Carga y Exploración del Dataset
Primero, cargamos el dataset mtcars en R y lo almacenamos en un dataframe llamado df:
# Cargar el dataset
data(mtcars)
# Guardar los datos en un dataframe
df <- mtcars
# Imprimir el dataframe
print(df)
El resultado del dataset mtcars es el siguiente:
| Modelo | mpg | cyl | disp | hp | drat | wt | qsec | vs | am | gear | carb |
|---|---|---|---|---|---|---|---|---|---|---|---|
| Mazda RX4 | 21.0 | 6 | 160.0 | 110 | 3.90 | 2.620 | 16.46 | 0 | 1 | 4 | 4 |
| Mazda RX4 Wag | 21.0 | 6 | 160.0 | 110 | 3.90 | 2.875 | 17.02 | 0 | 1 | 4 | 4 |
| Datsun 710 | 22.8 | 4 | 108.0 | 93 | 3.85 | 2.320 | 18.61 | 1 | 1 | 4 | 1 |
| Hornet 4 Drive | 21.4 | 6 | 258.0 | 110 | 3.08 | 3.215 | 19.44 | 1 | 0 | 3 | 1 |
| Hornet Sportabout | 18.7 | 8 | 360.0 | 175 | 3.15 | 3.440 | 17.02 | 0 | 0 | 3 | 2 |
| Valiant | 18.1 | 6 | 225.0 | 105 | 2.76 | 3.460 | 20.22 | 1 | 0 | 3 | 1 |
| Duster 360 | 14.3 | 8 | 360.0 | 245 | 3.21 | 3.570 | 15.84 | 0 | 0 | 3 | 4 |
| Merc 240D | 24.4 | 4 | 146.7 | 62 | 3.69 | 3.190 | 20.00 | 1 | 0 | 4 | 2 |
| Merc 230 | 22.8 | 4 | 140.8 | 95 | 3.92 | 3.150 | 22.90 | 1 | 0 | 4 | 2 |
| Merc 280 | 19.2 | 6 | 167.6 | 123 | 3.92 | 3.440 | 18.30 | 1 | 0 | 4 | 4 |
| Merc 280C | 17.8 | 6 | 167.6 | 123 | 3.92 | 3.440 | 18.90 | 1 | 0 | 4 | 4 |
| Merc 450SE | 16.4 | 8 | 275.8 | 180 | 3.07 | 4.070 | 17.40 | 0 | 0 | 3 | 3 |
| Merc 450SL | 17.3 | 8 | 275.8 | 180 | 3.07 | 3.730 | 17.60 | 0 | 0 | 3 | 3 |
| Merc 450SLC | 15.2 | 8 | 275.8 | 180 | 3.07 | 3.780 | 18.00 | 0 | 0 | 3 | 3 |
| Cadillac Fleetwood | 10.4 | 8 | 472.0 | 205 | 2.93 | 5.250 | 17.98 | 0 | 0 | 3 | 4 |
| Lincoln Continental | 10.4 | 8 | 460.0 | 215 | 3.00 | 5.424 | 17.82 | 0 | 0 | 3 | 4 |
| Chrysler Imperial | 14.7 | 8 | 440.0 | 230 | 3.23 | 5.345 | 17.42 | 0 | 0 | 3 | 4 |
| Fiat 128 | 32.4 | 4 | 78.7 | 66 | 4.08 | 2.200 | 19.47 | 1 | 1 | 4 | 1 |
| Honda Civic | 30.4 | 4 | 75.7 | 52 | 4.93 | 1.615 | 18.52 | 1 | 1 | 4 | 2 |
| Toyota Corolla | 33.9 | 4 | 71.1 | 65 | 4.22 | 1.835 | 19.90 | 1 | 1 | 4 | 1 |
| Toyota Corona | 21.5 | 4 | 120.1 | 97 | 3.70 | 2.465 | 20.01 | 1 | 0 | 3 | 1 |
| Dodge Challenger | 15.5 | 8 | 318.0 | 150 | 2.76 | 3.520 | 16.87 | 0 | 0 | 3 | 2 |
| AMC Javelin | 15.2 | 8 | 304.0 | 150 | 3.15 | 3.435 | 17.30 | 0 | 0 | 3 | 2 |
| Camaro Z28 | 13.3 | 8 | 350.0 | 245 | 3.73 | 3.840 | 15.41 | 0 | 0 | 3 | 4 |
| Pontiac Firebird | 19.2 | 8 | 400.0 | 175 | 3.08 | 3.845 | 17.05 | 0 | 0 | 3 | 2 |
| Fiat X1-9 | 27.3 | 4 | 79.0 | 66 | 4.08 | 1.935 | 18.90 | 1 | 1 | 4 | 1 |
| Porsche 914-2 | 26.0 | 4 | 120.3 | 91 | 4.43 | 2.140 | 16.70 | 0 | 1 | 5 | 2 |
| Lotus Europa | 30.4 | 4 | 95.1 | 113 | 3.77 | 1.513 | 16.90 | 1 | 1 | 5 | 2 |
| Ford Pantera L | 15.8 | 8 | 351.0 | 264 | 4.22 | 3.170 | 14.50 | 0 | 1 | 5 | 4 |
| Ferrari Dino | 19.7 | 6 | 145.0 | 175 | 3.62 | 2.770 | 15.50 | 0 | 1 | 5 | 6 |
| Maserati Bora | 15.0 | 8 | 301.0 | 335 | 3.54 | 3.570 | 14.60 | 0 | 1 | 5 | 8 |
| Volvo 142E | 21.4 | 4 | 121.0 | 109 | 4.11 | 2.780 | 18.60 | 1 | 1 | 4 | 2 |
El dataset mtcars contiene varias variables relacionadas con el rendimiento y las especificaciones
de diferentes modelos de autos.
Cálculo de las Medias de las Variables
# Calcular el vector de medias de las variables
medias <- colMeans(df)
# Imprimir las medias
print(medias)
mpg
cyl
disp
hp
drat
wt
qsec
vs
am
gear
carb
20.090625
6.187500
230.721875
146.687500
3.596563
3.217250
17.848750
0.437500
0.406250
3.687500
2.812500
Matriz de Covarianzas
A continuación, se calcula la matriz de covarianza para todas las variables del dataset.
# Calcular la matriz de covarianza
matriz_covarianzas <- cov(df)
# Imprimir la matriz de covarianza
print(matriz_covarianzas)
mpg
cyl
disp
hp
drat
wt
qsec
vs
am
gear
carb
mpg 36.324103 -9.172379 -633.0972 -320.7321 2.19506351 -5.1166847 4.50914919 2.01713710 1.80393145 2.1356855 -5.36310484 cyl -9.172379 3.1895161 199.66028 101.931452 -0.66836694 1.3673710 -1.88685484 -0.72983871 -0.46572581 -0.6491935 1.52016129 disp -633.09721 199.6602823 15360.79983 6721.158669 -47.06401915 107.6842040 -96.05168145 -44.37762097 -36.56401210 -50.8026210 79.06875000 hp -320.732056 101.9314516 6721.15867 4700.866935 -16.45110887 44.1926613 -86.77008065 -24.98790323 -8.32056452 -6.3588710 83.03629032 drat 2.19506351 -0.66836694 -47.064019 -16.45110887 0.28588135 -0.3727207 0.08714073 0.11864919 0.19015121 0.2759879 -0.07840726 wt -5.1166847 1.3673710 107.6842040 44.1926613 -0.3727207 0.9573790 -0.30548161 -0.27366129 -0.33810484 -0.4210806 0.67579032 qsec 4.50914919 -1.88685484 -96.05168145 -86.77008065 0.08714073 -0.30548161 3.19316613 0.67056452 -0.20495968 -0.2804032 -1.89411290 vs 2.01713710 -0.72983871 -44.37762097 -24.98790323 0.11864919 -0.27366129 0.67056452 0.25403226 0.04233871 0.0766129 -0.46370968 am 1.80393145 -0.46572581 -36.56401210 -8.32056452 0.19015121 -0.33810484 -0.20495968 0.04233871 0.24899194 0.2923387 0.04637097 gear 2.1356855 -0.6491935 -50.8026210 -6.3588710 0.2759879 -0.4210806 -0.2804032 0.07661290 0.29233871 0.5443548 0.32661290 carb -5.36310484 1.52016129 79.06875000 83.03629032 -0.07840726 0.67579032 -1.89411290 -0.46370968 0.04637097 0.3266129 2.60887097
Distancia de Mahalanobis
La distancia de Mahalanobis es una medida útil para identificar casos atípicos en datos multivariantes. Se define de la siguiente manera:
Fórmula:
D^2 = diag{ (X - X̄) S^(-1) (X - X̄)' }
X = Vector de valores observados.
X̄ = Media (centroide) del conjunto de datos.
S^(-1) = Matriz de covarianzas inversa.
diag = Diagonal principal de la matriz resultante.
# Calcular distancia de Mahalanobis
D2 <- mahalanobis(df, medias, matriz_covarianzas, inverted = FALSE)
# Imprimir las distancias
print(D2)
Modelo
Distancia (D2)
Mazda RX4 8.946673 Mazda RX4 Wag 8.287933 Datsun 710 8.937150 Hornet 4 Drive 6.096726 Hornet Sportabout 5.429061 Valiant 8.877558 Duster 360 9.136276 Merc 240D 10.030345 Merc 230 22.593116 Merc 280 12.393107 Merc 280C 11.058878 Merc 450SE 9.476126 Merc 450SL 5.594527 Merc 450SLC 6.026462 Cadillac Fleetwood 11.201310 Lincoln Continental 8.672093 Chrysler Imperial 12.257618 Fiat 128 9.078630 Honda Civic 14.954377 Toyota Corolla 10.296463 Toyota Corona 13.432391 Dodge Challenger 6.227235 AMC Javelin 5.786691 Camaro Z28 11.681526 Pontiac Firebird 6.718085 Fiat X1-9 3.645789 Porsche 914-2 18.356164 Lotus Europa 14.000669 Ford Pantera L 21.573003 Ferrari Dino 11.152850 Maserati Bora 19.192384 Volvo 142E 9.888781
Cálculo de la Significación Estadística
# Calcular los p-valores
pvalores <- pchisq(D2, df = ncol(mtcars), lower.tail = FALSE)
# Inicializar variable para casos atípicos
hay_casos_atipicos <- FALSE
# Verificar cada p-valor
for (i in seq_along(pvalores)) {
print(paste("p-valor para elemento", i, ":", pvalores[i]))
if (pvalores[i] < 0.001) {
print(paste("Caso atípico detectado en el elemento", i, "con p-valor:", pvalores[i]))
hay_casos_atipicos <- TRUE
}
}
# Comprobar si se encontraron casos atípicos
if (!hay_casos_atipicos) {
print("No se encuentra ningún caso atípico.")
}
Resultados: p-valores
Elemento
p-valor
1 0.626814793272655 2 0.68730255517125 3 0.627693724281809 4 0.866832982960977 5 0.908622503177606 6 0.633193297550124 7 0.609314972177258 8 0.527659693078912 9 0.0201605479950018 10 0.334831368429065 11 0.438343906903962 12 0.578031268831444 13 0.899003940235217 14 0.871592789926057 15 0.426555606400835 16 0.652130411193711 17 0.344593779241114 18 0.614634517721478 19 0.184595108733901 20 0.503933836845313 21 0.266005907723727 22 0.857780183459677 23 0.887210816321477 24 0.388051627213303 25 0.821432466651442 26 0.979157957182606 27 0.0736758537409455 28 0.2329564460777 29 0.0278979309345972 30 0.430548297420681 31 0.0577263701654486 32 0.540418370341476
Conclusiones
- La matriz de covarianzas y las medias fueron calculadas para explorar la distribución y la variabilidad de
las diferentes variables del dataset
mtcars. - Se utilizó la distancia de Mahalanobis para identificar posibles observaciones atípicas.
- Tras calcular los p-valores, no se encontraron casos atípicos significativos (con un nivel de significancia menor a 0.001).
Por lo tanto, la hipótesis nula de que no hay casos atípicos fue validada. Todos los datos observados parecen
ser consistentes con los valores esperados en el contexto multivariante del dataset mtcars.