Matriz de informacion
V1	V2	V3	V4	V5	V6	V7	V8	V9	V10
4	1	4	3	3	2	4	4	4	4
5	5	4	4	3	3	4	1	1	3
2	1	3	1	4	2	1	5	4	5
1	1	1	1	4	4	2	5	5	4
1	1	2	1	5	5	4	3	3	2
5	5	5	5	3	3	4	2	2	1

a) Calcula la matriz de varianza covarianza para la batería de indicadores:

1. De forma “manual”

library(dplyr)

## 
## Attaching package: 'dplyr'

## The following object is masked from 'package:kableExtra':
## 
##     group_rows

## The following objects are masked from 'package:stats':
## 
##     filter, lag

## The following objects are masked from 'package:base':
## 
##     intersect, setdiff, setequal, union

library(kableExtra)
centrado <- function(x){x-mean(x)
}
xcentrada <- apply(X = X6_2,MARGIN = 2,centrado)
xcentrada %>% head() %>% 
  kable(caption = "Matriz de Variables Centradas",
        align = "c",
        digits = 2) %>%
  kable_material(html_font = "sans-serif")

Matriz de Variables Centradas
V1	V2	V3	V4	V5	V6	V7	V8	V9	V10
0.3	-2.4	0.5	0.2	-0.7	-1.7	0.35	1.15	1.2	1.35
1.3	1.6	0.5	1.2	-0.7	-0.7	0.35	-1.85	-1.8	0.35
-1.7	-2.4	-0.5	-1.8	0.3	-1.7	-2.65	2.15	1.2	2.35
-2.7	-2.4	-2.5	-1.8	0.3	0.3	-1.65	2.15	2.2	1.35
-2.7	-2.4	-1.5	-1.8	1.3	1.3	0.35	0.15	0.2	-0.65
1.3	1.6	1.5	2.2	-0.7	-0.7	0.35	-0.85	-0.8	-1.65

n_obs <- nrow(X6_2)
mat_V <- t(xcentrada) %*% xcentrada/ (n_obs-1)
mat_V %>% head(n=10) %>% 
  kable(caption = "Calculo de V(x) forma manual",
        align = "c",
        digits = 2) %>%
  kable_material(html_font = "sans-serif") %>%
  kable_styling(bootstrap_options = c("striped","hover"))

Calculo de V(x) forma manual
	V1	V2	V3	V4	V5	V6	V7	V8	V9	V10
V1	1.80	1.92	1.32	1.73	-0.62	-0.31	0.36	-1.21	-1.27	-0.90
V2	1.92	2.67	1.42	2.14	-0.66	-0.14	0.52	-1.78	-1.81	-1.54
V3	1.32	1.42	1.42	1.53	-0.53	-0.32	0.29	-0.92	-1.11	-0.87
V4	1.73	2.14	1.53	2.48	-0.80	-0.48	0.35	-1.61	-1.83	-1.39
V5	-0.62	-0.66	-0.53	-0.80	0.85	0.80	0.21	0.37	0.46	0.15
V6	-0.31	-0.14	-0.32	-0.48	0.80	1.38	0.63	0.22	0.09	-0.37
V7	0.36	0.52	0.29	0.35	0.21	0.63	1.61	-0.53	-0.34	-0.71
V8	-1.21	-1.78	-0.92	-1.61	0.37	0.22	-0.53	1.92	1.81	1.37
V9	-1.27	-1.81	-1.11	-1.83	0.46	0.09	-0.34	1.81	2.17	1.56
V10	-0.90	-1.54	-0.87	-1.39	0.15	-0.37	-0.71	1.37	1.56	1.82

2. Usando el comando cov de R base

library(dplyr)
library(kableExtra)
cov(X6_2) %>%
  kable(caption = "Calculo de V(x) a traves de R base",
        align = "c",
        digits = 2) %>%
  kable_material(html_font = "sans-serif") %>% 
  kable_styling(bootstrap_options = c("striped","hover"))

Calculo de V(x) a traves de R base
	V1	V2	V3	V4	V5	V6	V7	V8	V9	V10
V1	1.80	1.92	1.32	1.73	-0.62	-0.31	0.36	-1.21	-1.27	-0.90
V2	1.92	2.67	1.42	2.14	-0.66	-0.14	0.52	-1.78	-1.81	-1.54
V3	1.32	1.42	1.42	1.53	-0.53	-0.32	0.29	-0.92	-1.11	-0.87
V4	1.73	2.14	1.53	2.48	-0.80	-0.48	0.35	-1.61	-1.83	-1.39
V5	-0.62	-0.66	-0.53	-0.80	0.85	0.80	0.21	0.37	0.46	0.15
V6	-0.31	-0.14	-0.32	-0.48	0.80	1.38	0.63	0.22	0.09	-0.37
V7	0.36	0.52	0.29	0.35	0.21	0.63	1.61	-0.53	-0.34	-0.71
V8	-1.21	-1.78	-0.92	-1.61	0.37	0.22	-0.53	1.92	1.81	1.37
V9	-1.27	-1.81	-1.11	-1.83	0.46	0.09	-0.34	1.81	2.17	1.56
V10	-0.90	-1.54	-0.87	-1.39	0.15	-0.37	-0.71	1.37	1.56	1.82

b)Calcula la matriz de correlación para la batería de indicadores

1. De forma “manual”

zx <- scale(x = X6_2,center = TRUE)
zx %>%
  kable(caption = "Matriz de Variables Estandarizadas",
        align = "c",
        digits = 2) %>%
  kable_material(html_font = "sans-serif")

Matriz de Variables Estandarizadas
V1	V2	V3	V4	V5	V6	V7	V8	V9	V10
0.22	-1.47	0.42	0.13	-0.76	-1.45	0.28	0.83	0.81	1.00
0.97	0.98	0.42	0.76	-0.76	-0.60	0.28	-1.33	-1.22	0.26
-1.27	-1.47	-0.42	-1.14	0.32	-1.45	-2.09	1.55	0.81	1.74
-2.01	-1.47	-2.10	-1.14	0.32	0.26	-1.30	1.55	1.49	1.00
-2.01	-1.47	-1.26	-1.14	1.41	1.11	0.28	0.11	0.14	-0.48
0.97	0.98	1.26	1.40	-0.76	-0.60	0.28	-0.61	-0.54	-1.22
0.22	0.98	0.42	0.76	-1.84	-1.45	1.06	-1.33	-1.22	-1.22
-0.52	-0.86	-0.42	-1.14	0.32	0.26	-1.30	1.55	1.49	1.74
0.22	0.37	0.42	0.13	0.32	0.26	-0.51	-1.33	-1.22	-1.22
0.97	0.98	1.26	1.40	-1.84	-1.45	-0.51	-0.61	-0.54	-0.48
-1.27	-0.86	-1.26	-1.14	1.41	0.26	0.28	0.11	0.81	0.26
0.22	0.37	1.26	1.40	0.32	1.11	1.06	-0.61	-1.22	-0.48
-0.52	-0.86	-1.26	-1.14	0.32	1.11	0.28	0.83	0.14	0.26
0.97	0.98	0.42	0.76	1.41	0.26	0.28	-1.33	-0.54	-0.48
0.22	-0.24	-0.42	-1.14	0.32	0.26	1.06	0.11	0.81	1.00
0.97	0.98	0.42	0.76	0.32	1.11	0.28	-0.61	-1.22	-1.22
0.22	0.37	1.26	-0.51	0.32	1.11	1.06	0.83	0.81	-0.48
0.97	0.98	0.42	0.76	-1.84	-1.45	-2.09	-0.61	-0.54	0.26
-0.52	-0.24	-1.26	-0.51	0.32	0.26	1.06	0.83	1.49	1.00
0.97	0.98	0.42	0.76	0.32	1.11	0.28	0.11	-0.54	-1.22

n__obs <- nrow(X6_2)
mat_R <- t(zx)%*%zx/(n__obs-1)
mat_R %>%
  kable(caption = "Calculo R(x) forma manual",
        align = "c",
        digits = 2) %>%
  kable_material(html_font = "sans-serif") %>% 
  kable_styling(bootstrap_options = c("striped","hover"))

Calculo R(x) forma manual
	V1	V2	V3	V4	V5	V6	V7	V8	V9	V10
V1	1.00	0.87	0.82	0.82	-0.50	-0.19	0.21	-0.65	-0.64	-0.50
V2	0.87	1.00	0.73	0.83	-0.44	-0.07	0.25	-0.78	-0.75	-0.70
V3	0.82	0.73	1.00	0.81	-0.48	-0.23	0.19	-0.56	-0.63	-0.54
V4	0.82	0.83	0.81	1.00	-0.55	-0.26	0.17	-0.74	-0.79	-0.65
V5	-0.50	-0.44	-0.48	-0.55	1.00	0.74	0.18	0.29	0.34	0.12
V6	-0.19	-0.07	-0.23	-0.26	0.74	1.00	0.42	0.13	0.05	-0.24
V7	0.21	0.25	0.19	0.17	0.18	0.42	1.00	-0.30	-0.18	-0.41
V8	-0.65	-0.78	-0.56	-0.74	0.29	0.13	-0.30	1.00	0.89	0.73
V9	-0.64	-0.75	-0.63	-0.79	0.34	0.05	-0.18	0.89	1.00	0.78
V10	-0.50	-0.70	-0.54	-0.65	0.12	-0.24	-0.41	0.73	0.78	1.00

2. Usando el comando cor R base

library(dplyr)
library(kableExtra)
cor(X6_2) %>% 
  kable(caption = "Calculo R(x) utilizando cor",
        align = "c",
        digits = 2) %>%
  kable_material(html_font = "sans-serif") %>% 
  kable_styling(bootstrap_options = c("striped","hover"))

Calculo R(x) utilizando cor
	V1	V2	V3	V4	V5	V6	V7	V8	V9	V10
V1	1.00	0.87	0.82	0.82	-0.50	-0.19	0.21	-0.65	-0.64	-0.50
V2	0.87	1.00	0.73	0.83	-0.44	-0.07	0.25	-0.78	-0.75	-0.70
V3	0.82	0.73	1.00	0.81	-0.48	-0.23	0.19	-0.56	-0.63	-0.54
V4	0.82	0.83	0.81	1.00	-0.55	-0.26	0.17	-0.74	-0.79	-0.65
V5	-0.50	-0.44	-0.48	-0.55	1.00	0.74	0.18	0.29	0.34	0.12
V6	-0.19	-0.07	-0.23	-0.26	0.74	1.00	0.42	0.13	0.05	-0.24
V7	0.21	0.25	0.19	0.17	0.18	0.42	1.00	-0.30	-0.18	-0.41
V8	-0.65	-0.78	-0.56	-0.74	0.29	0.13	-0.30	1.00	0.89	0.73
V9	-0.64	-0.75	-0.63	-0.79	0.34	0.05	-0.18	0.89	1.00	0.78
V10	-0.50	-0.70	-0.54	-0.65	0.12	-0.24	-0.41	0.73	0.78	1.00

3. Presenta la matriz de correlación de forma gráfica (las dos versiones propuestas en clase)

library(PerformanceAnalytics)
chart.Correlation(as.matrix(X6_2),histogram = TRUE,pch=12)

library(corrplot)

## corrplot 0.92 loaded

library(grDevices)
library(Hmisc)

## 
## Attaching package: 'Hmisc'

## The following objects are masked from 'package:dplyr':
## 
##     src, summarize

## The following objects are masked from 'package:base':
## 
##     format.pval, units

mat_R <- rcorr(as.matrix(X6_2))
corrplot(mat_R$r,
         p.mat = mat_R$r,
         type = "upper",
         tl.col = "black",
         tl.srt = 20,
         pch.col = "blue",
         insig = "p-value",
         sig.level = -1,
         col = terrain.colors(100))

c) Realiza un análisis de componentes principales, y con base en los criterios vistos en clase:

a. ¿Cuántas Componentes habría que retener?

Criterio del porcentaje acumulado de la varianza

En este criterio deberia de retener tantos componentes, a manera que se explique alrededor de el 75% de la varianza de los datos originales. Por tal motivo,este criterio al momento de realizar la evaluación de la trabla resumen, observamos que las primeras dimensiones de “variance.percent” se obtiene un total del 77.70% por lo que ya se cumple el criterio por lo que se deberia seleccionar dichas dimensiones.

Criterio de Autovalores (Criterio de raiz latente)

Se basa únicamente en que se retengan aquellos componentes cuyo autovalor sea maryor que uno o que sea por lo menos de uno. En la tabla “Resumen de PCA” se observa la dimensión 1 y 2 son los que superan el autovalor de valor de 1. Por consiguiente las dos primeras dimensiones antes mencionadas, bajo este criterio, se retendrán.

Criterio del codo

la cantidad de componentes que se deberán retener ubicados antes del codo, otro sector hace hincapie sobre la componentes a retener serán aquellos localizados donde se de el codo. Una mayor explicación, el último gráfico de color verde podemos ver que las primeras tres dimenciones son las que cumplen con dicho criterio y por tal motivo se retendrán.

b. Incluye las tablas y gráficos vistos en clase.

library(kableExtra)
library(dplyr)
library(Hmisc)
Rx<-X6_2 %>% as.matrix() %>% rcorr()
Rx$r %>% kable(caption="Matriz R(X)",
        align = "c",
        digits = 2) %>% 
  kable_material(html_font = "sans-serif") %>% 
  kable_styling(bootstrap_options = c("striped", "hover"))

Matriz R(X)
	V1	V2	V3	V4	V5	V6	V7	V8	V9	V10
V1	1.00	0.87	0.82	0.82	-0.50	-0.19	0.21	-0.65	-0.64	-0.50
V2	0.87	1.00	0.73	0.83	-0.44	-0.07	0.25	-0.78	-0.75	-0.70
V3	0.82	0.73	1.00	0.81	-0.48	-0.23	0.19	-0.56	-0.63	-0.54
V4	0.82	0.83	0.81	1.00	-0.55	-0.26	0.17	-0.74	-0.79	-0.65
V5	-0.50	-0.44	-0.48	-0.55	1.00	0.74	0.18	0.29	0.34	0.12
V6	-0.19	-0.07	-0.23	-0.26	0.74	1.00	0.42	0.13	0.05	-0.24
V7	0.21	0.25	0.19	0.17	0.18	0.42	1.00	-0.30	-0.18	-0.41
V8	-0.65	-0.78	-0.56	-0.74	0.29	0.13	-0.30	1.00	0.89	0.73
V9	-0.64	-0.75	-0.63	-0.79	0.34	0.05	-0.18	0.89	1.00	0.78
V10	-0.50	-0.70	-0.54	-0.65	0.12	-0.24	-0.41	0.73	0.78	1.00

Rx$P %>% kable(caption="p-values de R(X)",
        align = "c",
        digits = 2) %>% 
  kable_classic_2(html_font = "sans-serif") %>% 
  kable_styling(bootstrap_options = c("striped", "hover"))

p-values de R(X)
	V1	V2	V3	V4	V5	V6	V7	V8	V9	V10
V1	NA	0.00	0.00	0.00	0.02	0.41	0.37	0.00	0.00	0.03
V2	0.00	NA	0.00	0.00	0.05	0.77	0.29	0.00	0.00	0.00
V3	0.00	0.00	NA	0.00	0.03	0.34	0.42	0.01	0.00	0.01
V4	0.00	0.00	0.00	NA	0.01	0.27	0.46	0.00	0.00	0.00
V5	0.02	0.05	0.03	0.01	NA	0.00	0.46	0.21	0.14	0.61
V6	0.41	0.77	0.34	0.27	0.00	NA	0.06	0.58	0.82	0.32
V7	0.37	0.29	0.42	0.46	0.46	0.06	NA	0.20	0.45	0.07
V8	0.00	0.00	0.01	0.00	0.21	0.58	0.20	NA	0.00	0.00
V9	0.00	0.00	0.00	0.00	0.14	0.82	0.45	0.00	NA	0.00
V10	0.03	0.00	0.01	0.00	0.61	0.32	0.07	0.00	0.00	NA

Descomposición de autovalores y autovectores

library(stargazer)

## 
## Please cite as:

##  Hlavac, Marek (2022). stargazer: Well-Formatted Regression and Summary Statistics Tables.

##  R package version 5.2.3. https://CRAN.R-project.org/package=stargazer

descomposicion<-eigen(Rx$r)
t(descomposicion$values) %>% kable(caption="Autovalores de R(X)",
        align = "c",
        digits = 2) %>% 
  kable_classic_2(html_font = "sans-serif") %>% 
  kable_styling(bootstrap_options = c("striped", "hover"))

Autovalores de R(X)
5.7	2.07	0.72	0.55	0.32	0.27	0.15	0.13	0.07	0.03

descomposicion$vectors %>% kable(caption="Autovectores de R(X)",
        align = "c",
        digits = 2) %>% 
  kable_classic_2(html_font = "sans-serif") %>% 
  kable_styling(bootstrap_options = c("striped", "hover"))

Autovectores de R(X)
-0.37	-0.07	-0.31	-0.34	0.38	-0.13	0.05	0.14	0.67	-0.10
-0.39	0.04	-0.04	-0.19	0.28	-0.47	-0.33	-0.06	-0.48	0.41
-0.35	-0.08	-0.32	-0.36	-0.31	0.56	-0.11	0.34	-0.32	-0.09
-0.39	-0.08	-0.02	-0.09	-0.14	0.12	0.26	-0.85	-0.03	-0.11
0.22	0.50	0.18	-0.30	0.34	0.50	-0.26	-0.23	0.12	0.27
0.08	0.63	0.00	-0.40	-0.13	-0.33	0.38	0.09	-0.18	-0.36
-0.12	0.47	-0.63	0.57	0.04	0.07	0.08	-0.02	0.00	0.16
0.36	-0.11	-0.33	-0.34	-0.47	-0.17	0.10	-0.10	0.19	0.58
0.37	-0.10	-0.43	-0.10	0.03	-0.14	-0.56	-0.28	-0.08	-0.49
0.32	-0.32	-0.27	-0.11	0.56	0.15	0.51	-0.01	-0.35	0.04

library(dplyr)
library(factoextra)

## Loading required package: ggplot2

## Welcome! Want to learn more? See two factoextra-related books at https://goo.gl/ve3WBa

library(kableExtra)
library(stargazer)
library(ggplot2)
options(scipen = 999999)
pc <- princomp(x = X6_2,cor = TRUE,fix_sign = FALSE)
factoextra::get_eig(pc) %>% 
    kable(caption = "Resumen de PCA",
        align = "c",
        digits = 2) %>%
  kable_material(html_font = "sans-serif") %>% 
  kable_styling(bootstrap_options = c("hover"))

Resumen de PCA
	eigenvalue	variance.percent	cumulative.variance.percent
Dim.1	5.70	57.01	57.01
Dim.2	2.07	20.69	77.70
Dim.3	0.72	7.20	84.91
Dim.4	0.55	5.48	90.39
Dim.5	0.32	3.16	93.54
Dim.6	0.27	2.71	96.25
Dim.7	0.15	1.46	97.72
Dim.8	0.13	1.28	99.00
Dim.9	0.07	0.68	99.68
Dim.10	0.03	0.32	100.00

Graficos

fviz_eig(pc,
         choice = "eigenvalue",
         barcolor = "blue",
         barfill = "blue",
         addlabels = TRUE,
         )+labs(title= "Grafico de sedimentacion", subtitle="Usando princomp, con autovalor")+
  xlab(label = "Componentes")+
  ylab(label = "Autovalores")+
  geom_hline(yintercept = 1)

fviz_eig(pc,
         choice = "variance",
         barcolor = "green",
         barfill = "green",
         addlabels = TRUE,
       )+labs(title = "Gráfico de Sedimentación",
              subtitle = "Usando princomp, con %Varianza Explicada")+
  xlab(label = "Componentes")+
  ylab(label = "%Varianza")

Componentes principales

FERNANDO JAVIER ASCENCIO ORELLANA-AO21017

2023-10-26

a) Calcula la matriz de varianza covarianza para la batería de indicadores:

1. De forma “manual”

2. Usando el comando cov de R base

b)Calcula la matriz de correlación para la batería de indicadores

1. De forma “manual”

2. Usando el comando cor R base

3. Presenta la matriz de correlación de forma gráfica (las dos versiones propuestas en clase)

c) Realiza un análisis de componentes principales, y con base en los criterios vistos en clase:

a. ¿Cuántas Componentes habría que retener?

b. Incluye las tablas y gráficos vistos en clase.

Descomposición de autovalores y autovectores

Graficos

V1	V2	V3	V4	V5	V6	V7	V8	V9	V10
4	1	4	3	3	2	4	4	4	4
5	5	4	4	3	3	4	1	1	3
2	1	3	1	4	2	1	5	4	5
1	1	1	1	4	4	2	5	5	4
1	1	2	1	5	5	4	3	3	2
5	5	5	5	3	3	4	2	2	1

V1	V2	V3	V4	V5	V6	V7	V8	V9	V10
4	1	4	3	3	2	4	4	4	4
5	5	4	4	3	3	4	1	1	3
2	1	3	1	4	2	1	5	4	5
1	1	1	1	4	4	2	5	5	4
1	1	2	1	5	5	4	3	3	2
5	5	5	5	3	3	4	2	2	1

V1	V2	V3	V4	V5	V6	V7	V8	V9	V10
4	1	4	3	3	2	4	4	4	4
5	5	4	4	3	3	4	1	1	3
2	1	3	1	4	2	1	5	4	5
1	1	1	1	4	4	2	5	5	4
1	1	2	1	5	5	4	3	3	2
5	5	5	5	3	3	4	2	2	1