picture might result.")
- PICTURE (4.0, 4.0, drawLetterRConfigurationExample2)
- NORMAL (U"The following script summarizes:")
- CODE (U"dissimilarity = Create letter R example: 32.5")
- CODE (U"configuration = To Configuration (monotone mds): 2, \"Primary approach\", 0.00001, 50, 1")
- CODE (U"Draw: 1, 2, -0.8, 1.2, -0.8, 0.7, \"yes\"")
- ENTRY (U"Obtaining the stress value")
- NORMAL (U"Select the Dissimilarity and the Configuration together and query for "
- "the @stress value with: "
- "@@Dissimilarity & Configuration: Get stress (monotone mds)...|"
- "Get stress (monotone mds)...@. ")
- NORMAL (U"The following script summarizes:")
- CODE (U"selectObject: dissimilarity, configuration")
- CODE (U"Get stress (monotone mds): \"Primary approach\", \"Kruskals's "
- "stress-1\"")
- ENTRY (U"The Shepard diagram")
- NORMAL (U"Select the Dissimilarity and the Configuration together to "
- "@@Dissimilarity & Configuration: Draw Shepard diagram...|"
- "draw the Shepard diagram@.")
- PICTURE (4.0, 4.0, drawLetterRShepard)
- NORMAL (U"The following script summarizes:")
- CODE (U"selectObject: dissimilarity, configuration")
- CODE (U"Draw Shepard diagram: 0, 200, 0, 2.2, 1, \"+\", \"yes\"")
- ENTRY (U"The (monotone) regression")
- NORMAL (U"Select the Dissimilarity and the Configuration together to "
- "@@Dissimilarity & Configuration: Draw regression (monotone mds)...|"
- "draw the monotone regression@ of distances on dissimilarities.")
- PICTURE (4.0, 4.0, drawLetterRRegression)
- NORMAL (U"The following script summarizes:")
- CODE (U"selectObject: dissimilarity, configuration")
- CODE (U"Draw monotone regresion: \"Primary approach\", 0, 200, 0, 2.2, 1, \"+\", \"yes\"")
- NORMAL (U"When you enter %noiseRange = 0 in the form for the letter #R, perfect "
- "reconstruction is possible. The Shepard diagram then will show "
- "a perfectly smooth monotonically increasing function.")
- ENTRY (U"Weighing the dissimilarities")
- NORMAL (U"When you can't have equal confidence in all the number in the "
- "Dissimilarity object, you can give different weights to these numbers by "
- "associating a @Weight object with the Dissimilarity object. "
- "An easy way to do this is to select the Dissimilarity object and first "
- "choose @@Dissimilarity: To Weight|To Weight@. Then you might change the "
- "individual weights in the Weight object with the @@TableOfReal: Set "
- "value...| Set value...@ command (remember: make %w__%ij_ = %w__%ji_).")
- NORMAL (U"The following script summarizes:")
- CODE (U"selectObject: dissimilarity")
- CODE (U"weight = To Weight")
- CODE (U"! Change [i][j] and [j][i] cells in the Weight object")
- CODE (U"Set value: i, j, val")
- CODE (U"Set value: j, i, val")
- CODE (U"...")
- CODE (U"! now we can do a weighed analysis.")
- CODE (U"selectObject: dissimilarity, weight")
- CODE (U"To Configuration (monotone mds): 2, \"Primary approach\", 0.00001, 50, 1)")
- NORMAL (U"You can also query the @stress values with three objects selected. "
- "The following script summarizes:")
- CODE (U"selectObject: dissimilarity, weight, configuration")
- CODE (U"Get stress (monotone mds): \"Primary approach\", \"Kruskals's stress-1\"")
- ENTRY (U"Using a start Configuration")
- NORMAL (U"You could also use a Configuration object as a starting "
- "configuration in the minimization process. "
- "Let's assume that you are not satisfied with the stress value from the "
- "Configuration object that you obtained in the previous analysis. "
- "You can than use this Configuration object as a "
- "starting point for further analysis:")
- NORMAL (U"The following script summarizes:")
- CODE (U"selectObject: dissimilarity, configuration, weight")
- CODE (U"To Configuration (monotone mds): 2, \"Primary approach\", 0.00001, 50, 1")
- ENTRY (U"Multiple Dissimilarity's (INDSCAL)")
- NORMAL (U"When you have multiple Dissimilarity objects you can also perform "
- "@@individual difference scaling@ (often called @@INDSCAL analysis@). ")
- NORMAL (U"As an example we can use an @@Create INDSCAL Carroll & Wish "
- "example...| example taken from Carrol & Wish@. "
- "Because INDSCAL only works on metrical data, we cannot use Dissimilarity "
- "objects directly. We have to transform them first @@Dissimilarity: To "
- "Distance...|to Distance@ objects.")
- NORMAL (U"This type of analysis on multiple objects results in two new objects: "
- "a Configuration and a @Salience.")
- MAN_END
- MAN_BEGIN (U"Procrustes", U"djmw", 20010927)
- INTRO (U"One of the @@types of objects@ in P\\s{RAAT}.")
- NORMAL (U"An object of type Procrustes represents the special @@AffineTransform|"
- "affine transform@ that consists of a "
- "combination of a translation, a shape preserving transformation and a scaling (this scaling is often called %dilation). "
- "Because the transformation has to be shape preserving, only a combination of a rotation and a reflection is allowed. "
- "A configuration matrix #%X is transformed in the following way to a new configuration matrix #%Y: ")
- FORMULA (U"#%Y = %s #%X #%T+ #1#%t',")
- NORMAL (U"where %s is the scaling factor, #%T is the shape preserving transformation matrix, #%t is the translation vector, "
- "and #1 is the vector with only ones as its elements.")
- NORMAL (U"For more information about the Procrustes transform and its algorithm "
- "see chapter 19 in @@Borg & Groenen (1997)@.")
- MAN_END
- MAN_BEGIN (U"Procrustes transform", U"djmw", 19980119)
- INTRO (U"A transformation that only uses a combination of a translation, "
- "a scaling and a rigid transformation to transform one Configuration such that it "
- "matches as closely as possible another Configuration. ")
- NORMAL (U"We speak of %%orthogonal Procrustes transform% when only the rigid "
- "transformation is allowed but no scaling or translation.")
- NORMAL (U"For more information about the Procrustes transform and its algorithm "
- "see chapter 19 in @@Borg & Groenen (1997)@.")
- MAN_END
- MAN_BEGIN (U"Proximity", U"djmw", 19961008)
- INTRO (U"One of the @@types of objects@ in P\\s{RAAT}.")
- NORMAL (U"An object of type #Proximity represents proximities between objects.")
- ENTRY (U"Inside a Proximity")
- NORMAL (U"With @Inspect you will see the following attributes:")
- TAG (U"%numberOfRows, %numberOfColumns")
- DEFINITION (U"the number of objects (%numberOfRows and %numberOfColumns are "
- "equal and \\>_1).")
- TAG (U"%rowLabels, %columnLabels")
- DEFINITION (U"the names associated with the objects (%rowLabels and "
- "%columnLabels are equal.")
- TAG (U"%data [1..%numberOfRows][1..%numberOfColumns]")
- DEFINITION (U"the proximities between the objects.")
- MAN_END
- MAN_BEGIN (U"Salience", U"djmw", 19980112)
- INTRO (U"One of the @@types of objects@ in P\\s{RAAT}.")
- NORMAL (U"Elements %s__%ij_ in the "
- "Salience matrix represent the importance of dimension %j (in the "
- "@Configuration) for data source %i.")
- ENTRY (U"Commands")
- NORMAL (U"Creation, as a by-product of:")
- LIST_ITEM (U"\\bu @@Distance: To Configuration (indscal)...")
- LIST_ITEM (U"\\bu @@Distance: To Configuration (ytl)...")
- MAN_END
- MAN_BEGIN (U"ScalarProduct", U"djmw", 19980125)
- INTRO (U"One of the @@types of objects@ in P\\s{RAAT}.")
- NORMAL (U"An object of type ScalarProduct represents scalar products %b__%ij_ "
- "between objects %i and %j in a metrical space.")
- FORMULA (U"%b__%ij_ = \\su__%k=1..%numberOfDimensions_ %x__%ik_%x__%jk_,")
- NORMAL (U"where %x__%ik_ and %x__%jk_ are the coordinates of the %k-th "
- "dimension of points %i and %j, respectively. From this definition one "
- "can see that scalar products, in contrast to distances, "
- "do change when the origin is shifted.")
- NORMAL (U"Creation")
- LIST_ITEM (U"\\bu @@Distance: To ScalarProduct...@")
- MAN_END
- MAN_BEGIN (U"Similarity", U"djmw", 19961008)
- INTRO (U"One of the @@types of objects@ in P\\s{RAAT}.")
- NORMAL (U"An object of type Similarity represent a one-way "
- "table of similarities between \"objects\".")
- ENTRY (U"Commands")
- NORMAL (U"Creation")
- LIST_ITEM (U"\\bu @@Confusion: To Similarity...")
- NORMAL (U"Drawing")
- LIST_ITEM (U"\\bu ##Draw as numbers...")
- LIST_ITEM (U"\\bu ##Draw as squares...")
- NORMAL (U"Query")
- LIST_ITEM (U"\\bu ##Get column mean (index)...")
- LIST_ITEM (U"\\bu ##Get column mean (label)...")
- LIST_ITEM (U"\\bu ##Get column stdev (index)...")
- LIST_ITEM (U"\\bu ##Get column stdev (label)...")
- NORMAL (U"Modification")
- LIST_ITEM (U"\\bu ##Formula...")
- LIST_ITEM (U"\\bu ##Remove column (index)...")
- LIST_ITEM (U"\\bu ##Insert column (index)...")
- LIST_ITEM (U"\\bu ##Set row label (index)...")
- LIST_ITEM (U"\\bu ##Set row label (label)...")
- LIST_ITEM (U"\\bu ##Set column label (index)...")
- LIST_ITEM (U"\\bu ##Set column label (label)...")
- NORMAL (U"Analysis")
- LIST_ITEM (U"\\bu @@Similarity: To Dissimilarity...")
- MAN_END
- MAN_BEGIN (U"Similarity: To Dissimilarity...", U"djmw", 20040407)
- INTRO (U"A command that creates a @Dissimilarity from every selected "
- "@Similarity.")
- ENTRY (U"Setting")
- TAG (U"##Maximum dissimilarity")
- DEFINITION (U"determines the maximum dissimilarity possible. When the default "
- "value, 0.0, is chchosen%maximumDissimilarity "
- "is calculated as the maximum element in the Similarity object.")
- ENTRY (U"Algorithm")
- NORMAL (U"To obtain dissimilarities we 'reverse' similarities:")
- FORMULA (U"%%dissimilarity__%ij_ = %maximumDissimilarity \\-- %similarity__%ij_")
- NORMAL (U"In this way the order of dissimilarities is the reverse of the order "
- "of the similarities.")
- MAN_END
- MAN_BEGIN (U"smacof", U"djmw", 19980119)
- INTRO (U"Scaling by Majorizing a Complicated Function, the iterative algorithm "
- "to find an optimal Configuration.")
- LIST_ITEM (U"1. Initialize")
- LIST_ITEM (U" 1.a. Get initial Configuration #Z")
- LIST_ITEM (U" 1.b. Set stress %\\si__%n_^^[0]^ to a very large value.")
- LIST_ITEM (U" 1.c. Set iteration counter %k = 0")
- LIST_ITEM (U"2. Increase iteration counter by one: %k = %k + 1")
- LIST_ITEM (U"3. Calculate distances %d__%ij_(#Z).")
- LIST_ITEM (U"4. Transform dissimilarities %\\de__%ij_ into disparities "
- "%d\\'p__%ij_.")
- LIST_ITEM (U"5. Standardize the disparities so that %\\et__%d\\'p_^2 = "
- "%n(%n\\--1)/2.")
- LIST_ITEM (U"6. Compute the Guttman transform #X^^[%k]^ of #Z.")
- LIST_ITEM (U"7. Compute new distances %d__%ij_(#X^^[%k]^).")
- LIST_ITEM (U"8. Compute normalized stress %\\si__%n_ (#d\\'p, #X^^[%k]^)")
- LIST_ITEM (U"9. If |%\\si__%n_^^[%k]^ \\-- %\\si__%n_^^[%k\\--1]^| / %\\si__%n_"
- "^^[%k\\--1]^ < %\\ep or %k > %maximumNumberOfIterations, then stop")
- LIST_ITEM (U"10. Set #Z = #X^^[%k]^, and go to 2.")
- NORMAL (U"This algorithm goes back to @@De Leeuw (1977)@.")
- MAN_END
- MAN_BEGIN (U"spline", U"djmw", 20121101)
- INTRO (U"A spline function %f is a piecewise polynomial function defined on "
- "an interval [%x__%min_, %x__%max_] "
- "with specified continuity constraints, i.e., when the interval [%x__%min_,"
- " %x__%max_] is subdivided by points %\\xi__%i_ such that %x__%min_ = "
- "%\\xi__%1_ < ... < %\\xi__%q_ = %%x__%max_, then within each subinterval "
- "[%\\xi__%j_, %\\xi__%j+1_) the function is a polynomial %P__%j_ of "
- "specified degree %k.")
- NORMAL (U"A %%knot sequence% %t = {%t__1_, ..., %t__%n+%k_}, where %n is the "
- "number of free parameters that specify the spline function, is derived "
- "from the %\\xi__%i_ by placing knots at the boundary values %\\xi__%i_ "
- "according to the order of continuity at that boundary. The most common "
- "continuity characteristics imposed on %f request that for adjacent "
- "polynomials the derivatives up to order %k\\--2 match. For example, "
- "the knot sequence of a spline of order %k for a partition of [%x__%min_,"
- " %%x__%max_] into three intervals (%q = 4) will be %t__1_ = ... = "
- "%t__%k_ = %x__%min_ (=%\\xi__1_), %t__%k+1_ = %\\xi__2_, %t__%k+2_ = "
- "%\\xi__3_ , %t__%k+3_ = ... = %t__%k+%k+2_ = %x__%max_ (= %\\xi__4_). "
- "This is called a %simple knot sequence, because all interior knots are "
- "simple. The number of free parameters %n for this case obeys a simple "
- "formula:")
- FORMULA (U"%n = %numberOfInteriorKnots + %order.")
- NORMAL (U"With suitable basis functions, for example, the M-spline family "
- "%M__%i_(%x|%k, %t), %i=1..%n, we can write any spline %f in the form:")
- FORMULA (U"%f = \\su__%i=1..%n_ %a__%i_%M__%i_,")
- NORMAL (U"where the %M__%i_ are defined by the following recursive formula's:")
- FORMULA (U"%M__%i_(%x|1,%t) = 1 / (%t__%i+1_ \\-- %t__%i_), %t__%i_ "
- "\\<_ %x < %t__%i+1_, 0 otherwise")
- FORMULA (U"%M__%i_(%x|%k,%t) = %k [(%x\\--%t__%i_)%M__%i_(%x|%k\\--1,%t) + "
- "(%t__%i+%k_\\--%x)%M__%i+1_(%x|%k\\--1,%t)] / "
- "((%k\\--1)(%t__%i+%k_\\--%t__%i_))")
- NORMAL (U"These %M__%i_ are localized because %M__%i_(%x|%k,%t) > 0 only when "
- "%t__%i_ \\<_ %x < %t__%i+%k_ and zero otherwise. Also, we have \\in"
- " M__%i_(%x)%dx = 1. Because of this localization a change in coefficient "
- "%a__%i_ will only effect %f within this interval.")
- NORMAL (U"The following picture shows an M-spline of order 3 on the interval "
- "[0, 1], with three interior knots at 0.3, 0.5 and 0.6.")
- PICTURE (5.0, 5.0, drawMsplineExample)
- NORMAL (U"Because the M-splines are nonnegative, %monotone splines% can be "
- "derived from them by %integration:")
- FORMULA (U"%I__%i_(%x|%k,%t) = \\in__%xmin_^%x %M__%i_(%u|%k,%t) %du")
- NORMAL (U"Because each %M__%i_(%x|%k, %t) is a piecewise polynomial of degree"
- " %k\\--1, each %I__%i_ will be of degree %k. Now we can write:")
- FORMULA (U"%f = \\su__%i=1..%n_ %b__%i_%I__%i_(%x|%k,%t)")
- NORMAL (U"We can use an M-spline of order %k+1 with a simple knot sequence %t, "
- "for which %t__%j_ \\<_ x < %t__%j+1_, to put "
- "the I-spline of order %k into a more convenient form:")
- FORMULA (U"%I__%i_(%x|%k,%t) = 0, %i > %j")
- FORMULA (U"%I__%i_(%x|%k,%t) = \\su__%m=%i+1..%j_ (%t__%m+%k+1_\\--"
- "%t__%m_)%M__%m_(%x|%k+1,%t)/(%k+1), %j\\--%k \\<_ %i \\<_ %j")
- FORMULA (U"%I__%i_(%x|%k,%t) = 1, %i < %j\\--%k")
- NORMAL (U"The following figure shows the I-splines that were derived from "
- "the M-splines above.")
- PICTURE (5.0, 5.0, drawIsplineExample)
- NORMAL (U"These spline formula's were taken from @@Ramsay (1988)@ and the "
- "errors in his I-spline formulas were corrected.")
- MAN_END
- MAN_BEGIN (U"stress", U"djmw", 19980108)
- INTRO (U"A badness-of-fit measure for the entire MDS representation.")
- NORMAL (U"Several measures exist.")
- ENTRY (U"Raw stress")
- FORMULA (U"%\\si__%r_ (#d\\'p, #X) = \\su__%i<%j_ %w__%ij_(%d\\'p__%ij_ \\-- "
- "%d__%ij_(#X))^2")
- FORMULA (U"= \\su__%i<%j_ %w__%ij_%d\\'p__%ij_^2 + \\su__%i<%j_ "
- "%w__%ij_%d__%ij_^2(#X) \\-- "
- "2 \\su__%i<%j_ %w__%ij_%d\\'p__%ij_%d__%ij_(#X)")
- FORMULA (U"= %\\et__%d\\'p_^2 + %\\et^2(#X) \\-- 2%\\ro(#d\\'p, #X)")
- NORMAL (U"where the %d\\'p__%ij_ are the @disparities that are the result "
- "from the transformation of the dissimilarities, i.e., %f(%\\de__%ij_). "
- "Raw stress can be misleading because it is dependent on the normalization "
- "of the disparities. The following measure tries to circumvent this "
- "inconvenience.")
- ENTRY (U"Normalized stress")
- FORMULA (U"%\\si__%n_ = \\si__%r_ / %\\et__%d\\'p_^2")
- NORMAL (U"This is the stress function that we minimize by iterative "
- "majorization. It goes back to @@De Leeuw (1977)@.")
- ENTRY (U"Kruskal's stress-1")
- FORMULA (U"%\\si__1_ = \\Vr (\\su__%i<%j_ %w__%ij_(%d\\'p__%ij_ \\-- "
- "%d__%ij_(#X))^2 / \\su__%i<%j_ %w__%ij_%d__%ij_^2(#X))^^1/2^")
- NORMAL (U"In this measure, which is due to @@Kruskal (1964)@, stress is "
- "expressed in relation to the size of #X.")
- ENTRY (U"Kruskal's stress-2")
- FORMULA (U"%\\si__2_ = \\Vr (\\su__%i<%j_ %w__%ij_(%d\\'p__%ij_ \\-- "
- "%d__%ij_(#X))^2 / \\su__%i<%j_ %w__%ij_(%d__%ij_(#X) - "
- "%averageDistance)^2)^^1/2^.")
- NORMAL (U"In general, this measure results in a stress value that is "
- "approximately twice the value for stress-1.")
- ENTRY (U"Relation between %\\si__1_ and %\\si__n_")
- NORMAL (U"When we have calculated %\\si__%n_ for Configuration #X, "
- "disparities #d\\'p and Weight #W we cannot "
- "directly use #X, #d\\'p and #W to calculate %\\si__1_ because the "
- "scale of #X is not necessarily optimal "
- "for %\\si__1_. We allow therefore a scale factor %b > 0 and "
- "try to calculate %\\si__1_ (#d\\'p, %b #X). We minimize the resulting "
- "expression for %b and substitute "
- "the result back into the formula for stress, i.e.,")
- FORMULA (U"%\\si__1_^2 (#d\\'p, %b #X) = (%\\et__%d\\'p_^2 + %b^2 %\\et^2(#X) "
- "\\-- 2 %b %\\ro(#d\\'p, #X)) / %b^2 %\\et^2(#X)")
- FORMULA (U"d%\\si__1_^2 (%b) / d%b == 0, gives")
- FORMULA (U"%b = %\\et__%d\\'p_^2 / %\\ro")
- FORMULA (U"%\\si__1_^2 = (1 - %\\ro^2 / (%\\et__%d\\'p_^2\\.c%\\et^2(#X)))")
- NORMAL (U"This means that %\\si__1_ = \\Vr %\\si__%n_.")
- ENTRY (U"Relation between %\\si__2_ and %\\si__n_")
- NORMAL (U"We can do the same trick as before for %\\si__2_:")
- FORMULA (U"%\\si__2_^2 (#d\\'p, %b #X) = (%\\et__%d\\'p_^2 + %b^2 %\\et^2(#X) "
- "\\-- 2 %b %\\ro(#d\\'p, #X)) / "
- "(%b^2 \\su__%i<%j_ %w__%ij_(%d__%ij_(#X) - %averageDistance)^2)")
- NORMAL (U"From which we derive:")
- FORMULA (U"%\\si__2_ = \\Vr ((%\\et__%d\\'p_^2 \\.c %\\et^2(#X) - "
- "%\\ro^2(#d\\'p, #X)) / (%\\et__%d\\'p_^2 \\.c \\su__%i<%j_ "
- "%w__%ij_(%d__%ij_(#X) - %averageDistance)^2))")
- MAN_END
- MAN_BEGIN (U"TableOfReal: Centre columns", U"djmw", 19980422)
- INTRO (U"A command that centres the columns in the selected @TableOfReal "
- "objects.")
- ENTRY (U"Algorithm")
- NORMAL (U"The new values in the table, %x\\'p__%ij_, will be:")
- FORMULA (U"%x\\'p__%ij_ = %x__%ij_ \\-- %x__\\.c%j_,")
- NORMAL (U"where")
- FORMULA (U"%x__\\.c%j_ = \\su__%i=1..%numberOfRows_ %x__%ij_ / %numberOfRows,")
- NORMAL (U"the average of column %j.")
- MAN_END
- MAN_BEGIN (U"TableOfReal: Centre rows", U"djmw", 19980422)
- INTRO (U"A command that centres the rows in the selected @TableOfReal objects.")
- ENTRY (U"Algorithm")
- NORMAL (U"The new values in the table, %x\\'p__%ij_, will be:")
- FORMULA (U"%x\\'p__%ij_ = %x__%ij_ \\-- %x__%i\\.c_,")
- NORMAL (U"where")
- FORMULA (U"%x__%i\\.c_ = \\su__%j=1..%numberOfColumns_ %x__%ij_ / "
- "%numberOfColumns,")
- NORMAL (U"the average of row %i.")
- MAN_END
- MAN_BEGIN (U"TableOfReal: Get table norm", U"djmw", 19980422)
- INTRO (U"A command that returns the norm of the selected @TableOfReal object.")
- ENTRY (U"Algorithm")
- NORMAL (U"Returns: sqrt (\\su__%i=1..%numberOfRows_ \\su__%j=1..%numberOfColumns"
- "_ %x__%ij_^2).")
- MAN_END
- MAN_BEGIN (U"TableOfReal: Normalize columns...", U"djmw", 19980422)
- INTRO (U"A command that normalizes the columns in the selected @TableOfReal "
- "objects.")
- ENTRY (U"Setting")
- TAG (U"##Norm")
- DEFINITION (U"determines the sum of the squared elements in each column after "
- "normalization.")
- ENTRY (U"Algorithm")
- NORMAL (U"All elements %x__%ij_ in each column %j=1..%numberOfColumns will be "
- "multiplied by sqrt (%norm / \\su__%i=1..%numberOfRows_ %x__%ij_^2).")
- MAN_END
- MAN_BEGIN (U"TableOfReal: Normalize rows...", U"djmw", 19980422)
- INTRO (U"A command that normalizes the rows in the selected @TableOfReal "
- "objects.")
- ENTRY (U"Setting")
- TAG (U"##Norm")
- DEFINITION (U"determines the sum of the squared elements in each row after "
- "normalization.")
- ENTRY (U"Algorithm")
- NORMAL (U"All elements %x__%ij_ in each row %i=1..%numberOfRows will be "
- "multiplied by sqrt (%norm / \\su__%j=1..%numberOfColumns_ %x__%ij_^2).")
- MAN_END
- MAN_BEGIN (U"TableOfReal: Normalize table...", U"djmw", 19980422)
- INTRO (U"A command that normalizes the elements in the selected @TableOfReal "
- "objects.")
- ENTRY (U"Setting")
- TAG (U"##Norm")
- DEFINITION (U"determines the sum of the squared elements after normalization.")
- ENTRY (U"Algorithm")
- NORMAL (U"All elements %x__%ij_ will be multiplied by "
- "sqrt (%norm / \\su__%i=1..%numberOfRows_ \\su__%j=1..%numberOfColumns_"
- " %x__%ij_^2.")
- MAN_END
- MAN_BEGIN (U"Weight", U"djmw", 19980108)
- INTRO (U"One of the @@types of objects@ in P\\s{RAAT}.")
- NORMAL (U"An object of type Weight represents a matrix "
- "with weights %w__%ij_.")
- NORMAL (U"An object of type Weight selected together with an onject of type"
- " @Dissimilarity can be used to make distinctions in the importance of the"
- " contribution of each individual dissimilarity %\\de__%ij_ to "
- "@stress and therefore to the final configuration. Weights can be "
- "used for instance to code for missing values, i.e., take %w__%ij_ = 0 if "
- "dissimilarity %\\de__%ij_ is missing and %w__%ij_ = 1 if %\\de__%ij_ is"
- " known.")
- ENTRY (U"Commands")
- NORMAL (U"Creation")
- LIST_ITEM (U"\\bu @@Dissimilarity: To Weight")
- NORMAL (U"Analysis")
- NORMAL (U"See @@Dissimilarity & Weight: To Configuration...@ for help on the "
- "following analysis items: ")
- LIST_ITEM (U"\\bu ##Dissimilarity & Weight: To Configuration (monotone mds)...")
- LIST_ITEM (U"\\bu ##Dissimilarity & Weight: To Configuration (i-spline mds)...")
- LIST_ITEM (U"\\bu ##Dissimilarity & Weight: To Configuration (interval mds)...")
- LIST_ITEM (U"\\bu ##Dissimilarity & Weight: To Configuration (ratio mds)...")
- LIST_ITEM (U"\\bu ##Dissimilarity & Weight: To Configuration (absolute mds)...")
- NORMAL (U"Query")
- NORMAL (U"See @@Dissimilarity & Configuration & Weight: Get stress...@ for help "
- "on the following query items: ")
- LIST_ITEM (U"\\bu ##Dissimilarity & Configuration & Weight: Get stress "
- "(monotone mds)...")
- LIST_ITEM (U"\\bu ##Dissimilarity & Configuration & Weight: Get stress "
- "(i-spline mds)...")
- LIST_ITEM (U"\\bu ##Dissimilarity & Configuration & Weight: Get stress "
- "(interval mds)...")
- LIST_ITEM (U"\\bu ##Dissimilarity & Configuration & Weight: Get stress "
- "(ratio mds)...")
- LIST_ITEM (U"\\bu ##Dissimilarity & Configuration & Weight: Get stress "
- "(absolute mds)...")
- MAN_END
- /************ references ***********************************************/
- MAN_BEGIN (U"Abramowitz & Stegun (1970)", U"djmw", 19971201)
- NORMAL (U"M. Abramowitz & I. Stegun (1970): %%Handbook of mathematical "
- "functions%. New York: Dover Publications.")
- MAN_END
- MAN_BEGIN (U"Borg & Groenen (1997)", U"djmw", 19971219)
- NORMAL (U"I. Borg & P. Groenen (1997): %%Modern multidimensional scaling: "
- "theory and applications%. Springer.")
- MAN_END
- MAN_BEGIN (U"Brokken (1983)", U"djmw", 19980406)
- NORMAL (U" F.B. Brokken (1983): \"Orthogonal Procrustes rotation maximizing "
- "congruence.\" %Psychometrika #48: 343\\--352.")
- MAN_END
- MAN_BEGIN (U"Cailliez (1983)", U"djmw", 19971201)
- NORMAL (U" F. Cailliez (1983): \"The analytical solution of the additive "
- "constant problem.\" %Psychometrika #48, 305-308.")
- MAN_END
- MAN_BEGIN (U"Carroll & Chang (1970)", U"djmw", 19971201)
- NORMAL (U"J.D. Carroll & J.-J. Chang, (1970): \"Analysis of Individual "
- "Differences in Multidimensional scaling via an N-way generalization of "
- "\"Eckart-Young\" Decomposition.\" %Psychometrika #35: 283\\--319.")
- MAN_END
- MAN_BEGIN (U"Carroll & Wish (1974)", U"djmw", 19971201)
- NORMAL (U"J.D. Carroll & M. Wish, (1974): \"Models and methods for three-way "
- "multidimensional scaling.\" In D.H. Krantz, R.C. Atkinson, R.D. Luce & "
- "P. Suppes (eds.): %%Contemporary developments in mathematical psychology: "
- "Vol. 2 Measurement, psychophysics, and neural "
- "information processing%, 283\\--319. New York: Academic Press.")
- MAN_END
- MAN_BEGIN (U"De Leeuw (1977)", U"djmw", 19971201)
- NORMAL (U"J. de Leeuw (1977): \"Applications of convex analysis to "
- "multidimensional scaling.\" In J.R. Barra, F. Brodeau, G. Romier & "
- "B. van Cutsem (eds.): %%Recent developments in statistics%. Amsterdam: "
- "North-Holland. 133\\--145.")
- MAN_END
- MAN_BEGIN (U"De Leeuw & Pruzansky (1978)", U"djmw", 19971201)
- NORMAL (U"J. de Leeuw & S. Pruzansky (1978): \"A new computational method to "
- "fit the weighted Euclidean distance model.\" %Psychometrika #43: 479\\--490.")
- MAN_END
- MAN_BEGIN (U"Gifi (1990)", U"djmw", 19971207)
- NORMAL (U"A. Gifi (1990): %%Nonlinear multivariate analysis%. John Wiley & "
- "Sons Ltd., reprint 1996.")
- MAN_END
- MAN_BEGIN (U"Golub & van Loan (1996)", U"djmw", 19971207)
- NORMAL (U"G. Golub & C. van Loan (1996): %%Matrix computations%. Third edition. "
- "London: The Johns Hopkins University Press.") // ??
- MAN_END
- MAN_BEGIN (U"Green, Carmone & Smith (1989)", U"djmw", 19971201)
- NORMAL (U"P. Green, F. Carmone, S. Smith (1989): "
- "%%Multidimensional scaling: concepts and applications%. Section 3. Allyn and Bacon.")
- MAN_END
- MAN_BEGIN (U"Kaiser (1958)", U"djmw", 19980404)
- NORMAL (U" H.F. Kaiser (1958): \"The varimax criterion for analytic rotation "
- "in factor analysis.\" %Psychometrika #23: 187\\--200.")
- MAN_END
- MAN_BEGIN (U"Kiers & Groenen (1996)", U"djmw", 19971219)
- NORMAL (U"H.A.L. Kiers & P. Groenen (1996): \"A monotonically convergent "
- "algorithm for orthogonal congruence rotation.\" %Psychometrika #61: "
- "375\\--389.")
- MAN_END
- MAN_BEGIN (U"Klein, Plomp & Pols (1970)", U"djmw", 19971201)
- NORMAL (U" W. Klein, R. Plomp, & L.C.W. Pols (1970): \"Vowel Spectra, "
- "Vowel Spaces, and Vowel Identification.\" %%Journal of the Acoustical Society of America% #48: 999\\--1009.")
- MAN_END
- MAN_BEGIN (U"Kruskal (1964)", U"djmw", 19971201)
- NORMAL (U"J.B. Kruskal (1964): \"Nonmetric multidimensional scaling: a "
- "numerical method.\" %Psychometrika #29: 115\\--129.")
- MAN_END
- MAN_BEGIN (U"Ramsay (1988)", U"djmw", 19980106)
- NORMAL (U"J.O. Ramsay (1988): \"Monotone regression splines in action.\" "
- "%%Statistical Science% #3: 425\\--461.")
- MAN_END
- MAN_BEGIN (U"Stevens (1951)", U"djmw", 19971201)
- NORMAL (U"S.S. Stevens (1951): \"Mathematics, measurement, and psychophysics.\" "
- "In S.S. Stevens (ed.): %%Handbook of experimental psychology%. New York: "
- "Wiley.")
- MAN_END
- MAN_BEGIN (U"Takane, Young & de Leeuw (1976)", U"djmw", 19971201)
- NORMAL (U"Y. Takane, F. Young, J. de Leeuw (1976): \"Non-metric individual "
- "differences multidimensional scaling: an alternating least squares method "
- "with optimal scaling features.\" %Psychometrika #42: 7\\--67.")
- MAN_END
- MAN_BEGIN (U"Ten Berge (1995)", U"djmw", 19980404)
- NORMAL (U"J.M.F. ten Berge (1995): \"Suppressing permutations or rigid planar "
- "rotations: a remedy against nonoptimal varimax rotations.\" "
- "%Psychometrika #60, 437\\--446.")
- MAN_END
- MAN_BEGIN (U"Ten Berge, Kiers & Krijnen (1993)", U"djmw", 19971207)
- NORMAL (U"J.M.F. ten Berge, H.A.L. Kiers & W.P. Krijnen (1993): \"Computational "
- "solutions for the problem of negative saliences and nonsymmetry in "
- "INDSCAL.\" %%Journal of Classification% #10: 115\\--124.")
- MAN_END
- MAN_BEGIN (U"Torgerson (1958)", U"djmw", 19971201)
- NORMAL (U"W.S. Torgerson (1958): %%Theory and methods of scaling%. New York: Wiley.")
- MAN_END
- MAN_BEGIN (U"Young, Takane & Lewyckyj (1978)", U"djmw", 19971201)
- NORMAL (U"F.W. Young, Y. Takane & R. Lewyckyj (1978): "
- "\"Three notes on ALSCAL.\" %Psychometrika #43: 433\\--435.")
- MAN_END
- MAN_BEGIN (U"Weller & Romney (1990)", U"djmw", 19971216)
- NORMAL (U"S.C. Weller & A.K. Romney (1990): %%Metric Scaling: "
- "correspondence analysis%. Sage University Paper Series on Quantitative "
- "Applications in the Social Sciences 07-075. Newbury Park, CA: Sage.")
- MAN_END
- }
- /* End of file manual_MDS.cpp */
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