CILJEVI PREDAVANJA

• Definicija FA
• Primjeri FA
• Izvođenje FA

DEFINICIJA (FA)

• Charles Spearman (1904) je zaslužan za uvođenje FA
• FA ima mnogo sičnosti sa PCA
• Cilj FA je reducirati veliki broj (koreliranih) varijabli na mali broj indeksa (faktora)
• FA je statistički model (za razliku od PCA koja je statistička metoda)
• Procedura provođenja FA:

1. Izračunaj faktorska opterećenja (~PCA)
2. Faktorska rotacija (ortogonalna/zaobljena)
3. Izračunaj faktorske koeficijente (za pripadajuće individualne elemente)

PRIMJERI

1. Anksioznost oko učenja statistike

• Pregled podataka:

Dimenzije mjerenja anksioznosti vezane uz učenje statistike.

Korelacija između dimenzija anksioznosti.

Struktura varijance kod FA.

Razlika između PCA i FA.

• Dva su moguća načina ekstrakcije faktora:

1. PCA procedura

Struktura varijance kod PCA.

Opterećenja po komponentama na osnovi PCA.

Opterećenja po komponentama na osnovi PCA.

Scree grafikon na osnovi PCA.

Zajednička varijanca (communalities) kod PCA.

2. Standardna FA

Zajednička varijanca (communalities) kod SFA pristupa.

Ukupna varijanca kod SFA pristupa.

Varijanca po faktorima kod SFA pristupa.

• Rotacija:

• Cilj rotacije je postići jednostavnu strukturu(simple structure) kod faktorskih opterećenja:

1. Svaki red mora imati barem jednu nulu
2. za n faktora treba biti barem n nula u svakoj koloni
3. Veći dio individulanih elemenata treba težiti 0
4. Manji dio individualnih elemenata treba biti različit od nula

Primjer jednostavne strukture.

• Ortogonalna rotacija

Opterećenja kod Varimax rotacije.

Opterećenja kod Varimax rotacije sa Keiser korekcijom.

Transformacijska matrica.

Grafički prikaz rotacije.

Ukupna objašnjena varijanca.

Ukupna objašnjene varijance kod različitih rotacija.

• Zaobljena rotacija

Matrica faktorskih obrazaca.

Strukturna matrica.

Strukturna matrica.

Grafički prikaz zaobljene rotacije.

Grafički prikaz zaobljene rotacije u faktorskom prostoru.

Ukupna objašnjena varijanca.

Interpretacija rezultata.

2. Anksioznost oko pisanja ispita

Pitanja za provedbu istraživanja o anksioznosti studenata oko pisanja ispita.

Korelacija izmedju pitanja.

Varijanca za PCA komponente.

Opterećenja za dva najvažnija faktora(ne-rotirano i OBLIMIN rotirano).

Vizualizacija faktorskih opterećenja.

3. Psihomotorne sposobnosti

Varijable u istraživanju psihomotornih sposobnosti pilota.

Korelacija izmedju varijabli.

Scree grafikon za komponente.

Zajednički dio varijance/comunalities.

Zajednički dio varijance/comunalities.

IZVOĐENJE FA

1. Primjer

library(psych)
library(GPArotation)
## UČITAJ I UREDI PODATKE ##

bfi_dta <- bfi # Učitaj podatke
head(bfi_dta, 10) # Pogledaj podatke

##       A1 A2 A3 A4 A5 C1 C2 C3 C4 C5 E1 E2 E3 E4 E5 N1 N2 N3 N4 N5 O1 O2 O3 O4
## 61617  2  4  3  4  4  2  3  3  4  4  3  3  3  4  4  3  4  2  2  3  3  6  3  4
## 61618  2  4  5  2  5  5  4  4  3  4  1  1  6  4  3  3  3  3  5  5  4  2  4  3
## 61620  5  4  5  4  4  4  5  4  2  5  2  4  4  4  5  4  5  4  2  3  4  2  5  5
## 61621  4  4  6  5  5  4  4  3  5  5  5  3  4  4  4  2  5  2  4  1  3  3  4  3
## 61622  2  3  3  4  5  4  4  5  3  2  2  2  5  4  5  2  3  4  4  3  3  3  4  3
## 61623  6  6  5  6  5  6  6  6  1  3  2  1  6  5  6  3  5  2  2  3  4  3  5  6
## 61624  2  5  5  3  5  5  4  4  2  3  4  3  4  5  5  1  2  2  1  1  5  2  5  6
## 61629  4  3  1  5  1  3  2  4  2  4  3  6  4  2  1  6  3  2  6  4  3  2  4  5
## 61630  4  3  6  3  3  6  6  3  4  5  5  3 NA  4  3  5  5  2  3  3  6  6  6  6
## 61633  2  5  6  6  5  6  5  6  2  1  2  2  4  5  5  5  5  5  2  4  5  1  5  5
##       O5 gender education age
## 61617  3      1        NA  16
## 61618  3      2        NA  18
## 61620  2      2        NA  17
## 61621  5      2        NA  17
## 61622  3      1        NA  17
## 61623  1      2         3  21
## 61624  1      1        NA  18
## 61629  3      1         2  19
## 61630  1      1         1  19
## 61633  2      2        NA  17

summary(bfi_dta) # Pregledaj podatke

##        A1              A2              A3              A4            A5      
##  Min.   :1.000   Min.   :1.000   Min.   :1.000   Min.   :1.0   Min.   :1.00  
##  1st Qu.:1.000   1st Qu.:4.000   1st Qu.:4.000   1st Qu.:4.0   1st Qu.:4.00  
##  Median :2.000   Median :5.000   Median :5.000   Median :5.0   Median :5.00  
##  Mean   :2.413   Mean   :4.802   Mean   :4.604   Mean   :4.7   Mean   :4.56  
##  3rd Qu.:3.000   3rd Qu.:6.000   3rd Qu.:6.000   3rd Qu.:6.0   3rd Qu.:5.00  
##  Max.   :6.000   Max.   :6.000   Max.   :6.000   Max.   :6.0   Max.   :6.00  
##  NA's   :16      NA's   :27      NA's   :26      NA's   :19    NA's   :16    
##        C1              C2             C3              C4              C5       
##  Min.   :1.000   Min.   :1.00   Min.   :1.000   Min.   :1.000   Min.   :1.000  
##  1st Qu.:4.000   1st Qu.:4.00   1st Qu.:4.000   1st Qu.:1.000   1st Qu.:2.000  
##  Median :5.000   Median :5.00   Median :5.000   Median :2.000   Median :3.000  
##  Mean   :4.502   Mean   :4.37   Mean   :4.304   Mean   :2.553   Mean   :3.297  
##  3rd Qu.:5.000   3rd Qu.:5.00   3rd Qu.:5.000   3rd Qu.:4.000   3rd Qu.:5.000  
##  Max.   :6.000   Max.   :6.00   Max.   :6.000   Max.   :6.000   Max.   :6.000  
##  NA's   :21      NA's   :24     NA's   :20      NA's   :26      NA's   :16     
##        E1              E2              E3              E4       
##  Min.   :1.000   Min.   :1.000   Min.   :1.000   Min.   :1.000  
##  1st Qu.:2.000   1st Qu.:2.000   1st Qu.:3.000   1st Qu.:4.000  
##  Median :3.000   Median :3.000   Median :4.000   Median :5.000  
##  Mean   :2.974   Mean   :3.142   Mean   :4.001   Mean   :4.422  
##  3rd Qu.:4.000   3rd Qu.:4.000   3rd Qu.:5.000   3rd Qu.:6.000  
##  Max.   :6.000   Max.   :6.000   Max.   :6.000   Max.   :6.000  
##  NA's   :23      NA's   :16      NA's   :25      NA's   :9      
##        E5              N1              N2              N3       
##  Min.   :1.000   Min.   :1.000   Min.   :1.000   Min.   :1.000  
##  1st Qu.:4.000   1st Qu.:2.000   1st Qu.:2.000   1st Qu.:2.000  
##  Median :5.000   Median :3.000   Median :4.000   Median :3.000  
##  Mean   :4.416   Mean   :2.929   Mean   :3.508   Mean   :3.217  
##  3rd Qu.:5.000   3rd Qu.:4.000   3rd Qu.:5.000   3rd Qu.:4.000  
##  Max.   :6.000   Max.   :6.000   Max.   :6.000   Max.   :6.000  
##  NA's   :21      NA's   :22      NA's   :21      NA's   :11     
##        N4              N5             O1              O2              O3       
##  Min.   :1.000   Min.   :1.00   Min.   :1.000   Min.   :1.000   Min.   :1.000  
##  1st Qu.:2.000   1st Qu.:2.00   1st Qu.:4.000   1st Qu.:1.000   1st Qu.:4.000  
##  Median :3.000   Median :3.00   Median :5.000   Median :2.000   Median :5.000  
##  Mean   :3.186   Mean   :2.97   Mean   :4.816   Mean   :2.713   Mean   :4.438  
##  3rd Qu.:4.000   3rd Qu.:4.00   3rd Qu.:6.000   3rd Qu.:4.000   3rd Qu.:5.000  
##  Max.   :6.000   Max.   :6.00   Max.   :6.000   Max.   :6.000   Max.   :6.000  
##  NA's   :36      NA's   :29     NA's   :22                      NA's   :28     
##        O4              O5           gender        education         age       
##  Min.   :1.000   Min.   :1.00   Min.   :1.000   Min.   :1.00   Min.   : 3.00  
##  1st Qu.:4.000   1st Qu.:1.00   1st Qu.:1.000   1st Qu.:3.00   1st Qu.:20.00  
##  Median :5.000   Median :2.00   Median :2.000   Median :3.00   Median :26.00  
##  Mean   :4.892   Mean   :2.49   Mean   :1.672   Mean   :3.19   Mean   :28.78  
##  3rd Qu.:6.000   3rd Qu.:3.00   3rd Qu.:2.000   3rd Qu.:4.00   3rd Qu.:35.00  
##  Max.   :6.000   Max.   :6.00   Max.   :2.000   Max.   :5.00   Max.   :86.00  
##  NA's   :14      NA's   :20                     NA's   :223

sum(is.na(bfi_dta)) # Broj NA elemenata

## [1] 731

clean_bfd <- bfi_dta[complete.cases(bfi_dta),]
sum(is.na(clean_bfd)) # Broj NA elemenata

## [1] 0

## PROVEDI ANALIZU ##

cor_bfd <- cor(clean_bfd) # Napravi korelacijsku tablicu
fa_bfd <- fa(cor_bfd, nfactors = 6) # Provedi FA
summary(fa_bfd)

## 
## Factor analysis with Call: fa(r = cor_bfd, nfactors = 6)
## 
## Test of the hypothesis that 6 factors are sufficient.
## The degrees of freedom for the model is 225  and the objective function was  0.57 
## 
## The root mean square of the residuals (RMSA) is  0.02 
## The df corrected root mean square of the residuals is  0.03 
## 
##  With factor correlations of 
##       MR2   MR3   MR1   MR5   MR4   MR6
## MR2  1.00 -0.18  0.24 -0.05 -0.01  0.10
## MR3 -0.18  1.00 -0.23  0.16  0.19  0.04
## MR1  0.24 -0.23  1.00 -0.28 -0.19 -0.15
## MR5 -0.05  0.16 -0.28  1.00  0.18  0.17
## MR4 -0.01  0.19 -0.19  0.18  1.00  0.05
## MR6  0.10  0.04 -0.15  0.17  0.05  1.00

2. Primjer

library(FactoMineR)
library(factoextra)

## UČITAJ I PREGLEDAJ PODATKE ##

data(wine)
df <- wine[,c(1,2,16,22,29,28,30,31)]
head(df[, 1:7], 10)

##           Label      Soil Plante Acidity Harmony Intensity Overall.quality
## 2EL      Saumur      Env1  2.000   2.107   3.143     2.857           3.393
## 1CHA     Saumur      Env1  2.000   2.107   2.964     2.893           3.214
## 1FON Bourgueuil      Env1  1.750   2.179   3.143     3.074           3.536
## 1VAU     Chinon      Env2  2.304   3.179   2.038     2.462           2.464
## 1DAM     Saumur Reference  1.762   2.571   3.643     3.643           3.741
## 2BOU Bourgueuil Reference  1.750   2.393   3.500     3.464           3.643
## 1BOI Bourgueuil Reference  1.826   2.607   3.556     3.643           3.714
## 3EL      Saumur      Env1  2.080   2.179   3.296     3.321           3.393
## DOM1     Chinon      Env1  1.875   2.286   3.286     3.148           3.200
## 1TUR     Saumur      Env2  2.000   2.357   2.963     2.857           3.179

str(df)

## 'data.frame':    21 obs. of  8 variables:
##  $ Label          : Factor w/ 3 levels "Saumur","Bourgueuil",..: 1 1 2 3 1 2 2 1 3 1 ...
##  $ Soil           : Factor w/ 4 levels "Reference","Env1",..: 2 2 2 3 1 1 1 2 2 3 ...
##  $ Plante         : num  2 2 1.75 2.3 1.76 ...
##  $ Acidity        : num  2.11 2.11 2.18 3.18 2.57 ...
##  $ Harmony        : num  3.14 2.96 3.14 2.04 3.64 ...
##  $ Intensity      : num  2.86 2.89 3.07 2.46 3.64 ...
##  $ Overall.quality: num  3.39 3.21 3.54 2.46 3.74 ...
##  $ Typical        : num  3.25 3.04 3.18 2.25 3.44 ...

## PROVEDI FA ##

FA_df <- FAMD(df, graph = F)
print(FA_df)

## *The results are available in the following objects:
## 
##   name          description                             
## 1 "$eig"        "eigenvalues and inertia"               
## 2 "$var"        "Results for the variables"             
## 3 "$ind"        "results for the individuals"           
## 4 "$quali.var"  "Results for the qualitative variables" 
## 5 "$quanti.var" "Results for the quantitative variables"

## SVOJSTVENE VRIJEDNOSTI ##

FA_df_eig <- get_eigenvalue(FA_df)
head(FA_df_eig)

##       eigenvalue variance.percent cumulative.variance.percent
## Dim.1  4.8315174        43.922886                    43.92289
## Dim.2  1.8568797        16.880724                    60.80361
## Dim.3  1.5824794        14.386176                    75.18979
## Dim.4  1.1491200        10.446546                    85.63633
## Dim.5  0.6518053         5.925503                    91.56183

## SCREE PLOT ##
fviz_screeplot(FA_df)

## ANALIZA VARIJABLI ##

variables <- get_famd_var(FA_df)
variables

## FAMD results for variables 
##  ===================================================
##   Name       Description                      
## 1 "$coord"   "Coordinates"                    
## 2 "$cos2"    "Cos2, quality of representation"
## 3 "$contrib" "Contributions"

head(variables$coord)   # Kordinate za faktore (dimenzija)

##                     Dim.1       Dim.2       Dim.3       Dim.4        Dim.5
## Plante          0.7344160 0.060551966 0.105902048 0.004011299 0.0010340559
## Acidity         0.1732738 0.491118153 0.126394029 0.115376784 0.0045862935
## Harmony         0.8943968 0.023628146 0.040124469 0.003653813 0.0086624633
## Intensity       0.6991811 0.134639254 0.065382234 0.023214984 0.0064730431
## Overall.quality 0.9115699 0.005246728 0.009336677 0.005445276 0.0007961880
## Typical         0.7808611 0.027094327 0.001549791 0.083446627 0.0005912942

head(variables$cos2)    # Reprezentativnost varijabli 

##                      Dim.1        Dim.2        Dim.3        Dim.4        Dim.5
## Plante          0.53936692 3.666541e-03 1.121524e-02 1.609052e-05 1.069272e-06
## Acidity         0.03002381 2.411970e-01 1.597545e-02 1.331180e-02 2.103409e-05
## Harmony         0.79994566 5.582893e-04 1.609973e-03 1.335035e-05 7.503827e-05
## Intensity       0.48885427 1.812773e-02 4.274836e-03 5.389355e-04 4.190029e-05
## Overall.quality 0.83095973 2.752815e-05 8.717353e-05 2.965103e-05 6.339153e-07
## Typical         0.60974400 7.341026e-04 2.401853e-06 6.963340e-03 3.496288e-07

head(variables$contrib) # Doprinos dimenzijama (faktorima)

##                     Dim.1      Dim.2      Dim.3      Dim.4      Dim.5
## Plante          15.200526  3.2609526 6.69215972  0.3490757 0.15864490
## Acidity          3.586323 26.4485720 7.98708850 10.0404466 0.70362936
## Harmony         18.511716  1.2724651 2.53554453  0.3179662 1.32899551
## Intensity       14.471254  7.2508336 4.13163258  2.0202401 0.99309457
## Overall.quality 18.867156  0.2825562 0.59000304  0.4738648 0.12215119
## Typical         16.161818  1.4591321 0.09793437  7.2617850 0.09071638

# VIZUALIZACIJA #

fviz_famd_var(FA_df, repel = T) # Varijable u faktorskom prostoru

fviz_contrib(FA_df, "var", axes = 1) # Doprinos varijabli 1. dimenziji

fviz_contrib(FA_df, "var", axes = 2) # Doprinos varijabli 2. dimenziji

## ANALIZA KVANTITATIVNIH VARIJABLI ##

quant_variables <- get_famd_var(FA_df, "quanti.var")
quant_variables

## FAMD results for quantitative variables 
##  ===================================================
##   Name       Description                      
## 1 "$coord"   "Coordinates"                    
## 2 "$cos2"    "Cos2, quality of representation"
## 3 "$contrib" "Contributions"

head(quant_variables$coord)   # Kordinate za faktore (dimenzija)

##                      Dim.1     Dim.2       Dim.3       Dim.4       Dim.5
## Plante          -0.8569808 0.2460731  0.32542595 -0.06333481  0.03215674
## Acidity         -0.4162617 0.7007982 -0.35551938  0.33967158 -0.06772218
## Harmony          0.9457255 0.1537145  0.20031093 -0.06044678  0.09307236
## Intensity        0.8361705 0.3669322  0.25569950  0.15236464  0.08045522
## Overall.quality  0.9547617 0.0724343 -0.09662648 -0.07379211 -0.02821680
## Typical          0.8836634 0.1646035 -0.03936739 -0.28887130  0.02431654

head(quant_variables$cos2)    # Reprezentativnost varijabli 

##                     Dim.1       Dim.2       Dim.3       Dim.4        Dim.5
## Plante          0.7344160 0.060551966 0.105902048 0.004011299 0.0010340559
## Acidity         0.1732738 0.491118153 0.126394029 0.115376784 0.0045862935
## Harmony         0.8943968 0.023628146 0.040124469 0.003653813 0.0086624633
## Intensity       0.6991811 0.134639254 0.065382234 0.023214984 0.0064730431
## Overall.quality 0.9115699 0.005246728 0.009336677 0.005445276 0.0007961880
## Typical         0.7808611 0.027094327 0.001549791 0.083446627 0.0005912942

head(quant_variables$contrib) # Doprinos dimenzijama (faktorima)

##                     Dim.1      Dim.2      Dim.3      Dim.4      Dim.5
## Plante          15.200526  3.2609526 6.69215972  0.3490757 0.15864490
## Acidity          3.586323 26.4485720 7.98708850 10.0404466 0.70362936
## Harmony         18.511716  1.2724651 2.53554453  0.3179662 1.32899551
## Intensity       14.471254  7.2508336 4.13163258  2.0202401 0.99309457
## Overall.quality 18.867156  0.2825562 0.59000304  0.4738648 0.12215119
## Typical         16.161818  1.4591321 0.09793437  7.2617850 0.09071638

# VIZUALIZACIJA #

fviz_famd_var(FA_df, "quanti.var", repel = T, col.var = "black")

fviz_famd_var(FA_df, "quanti.var", repel = T, col.var = "contrib")

## ANALIZA KVALITATIVNIH VARIJABLI ##

quali_variables <- get_famd_var(FA_df, "quali.var")
quali_variables

## FAMD results for qualitative variable categories 
##  ===================================================
##   Name       Description                      
## 1 "$coord"   "Coordinates"                    
## 2 "$cos2"    "Cos2, quality of representation"
## 3 "$contrib" "Contributions"

head(quali_variables$coord)   # Kordinate za faktore (dimenzija)

##                  Dim.1      Dim.2      Dim.3      Dim.4      Dim.5
## Saumur      0.08238236  0.2041021  0.9985884 -0.3996689 -0.1453137
## Bourgueuil  0.72578122 -1.0944657 -0.8193199  1.0187071 -0.4092349
## Chinon     -1.31522332  1.0804177 -1.5171382 -0.4289712  1.0134650
## Reference   2.06918009  0.6112128 -0.2525731  0.6853384  0.0157536
## Env1       -0.30736790 -1.6929988 -0.1158917 -0.3395732  0.3152922
## Env2       -1.40724432  1.2078471 -0.5667802 -1.0387469 -0.6525117

head(quali_variables$cos2)    # Reprezentativnost varijabli 

##                  Dim.1      Dim.2       Dim.3      Dim.4        Dim.5
## Saumur     0.005417832 0.03325463 0.796031214 0.12751397 1.685660e-02
## Bourgueuil 0.144311261 0.32816561 0.183905958 0.28430691 4.588114e-02
## Chinon     0.269097795 0.18159107 0.358064613 0.02862644 1.597823e-01
## Reference  0.799196698 0.06973357 0.011907792 0.08767328 4.632514e-05
## Env1       0.029258912 0.88767590 0.004159544 0.03571150 3.078702e-02
## Env2       0.371052065 0.27335044 0.060190111 0.20216934 7.977604e-02

head(quali_variables$contrib) # Doprinos dimenzijama (faktorima)

##                 Dim.1      Dim.2      Dim.3     Dim.4       Dim.5
## Saumur     0.01522912  0.6328503 20.8578860  6.336410  2.60345729
## Bourgueuil 0.64472767  9.9258736  7.6588378 22.454315 11.26267552
## Chinon     1.41147307  6.4484676 17.5071254  2.654399 46.04923094
## Reference  6.11375771  3.6115700  0.8491350 11.856544  0.01947164
## Env1       0.13490519 27.7092332  0.1787751  2.910817  7.79954850
## Env2       2.01986625 10.0741200  3.0542451 19.455364 23.86115933

# VIZUALIZACIJA #

fviz_famd_var(FA_df, "quali.var", col.var = "contrib")

## ANALIZA INDIVIDUALNIH ELEMENATA ##


indvs <- get_famd_ind(FA_df)
indvs

## FAMD results for individuals 
##  ===================================================
##   Name       Description                      
## 1 "$coord"   "Coordinates"                    
## 2 "$cos2"    "Cos2, quality of representation"
## 3 "$contrib" "Contributions"

head(indvs$coord)   # Kordinate za faktore (dimenzija)

##           Dim.1      Dim.2      Dim.3      Dim.4       Dim.5
## 2EL  -0.1213241 -1.5797151  0.7615777 -1.1978981  0.28209940
## 1CHA -0.6536760 -1.6846472  0.7643619 -0.9923990  0.24345633
## 1FON  0.8701622 -2.2457285 -0.7788402  0.3339757 -0.16854021
## 1VAU -5.6883455  2.1640314 -2.2984115  0.2968312 -0.13215744
## 1DAM  2.4441041  1.2242463  0.2751576  0.4694187 -0.06885226
## 2BOU  2.2703576 -0.0768261 -0.8093529  1.4650543 -0.49582803

head(indvs$cos2)    # Reprezentativnost varijabli 

##            Dim.1        Dim.2      Dim.3       Dim.4        Dim.5
## 2EL  0.002868247 0.4862727662 0.11301885 0.279616085 0.0155069631
## 1CHA 0.079955815 0.5310591076 0.10932589 0.184288214 0.0110909081
## 1FON 0.102535586 0.6829496072 0.08214305 0.015104410 0.0038466371
## 1VAU 0.747872962 0.1082388092 0.12209882 0.002036456 0.0004036818
## 1DAM 0.629756926 0.1580051034 0.00798172 0.023230275 0.0004997697
## 2BOU 0.612506755 0.0007013583 0.07783918 0.255052598 0.0292135595

head(indvs$contrib) # Doprinos dimenzijama (faktorima)

##            Dim.1       Dim.2      Dim.3     Dim.4      Dim.5
## 2EL   0.01450746  6.39962409  1.7453040 5.9464124 0.58138944
## 1CHA  0.42113589  7.27804699  1.7580884 4.0812003 0.43301702
## 1FON  0.74627279 12.93336227  1.8253213 0.4622159 0.20752473
## 1VAU 31.89106950 12.00947579 15.8963787 0.3651190 0.12759864
## 1DAM  5.88757653  3.84356758  0.2278272 0.9131374 0.03463372
## 2BOU  5.08025725  0.01513612  1.9711449 8.8945243 1.79607702

# VIZUALIZACIJA #

fviz_famd_ind(FA_df, col.ind = "cos2", repel = T)

fviz_mfa_ind(FA_df,
             habillage = "Label", 
             palette = c("#00AFBB", "#E7B800", "#FC4E07"),
             addEllipses = TRUE, ellipse.type = "confidence", 
             repel = TRUE )

fviz_ellipses(FA_df, c("Label", "Soil"), repel = TRUE)

3. Primjer

# PREGLEDAJ PODATKE #
head(wine, 12)

##           Label      Soil Odor.Intensity.before.shaking
## 2EL      Saumur      Env1                         3.074
## 1CHA     Saumur      Env1                         2.964
## 1FON Bourgueuil      Env1                         2.857
## 1VAU     Chinon      Env2                         2.808
## 1DAM     Saumur Reference                         3.607
## 2BOU Bourgueuil Reference                         2.857
## 1BOI Bourgueuil Reference                         3.214
## 3EL      Saumur      Env1                         3.120
## DOM1     Chinon      Env1                         2.857
## 1TUR     Saumur      Env2                         2.893
## 4EL      Saumur      Env2                         3.250
## PER1     Saumur      Env2                         3.393
##      Aroma.quality.before.shaking Fruity.before.shaking Flower.before.shaking
## 2EL                         3.000                 2.714                 2.280
## 1CHA                        2.821                 2.375                 2.280
## 1FON                        2.929                 2.560                 1.960
## 1VAU                        2.593                 2.417                 1.913
## 1DAM                        3.429                 3.154                 2.154
## 2BOU                        3.111                 2.577                 2.040
## 1BOI                        3.222                 2.962                 2.115
## 3EL                         2.852                 2.500                 2.200
## DOM1                        2.815                 2.808                 1.923
## 1TUR                        3.000                 2.571                 1.846
## 4EL                         3.286                 2.714                 1.926
## PER1                        3.179                 2.769                 2.038
##      Spice.before.shaking Visual.intensity Nuance Surface.feeling
## 2EL                 1.960            4.321  4.000           3.269
## 1CHA                1.680            3.222  3.000           2.808
## 1FON                2.077            3.536  3.393           3.000
## 1VAU                2.160            2.893  2.786           2.538
## 1DAM                2.040            4.393  4.036           3.385
## 2BOU                2.077            4.464  4.259           3.407
## 1BOI                2.040            4.143  3.929           3.250
## 3EL                 2.185            4.214  3.857           3.077
## DOM1                2.074            4.037  3.893           3.280
## 1TUR                1.680            3.704  3.407           3.111
## 4EL                 1.962            3.857  3.643           3.259
## PER1                1.920            4.714  4.500           3.321
##      Odor.Intensity Quality.of.odour Fruity Flower Spice Plante Phenolic
## 2EL           3.407            3.308  2.885  2.320 1.840  2.000    1.650
## 1CHA          3.370            3.000  2.560  2.440 1.739  2.000    1.381
## 1FON          3.250            2.929  2.769  2.192 2.250  1.750    1.250
## 1VAU          3.160            2.880  2.391  2.083 2.167  2.304    1.476
## 1DAM          3.536            3.360  3.160  2.231 2.148  1.762    1.600
## 2BOU          3.179            3.385  2.800  2.240 2.148  1.750    1.476
## 1BOI          3.429            3.500  3.038  2.200 2.385  1.826    1.476
## 3EL           3.654            3.077  2.520  2.320 2.444  2.080    1.905
## DOM1          3.357            3.346  3.000  2.040 2.125  1.875    1.524
## 1TUR          3.222            3.259  2.926  2.040 2.042  2.000    1.773
## 4EL           3.607            3.385  2.889  2.115 2.160  1.955    1.571
## PER1          3.481            3.385  2.962  2.000 2.200  2.042    1.545
##      Aroma.intensity Aroma.persistency Aroma.quality Attack.intensity Acidity
## 2EL            3.259             2.963         3.200            2.963   2.107
## 1CHA           2.962             2.808         2.926            3.036   2.107
## 1FON           3.077             2.800         3.077            3.222   2.179
## 1VAU           2.542             2.583         2.478            2.704   3.179
## 1DAM           3.615             3.296         3.462            3.464   2.571
## 2BOU           3.214             3.148         3.321            3.286   2.393
## 1BOI           3.250             3.222         3.385            3.393   2.607
## 3EL            3.280             3.160         2.962            3.250   2.179
## DOM1           3.148             2.893         3.308            3.286   2.286
## 1TUR           3.077             2.704         2.778            2.893   2.357
## 4EL            3.286             3.036         3.222            3.321   2.429
## PER1           3.321             3.071         3.143            3.357   2.429
##      Astringency Alcohol Balance Smooth Bitterness Intensity Harmony
## 2EL        2.429   2.500   3.250  2.731      1.926     2.857   3.143
## 1CHA       2.179   2.654   2.926  2.500      1.926     2.893   2.964
## 1FON       2.250   2.643   3.321  2.679      2.000     3.074   3.143
## 1VAU       2.185   2.500   2.333  1.680      1.963     2.462   2.038
## 1DAM       2.536   2.786   3.464  3.036      2.071     3.643   3.643
## 2BOU       2.643   2.857   3.286  2.857      2.179     3.464   3.500
## 1BOI       2.607   2.778   3.464  2.857      1.929     3.643   3.556
## 3EL        2.630   2.778   3.179  2.786      2.000     3.321   3.296
## DOM1       2.407   2.741   3.143  2.821      1.964     3.148   3.286
## 1TUR       2.250   2.704   3.214  2.500      2.185     2.857   2.963
## 4EL        2.571   2.893   3.192  2.857      2.214     3.357   3.071
## PER1       2.607   2.821   3.107  2.889      2.037     3.250   3.393
##      Overall.quality Typical
## 2EL            3.393   3.250
## 1CHA           3.214   3.036
## 1FON           3.536   3.179
## 1VAU           2.464   2.250
## 1DAM           3.741   3.444
## 2BOU           3.643   3.393
## 1BOI           3.714   3.357
## 3EL            3.393   3.071
## DOM1           3.200   3.500
## 1TUR           3.179   2.964
## 4EL            3.571   3.500
## PER1           3.148   3.556

colnames(wine)

##  [1] "Label"                         "Soil"                         
##  [3] "Odor.Intensity.before.shaking" "Aroma.quality.before.shaking" 
##  [5] "Fruity.before.shaking"         "Flower.before.shaking"        
##  [7] "Spice.before.shaking"          "Visual.intensity"             
##  [9] "Nuance"                        "Surface.feeling"              
## [11] "Odor.Intensity"                "Quality.of.odour"             
## [13] "Fruity"                        "Flower"                       
## [15] "Spice"                         "Plante"                       
## [17] "Phenolic"                      "Aroma.intensity"              
## [19] "Aroma.persistency"             "Aroma.quality"                
## [21] "Attack.intensity"              "Acidity"                      
## [23] "Astringency"                   "Alcohol"                      
## [25] "Balance"                       "Smooth"                       
## [27] "Bitterness"                    "Intensity"                    
## [29] "Harmony"                       "Overall.quality"              
## [31] "Typical"

## PROCIJENI MODEL ##

FA2_df <- MFA(wine,
              group = c(2, 5, 3, 10, 9, 2), 
               type = c("n", "s", "s", "s", "s", "s"),
               name.group = c("origin","odor","visual",
                              "odor.after.shaking",
                              "taste","overall"),
               num.group.sup = c(1, 6),
               graph = FALSE)

print(FA2_df)

## **Results of the Multiple Factor Analysis (MFA)**
## The analysis was performed on 21 individuals, described by 31 variables
## *Results are available in the following objects :
## 
##    name                 description                                           
## 1  "$eig"               "eigenvalues"                                         
## 2  "$separate.analyses" "separate analyses for each group of variables"       
## 3  "$group"             "results for all the groups"                          
## 4  "$partial.axes"      "results for the partial axes"                        
## 5  "$inertia.ratio"     "inertia ratio"                                       
## 6  "$ind"               "results for the individuals"                         
## 7  "$quanti.var"        "results for the quantitative variables"              
## 8  "$quanti.var.sup"    "results for the quantitative supplementary variables"
## 9  "$quali.var.sup"     "results for the categorical supplementary variables" 
## 10 "$summary.quanti"    "summary for the quantitative variables"              
## 11 "$summary.quali"     "summary for the categorical variables"               
## 12 "$global.pca"        "results for the global PCA"

# SVOJSTVENE VRIJEDNOSTI #

eig_FA2_df <- get_eigenvalue(FA2_df)
head(eig_FA2_df,10)

##        eigenvalue variance.percent cumulative.variance.percent
## Dim.1   3.4619504        49.378382                    49.37838
## Dim.2   1.3667683        19.494446                    68.87283
## Dim.3   0.6154291         8.777969                    77.65080
## Dim.4   0.3721997         5.308747                    82.95954
## Dim.5   0.2703825         3.856511                    86.81605
## Dim.6   0.2024033         2.886912                    89.70297
## Dim.7   0.1757134         2.506230                    92.20920
## Dim.8   0.1258987         1.795714                    94.00491
## Dim.9   0.1052755         1.501563                    95.50647
## Dim.10  0.0787912         1.123812                    96.63029

fviz_screeplot(FA2_df)

## GRUPE VARIJABLII ##

var_grupa <- get_mfa_var(FA2_df, "group")
var_grupa

## Multiple Factor Analysis results for variable groups 
##  ===================================================
##   Name           Description                                          
## 1 "$coord"       "Coordinates"                                        
## 2 "$cos2"        "Cos2, quality of representation"                    
## 3 "$contrib"     "Contributions"                                      
## 4 "$correlation" "Correlation between groups and principal dimensions"

head(var_grupa$coord)

##                        Dim.1      Dim.2      Dim.3      Dim.4      Dim.5
## odor               0.7820738 0.61977283 0.37353451 0.17260092 0.08553276
## visual             0.8546846 0.04014481 0.01438360 0.04550736 0.02966750
## odor.after.shaking 0.9247734 0.46892047 0.18009116 0.10139051 0.11589439
## taste              0.9004187 0.23793016 0.04741982 0.05270088 0.03928784

head(var_grupa$cos2)

##                        Dim.1       Dim.2        Dim.3       Dim.4        Dim.5
## odor               0.3799491 0.238613517 0.0866745169 0.018506155 0.0045445922
## visual             0.7284016 0.001607007 0.0002062976 0.002065011 0.0008776492
## odor.after.shaking 0.6245535 0.160582210 0.0236855692 0.007507471 0.0098089810
## taste              0.7222292 0.050429542 0.0020031144 0.002474125 0.0013749986

head(var_grupa$contrib)

##                       Dim.1     Dim.2     Dim.3    Dim.4    Dim.5
## odor               22.59055 45.345861 60.694972 46.37321 31.63399
## visual             24.68795  2.937207  2.337166 12.22660 10.97242
## odor.after.shaking 26.71250 34.308703 29.262699 27.24089 42.86313
## taste              26.00900 17.408230  7.705163 14.15930 14.53047

# VIZUALIZIRAJ GRUPE #

fviz_mfa_var(FA2_df, "group")

fviz_contrib(FA2_df, "group", axes = 1)

fviz_contrib(FA2_df, "group", axes = 2)

## KVANTITATIVNE VARIJABLE ##

quanti_vars <- get_mfa_var(FA2_df, "quanti.var")
quanti_vars

## Multiple Factor Analysis results for quantitative variables 
##  ===================================================
##   Name       Description                      
## 1 "$coord"   "Coordinates"                    
## 2 "$cos2"    "Cos2, quality of representation"
## 3 "$contrib" "Contributions"

head(quanti_vars$coord)

##                                   Dim.1       Dim.2       Dim.3      Dim.4
## Odor.Intensity.before.shaking 0.5908036  0.66723783 -0.02326175  0.3287015
## Aroma.quality.before.shaking  0.8352510 -0.07539908 -0.35417877  0.1414425
## Fruity.before.shaking         0.7160259 -0.15069626 -0.53748761  0.2517063
## Flower.before.shaking         0.4387181 -0.40937751  0.63731284  0.4029075
## Spice.before.shaking          0.0380525  0.86501993  0.12795122 -0.1822298
## Visual.intensity              0.8811873  0.23833245  0.14099033 -0.2128871
##                                     Dim.5
## Odor.Intensity.before.shaking  0.05786231
## Aroma.quality.before.shaking   0.04992114
## Fruity.before.shaking          0.18981578
## Flower.before.shaking          0.12200773
## Spice.before.shaking           0.36741971
## Visual.intensity              -0.17676282

head(quanti_vars$cos2)

##                                     Dim.1       Dim.2       Dim.3      Dim.4
## Odor.Intensity.before.shaking 0.349048863 0.445206325 0.000541109 0.10804466
## Aroma.quality.before.shaking  0.697644264 0.005685021 0.125442602 0.02000597
## Fruity.before.shaking         0.512693037 0.022709361 0.288892928 0.06335608
## Flower.before.shaking         0.192473567 0.167589944 0.406167661 0.16233443
## Spice.before.shaking          0.001447992 0.748259477 0.016371514 0.03320769
## Visual.intensity              0.776491025 0.056802358 0.019878273 0.04532093
##                                     Dim.5
## Odor.Intensity.before.shaking 0.003348047
## Aroma.quality.before.shaking  0.002492121
## Fruity.before.shaking         0.036030029
## Flower.before.shaking         0.014885886
## Spice.before.shaking          0.134997242
## Visual.intensity              0.031245096

head(quanti_vars$contrib)

##                                    Dim.1      Dim.2       Dim.3     Dim.4
## Odor.Intensity.before.shaking 4.49733206 14.5296787  0.03921898 12.948424
## Aroma.quality.before.shaking  8.98882147  0.1855354  9.09194110  2.397581
## Fruity.before.shaking         6.60581103  0.7411389 20.93864000  7.592798
## Flower.before.shaking         2.47993227  5.4694372 29.43858302 19.454686
## Spice.before.shaking          0.01865671 24.4200703  1.18658923  3.979718
## Visual.intensity              7.91221841  1.4660681  1.13941864  4.295418
##                                    Dim.5
## Odor.Intensity.before.shaking  0.5523351
## Aroma.quality.before.shaking   0.4111309
## Fruity.before.shaking          5.9439566
## Flower.before.shaking          2.4557588
## Spice.before.shaking          22.2708049
## Visual.intensity               4.0764862

fviz_mfa_var(FA2_df,"quanti.var", palette = "jco", 
             col.var.sup = "violet", repel = TRUE)

fviz_contrib(FA2_df, choice = "quanti.var",
             axes = 1,
             top = 20,
             palette = "jco")

fviz_contrib(FA2_df, choice = "quanti.var",
             axes = 2,
             top = 20,
             palette = "jco")

## INDIVIDUALNI ELEMENTI ##

fviz_mfa_ind(FA2_df, col.ind = "cos2",
             gradient.cols = c("#00AFBB", "#E7B800", "#FC4E07"),
             repel = TRUE)

fviz_mfa_ind(FA2_df,
             habillage = "Label",
             palette = c("#00AFBB", "#E7B800", "#FC4E07"),
             addEllipses = TRUE, ellipse.type = "confidence", 
             repel = TRUE 
             )

fviz_ellipses(FA2_df, c("Label", "Soil"), repel = T)