57     Typology and Ascending Classification

57.1  Types of Variables Used

A set of active variables is denoted here X_a, and a set of passive variables X_p.

57.2  Case Profile

Pk = (xk1, xk2, ..., xkv, ..., xka) = (xkv)

where all x_v Î X_a.

If the active variables are requested to be standardized the k^th case profile becomes

Pk = æ ç è xkv sv ö ÷ ø

where s_v is the standard deviation of the variable x_v (see 7.b below).

57.3  Group Profile

Pi = ( x   i1  , x   i2  , ..., x   iv  , ..., x   ia  ) = ( x   iv  )

and in the case of standardized data it becomes

Pi = ( x   iv   /  sv )

where the numerator is the mean of the variable x_v for the cases belonging to the group i and denominator is the overall standard deviation of this variable.

57.4  Distances Used

a)  City block distance.

dij = d (Pi, Pj) = a å v = 1  av  | x   iv  - x   jv  | a å v = 1  av

b)  Euclidean distance.

dij = d (Pi, Pj) =   æ  ú  ú  ú Ö a å v = 1  av  ( x   iv  - x   jv  )2 a å v = 1  av

c)  Chi-square distance.

dij = d (Pi, Pj) =   æ  ú Ö a å v = 1  1 pv æ è piv pi - pjv pj ö ø 2

where

pv = t å g = 1  x   gv    ,         pi = a å v = 1  x   iv    ,         pj = a å v = 1  x   jv

piv = x   iv   /   é ë t å g = 1  a å v = 1  x   gv  ù û   ,         pjv = x   jv   /   é ë t å g = 1  a å v = 1  x   gv  ù û

Moreover, the program provides a possibility of using "weighted" distance, called displacement, which is defined as follows:

Dij = D (Pi, Pj) = 2 Ni Nj Ni + Nj  dij

Note that displacement between two case profiles is equal to their distance since N_i = N_j = 1.

57.5  Building of an Initial Typology

• case profiles of t randomly selected cases (using random numbers) constitute the starting configuration; in order to obtain the initial configuration, the remaining cases are distributed into t groups as described below;
• case profiles of t cases selected in a stepwise manner constitute the starting configuration; in order to obtain the initial configuration, the remaining cases are distributed into t groups as described below;
• the initial configuration is a set of group profiles calculated for cases distributed across categories of a key variable;
• the initial configuration is a set of 'a priori' group profiles provided by the user.

When the construction starts from t case profiles, the program considers this set of t vectors as a set of t 'starting cases' and distributes the remaining cases according to their distance to each of the starting case.

Let denote the set of t starting cases by

Pstarting = ì í î Pk1, Pk2, ..., Pkt ü ý þ

and the distance between groups and/or cases i and j by D (P_i, P_j).

Note that D (P_i, P_j) can be any distance defined in the section 4 above.

For each case i ¬ Î P_starting the program calculates

b = min 1 £ j £ t     é ë D (Pi, Pkj) ù û

g = min é ë D (Pk1,Pk2),D (Pk1,Pk3), ... ,D (Pkt-1,Pkt) ù û

There are two possibilities:

• b £ g : case i is assigned to the closest group P_kj and the profile of this group is recalculated

  
  

  Pkj = æ è Pkj + Pi ö ø / 2

• b > g : case i forms a new group which is added to the set P_starting, and the two closest profiles P_kj and P_kj¢ are aggregated forming one group with the new profile

  
  

  Pkj = æ è Pkj + Pkj¢ ö ø / 2

At the end of this procedure, the initial configuration is a set of t profiles

Pinitial = ì í î P1, P2, ..., Pj, ..., Pt ü ý þ

where P_j is a mean profile of all the cases belonging to the group j.

At this stage the program does not take into account weighting of cases, if any.

b)  Stabilization of the initial configuration. The initial configuration is stabilized by an iteration process. During each iteration, the program redistributes the cases among initial groups taking into account their distances to each group profile.

Here again there are two possibilities:

• when case i Î P_j and

  
  

  D (Pi, Pj) = min 1 £ g £ t  é ë D (Pi, Pg) ù û

  then this case remains in the group P_j;

• when case i Î P_j but

  
  

  D (Pi, Pj¢) = min 1 £ g £ t  é ë D (Pi, Pg) ù û

  then the case i is moved from the group P_j to the group P_j¢, and the profiles of those two groups are recalculated as follows:

  
  

  Pj = ( Nj Pj - Pi ) / (Nj - 1)

  
  

  Pj¢ = ( Nj¢ Pj¢ + Pi ) / (Nj¢ + 1)

After this operation, the group P_j contains N_j - 1 cases and the group P_j¢ contains N_j¢ + 1 cases.

Note that if the cases are weighted, then

Nj = Nj - wi

Nj¢ = Nj¢ + wi

Pi = wi  Pi

where w_i is the weight of the case i, and N_j and N_j¢ are the weighted number of cases in the groups P_j and P_j¢ respectively.

Stability of groups is measured by the percentage of cases that do not change groups between two subsequent iterations.

The procedure is repeated until the groups are stabilized or when the number of iterations fixed by the user is reached.

57.6  Characteristics of Distances by Groups

b)  Mean. Mean distance for each group, i.e. the mean of distances from the group profile over all cases belonging to this group.

c)  SD. Standard deviation of distance for each group.

d)  Classification of distances. Distribution of cases, both in terms of frequency and percentages, across 15 continuous intervals, which are different for each group.

e)  Total count. Total number of cases participating in the building of the initial typology.

f)  Mean. Overall mean distance.

g)  SD. Overall standard deviation of distance.

h)  Classification of distances (same limits for each group). Same as 6.d above except that the 15 intervals are of the same range for all groups.

57.7  Summary Statistics for Quantitative Variables and for Qualitative Active Variables

x   v  = æ è å k  wk xkv ö ø  /  W

b)  S. D. Standard deviation.

sv =   æ Ö é ë   W   å k  wk xkv2 - æ è å k  wk xkv ö ø 2   ù û  /  W2

c)  Weight. The value of variable weight calculated for each variable as follows:

av = ì ï ï ï ï ï ï í ï ï ï ï ï ï î 0 for quantitative passive variables 1 for quantitative active variables (   ______ Ö(c+1)/3   ) / c for categories of a qualitative active variable, where c is the number of non-empty categories of the variable under consideration 1 for categories of a qualitative active variable if Chi-square distance is used.

57.8  Description of Resulting Typology

a)  Proportion of cases. Percentage, multiplied by 1000, of cases belonging to each group of the typology.

b)  Explained variance.

EV (xv) = 1000  ×  tg å i = 1  Ni æ è x   iv  - x   v  ö ø 2   å k  wk æ è xkv - x   v  ö ø 2

where

tg = number of groups in the typology x   iv  = mean of the variable v in group i x   v  = grand mean of the variable v.

c)  Grand mean.

For quantitative variables, mean values as described under 7.a above.

For each category of qualitative variables, percentage of cases in this category.

d)  Statistics for each group of the typology.

For quantitative variables:
first line: mean values as described under 7.a above;
second line: standard deviations as described under 7.b above.

For each category of qualitative variables:
first line: column percentage of cases;
second line: row percentage of cases.

57.9  Summary Table of the Amount of Variance Explained by the Typology

a)  Variables explaining 80 % of the variance. List of the most discriminating variables, i.e. those variables which - taken altogether - are responsible for at least 80 % of the explained variance, together with the amount of variance explained by each of them individually (see 8.b above).

b)  Mean variance explained by active variables.

EV   active  = a å v = 1  av  EV (xv) a å v = 1  av

c)  Mean variance explained by all variables.

EV   all  = a+p å v = 1  av  EV (xv) a+p å v = 1  av

d)  Mean variance explained by the variables which explain 80 % of the total variance. After each regrouping, the program looks for variables which explain at least 80 % of the total variance (see 9.a above) and prints mean variance explained by those variables before and after regrouping, and the percentage of such variables.

57.10  Hierarchical Ascending Classification

a)  Group i + j. Profile of the new group, printed for up to 15 active variables in descending order of their deviation (see 10.d below). Note that if there are less than 15 active variables, or less than 15 variables with valid cases in aggregated groups, the program completes the list using passive variables.

b)  Group i. Profile of the group i, printed for the same variables as above.

c)   Group j. Profile of the group j, printed for the same variables as above.

d)  Dev. Absolute value of the difference between profiles of groups i and j, printed for the same variables as above.

Dev (xv) = | x   iv  - x   jv  |

e)  Weighted deviation. Deviation weighted by the variable weight and the variable standard deviation, printed for the same variables as above.

WDev (xv) = Dev (xv)    av sv

57.11  References

Diday, E., Optimisation en classification automatique, RAIRO, Vol. 3, 1972.

Hall & Ball, A clustering technique for summarizing multivariate data, Behavioral Sciences, Vol. 12, No 2, 1967.