Introducción: Presentación de la serie a analizar.

La serie ha sido obtenida desde el INE y son datos reales de la tasa de varuiación mensual del Índice de Precios de Consumo (IPC) de España. Tiene 200 meses y el último mes con datos ha sido Febrero de 2022.

El IPC tiene como objetivo proporcionar una medida estadística de la evolución del conjunto de precios de los bienes y servicios que consume la población residente en viviendas familiares en España.

Representación grafica y descomposición estacional (si tuviera comportamiento estacional).

Descomposición de las series

Representacion de la descomposición

Coeficientes debidos a la estacionalidad

##  [1]  6.746313e-15 -1.292364e-14  2.635436e-14 -7.448438e-15 -2.266360e-14
##  [6]  2.538843e+01 -7.093364e-15 -1.338843e+01  1.258286e-14 -6.658688e-15
## [11] -7.635873e-14  2.450847e-14

Análisis gráfico de la estacionalidad. Representación por año

Representacion de residuos

## [1] -3.02742e+14

## [1] 1.590911e+15

Contraste de normalidad de los residuos

## 
##  One-sample Kolmogorov-Smirnov test
## 
## data:  v_ipc_desc$random
## D = 0.42308, p-value < 2.2e-16
## alternative hypothesis: two-sided

## 
##  Shapiro-Wilk normality test
## 
## data:  v_ipc_desc$random
## W = 0.54181, p-value < 2.2e-16

Construcción del periodograma

Eliminar la heterocedasticidad. Estabilización de la varianza

Una función para pintar…hay más..

Eliminar tendencia

Representación

Eliminar estacionalidad

Contraste de normalidad de los residuos

## 
##  One-sample Kolmogorov-Smirnov test
## 
## data:  ipcLog.diff_1_12
## D = 0.20635, p-value = 0.009354
## alternative hypothesis: two-sided

## 
##  Shapiro-Wilk normality test
## 
## data:  ipcLog.diff_1_12
## W = NaN, p-value = NA

Ventanas de ajuste y evaluación

Suavizado exponencial

Distribución de residuos

##          Point Forecast      Lo 80     Hi 80     Lo 95    Hi 95
## Mar 2022      0.1433328 -0.6443021 0.9309676 -1.061251 1.347916
## Apr 2022      0.1433328 -0.6443021 0.9309676 -1.061251 1.347916
## May 2022      0.1433328 -0.6443021 0.9309676 -1.061251 1.347916
## Jun 2022      0.1433328 -0.6443021 0.9309676 -1.061251 1.347916
## Jul 2022      0.1433328 -0.6443021 0.9309676 -1.061251 1.347916
## Aug 2022      0.1433328 -0.6443021 0.9309676 -1.061251 1.347916
## Sep 2022      0.1433328 -0.6443021 0.9309676 -1.061251 1.347916
## Oct 2022      0.1433328 -0.6443021 0.9309676 -1.061251 1.347916

## Simple exponential smoothing 
## 
## Call:
##  ses(y = ipc_tr, h = 8) 
## 
##   Smoothing parameters:
##     alpha = 1e-04 
## 
##   Initial states:
##     l = 0.1433 
## 
##   sigma:  0.6146
## 
##      AIC     AICc      BIC 
## 837.0787 837.2050 846.8823

#Representamos los valores observados y los suavizados con la predicción

Suavizado Exponencial doble de Holt

Inspección del objeto creado y Distribución de residuos

##          Point Forecast      Lo 80     Hi 80     Lo 95    Hi 95
## Mar 2022      0.2107487 -0.5847585 1.0062560 -1.005875 1.427372
## Apr 2022      0.2088434 -0.5866708 1.0043575 -1.007791 1.425477
## May 2022      0.2069380 -0.5885916 1.0024676 -1.009720 1.423596
## Jun 2022      0.2050327 -0.5905244 1.0005897 -1.011667 1.421732
## Jul 2022      0.2031273 -0.5924727 0.9987273 -1.013638 1.419892
## Aug 2022      0.2012219 -0.5944399 0.9968837 -1.015638 1.418082
## Sep 2022      0.1993166 -0.5964293 0.9950625 -1.017672 1.416305
## Oct 2022      0.1974112 -0.5984445 0.9932670 -1.019745 1.414568

## Holt's method 
## 
## Call:
##  holt(y = ipc_tr, h = 8) 
## 
##   Smoothing parameters:
##     alpha = 0.0021 
##     beta  = 0.0021 
## 
##   Initial states:
##     l = 0.2765 
##     b = -0.013 
## 
##   sigma:  0.6207
## 
##      AIC     AICc      BIC 
## 842.9061 843.2253 859.2454

Representamos los valores observados y los suavizados con la predicción

Ajuste de modelo

##Visualización

Predicciones utilizando el paquete forecast (ETS)

Se comprueba la precisión de las distintas predicciones.

Parece que los distintos metodos se adaptan bastantes bien tras comprobar los resultados con datos reales, excepto HOLT-WINTERS multiplicativo ya que no es compatible con los valores negativos de la tasa de variación. Parece que se producirá una caida de la tasa de variación del IPC este mes, aunque es un poco improbable con la coyuntura actual económica.

Ciudades con más de 100000 hab

Selecciono las numéricas

##  [1] "AbstentionPtge"               "AbstencionAlta"              
##  [3] "Izda_Pct"                     "Dcha_Pct"                    
##  [5] "Otros_Pct"                    "Izquierda"                   
##  [7] "Derecha"                      "Age_0-4_Ptge"                
##  [9] "Age_under19_Ptge"             "Age_19_65_pct"               
## [11] "Age_over65_pct"               "WomanPopulationPtge"         
## [13] "ForeignersPtge"               "SameComAutonPtge"            
## [15] "SameComAutonDiffProvPtge"     "DifComAutonPtge"             
## [17] "UnemployLess25_Ptge"          "Unemploy25_40_Ptge"          
## [19] "UnemployMore40_Ptge"          "AgricultureUnemploymentPtge" 
## [21] "IndustryUnemploymentPtge"     "ConstructionUnemploymentPtge"
## [23] "ServicesUnemploymentPtge"     "totalEmpresas"               
## [25] "Industria"                    "Construccion"                
## [27] "ComercTTEHosteleria"          "Servicios"                   
## [29] "inmuebles"                    "SUPERFICIE"                  
## [31] "PobChange_pct"                "PersonasInmueble"            
## [33] "Explotaciones"

Gráfico de correlación y comunidades

Comprobar matriz de correlaciones

Calular determinante de la matriz

## [1] 1.140791e-31

Gráfico de correlaciones

Test de esferidad de Bartlett

## $chisq
## [1] 5692.248
## 
## $p.value
## [1] 0
## 
## $df
## [1] 630

Indice de Adecuacion Muestral KMO

## Error in solve.default(r) : 
##   sistema es computacionalmente singular: número de condición recíproco = 3.96215e-18

## Kaiser-Meyer-Olkin factor adequacy
## Call: psych::KMO(r = elec_r)
## Overall MSA =  0.5
## MSA for each item = 
##        Pop       Cens        Abs       AbsA       Izda       Dcha        Otr 
##        0.5        0.5        0.5        0.5        0.5        0.5        0.5 
##       IzqA      DechA       Age4      Age19   Age19_65      Age65       WomP 
##        0.5        0.5        0.5        0.5        0.5        0.5        0.5 
##      Forei    SameCom  SCDifProv     DifCom      UnL25    Un25_40      UnM40 
##        0.5        0.5        0.5        0.5        0.5        0.5        0.5 
##       AgrU       IndU      ConsU      ServU       Empr      Indus      Const 
##        0.5        0.5        0.5        0.5        0.5        0.5        0.5 
##    ComHost      Servi  inmuebles    Pob2010 SUPERFICIE  PobChange    PersInm 
##        0.5        0.5        0.5        0.5        0.5        0.5        0.5 
##     Explot 
##        0.5

Selecciono un subconjunto de variables

##  [1] "Pop"        "Cens"       "Abs"        "AbsA"       "Izda"      
##  [6] "Dcha"       "Otr"        "IzqA"       "DechA"      "Age4"      
## [11] "Age19"      "Age19_65"   "Age65"      "WomP"       "Forei"     
## [16] "SameCom"    "SCDifProv"  "DifCom"     "UnL25"      "Un25_40"   
## [21] "UnM40"      "AgrU"       "IndU"       "ConsU"      "ServU"     
## [26] "Empr"       "Indus"      "Const"      "ComHost"    "Servi"     
## [31] "inmuebles"  "Pob2010"    "SUPERFICIE" "PobChange"  "PersInm"   
## [36] "Explot"

Indice de Adecuacion Muestral KMO

## Kaiser-Meyer-Olkin factor adequacy
## Call: psych::KMO(r = elec_red)
## Overall MSA =  0.49
## MSA for each item = 
##     Izda     Dcha      Otr     IzqA    DechA     Age4    Age19 Age19_65 
##     0.41     0.43     0.48     0.58     0.61     0.48     0.50     0.35 
##    UnM40     AgrU     IndU    ConsU    ServU     Empr    Indus    Const 
##     0.57     0.28     0.36     0.21     0.28     0.65     0.68     0.66 
##  ComHost 
##     0.67

Gráfico de correlaciones

Indice de Adecuacion Muestral KMO

## Kaiser-Meyer-Olkin factor adequacy
## Call: psych::KMO(r = elec_red[, -c(3, 14)])
## Overall MSA =  0.44
## MSA for each item = 
##     Izda     Dcha     IzqA    DechA     Age4    Age19 Age19_65     AgrU 
##     0.50     0.56     0.42     0.43     0.55     0.51     0.30     0.22 
##     IndU    ConsU    ServU     Empr    Const  ComHost 
##     0.23     0.20     0.28     0.59     0.63     0.67

PCA

## Call:
## princomp(x = elec_red, cor = T)
## 
## Standard deviations:
##    Comp.1    Comp.2    Comp.3    Comp.4    Comp.5    Comp.6    Comp.7    Comp.8 
## 1.8627167 1.5621482 1.5162700 1.3785199 1.1224419 0.9592571 0.8044901 0.7142493 
##    Comp.9   Comp.10   Comp.11   Comp.12   Comp.13   Comp.14 
## 0.4615783 0.3861494 0.2865314 0.2413469 0.2003032 0.1026930 
## 
##  14  variables and  54 observations.

Información del modelo

## Importance of components:
##                           Comp.1    Comp.2    Comp.3    Comp.4     Comp.5
## Standard deviation     1.8627167 1.5621482 1.5162700 1.3785199 1.12244190
## Proportion of Variance 0.2478367 0.1743076 0.1642196 0.1357369 0.08999113
## Cumulative Proportion  0.2478367 0.4221443 0.5863639 0.7221009 0.81209199
##                            Comp.6     Comp.7     Comp.8     Comp.9    Comp.10
## Standard deviation     0.95925714 0.80449005 0.71424929 0.46157831 0.38614938
## Proportion of Variance 0.06572673 0.04622887 0.03643943 0.01521818 0.01065081
## Cumulative Proportion  0.87781872 0.92404760 0.96048703 0.97570521 0.98635602
##                            Comp.11     Comp.12     Comp.13      Comp.14
## Standard deviation     0.286531360 0.241346866 0.200303151 0.1026929802
## Proportion of Variance 0.005864301 0.004160594 0.002865811 0.0007532749
## Cumulative Proportion  0.992220321 0.996380914 0.999246725 1.0000000000

Sedimentación

Cargas

## 
## Loadings:
##          Comp.1 Comp.2 Comp.3 Comp.4 Comp.5 Comp.6 Comp.7 Comp.8 Comp.9 Comp.10
## Izda      0.109  0.261  0.404  0.109  0.395         0.289  0.404  0.444        
## Dcha     -0.122 -0.375  0.426         0.109  0.172         0.259 -0.685        
## IzqA      0.168  0.499                0.425                      -0.416        
## DechA    -0.239 -0.431  0.317        -0.110  0.139  0.190  0.141  0.308        
## Age4      0.303 -0.236 -0.138  0.378  0.183  0.159 -0.368  0.227  0.151 -0.449 
## Age19     0.267 -0.356         0.376  0.182        -0.315                0.542 
## Age19_65  0.109         0.237  0.494         0.156  0.426 -0.654               
## AgrU            -0.296        -0.263  0.500 -0.498        -0.330        -0.342 
## IndU                   -0.470                0.639  0.285               -0.187 
## ConsU     0.131        -0.149  0.429 -0.377 -0.455  0.392  0.324 -0.190 -0.149 
## ServU    -0.137  0.277  0.417  0.173 -0.256  0.122 -0.468 -0.210        -0.295 
## Empr     -0.494        -0.103  0.184  0.196                                    
## Const    -0.443        -0.177  0.295  0.160                             -0.380 
## ComHost  -0.477        -0.134  0.196  0.215                              0.287 
##          Comp.11 Comp.12 Comp.13 Comp.14
## Izda      0.304   0.190                 
## Dcha              0.183   0.191         
## IzqA     -0.277  -0.395  -0.350         
## DechA    -0.259  -0.440  -0.449         
## Age4     -0.420   0.202                 
## Age19     0.395  -0.247  -0.103         
## Age19_65 -0.173           0.141         
## AgrU      0.210          -0.240         
## IndU      0.351   0.112  -0.299         
## ConsU     0.153          -0.285         
## ServU     0.335   0.101  -0.381         
## Empr                              0.803 
## Const     0.244  -0.453   0.397  -0.297 
## ComHost  -0.190   0.487  -0.253  -0.496 
## 
##                Comp.1 Comp.2 Comp.3 Comp.4 Comp.5 Comp.6 Comp.7 Comp.8 Comp.9
## SS loadings     1.000  1.000  1.000  1.000  1.000  1.000  1.000  1.000  1.000
## Proportion Var  0.071  0.071  0.071  0.071  0.071  0.071  0.071  0.071  0.071
## Cumulative Var  0.071  0.143  0.214  0.286  0.357  0.429  0.500  0.571  0.643
##                Comp.10 Comp.11 Comp.12 Comp.13 Comp.14
## SS loadings      1.000   1.000   1.000   1.000   1.000
## Proportion Var   0.071   0.071   0.071   0.071   0.071
## Cumulative Var   0.714   0.786   0.857   0.929   1.000

Con la función PCA()

Información del modelo

## 
## Call:
## FactoMineR::PCA(X = elec_red, scale.unit = T) 
## 
## 
## Eigenvalues
##                        Dim.1   Dim.2   Dim.3   Dim.4   Dim.5   Dim.6   Dim.7
## Variance               3.470   2.440   2.299   1.900   1.260   0.920   0.647
## % of var.             24.784  17.431  16.422  13.574   8.999   6.573   4.623
## Cumulative % of var.  24.784  42.214  58.636  72.210  81.209  87.782  92.405
##                        Dim.8   Dim.9  Dim.10  Dim.11  Dim.12  Dim.13  Dim.14
## Variance               0.510   0.213   0.149   0.082   0.058   0.040   0.011
## % of var.              3.644   1.522   1.065   0.586   0.416   0.287   0.075
## Cumulative % of var.  96.049  97.571  98.636  99.222  99.638  99.925 100.000
## 
## Individuals (the 10 first)
##                                                  Dist    Dim.1    ctr   cos2  
## Algeciras                                    |  3.250 | -2.092  2.336  0.414 |
## Cádiz                                        |  4.053 | -1.290  0.888  0.101 |
## Jerez de la Frontera                         |  2.512 | -1.046  0.584  0.174 |
## Castellón de la Plana / Castelló de la Plana |  2.507 | -0.016  0.000  0.000 |
## Córdoba                                      |  2.639 |  1.719  1.576  0.424 |
## A Coruña                                     |  3.540 |  3.051  4.970  0.743 |
## Granada                                      |  2.352 |  1.538  1.263  0.428 |
## Donostia / San Sebastián                     |  4.220 |  0.845  0.381  0.040 |
## Huelva                                       |  5.430 | -2.738  4.001  0.254 |
## Jaén                                         |  3.847 | -1.226  0.802  0.102 |
##                                               Dim.2    ctr   cos2    Dim.3
## Algeciras                                     1.597  1.936  0.242 |  1.018
## Cádiz                                        -2.793  5.919  0.475 |  1.161
## Jerez de la Frontera                          1.440  1.574  0.329 |  0.591
## Castellón de la Plana / Castelló de la Plana  1.685  2.154  0.452 | -1.103
## Córdoba                                       1.140  0.987  0.187 |  0.148
## A Coruña                                     -0.275  0.058  0.006 | -0.105
## Granada                                       0.569  0.246  0.059 |  1.073
## Donostia / San Sebastián                     -3.447  9.014  0.667 | -0.562
## Huelva                                        0.424  0.136  0.006 |  0.113
## Jaén                                          1.979  2.972  0.265 |  1.455
##                                                 ctr   cos2  
## Algeciras                                     0.834  0.098 |
## Cádiz                                         1.086  0.082 |
## Jerez de la Frontera                          0.282  0.055 |
## Castellón de la Plana / Castelló de la Plana  0.979  0.193 |
## Córdoba                                       0.018  0.003 |
## A Coruña                                      0.009  0.001 |
## Granada                                       0.927  0.208 |
## Donostia / San Sebastián                      0.254  0.018 |
## Huelva                                        0.010  0.000 |
## Jaén                                          1.704  0.143 |
## 
## Variables (the 10 first)
##                                                 Dim.1    ctr   cos2    Dim.2
## Izda                                         | -0.204  1.196  0.041 | -0.408
## Dcha                                         |  0.227  1.484  0.052 |  0.585
## IzqA                                         | -0.313  2.831  0.098 | -0.779
## DechA                                        |  0.444  5.692  0.198 |  0.673
## Age4                                         | -0.564  9.159  0.318 |  0.369
## Age19                                        | -0.498  7.149  0.248 |  0.556
## Age19_65                                     | -0.203  1.186  0.041 |  0.051
## AgrU                                         | -0.179  0.918  0.032 |  0.462
## IndU                                         | -0.022  0.014  0.000 |  0.139
## ConsU                                        | -0.245  1.724  0.060 |  0.013
##                                                 ctr   cos2    Dim.3    ctr
## Izda                                          6.823  0.167 |  0.612 16.302
## Dcha                                         14.043  0.343 |  0.645 18.114
## IzqA                                         24.875  0.607 |  0.004  0.001
## DechA                                        18.551  0.453 |  0.481 10.050
## Age4                                          5.582  0.136 | -0.209  1.904
## Age19                                        12.683  0.310 | -0.081  0.285
## Age19_65                                      0.107  0.003 |  0.360  5.632
## AgrU                                          8.734  0.213 |  0.031  0.043
## IndU                                          0.793  0.019 | -0.712 22.043
## ConsU                                         0.007  0.000 | -0.226  2.216
##                                                cos2  
## Izda                                          0.375 |
## Dcha                                          0.416 |
## IzqA                                          0.000 |
## DechA                                         0.231 |
## Age4                                          0.044 |
## Age19                                         0.007 |
## Age19_65                                      0.129 |
## AgrU                                          0.001 |
## IndU                                          0.507 |
## ConsU                                         0.051 |

AF

## 
## Call:
## factanal(x = elec_red, factors = 3, scores = "regression", rotation = "none")
## 
## Uniquenesses:
##     Izda     Dcha     IzqA    DechA     Age4    Age19 Age19_65     AgrU 
##    0.216    0.296    0.091    0.005    0.877    0.894    0.917    0.958 
##     IndU    ConsU    ServU     Empr    Const  ComHost 
##    0.846    0.890    0.849    0.005    0.110    0.039 
## 
## Loadings:
##          Factor1 Factor2 Factor3
## Izda     -0.149           0.871 
## Dcha      0.486  -0.631   0.264 
## IzqA     -0.510   0.520   0.615 
## DechA     0.722  -0.688         
## Age4     -0.309          -0.139 
## Age19    -0.178  -0.203  -0.181 
## Age19_65         -0.131   0.256 
## AgrU             -0.201         
## IndU              0.144  -0.359 
## ConsU    -0.181          -0.262 
## ServU     0.135           0.364 
## Empr      0.855   0.514         
## Const     0.751   0.568         
## ComHost   0.817   0.542         
## 
##                Factor1 Factor2 Factor3
## SS loadings      3.186   2.166   1.657
## Proportion Var   0.228   0.155   0.118
## Cumulative Var   0.228   0.382   0.501
## 
## Test of the hypothesis that 3 factors are sufficient.
## The chi square statistic is 219.55 on 52 degrees of freedom.
## The p-value is 1.81e-22

## Factor Analysis using method =  wls
## Call: psych::fa(r = elec_red, nfactors = 3, rotate = "promax", fm = "wls")
## Standardized loadings (pattern matrix) based upon correlation matrix
##           WLS1  WLS2  WLS3    h2   u2 com
## Izda                  0.66 0.472 0.53 1.4
## Dcha            0.78  0.35 0.702 0.30 1.4
## IzqA           -0.68       0.563 0.44 1.4
## DechA           0.84       0.740 0.26 1.2
## Age4     -0.53             0.418 0.58 1.6
## Age19    -0.54             0.487 0.51 1.9
## Age19_65                   0.108 0.89 2.3
## AgrU                       0.136 0.86 2.4
## IndU                 -0.59 0.358 0.64 1.1
## ConsU                      0.064 0.94 2.8
## ServU                 0.66 0.511 0.49 1.2
## Empr      0.85             0.728 0.27 1.2
## Const     0.77             0.598 0.40 1.2
## ComHost   0.83             0.687 0.31 1.2
## 
##                       WLS1 WLS2 WLS3
## SS loadings           2.80 2.01 1.76
## Proportion Var        0.20 0.14 0.13
## Cumulative Var        0.20 0.34 0.47
## Proportion Explained  0.43 0.31 0.27
## Cumulative Proportion 0.43 0.73 1.00
## 
##  With factor correlations of 
##       WLS1  WLS2  WLS3
## WLS1  1.00 -0.07  0.15
## WLS2 -0.07  1.00 -0.06
## WLS3  0.15 -0.06  1.00
## 
## Mean item complexity =  1.6
## Test of the hypothesis that 3 factors are sufficient.
## 
## The degrees of freedom for the null model are  91  and the objective function was  13.91 with Chi Square of  660.71
## The degrees of freedom for the model are 52  and the objective function was  8.31 
## 
## The root mean square of the residuals (RMSR) is  0.13 
## The df corrected root mean square of the residuals is  0.17 
## 
## The harmonic number of observations is  54 with the empirical chi square  168.78  with prob <  2.9e-14 
## The total number of observations was  54  with Likelihood Chi Square =  377.94  with prob <  5.1e-51 
## 
## Tucker Lewis Index of factoring reliability =  -0.053
## RMSEA index =  0.34  and the 90 % confidence intervals are  0.312 0.377
## BIC =  170.52
## Fit based upon off diagonal values = 0.81
## Measures of factor score adequacy             
##                                                   WLS1 WLS2 WLS3
## Correlation of (regression) scores with factors   0.92 0.90 0.89
## Multiple R square of scores with factors          0.84 0.81 0.80
## Minimum correlation of possible factor scores     0.68 0.63 0.59

Biplot para AF con la función biplot()

CLustering jerárquico

## Indice VI  single = 0.9798834 
## Indice Rand  single = 0.2974756 
## Silueta media  single = 0.2788331 
## Within SS  single = 697.67

## Indice VI  complete = 0.4255589 
## Indice Rand  complete = 0.8267175 
## Silueta media  complete = 0.5815634 
## Within SS  complete = 154.2363

## Indice VI  average = 0.4232592 
## Indice Rand  average = 0.7900927 
## Silueta media  average = 0.5576904 
## Within SS  average = 169.7858

## Indice VI  mcquitty = 0.4779669 
## Indice Rand  mcquitty = 0.8075494 
## Silueta media  mcquitty = 0.5591647 
## Within SS  mcquitty = 164.5828

## Indice VI  ward.D2 = 0.4594814 
## Indice Rand  ward.D2 = 0.75846 
## Silueta media  ward.D2 = 0.5360699 
## Within SS  ward.D2 = 184.4635

## Accuracy K-means = 0.94

## Indice VI  K-means = 0.3383608 
## Indice Rand  K-means = 0.8572115 
## Silueta media  K-means = 0.5851126 
## Within SS  K-means = 151.9651

##                 vi      rand silhouette      wss  Acc
## single   0.9798834 0.2974756  0.2788331 697.6700 0.51
## complete 0.4255589 0.8267175  0.5815634 154.2363 0.93
## average  0.4232592 0.7900927  0.5576904 169.7858 0.91
## mcquitty 0.4779669 0.8075494  0.5591647 164.5828 0.92
## ward.D2  0.4594814 0.7584600  0.5360699 184.4635 0.89
## med.km   0.3383608 0.8572115  0.5851126 151.9651 0.94