predictive analytics final
Nelson White
12/13/2020
DengAI, Predicting the Spread of Desease
Libraries
library(neuralnet)
library(ggplot2)## Warning: package 'ggplot2' was built under R version 3.6.2library(ggfortify)## Warning: package 'ggfortify' was built under R version 3.6.2library(forecast)## Warning: package 'forecast' was built under R version 3.6.2## Registered S3 method overwritten by 'xts':
## method from
## as.zoo.xts zoo## Registered S3 method overwritten by 'quantmod':
## method from
## as.zoo.data.frame zoo## Registered S3 methods overwritten by 'forecast':
## method from
## autoplot.Arima ggfortify
## autoplot.acf ggfortify
## autoplot.ar ggfortify
## autoplot.bats ggfortify
## autoplot.decomposed.ts ggfortify
## autoplot.ets ggfortify
## autoplot.forecast ggfortify
## autoplot.stl ggfortify
## autoplot.ts ggfortify
## fitted.ar ggfortify
## fortify.ts ggfortify
## residuals.ar ggfortifylibrary(psych)## Warning: package 'psych' was built under R version 3.6.2##
## Attaching package: 'psych'## The following objects are masked from 'package:ggplot2':
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## %+%, alphalibrary(stargazer)##
## Please cite as:## Hlavac, Marek (2018). stargazer: Well-Formatted Regression and Summary Statistics Tables.## R package version 5.2.2. https://CRAN.R-project.org/package=stargazerlibrary(vars)## Loading required package: MASS## Warning: package 'MASS' was built under R version 3.6.2## Loading required package: strucchange## Loading required package: zoo## Warning: package 'zoo' was built under R version 3.6.2##
## Attaching package: 'zoo'## The following objects are masked from 'package:base':
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## as.Date, as.Date.numeric## Loading required package: sandwich## Loading required package: urca## Loading required package: lmtestlibrary(dplyr)##
## Attaching package: 'dplyr'## The following object is masked from 'package:MASS':
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## select## The following object is masked from 'package:neuralnet':
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## compute## The following objects are masked from 'package:stats':
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## filter, lag## The following objects are masked from 'package:base':
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## intersect, setdiff, setequal, unionImport Data
features.test = read.csv("/Users/nelsonwhite/Documents/ms applied economics/Predictive Analytics:Forecasting/final/data/dengue_features_test.csv")
features.train = read.csv("/Users/nelsonwhite/Documents/ms applied economics/Predictive Analytics:Forecasting/final/data/dengue_features_train.csv")
labels.train = read.csv("/Users/nelsonwhite/Documents/ms applied economics/Predictive Analytics:Forecasting/final/data/dengue_labels_train.csv")Descriptive Statistics
head(features.test)## city year weekofyear week_start_date ndvi_ne ndvi_nw ndvi_se
## 1 sj 2008 18 2008-04-29 -0.0189 -0.01890000 0.10272860
## 2 sj 2008 19 2008-05-06 -0.0180 -0.01240000 0.08204286
## 3 sj 2008 20 2008-05-13 -0.0015 NA 0.15108330
## 4 sj 2008 21 2008-05-20 NA -0.01986667 0.12432860
## 5 sj 2008 22 2008-05-27 0.0568 0.03983333 0.06226667
## 6 sj 2008 23 2008-06-03 -0.0440 -0.03046667 0.13200000
## ndvi_sw precipitation_amt_mm reanalysis_air_temp_k
## 1 0.09120000 78.60 298.4929
## 2 0.07231429 12.56 298.4757
## 3 0.09152857 3.66 299.4557
## 4 0.12568570 0.00 299.6900
## 5 0.07591429 0.76 299.7800
## 6 0.08352857 71.17 299.7686
## reanalysis_avg_temp_k reanalysis_dew_point_temp_k
## 1 298.5500 294.5271
## 2 298.5571 294.3957
## 3 299.3571 295.3086
## 4 299.7286 294.4029
## 5 299.6714 294.7600
## 6 299.7286 295.3143
## reanalysis_max_air_temp_k reanalysis_min_air_temp_k
## 1 301.1 296.4
## 2 300.8 296.7
## 3 302.2 296.4
## 4 303.0 296.9
## 5 302.3 297.3
## 6 301.9 297.6
## reanalysis_precip_amt_kg_per_m2 reanalysis_relative_humidity_percent
## 1 25.37 78.78143
## 2 21.83 78.23000
## 3 4.12 78.27000
## 4 2.20 73.01571
## 5 4.36 74.08429
## 6 22.55 76.55714
## reanalysis_sat_precip_amt_mm reanalysis_specific_humidity_g_per_kg
## 1 78.60 15.91857
## 2 12.56 15.79143
## 3 3.66 16.67429
## 4 0.00 15.77571
## 5 0.76 16.13714
## 6 71.17 16.66714
## reanalysis_tdtr_k station_avg_temp_c station_diur_temp_rng_c
## 1 3.128571 26.52857 7.057143
## 2 2.571429 26.07143 5.557143
## 3 4.428571 27.92857 7.785714
## 4 4.342857 28.05714 6.271429
## 5 3.542857 27.61429 7.085714
## 6 2.857143 28.00000 5.171429
## station_max_temp_c station_min_temp_c station_precip_mm
## 1 33.3 21.7 75.2
## 2 30.0 22.2 34.3
## 3 32.8 22.8 3.0
## 4 33.3 24.4 0.3
## 5 33.3 23.3 84.1
## 6 32.8 25.0 27.7summary(features.test)## city year weekofyear week_start_date
## iq:156 Min. :2008 Min. : 1.00 2010-07-02: 2
## sj:260 1st Qu.:2010 1st Qu.:13.75 2010-07-09: 2
## Median :2011 Median :26.00 2010-07-16: 2
## Mean :2011 Mean :26.44 2010-07-23: 2
## 3rd Qu.:2012 3rd Qu.:39.00 2010-07-30: 2
## Max. :2013 Max. :53.00 2010-08-06: 2
## (Other) :404
## ndvi_ne ndvi_nw ndvi_se ndvi_sw
## Min. :-0.4634 Min. :-0.21180 Min. :0.0062 Min. :-0.01467
## 1st Qu.:-0.0015 1st Qu.: 0.01597 1st Qu.:0.1487 1st Qu.: 0.13408
## Median : 0.1101 Median : 0.08870 Median :0.2042 Median : 0.18647
## Mean : 0.1260 Mean : 0.12680 Mean :0.2077 Mean : 0.20172
## 3rd Qu.: 0.2633 3rd Qu.: 0.24240 3rd Qu.:0.2549 3rd Qu.: 0.25324
## Max. : 0.5004 Max. : 0.64900 Max. :0.4530 Max. : 0.52904
## NA's :43 NA's :11 NA's :1 NA's :1
## precipitation_amt_mm reanalysis_air_temp_k reanalysis_avg_temp_k
## Min. : 0.000 Min. :294.6 Min. :295.2
## 1st Qu.: 8.175 1st Qu.:297.8 1st Qu.:298.3
## Median : 31.455 Median :298.5 Median :299.3
## Mean : 38.354 Mean :298.8 Mean :299.4
## 3rd Qu.: 57.773 3rd Qu.:300.2 3rd Qu.:300.5
## Max. :169.340 Max. :301.9 Max. :303.3
## NA's :2 NA's :2 NA's :2
## reanalysis_dew_point_temp_k reanalysis_max_air_temp_k
## Min. :290.8 Min. :298.2
## 1st Qu.:294.3 1st Qu.:301.4
## Median :295.8 Median :302.8
## Mean :295.4 Mean :303.6
## 3rd Qu.:296.6 3rd Qu.:305.8
## Max. :297.8 Max. :314.1
## NA's :2 NA's :2
## reanalysis_min_air_temp_k reanalysis_precip_amt_kg_per_m2
## Min. :286.2 Min. : 0.00
## 1st Qu.:293.5 1st Qu.: 9.43
## Median :296.3 Median : 25.85
## Mean :295.7 Mean : 42.17
## 3rd Qu.:298.3 3rd Qu.: 56.48
## Max. :299.7 Max. :301.40
## NA's :2 NA's :2
## reanalysis_relative_humidity_percent reanalysis_sat_precip_amt_mm
## Min. :64.92 Min. : 0.000
## 1st Qu.:77.40 1st Qu.: 8.175
## Median :80.33 Median : 31.455
## Mean :82.50 Mean : 38.354
## 3rd Qu.:88.33 3rd Qu.: 57.773
## Max. :97.98 Max. :169.340
## NA's :2 NA's :2
## reanalysis_specific_humidity_g_per_kg reanalysis_tdtr_k
## Min. :12.54 Min. : 1.486
## 1st Qu.:15.79 1st Qu.: 2.446
## Median :17.34 Median : 2.914
## Mean :16.93 Mean : 5.125
## 3rd Qu.:18.17 3rd Qu.: 8.171
## Max. :19.60 Max. :14.486
## NA's :2 NA's :2
## station_avg_temp_c station_diur_temp_rng_c station_max_temp_c
## Min. :24.16 Min. : 4.043 Min. :27.20
## 1st Qu.:26.51 1st Qu.: 5.929 1st Qu.:31.10
## Median :27.48 Median : 6.643 Median :32.80
## Mean :27.37 Mean : 7.811 Mean :32.53
## 3rd Qu.:28.32 3rd Qu.: 9.812 3rd Qu.:33.90
## Max. :30.27 Max. :14.725 Max. :38.40
## NA's :12 NA's :12 NA's :3
## station_min_temp_c station_precip_mm
## Min. :14.20 Min. : 0.00
## 1st Qu.:21.20 1st Qu.: 9.10
## Median :22.20 Median : 23.60
## Mean :22.37 Mean : 34.28
## 3rd Qu.:23.30 3rd Qu.: 47.75
## Max. :26.70 Max. :212.00
## NA's :9 NA's :5head(features.train)## city year weekofyear week_start_date ndvi_ne ndvi_nw ndvi_se
## 1 sj 1990 18 1990-04-30 0.1226000 0.1037250 0.1984833
## 2 sj 1990 19 1990-05-07 0.1699000 0.1421750 0.1623571
## 3 sj 1990 20 1990-05-14 0.0322500 0.1729667 0.1572000
## 4 sj 1990 21 1990-05-21 0.1286333 0.2450667 0.2275571
## 5 sj 1990 22 1990-05-28 0.1962000 0.2622000 0.2512000
## 6 sj 1990 23 1990-06-04 NA 0.1748500 0.2543143
## ndvi_sw precipitation_amt_mm reanalysis_air_temp_k
## 1 0.1776167 12.42 297.5729
## 2 0.1554857 22.82 298.2114
## 3 0.1708429 34.54 298.7814
## 4 0.2358857 15.36 298.9871
## 5 0.2473400 7.52 299.5186
## 6 0.1817429 9.58 299.6300
## reanalysis_avg_temp_k reanalysis_dew_point_temp_k
## 1 297.7429 292.4143
## 2 298.4429 293.9514
## 3 298.8786 295.4343
## 4 299.2286 295.3100
## 5 299.6643 295.8214
## 6 299.7643 295.8514
## reanalysis_max_air_temp_k reanalysis_min_air_temp_k
## 1 299.8 295.9
## 2 300.9 296.4
## 3 300.5 297.3
## 4 301.4 297.0
## 5 301.9 297.5
## 6 302.4 298.1
## reanalysis_precip_amt_kg_per_m2 reanalysis_relative_humidity_percent
## 1 32.00 73.36571
## 2 17.94 77.36857
## 3 26.10 82.05286
## 4 13.90 80.33714
## 5 12.20 80.46000
## 6 26.49 79.89143
## reanalysis_sat_precip_amt_mm reanalysis_specific_humidity_g_per_kg
## 1 12.42 14.01286
## 2 22.82 15.37286
## 3 34.54 16.84857
## 4 15.36 16.67286
## 5 7.52 17.21000
## 6 9.58 17.21286
## reanalysis_tdtr_k station_avg_temp_c station_diur_temp_rng_c
## 1 2.628571 25.44286 6.900000
## 2 2.371429 26.71429 6.371429
## 3 2.300000 26.71429 6.485714
## 4 2.428571 27.47143 6.771429
## 5 3.014286 28.94286 9.371429
## 6 2.100000 28.11429 6.942857
## station_max_temp_c station_min_temp_c station_precip_mm
## 1 29.4 20.0 16.0
## 2 31.7 22.2 8.6
## 3 32.2 22.8 41.4
## 4 33.3 23.3 4.0
## 5 35.0 23.9 5.8
## 6 34.4 23.9 39.1summary(features.train)## city year weekofyear week_start_date
## iq:520 Min. :1990 Min. : 1.00 2000-07-01: 2
## sj:936 1st Qu.:1997 1st Qu.:13.75 2000-07-08: 2
## Median :2002 Median :26.50 2000-07-15: 2
## Mean :2001 Mean :26.50 2000-07-22: 2
## 3rd Qu.:2005 3rd Qu.:39.25 2000-07-29: 2
## Max. :2010 Max. :53.00 2000-08-05: 2
## (Other) :1444
## ndvi_ne ndvi_nw ndvi_se
## Min. :-0.40625 Min. :-0.45610 Min. :-0.01553
## 1st Qu.: 0.04495 1st Qu.: 0.04922 1st Qu.: 0.15509
## Median : 0.12882 Median : 0.12143 Median : 0.19605
## Mean : 0.14229 Mean : 0.13055 Mean : 0.20378
## 3rd Qu.: 0.24848 3rd Qu.: 0.21660 3rd Qu.: 0.24885
## Max. : 0.50836 Max. : 0.45443 Max. : 0.53831
## NA's :194 NA's :52 NA's :22
## ndvi_sw precipitation_amt_mm reanalysis_air_temp_k
## Min. :-0.06346 Min. : 0.00 Min. :294.6
## 1st Qu.: 0.14421 1st Qu.: 9.80 1st Qu.:297.7
## Median : 0.18945 Median : 38.34 Median :298.6
## Mean : 0.20231 Mean : 45.76 Mean :298.7
## 3rd Qu.: 0.24698 3rd Qu.: 70.23 3rd Qu.:299.8
## Max. : 0.54602 Max. :390.60 Max. :302.2
## NA's :22 NA's :13 NA's :10
## reanalysis_avg_temp_k reanalysis_dew_point_temp_k
## Min. :294.9 Min. :289.6
## 1st Qu.:298.3 1st Qu.:294.1
## Median :299.3 Median :295.6
## Mean :299.2 Mean :295.2
## 3rd Qu.:300.2 3rd Qu.:296.5
## Max. :302.9 Max. :298.4
## NA's :10 NA's :10
## reanalysis_max_air_temp_k reanalysis_min_air_temp_k
## Min. :297.8 Min. :286.9
## 1st Qu.:301.0 1st Qu.:293.9
## Median :302.4 Median :296.2
## Mean :303.4 Mean :295.7
## 3rd Qu.:305.5 3rd Qu.:297.9
## Max. :314.0 Max. :299.9
## NA's :10 NA's :10
## reanalysis_precip_amt_kg_per_m2 reanalysis_relative_humidity_percent
## Min. : 0.00 Min. :57.79
## 1st Qu.: 13.05 1st Qu.:77.18
## Median : 27.25 Median :80.30
## Mean : 40.15 Mean :82.16
## 3rd Qu.: 52.20 3rd Qu.:86.36
## Max. :570.50 Max. :98.61
## NA's :10 NA's :10
## reanalysis_sat_precip_amt_mm reanalysis_specific_humidity_g_per_kg
## Min. : 0.00 Min. :11.72
## 1st Qu.: 9.80 1st Qu.:15.56
## Median : 38.34 Median :17.09
## Mean : 45.76 Mean :16.75
## 3rd Qu.: 70.23 3rd Qu.:17.98
## Max. :390.60 Max. :20.46
## NA's :13 NA's :10
## reanalysis_tdtr_k station_avg_temp_c station_diur_temp_rng_c
## Min. : 1.357 Min. :21.40 Min. : 4.529
## 1st Qu.: 2.329 1st Qu.:26.30 1st Qu.: 6.514
## Median : 2.857 Median :27.41 Median : 7.300
## Mean : 4.904 Mean :27.19 Mean : 8.059
## 3rd Qu.: 7.625 3rd Qu.:28.16 3rd Qu.: 9.567
## Max. :16.029 Max. :30.80 Max. :15.800
## NA's :10 NA's :43 NA's :43
## station_max_temp_c station_min_temp_c station_precip_mm
## Min. :26.70 Min. :14.7 Min. : 0.00
## 1st Qu.:31.10 1st Qu.:21.1 1st Qu.: 8.70
## Median :32.80 Median :22.2 Median : 23.85
## Mean :32.45 Mean :22.1 Mean : 39.33
## 3rd Qu.:33.90 3rd Qu.:23.3 3rd Qu.: 53.90
## Max. :42.20 Max. :25.6 Max. :543.30
## NA's :20 NA's :14 NA's :22head(labels.train)## city year weekofyear total_cases
## 1 sj 1990 18 4
## 2 sj 1990 19 5
## 3 sj 1990 20 4
## 4 sj 1990 21 3
## 5 sj 1990 22 6
## 6 sj 1990 23 2summary(labels.train)## city year weekofyear total_cases
## iq:520 Min. :1990 Min. : 1.00 Min. : 0.00
## sj:936 1st Qu.:1997 1st Qu.:13.75 1st Qu.: 5.00
## Median :2002 Median :26.50 Median : 12.00
## Mean :2001 Mean :26.50 Mean : 24.68
## 3rd Qu.:2005 3rd Qu.:39.25 3rd Qu.: 28.00
## Max. :2010 Max. :53.00 Max. :461.00Combine the training datasets into one new data frame:
train <- left_join(x = labels.train, y = features.train, by = c("year", "weekofyear", "city"))head(train)## city year weekofyear total_cases week_start_date ndvi_ne ndvi_nw
## 1 sj 1990 18 4 1990-04-30 0.1226000 0.1037250
## 2 sj 1990 19 5 1990-05-07 0.1699000 0.1421750
## 3 sj 1990 20 4 1990-05-14 0.0322500 0.1729667
## 4 sj 1990 21 3 1990-05-21 0.1286333 0.2450667
## 5 sj 1990 22 6 1990-05-28 0.1962000 0.2622000
## 6 sj 1990 23 2 1990-06-04 NA 0.1748500
## ndvi_se ndvi_sw precipitation_amt_mm reanalysis_air_temp_k
## 1 0.1984833 0.1776167 12.42 297.5729
## 2 0.1623571 0.1554857 22.82 298.2114
## 3 0.1572000 0.1708429 34.54 298.7814
## 4 0.2275571 0.2358857 15.36 298.9871
## 5 0.2512000 0.2473400 7.52 299.5186
## 6 0.2543143 0.1817429 9.58 299.6300
## reanalysis_avg_temp_k reanalysis_dew_point_temp_k
## 1 297.7429 292.4143
## 2 298.4429 293.9514
## 3 298.8786 295.4343
## 4 299.2286 295.3100
## 5 299.6643 295.8214
## 6 299.7643 295.8514
## reanalysis_max_air_temp_k reanalysis_min_air_temp_k
## 1 299.8 295.9
## 2 300.9 296.4
## 3 300.5 297.3
## 4 301.4 297.0
## 5 301.9 297.5
## 6 302.4 298.1
## reanalysis_precip_amt_kg_per_m2 reanalysis_relative_humidity_percent
## 1 32.00 73.36571
## 2 17.94 77.36857
## 3 26.10 82.05286
## 4 13.90 80.33714
## 5 12.20 80.46000
## 6 26.49 79.89143
## reanalysis_sat_precip_amt_mm reanalysis_specific_humidity_g_per_kg
## 1 12.42 14.01286
## 2 22.82 15.37286
## 3 34.54 16.84857
## 4 15.36 16.67286
## 5 7.52 17.21000
## 6 9.58 17.21286
## reanalysis_tdtr_k station_avg_temp_c station_diur_temp_rng_c
## 1 2.628571 25.44286 6.900000
## 2 2.371429 26.71429 6.371429
## 3 2.300000 26.71429 6.485714
## 4 2.428571 27.47143 6.771429
## 5 3.014286 28.94286 9.371429
## 6 2.100000 28.11429 6.942857
## station_max_temp_c station_min_temp_c station_precip_mm
## 1 29.4 20.0 16.0
## 2 31.7 22.2 8.6
## 3 32.2 22.8 41.4
## 4 33.3 23.3 4.0
## 5 35.0 23.9 5.8
## 6 34.4 23.9 39.1summary(train)## city year weekofyear total_cases
## iq:520 Min. :1990 Min. : 1.00 Min. : 0.00
## sj:936 1st Qu.:1997 1st Qu.:13.75 1st Qu.: 5.00
## Median :2002 Median :26.50 Median : 12.00
## Mean :2001 Mean :26.50 Mean : 24.68
## 3rd Qu.:2005 3rd Qu.:39.25 3rd Qu.: 28.00
## Max. :2010 Max. :53.00 Max. :461.00
##
## week_start_date ndvi_ne ndvi_nw
## 2000-07-01: 2 Min. :-0.40625 Min. :-0.45610
## 2000-07-08: 2 1st Qu.: 0.04495 1st Qu.: 0.04922
## 2000-07-15: 2 Median : 0.12882 Median : 0.12143
## 2000-07-22: 2 Mean : 0.14229 Mean : 0.13055
## 2000-07-29: 2 3rd Qu.: 0.24848 3rd Qu.: 0.21660
## 2000-08-05: 2 Max. : 0.50836 Max. : 0.45443
## (Other) :1444 NA's :194 NA's :52
## ndvi_se ndvi_sw precipitation_amt_mm
## Min. :-0.01553 Min. :-0.06346 Min. : 0.00
## 1st Qu.: 0.15509 1st Qu.: 0.14421 1st Qu.: 9.80
## Median : 0.19605 Median : 0.18945 Median : 38.34
## Mean : 0.20378 Mean : 0.20231 Mean : 45.76
## 3rd Qu.: 0.24885 3rd Qu.: 0.24698 3rd Qu.: 70.23
## Max. : 0.53831 Max. : 0.54602 Max. :390.60
## NA's :22 NA's :22 NA's :13
## reanalysis_air_temp_k reanalysis_avg_temp_k reanalysis_dew_point_temp_k
## Min. :294.6 Min. :294.9 Min. :289.6
## 1st Qu.:297.7 1st Qu.:298.3 1st Qu.:294.1
## Median :298.6 Median :299.3 Median :295.6
## Mean :298.7 Mean :299.2 Mean :295.2
## 3rd Qu.:299.8 3rd Qu.:300.2 3rd Qu.:296.5
## Max. :302.2 Max. :302.9 Max. :298.4
## NA's :10 NA's :10 NA's :10
## reanalysis_max_air_temp_k reanalysis_min_air_temp_k
## Min. :297.8 Min. :286.9
## 1st Qu.:301.0 1st Qu.:293.9
## Median :302.4 Median :296.2
## Mean :303.4 Mean :295.7
## 3rd Qu.:305.5 3rd Qu.:297.9
## Max. :314.0 Max. :299.9
## NA's :10 NA's :10
## reanalysis_precip_amt_kg_per_m2 reanalysis_relative_humidity_percent
## Min. : 0.00 Min. :57.79
## 1st Qu.: 13.05 1st Qu.:77.18
## Median : 27.25 Median :80.30
## Mean : 40.15 Mean :82.16
## 3rd Qu.: 52.20 3rd Qu.:86.36
## Max. :570.50 Max. :98.61
## NA's :10 NA's :10
## reanalysis_sat_precip_amt_mm reanalysis_specific_humidity_g_per_kg
## Min. : 0.00 Min. :11.72
## 1st Qu.: 9.80 1st Qu.:15.56
## Median : 38.34 Median :17.09
## Mean : 45.76 Mean :16.75
## 3rd Qu.: 70.23 3rd Qu.:17.98
## Max. :390.60 Max. :20.46
## NA's :13 NA's :10
## reanalysis_tdtr_k station_avg_temp_c station_diur_temp_rng_c
## Min. : 1.357 Min. :21.40 Min. : 4.529
## 1st Qu.: 2.329 1st Qu.:26.30 1st Qu.: 6.514
## Median : 2.857 Median :27.41 Median : 7.300
## Mean : 4.904 Mean :27.19 Mean : 8.059
## 3rd Qu.: 7.625 3rd Qu.:28.16 3rd Qu.: 9.567
## Max. :16.029 Max. :30.80 Max. :15.800
## NA's :10 NA's :43 NA's :43
## station_max_temp_c station_min_temp_c station_precip_mm
## Min. :26.70 Min. :14.7 Min. : 0.00
## 1st Qu.:31.10 1st Qu.:21.1 1st Qu.: 8.70
## Median :32.80 Median :22.2 Median : 23.85
## Mean :32.45 Mean :22.1 Mean : 39.33
## 3rd Qu.:33.90 3rd Qu.:23.3 3rd Qu.: 53.90
## Max. :42.20 Max. :25.6 Max. :543.30
## NA's :20 NA's :14 NA's :22Check and remove NA’s in the data:
anyNA(train)## [1] TRUEtrain = na.omit(train)As with the previous project, we will re-format the date variables to make them easier to work with:
train$week_start_date <- as.Date(train$week_start_date, "%m/%d/%Y")
train$Month <- format(train$week_start_date, "%m")
train$Month <- as.numeric(train$Month)Since we are predicting the cases per week in each location (San Juan and Iquitos), let us compare them side-by-side:
ggplot(data = train, aes(x=weekofyear, y=total_cases, fill = city)) +
geom_bar(stat = "identity",) +
facet_wrap(.~city) +
labs(x="Week of the Year", y="Total Cases", title = "Cases of Dengue per Week") +
scale_fill_discrete(name = "City", labels = c("Iquitos (iq)", "San Juan (sj)")) +
theme(legend.position="bottom") 
We can see from the data that the cities vary in cases and in yearly patterns considerably. In Iquitos, cases tend to be higher in the beginning and end of the year. In San Juan, cases spike around week 30, when that is the calmest part of the year in Iquitos.
Split up the data by city:
sj.train = train[1:936,]
iq.train = train[937:1456,]head(sj.train)## city year weekofyear total_cases week_start_date ndvi_ne ndvi_nw
## 1 sj 1990 18 4 <NA> 0.1226000 0.1037250
## 2 sj 1990 19 5 <NA> 0.1699000 0.1421750
## 3 sj 1990 20 4 <NA> 0.0322500 0.1729667
## 4 sj 1990 21 3 <NA> 0.1286333 0.2450667
## 5 sj 1990 22 6 <NA> 0.1962000 0.2622000
## 7 sj 1990 24 4 <NA> 0.1129000 0.0928000
## ndvi_se ndvi_sw precipitation_amt_mm reanalysis_air_temp_k
## 1 0.1984833 0.1776167 12.42 297.5729
## 2 0.1623571 0.1554857 22.82 298.2114
## 3 0.1572000 0.1708429 34.54 298.7814
## 4 0.2275571 0.2358857 15.36 298.9871
## 5 0.2512000 0.2473400 7.52 299.5186
## 7 0.2050714 0.2102714 3.48 299.2071
## reanalysis_avg_temp_k reanalysis_dew_point_temp_k
## 1 297.7429 292.4143
## 2 298.4429 293.9514
## 3 298.8786 295.4343
## 4 299.2286 295.3100
## 5 299.6643 295.8214
## 7 299.2214 295.8657
## reanalysis_max_air_temp_k reanalysis_min_air_temp_k
## 1 299.8 295.9
## 2 300.9 296.4
## 3 300.5 297.3
## 4 301.4 297.0
## 5 301.9 297.5
## 7 301.3 297.7
## reanalysis_precip_amt_kg_per_m2 reanalysis_relative_humidity_percent
## 1 32.00 73.36571
## 2 17.94 77.36857
## 3 26.10 82.05286
## 4 13.90 80.33714
## 5 12.20 80.46000
## 7 38.60 82.00000
## reanalysis_sat_precip_amt_mm reanalysis_specific_humidity_g_per_kg
## 1 12.42 14.01286
## 2 22.82 15.37286
## 3 34.54 16.84857
## 4 15.36 16.67286
## 5 7.52 17.21000
## 7 3.48 17.23429
## reanalysis_tdtr_k station_avg_temp_c station_diur_temp_rng_c
## 1 2.628571 25.44286 6.900000
## 2 2.371429 26.71429 6.371429
## 3 2.300000 26.71429 6.485714
## 4 2.428571 27.47143 6.771429
## 5 3.014286 28.94286 9.371429
## 7 2.042857 27.41429 6.771429
## station_max_temp_c station_min_temp_c station_precip_mm Month
## 1 29.4 20.0 16.0 NA
## 2 31.7 22.2 8.6 NA
## 3 32.2 22.8 41.4 NA
## 4 33.3 23.3 4.0 NA
## 5 35.0 23.9 5.8 NA
## 7 32.2 23.3 29.7 NAsummary(sj.train)## city year weekofyear total_cases week_start_date
## iq:209 Min. :1990 Min. : 1.00 Min. : 0.00 Min. :NA
## sj:727 1st Qu.:1996 1st Qu.:14.00 1st Qu.: 5.00 1st Qu.:NA
## Median :2001 Median :27.00 Median : 14.00 Median :NA
## Mean :2000 Mean :26.63 Mean : 24.55 Mean :NA
## 3rd Qu.:2003 3rd Qu.:39.00 3rd Qu.: 30.00 3rd Qu.:NA
## Max. :2008 Max. :52.00 Max. :329.00 Max. :NA
## NA's :936
## ndvi_ne ndvi_nw ndvi_se
## Min. :-0.40625 Min. :-0.45610 Min. :-0.01553
## 1st Qu.: 0.02252 1st Qu.: 0.03360 1st Qu.: 0.14666
## Median : 0.08935 Median : 0.09045 Median : 0.18650
## Mean : 0.10488 Mean : 0.10436 Mean : 0.19292
## 3rd Qu.: 0.19296 3rd Qu.: 0.17539 3rd Qu.: 0.23171
## Max. : 0.49340 Max. : 0.43710 Max. : 0.45538
##
## ndvi_sw precipitation_amt_mm reanalysis_air_temp_k
## Min. :-0.06346 Min. : 0.000 Min. :294.6
## 1st Qu.: 0.13496 1st Qu.: 6.763 1st Qu.:297.8
## Median : 0.17810 Median : 33.400 Median :298.9
## Mean : 0.18841 Mean : 41.794 Mean :298.9
## 3rd Qu.: 0.22735 3rd Qu.: 64.112 3rd Qu.:300.0
## Max. : 0.54602 Max. :390.600 Max. :302.2
##
## reanalysis_avg_temp_k reanalysis_dew_point_temp_k
## Min. :294.9 Min. :289.6
## 1st Qu.:298.3 1st Qu.:294.0
## Median :299.4 Median :295.5
## Mean :299.2 Mean :295.1
## 3rd Qu.:300.2 3rd Qu.:296.4
## Max. :302.6 Max. :297.8
##
## reanalysis_max_air_temp_k reanalysis_min_air_temp_k
## Min. :297.8 Min. :287.3
## 1st Qu.:300.8 1st Qu.:294.8
## Median :302.0 Median :296.9
## Mean :302.7 Mean :296.3
## 3rd Qu.:303.3 3rd Qu.:298.1
## Max. :313.2 Max. :299.9
##
## reanalysis_precip_amt_kg_per_m2 reanalysis_relative_humidity_percent
## Min. : 0.00 Min. :57.79
## 1st Qu.: 11.59 1st Qu.:76.73
## Median : 23.93 Median :79.48
## Mean : 35.69 Mean :80.59
## 3rd Qu.: 42.82 3rd Qu.:82.61
## Max. :570.50 Max. :98.46
##
## reanalysis_sat_precip_amt_mm reanalysis_specific_humidity_g_per_kg
## Min. : 0.000 Min. :11.72
## 1st Qu.: 6.763 1st Qu.:15.45
## Median : 33.400 Median :16.87
## Mean : 41.794 Mean :16.62
## 3rd Qu.: 64.112 3rd Qu.:17.89
## Max. :390.600 Max. :19.44
##
## reanalysis_tdtr_k station_avg_temp_c station_diur_temp_rng_c
## Min. : 1.357 Min. :22.84 Min. : 4.529
## 1st Qu.: 2.243 1st Qu.:26.18 1st Qu.: 6.414
## Median : 2.657 Median :27.37 Median : 7.029
## Mean : 4.094 Mean :27.14 Mean : 7.641
## 3rd Qu.: 3.604 3rd Qu.:28.16 3rd Qu.: 8.134
## Max. :16.029 Max. :30.80 Max. :15.800
##
## station_max_temp_c station_min_temp_c station_precip_mm Month
## Min. :26.70 Min. :14.70 Min. : 0.000 Min. : NA
## 1st Qu.:31.10 1st Qu.:21.10 1st Qu.: 8.275 1st Qu.: NA
## Median :32.20 Median :22.20 Median : 21.100 Median : NA
## Mean :32.16 Mean :22.28 Mean : 34.067 Mean :NaN
## 3rd Qu.:33.30 3rd Qu.:23.30 3rd Qu.: 45.525 3rd Qu.: NA
## Max. :42.20 Max. :25.60 Max. :543.300 Max. : NA
## NA's :936head(iq.train)## city year weekofyear total_cases week_start_date ndvi_ne ndvi_nw
## 1163 iq 2004 45 22 <NA> 0.4148857 0.3245571
## 1164 iq 2004 46 37 <NA> 0.3466857 0.3154143
## 1165 iq 2004 47 33 <NA> 0.2278714 0.1945000
## 1166 iq 2004 48 18 <NA> 0.2153167 0.1672857
## 1167 iq 2004 49 83 <NA> 0.2019429 0.1593667
## 1169 iq 2004 51 32 <NA> 0.2611429 0.2033143
## ndvi_se ndvi_sw precipitation_amt_mm reanalysis_air_temp_k
## 1163 0.2727429 0.3656714 91.63 299.5043
## 1164 0.2708571 0.3742000 74.68 297.1686
## 1165 0.1909143 0.2095571 74.08 299.4186
## 1166 0.2026286 0.2191286 71.98 297.9586
## 1167 0.1590429 0.2410857 53.01 299.1771
## 1169 0.2437429 0.2363571 49.56 298.7014
## reanalysis_avg_temp_k reanalysis_dew_point_temp_k
## 1163 300.7857 296.1786
## 1164 297.7071 296.4843
## 1165 301.2357 296.0971
## 1166 299.2000 297.0200
## 1167 300.3786 296.4400
## 1169 300.0357 296.3857
## reanalysis_max_air_temp_k reanalysis_min_air_temp_k
## 1163 309.2 294.3
## 1164 304.2 292.4
## 1165 309.3 293.6
## 1166 306.1 293.6
## 1167 309.5 293.3
## 1169 308.5 293.8
## reanalysis_precip_amt_kg_per_m2 reanalysis_relative_humidity_percent
## 1163 67.00 84.82714
## 1164 53.90 96.35429
## 1165 26.80 84.61143
## 1166 56.80 95.25143
## 1167 36.83 87.13000
## 1169 38.00 88.78000
## reanalysis_sat_precip_amt_mm reanalysis_specific_humidity_g_per_kg
## 1163 91.63 17.80571
## 1164 74.68 18.12143
## 1165 74.08 17.73857
## 1166 71.98 18.71429
## 1167 53.01 18.11000
## 1169 49.56 18.04000
## reanalysis_tdtr_k station_avg_temp_c station_diur_temp_rng_c
## 1163 10.657143 28.350 12.35
## 1164 5.928571 27.200 8.90
## 1165 11.642857 28.250 11.00
## 1166 7.914286 28.000 10.36
## 1167 10.957143 27.900 11.00
## 1169 10.214286 28.375 9.65
## station_max_temp_c station_min_temp_c station_precip_mm Month
## 1163 34.9 21.8 101.4 NA
## 1164 33.7 22.4 184.2 NA
## 1165 35.8 22.5 107.4 NA
## 1166 35.3 22.6 77.0 NA
## 1167 34.1 22.4 64.0 NA
## 1169 34.0 22.6 128.3 NAsummary(iq.train)## city year weekofyear total_cases
## iq :263 Min. :2004 Min. : 1.00 Min. : 0.000
## sj : 0 1st Qu.:2006 1st Qu.:13.00 1st Qu.: 3.000
## NA's:257 Median :2007 Median :25.00 Median : 6.000
## Mean :2007 Mean :26.03 Mean : 9.262
## 3rd Qu.:2009 3rd Qu.:40.00 3rd Qu.:11.000
## Max. :2010 Max. :52.00 Max. :83.000
## NA's :257 NA's :257 NA's :257
## week_start_date ndvi_ne ndvi_nw ndvi_se
## Min. :NA Min. :0.06173 Min. :0.05895 Min. :0.0860
## 1st Qu.:NA 1st Qu.:0.19586 1st Qu.:0.18614 1st Qu.:0.1923
## Median :NA Median :0.26341 Median :0.23351 Median :0.2537
## Mean :NA Mean :0.26369 Mean :0.24113 Mean :0.2505
## 3rd Qu.:NA 3rd Qu.:0.32007 3rd Qu.:0.29122 3rd Qu.:0.2968
## Max. :NA Max. :0.50836 Max. :0.45443 Max. :0.5383
## NA's :520 NA's :257 NA's :257 NA's :257
## ndvi_sw precipitation_amt_mm reanalysis_air_temp_k
## Min. :0.06474 Min. : 0.00 Min. :294.9
## 1st Qu.:0.20166 1st Qu.: 43.45 1st Qu.:297.3
## Median :0.26046 Median : 64.03 Median :297.9
## Mean :0.26729 Mean : 68.17 Mean :298.0
## 3rd Qu.:0.32245 3rd Qu.: 88.19 3rd Qu.:298.7
## Max. :0.54573 Max. :210.83 Max. :300.7
## NA's :257 NA's :257 NA's :257
## reanalysis_avg_temp_k reanalysis_dew_point_temp_k
## Min. :295.4 Min. :290.1
## 1st Qu.:298.4 1st Qu.:295.2
## Median :299.2 Median :296.2
## Mean :299.2 Mean :295.9
## 3rd Qu.:300.2 3rd Qu.:296.8
## Max. :302.2 Max. :298.4
## NA's :257 NA's :257
## reanalysis_max_air_temp_k reanalysis_min_air_temp_k
## Min. :301.9 Min. :286.9
## 1st Qu.:305.2 1st Qu.:292.4
## Median :306.9 Median :293.5
## Mean :306.9 Mean :293.2
## 3rd Qu.:308.4 3rd Qu.:294.4
## Max. :313.2 Max. :296.0
## NA's :257 NA's :257
## reanalysis_precip_amt_kg_per_m2 reanalysis_relative_humidity_percent
## Min. : 0.10 Min. :67.76
## 1st Qu.: 32.90 1st Qu.:86.80
## Median : 53.25 Median :91.80
## Mean : 62.38 Mean :89.91
## 3rd Qu.: 72.00 3rd Qu.:94.75
## Max. :288.40 Max. :98.61
## NA's :257 NA's :257
## reanalysis_sat_precip_amt_mm reanalysis_specific_humidity_g_per_kg
## Min. : 0.00 Min. :12.11
## 1st Qu.: 43.45 1st Qu.:16.72
## Median : 64.03 Median :17.79
## Mean : 68.17 Mean :17.50
## 3rd Qu.: 88.19 3rd Qu.:18.41
## Max. :210.83 Max. :20.46
## NA's :257 NA's :257
## reanalysis_tdtr_k station_avg_temp_c station_diur_temp_rng_c
## Min. : 3.714 Min. :21.40 Min. : 5.20
## 1st Qu.: 7.279 1st Qu.:27.05 1st Qu.: 9.20
## Median : 8.600 Median :27.70 Median :10.42
## Mean : 8.847 Mean :27.56 Mean :10.41
## 3rd Qu.:10.657 3rd Qu.:28.22 3rd Qu.:11.50
## Max. :13.800 Max. :30.03 Max. :14.90
## NA's :257 NA's :257 NA's :257
## station_max_temp_c station_min_temp_c station_precip_mm Month
## Min. :31.00 Min. :16.40 Min. : 0.00 Min. : NA
## 1st Qu.:33.20 1st Qu.:21.00 1st Qu.: 16.40 1st Qu.: NA
## Median :33.90 Median :21.40 Median : 45.00 Median : NA
## Mean :34.02 Mean :21.34 Mean : 65.29 Mean :NaN
## 3rd Qu.:35.00 3rd Qu.:22.10 3rd Qu.: 92.50 3rd Qu.: NA
## Max. :38.60 Max. :24.20 Max. :350.90 Max. : NA
## NA's :257 NA's :257 NA's :257 NA's :520Adding variables for converting Celcius to Fahrenheit. This will come in handy later.
sj.train$avg_temp_f = sj.train$station_avg_temp_c * 2 + 30sj.train$max_temp_f = sj.train$station_max_temp_c * 2 + 30What kind of seasonality is present in the data?
sj.ts = ts(sj.train, start = c(1990, 04, 30), end = c(2004, 10, 28), frequency = 52)
iq.ts = ts(iq.train, start = c(2005, 11, 11), end = c(2010, 6, 25), frequency = 52)autoplot(sj.ts[,4], main="Total Cases, San Juan", xlab = "Date", ylab="Number of Cases") 
There is clear seasonality in the San Juan data. The cases tend to spike in the middle of the year.
autoplot(iq.ts[,4], main="Total Cases, Iquitos", xlab = "Date", ylab="Number of Cases") 
There is some seasonality in the data although it is not uniform. Cases tend to spike in the spring and summer; however, in some years, they spike in the very beginning or stay relatively the same throughout. This is interesting to note because both cities have similar populations.
As mentioned in the introduction, cases of Dengue are related to environmental variables such as temperature, precipitation, and vegetation. Now we will investigate for patterns between cases of the virus and environmental variables.
autoplot(sj.ts[,10], main = "Precipitation vs Dengue Cases in San Juan", series = "Precipitation, Millimeters", xlab = "Date", ylab = "Precipitation & Cases") +
autolayer(sj.ts[,04], series = "Dengue Cases")
autoplot(iq.ts[,10], main = "Precipitation vs Dengue Cases in Iquitos", series = "Precipitation, Millimeters", xlab = "Date", ylab = "Precipitation & Cases") +
autolayer(iq.ts[,04], series = "Dengue Cases")
autoplot(sj.ts[,17], main = "Humidity vs Dengue Cases in San Juan", series = "Humidity, Percent", xlab = "Date", ylab = "Humidity & Cases") +
autolayer(sj.ts[,04], series = "Dengue Cases")
autoplot(iq.ts[,17], main = "Humidity vs Dengue Cases in Iquitos", series = "Humidity, Percent", xlab = "Date", ylab = "Humidity & Cases") +
autolayer(iq.ts[,04], series = "Dengue Cases")
autoplot(sj.ts[,21], main = "Average Temperature vs Dengue Cases in San Juan", series = "Temperature, Celcius", xlab = "Date", ylab = "Temperature & Cases") +
autolayer(sj.ts[,04], series = "Dengue Cases")
Although I typically prefer to use metric in an academic context, the Farenheit scale is much dramatic and thus may be easier to interpret on a graph of this scale.
autoplot(sj.ts[,27], main = "Average Temperature vs Dengue Cases in San Juan", series = "Temperature, Fahrenheit", xlab = "Date", ylab = "Temperature & Cases") +
autolayer(sj.ts[,04], series = "Dengue Cases")
autoplot(sj.ts[,28], main = "Maximum Temperature vs Dengue Cases in San Juan", series = "Temperature, Fahrenheit", xlab = "Date", ylab = "Temperature & Cases") +
autolayer(sj.ts[,04], series = "Dengue Cases")
There does not appear to be any discernable correlation in the data, or the changes in temperature are too small to see in this format.
San Juan Models & Evaluation
ARIMA
sj.arima = auto.arima(sj.train[,4])
forecast(sj.arima)## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## 937 2.900022 -11.28639 17.08644 -18.79623 24.59627
## 938 2.607559 -18.26069 23.47581 -29.30768 34.52279
## 939 4.397989 -21.94048 30.73646 -35.88323 44.67921
## 940 2.634337 -27.53971 32.80838 -43.51288 48.78155
## 941 3.654582 -30.40089 37.71006 -48.42877 55.73793
## 942 3.411803 -33.92202 40.74562 -53.68535 60.50895
## 943 3.182942 -37.16859 43.53447 -58.52940 64.89529
## 944 3.542094 -39.69544 46.77963 -62.58401 69.66820
## 945 3.269543 -42.59224 49.13133 -66.87001 73.40909
## 946 3.387852 -45.00377 51.77948 -70.62075 77.39646autoplot(forecast(sj.arima))
Auto.arima chose Arima(2,1,2). The forecast is quite broad, and obviously, anything below 0 is unreasonable.
checkresiduals(forecast(sj.arima))
##
## Ljung-Box test
##
## data: Residuals from ARIMA(2,1,2)
## Q* = 12.739, df = 6, p-value = 0.04737
##
## Model df: 4. Total lags used: 10summary(sj.arima)## Series: sj.train[, 4]
## ARIMA(2,1,2)
##
## Coefficients:
## ar1 ar2 ma1 ma2
## -1.0630 -0.4773 1.1418 0.6150
## s.e. 0.1499 0.1765 0.1356 0.1599
##
## sigma^2 estimated as 122.5: log likelihood=-3572.69
## AIC=7155.37 AICc=7155.44 BIC=7179.57
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE ACF1
## Training set -0.0006230492 11.04011 6.717139 NaN Inf 1.014952 0.002718952The residuals plot is mostly white noise with a light pattern; the ACF graph only breaks at Lag = 7, and the graph is normally distrubuted.
STL
sj.stl = stl(sj.ts[,4], s.window = "periodic", robust = TRUE)
forecast(sj.stl)## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## 2004.192 1.2762026053 -14.57110 17.12350 -22.96015 25.51256
## 2004.212 -0.2006205288 -22.61096 22.20972 -34.47428 34.07304
## 2004.231 -0.2657960059 -27.71229 27.18070 -42.24159 41.70999
## 2004.250 -0.2336949435 -31.92591 31.45852 -48.70276 48.23537
## 2004.269 -4.7129591314 -40.14576 30.71984 -58.90274 49.47683
## 2004.288 -1.2173991707 -40.03195 37.59715 -60.57914 58.14434
## 2004.308 -2.4110659839 -44.33548 39.51334 -66.52892 61.70679
## 2004.327 -3.5301549844 -48.34915 41.28884 -72.07490 65.01459
## 2004.346 -0.4338847010 -47.97155 47.10378 -73.13646 72.26869
## 2004.365 -0.5046762011 -50.61371 49.60436 -77.13984 76.13048
## 2004.385 -1.9023880714 -54.45714 50.65237 -82.27795 78.47317
## 2004.404 0.3359784418 -54.55563 55.22759 -83.61349 84.28545
## 2004.423 0.4146497294 -56.71831 57.54761 -86.96267 87.79197
## 2004.442 -1.6004500047 -60.89009 57.68919 -92.27613 89.07523
## 2004.462 0.3275070610 -61.04307 61.69809 -93.53070 94.18571
## 2004.481 0.6948279598 -62.68841 64.07807 -96.24147 97.63113
## 2004.500 -3.7185097955 -69.05243 61.61541 -103.63812 96.20110
## 2004.519 -2.7692705972 -69.99730 64.45876 -105.58567 100.04713
## 2004.538 -2.4696046833 -71.53982 66.60061 -108.10338 103.16417
## 2004.558 -4.4303130683 -75.29484 66.43421 -112.80825 103.94763
## 2004.577 -4.2301264714 -76.84464 68.38439 -115.28444 106.82419
## 2004.596 -2.2615543471 -76.58486 72.06175 -115.92924 111.40614
## 2004.615 -0.8065209425 -76.80021 75.18717 -117.02884 115.41579
## 2004.635 -0.5954715195 -78.22360 77.03266 -119.31746 118.12651
## 2004.654 -3.5589718858 -82.78784 75.66990 -124.72907 117.61113
## 2004.673 -3.3599021628 -84.15780 77.43799 -126.92962 120.20982
## 2004.692 -2.2206258420 -84.55766 80.11641 -128.14425 123.70300
## 2004.712 -2.5409825785 -86.38890 81.30693 -130.77531 125.69334
## 2004.731 -1.7810436592 -87.11310 83.55101 -132.28516 128.72307
## 2004.750 -2.1069049537 -88.89772 84.68391 -134.84201 130.62820
## 2004.769 -2.8570543495 -91.08252 85.36841 -137.78626 132.07215
## 2004.788 -0.7470195920 -90.38417 88.89013 -137.83522 136.34118
## 2004.808 -0.8450731830 -91.87202 90.18188 -140.05878 138.36864
## 2004.827 -0.5708222184 -92.96667 91.82502 -141.87807 140.73643
## 2004.846 -0.5133059099 -94.25806 93.23145 -143.88353 142.85692
## 2004.865 -1.4219724762 -96.49650 93.65255 -146.82591 143.98197
## 2004.885 1.6906079998 -94.69534 98.07656 -145.71899 149.10020
## 2004.904 2.1200256679 -95.55975 99.79980 -147.26830 151.50835
## 2004.923 1.3328724098 -97.62381 100.28955 -150.00831 152.67406
## 2004.942 2.1276703373 -98.08965 102.34499 -151.14150 155.39684
## 2004.962 2.2608111715 -99.20149 103.72311 -152.91238 157.43401
## 2004.981 3.0622541604 -99.62993 105.75444 -153.99189 160.11640
## 2005.000 6.3178635804 -97.58965 110.22538 -152.59496 165.23069
## 2005.019 7.2386172808 -97.87017 112.34741 -153.51140 167.98864
## 2005.038 5.6169647553 -100.67953 111.91346 -156.94949 168.18342
## 2005.058 6.8765650374 -100.59450 114.34763 -157.48625 171.23938
## 2005.077 10.3305155222 -98.30243 118.96346 -155.80924 176.47027
## 2005.096 6.8939789518 -102.88855 116.67651 -161.00391 174.79187
## 2005.115 5.2671491434 -105.65305 116.18735 -164.37065 174.90495
## 2005.135 3.1325162610 -108.91380 115.17883 -168.22754 174.49257
## 2005.154 3.2472015085 -109.91403 116.40843 -169.81796 176.31236
## 2005.173 -0.0003260588 -114.26559 114.26493 -174.75396 174.75331
## 2005.192 1.2762026053 -114.08253 116.63493 -175.14975 177.70215
## 2005.212 -0.2006205288 -116.64255 116.24131 -178.28318 177.88194
## 2005.231 -0.2657960059 -117.78094 117.24935 -179.98970 179.45811
## 2005.250 -0.2336949435 -118.81234 118.34495 -181.58409 181.11670
## 2005.269 -4.7129591314 -124.34566 114.91974 -187.67538 178.24946
## 2005.288 -1.2173991707 -121.89494 119.46014 -185.77777 183.34297
## 2005.308 -2.4110659839 -124.12448 119.30235 -188.55567 183.73354
## 2005.327 -3.5301549844 -126.27070 119.21039 -191.24562 184.18531
## 2005.346 -0.4338847010 -124.19304 123.32527 -189.70718 188.83941
## 2005.365 -0.5046762011 -125.27412 124.26477 -191.32308 190.31373
## 2005.385 -1.9023880714 -127.67401 123.86924 -194.25349 190.44871
## 2005.404 0.3359784418 -126.42990 127.10186 -193.53570 194.20766
## 2005.423 0.4146497294 -127.33775 128.16705 -194.96578 195.79508
## 2005.442 -1.6004500047 -130.33180 127.13090 -198.47806 195.27716
## 2005.462 0.3275070610 -129.37541 130.03043 -198.03599 198.69101
## 2005.481 0.6948279598 -129.97244 131.36209 -199.14351 200.53317
## 2005.500 -3.7185097955 -135.34305 127.90604 -205.02088 197.58386
## 2005.519 -2.7692705972 -135.34418 129.80564 -205.52510 199.98656
## 2005.538 -2.4696046833 -135.98812 131.04891 -206.66855 201.72934
## 2005.558 -4.4303130683 -138.88581 130.02518 -210.06225 201.20162
## 2005.577 -4.2301264714 -139.61612 131.15587 -211.28513 202.82488
## 2005.596 -2.2615543471 -138.57169 134.04858 -210.72992 206.20681
## 2005.615 -0.8065209425 -138.03458 136.42154 -210.67873 209.06568
## 2005.635 -0.5954715195 -138.73535 137.54441 -211.86219 210.67124
## 2005.654 -3.5589718858 -142.60469 135.48675 -216.21105 209.09311
## 2005.673 -3.3599021628 -143.30560 136.58580 -217.38838 210.66858
## 2005.692 -2.2206258420 -143.06056 138.61930 -217.61671 213.17546
## 2005.712 -2.5409825785 -144.26950 139.18753 -219.29604 214.21408
## 2005.731 -1.7810436592 -144.39261 140.83052 -219.88661 216.32453
## 2005.750 -2.1069049537 -145.59609 141.38228 -221.55467 217.34086
## 2005.769 -2.8570543495 -147.21852 141.50441 -223.63886 217.92475
## 2005.788 -0.7470195920 -145.97552 144.48149 -222.85485 221.36081
## 2005.808 -0.8450731830 -146.93547 145.24533 -224.27106 222.58091
## 2005.827 -0.5708222184 -147.51806 146.37642 -225.30723 224.16559
## 2005.846 -0.5133059099 -148.31242 147.28581 -226.55255 225.52593
## 2005.865 -1.4219724762 -150.06808 147.22413 -228.75657 225.91263
## 2005.885 1.6906079998 -147.79769 151.17891 -226.93202 230.31323
## 2005.904 2.1200256679 -148.20575 152.44580 -227.78340 232.02346
## 2005.923 1.3328724098 -149.82573 152.49148 -229.84427 232.51001
## 2005.942 2.1276703373 -149.85921 154.11455 -230.31620 234.57154
## 2005.962 2.2608111715 -150.54985 155.07147 -231.44292 235.96455
## 2005.981 3.0622541604 -150.56777 156.69228 -231.89459 238.01910
## 2006.000 6.3178635804 -148.12718 160.76290 -229.88544 242.52117
## 2006.019 7.2386172808 -148.01716 162.49440 -230.20461 244.68184
## 2006.038 5.6169647553 -150.44534 161.67927 -233.05974 244.29367
## 2006.058 6.8765650374 -149.98812 163.74125 -233.02727 246.78040
## 2006.077 10.3305155222 -147.33247 167.99350 -230.79421 251.45524
## 2006.096 6.8939789518 -151.56328 165.35124 -235.44549 249.23344
## 2006.115 5.2671491434 -153.98042 164.51472 -238.28100 248.81530
## 2006.135 3.1325162610 -156.90147 163.16650 -241.61834 247.88337
## 2006.154 3.2472015085 -157.56935 164.06375 -242.70049 249.19489
## 2006.173 -0.0003260588 -161.59565 161.59500 -247.13905 247.13840autoplot(forecast(sj.stl))
autoplot(sj.stl)
The STL forecast seems to pick up on the off-season dengue pattern well enough, but does not predict any seasonal spikes. It also predicts negative Dengue cases which does not make sense for the data.
checkresiduals(forecast(sj.stl))## Warning in checkresiduals(forecast(sj.stl)): The fitted degrees of freedom
## is based on the model used for the seasonally adjusted data.
##
## Ljung-Box test
##
## data: Residuals from STL + ETS(A,N,N)
## Q* = 109.92, df = 102, p-value = 0.2786
##
## Model df: 2. Total lags used: 104summary(sj.stl)## Call:
## stl(x = sj.ts[, 4], s.window = "periodic", robust = TRUE)
##
## Time.series components:
## seasonal trend remainder
## Min. :-4.900388 Min. :-1.09067 Min. :-25.82910
## 1st Qu.:-2.408055 1st Qu.:13.47346 1st Qu.: -6.54055
## Median :-0.692105 Median :17.52147 Median : -0.07309
## Mean : 0.046851 Mean :19.66993 Mean : 10.16894
## 3rd Qu.: 1.932597 3rd Qu.:25.30298 3rd Qu.: 12.79968
## Max. :10.143086 Max. :45.66268 Max. :289.49539
## IQR:
## STL.seasonal STL.trend STL.remainder data
## 4.341 11.830 19.340 26.500
## % 16.4 44.6 73.0 100.0
##
## Weights:
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.0000 0.7868 0.9452 0.7982 0.9876 1.0000
##
## Other components: List of 5
## $ win : Named num [1:3] 7351 79 53
## $ deg : Named int [1:3] 0 1 1
## $ jump : Named num [1:3] 736 8 6
## $ inner: int 1
## $ outer: int 15There is a light pattern in the residual graph, and the ACF breaks at a few more lags than the previous model. However the residuals are still normally distrubuted with a slight left skew.
VAR
Some factors that may affect mosquito population include precipitation, temperature, dew point, and humidity.
Thus the linear model:
Total cases = Average temperature(k) + Dew point(k) + Precipitation + Humidity(%)
sj.tslm = tslm(sj.ts[,4]~sj.ts[,10]+sj.ts[,12]+sj.ts[,13]+sj.ts[,17])
summary(sj.tslm)##
## Call:
## tslm(formula = sj.ts[, 4] ~ sj.ts[, 10] + sj.ts[, 12] + sj.ts[,
## 13] + sj.ts[, 17])
##
## Residuals:
## Min 1Q Median 3Q Max
## -37.930 -20.534 -7.808 7.785 288.368
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -1.474e+03 3.471e+02 -4.246 2.46e-05 ***
## sj.ts[, 10] -5.930e-03 3.308e-02 -0.179 0.85778
## sj.ts[, 12] -1.549e+01 6.673e+00 -2.321 0.02054 *
## sj.ts[, 13] 2.163e+01 6.562e+00 3.296 0.00103 **
## sj.ts[, 17] -3.089e+00 1.355e+00 -2.281 0.02286 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 34.64 on 730 degrees of freedom
## Multiple R-squared: 0.08285, Adjusted R-squared: 0.07783
## F-statistic: 16.49 on 4 and 730 DF, p-value: 6.091e-13The linear model shows us that the Dew point, Precipitation, and Humidity are all significant variables for predicting the number of cases.
sj.tsint = ts.intersect(sj.ts[,12], sj.ts[,13], sj.ts[,17])
VARselect(sj.tsint)## $selection
## AIC(n) HQ(n) SC(n) FPE(n)
## 10 4 1 10
##
## $criteria
## 1 2 3 4 5
## AIC(n) -3.02200595 -3.07309439 -3.11107032 -3.14427779 -3.14793248
## HQ(n) -2.99271032 -3.02182703 -3.03783123 -3.04906699 -3.03074995
## SC(n) -2.94609690 -2.94025355 -2.92129770 -2.89757339 -2.84429629
## FPE(n) 0.04870344 0.04627782 0.04455348 0.04309854 0.04294174
## 6 7 8 9 10
## AIC(n) -3.17093374 -3.2030264 -3.1931879 -3.21343239 -3.21581363
## HQ(n) -3.03177948 -3.0419004 -3.0100902 -3.00836296 -2.98877247
## SC(n) -2.81036576 -2.7855267 -2.7187563 -2.68206905 -2.62751851
## FPE(n) 0.04196591 0.0406413 0.0410442 0.04022297 0.04012896The following models will be named according to the number of lags used.
VARselect suggests a wide range of lags. Therefore multiple will be tested, starting with 10 because it has the most support.
sj.var10 = VAR(sj.tsint, p = 10)
summary(sj.var10)##
## VAR Estimation Results:
## =========================
## Endogenous variables: sj.ts...12., sj.ts...13., sj.ts...17.
## Deterministic variables: const
## Sample size: 725
## Log Likelihood: -1827.459
## Roots of the characteristic polynomial:
## 0.9788 0.9788 0.926 0.926 0.9245 0.8608 0.8608 0.813 0.813 0.8099 0.8099 0.8093 0.8093 0.8046 0.8046 0.7958 0.7958 0.7926 0.7926 0.7708 0.7708 0.7574 0.7574 0.7559 0.7559 0.7316 0.7316 0.7243 0.7243 0.4138
## Call:
## VAR(y = sj.tsint, p = 10)
##
##
## Estimation results for equation sj.ts...12.:
## ============================================
## sj.ts...12. = sj.ts...12..l1 + sj.ts...13..l1 + sj.ts...17..l1 + sj.ts...12..l2 + sj.ts...13..l2 + sj.ts...17..l2 + sj.ts...12..l3 + sj.ts...13..l3 + sj.ts...17..l3 + sj.ts...12..l4 + sj.ts...13..l4 + sj.ts...17..l4 + sj.ts...12..l5 + sj.ts...13..l5 + sj.ts...17..l5 + sj.ts...12..l6 + sj.ts...13..l6 + sj.ts...17..l6 + sj.ts...12..l7 + sj.ts...13..l7 + sj.ts...17..l7 + sj.ts...12..l8 + sj.ts...13..l8 + sj.ts...17..l8 + sj.ts...12..l9 + sj.ts...13..l9 + sj.ts...17..l9 + sj.ts...12..l10 + sj.ts...13..l10 + sj.ts...17..l10 + const
##
## Estimate Std. Error t value Pr(>|t|)
## sj.ts...12..l1 0.455696 0.156746 2.907 0.00376 **
## sj.ts...13..l1 0.171377 0.164831 1.040 0.29884
## sj.ts...17..l1 -0.036273 0.034227 -1.060 0.28962
## sj.ts...12..l2 -0.333581 0.181847 -1.834 0.06702 .
## sj.ts...13..l2 0.496525 0.193441 2.567 0.01047 *
## sj.ts...17..l2 -0.104186 0.040390 -2.579 0.01010 *
## sj.ts...12..l3 0.134258 0.182732 0.735 0.46276
## sj.ts...13..l3 -0.042968 0.194740 -0.221 0.82544
## sj.ts...17..l3 0.002479 0.040599 0.061 0.95133
## sj.ts...12..l4 0.504420 0.185646 2.717 0.00675 **
## sj.ts...13..l4 -0.453594 0.197819 -2.293 0.02215 *
## sj.ts...17..l4 0.107496 0.041285 2.604 0.00942 **
## sj.ts...12..l5 0.180015 0.186656 0.964 0.33517
## sj.ts...13..l5 -0.110613 0.198659 -0.557 0.57785
## sj.ts...17..l5 0.034489 0.041521 0.831 0.40647
## sj.ts...12..l6 0.036953 0.187549 0.197 0.84386
## sj.ts...13..l6 0.016123 0.199779 0.081 0.93570
## sj.ts...17..l6 -0.005526 0.041746 -0.132 0.89473
## sj.ts...12..l7 0.145489 0.197729 0.736 0.46210
## sj.ts...13..l7 -0.255344 0.210249 -1.214 0.22498
## sj.ts...17..l7 0.037539 0.044132 0.851 0.39528
## sj.ts...12..l8 -0.328971 0.243696 -1.350 0.17748
## sj.ts...13..l8 0.373316 0.258828 1.442 0.14966
## sj.ts...17..l8 -0.077416 0.055070 -1.406 0.16024
## sj.ts...12..l9 0.514957 0.244438 2.107 0.03550 *
## sj.ts...13..l9 -0.606062 0.260154 -2.330 0.02011 *
## sj.ts...17..l9 0.127927 0.055339 2.312 0.02109 *
## sj.ts...12..l10 0.065753 0.240674 0.273 0.78478
## sj.ts...13..l10 -0.167580 0.250283 -0.670 0.50336
## sj.ts...17..l10 0.024457 0.053757 0.455 0.64928
## const 49.876953 10.516259 4.743 2.56e-06 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
##
## Residual standard error: 0.5362 on 694 degrees of freedom
## Multiple R-Squared: 0.8178, Adjusted R-squared: 0.8099
## F-statistic: 103.8 on 30 and 694 DF, p-value: < 2.2e-16
##
##
## Estimation results for equation sj.ts...13.:
## ============================================
## sj.ts...13. = sj.ts...12..l1 + sj.ts...13..l1 + sj.ts...17..l1 + sj.ts...12..l2 + sj.ts...13..l2 + sj.ts...17..l2 + sj.ts...12..l3 + sj.ts...13..l3 + sj.ts...17..l3 + sj.ts...12..l4 + sj.ts...13..l4 + sj.ts...17..l4 + sj.ts...12..l5 + sj.ts...13..l5 + sj.ts...17..l5 + sj.ts...12..l6 + sj.ts...13..l6 + sj.ts...17..l6 + sj.ts...12..l7 + sj.ts...13..l7 + sj.ts...17..l7 + sj.ts...12..l8 + sj.ts...13..l8 + sj.ts...17..l8 + sj.ts...12..l9 + sj.ts...13..l9 + sj.ts...17..l9 + sj.ts...12..l10 + sj.ts...13..l10 + sj.ts...17..l10 + const
##
## Estimate Std. Error t value Pr(>|t|)
## sj.ts...12..l1 0.37146 0.21587 1.721 0.085738 .
## sj.ts...13..l1 0.31595 0.22700 1.392 0.164422
## sj.ts...17..l1 0.01039 0.04714 0.221 0.825545
## sj.ts...12..l2 -0.15790 0.25044 -0.630 0.528583
## sj.ts...13..l2 0.37650 0.26640 1.413 0.158028
## sj.ts...17..l2 -0.05828 0.05562 -1.048 0.295138
## sj.ts...12..l3 -0.16786 0.25166 -0.667 0.504971
## sj.ts...13..l3 0.31745 0.26819 1.184 0.236944
## sj.ts...17..l3 -0.07598 0.05591 -1.359 0.174587
## sj.ts...12..l4 0.44225 0.25567 1.730 0.084115 .
## sj.ts...13..l4 -0.45009 0.27243 -1.652 0.098963 .
## sj.ts...17..l4 0.12883 0.05686 2.266 0.023768 *
## sj.ts...12..l5 -0.36886 0.25706 -1.435 0.151762
## sj.ts...13..l5 0.50685 0.27359 1.853 0.064367 .
## sj.ts...17..l5 -0.09269 0.05718 -1.621 0.105487
## sj.ts...12..l6 0.12155 0.25829 0.471 0.638084
## sj.ts...13..l6 -0.14291 0.27513 -0.519 0.603629
## sj.ts...17..l6 0.02357 0.05749 0.410 0.681980
## sj.ts...12..l7 0.08636 0.27231 0.317 0.751236
## sj.ts...13..l7 -0.19479 0.28955 -0.673 0.501336
## sj.ts...17..l7 0.02781 0.06078 0.458 0.647407
## sj.ts...12..l8 0.11357 0.33561 0.338 0.735169
## sj.ts...13..l8 -0.10379 0.35645 -0.291 0.771011
## sj.ts...17..l8 0.03432 0.07584 0.453 0.651028
## sj.ts...12..l9 -0.21146 0.33664 -0.628 0.530114
## sj.ts...13..l9 0.15963 0.35828 0.446 0.656057
## sj.ts...17..l9 -0.03491 0.07621 -0.458 0.647012
## sj.ts...12..l10 0.08717 0.33145 0.263 0.792638
## sj.ts...13..l10 -0.28116 0.34469 -0.816 0.414945
## sj.ts...17..l10 0.05984 0.07403 0.808 0.419238
## const 50.03539 14.48282 3.455 0.000584 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
##
## Residual standard error: 0.7384 on 694 degrees of freedom
## Multiple R-Squared: 0.7868, Adjusted R-squared: 0.7776
## F-statistic: 85.36 on 30 and 694 DF, p-value: < 2.2e-16
##
##
## Estimation results for equation sj.ts...17.:
## ============================================
## sj.ts...17. = sj.ts...12..l1 + sj.ts...13..l1 + sj.ts...17..l1 + sj.ts...12..l2 + sj.ts...13..l2 + sj.ts...17..l2 + sj.ts...12..l3 + sj.ts...13..l3 + sj.ts...17..l3 + sj.ts...12..l4 + sj.ts...13..l4 + sj.ts...17..l4 + sj.ts...12..l5 + sj.ts...13..l5 + sj.ts...17..l5 + sj.ts...12..l6 + sj.ts...13..l6 + sj.ts...17..l6 + sj.ts...12..l7 + sj.ts...13..l7 + sj.ts...17..l7 + sj.ts...12..l8 + sj.ts...13..l8 + sj.ts...17..l8 + sj.ts...12..l9 + sj.ts...13..l9 + sj.ts...17..l9 + sj.ts...12..l10 + sj.ts...13..l10 + sj.ts...17..l10 + const
##
## Estimate Std. Error t value Pr(>|t|)
## sj.ts...12..l1 2.30894 0.83614 2.761 0.00591 **
## sj.ts...13..l1 -2.12943 0.87927 -2.422 0.01570 *
## sj.ts...17..l1 0.81320 0.18258 4.454 9.82e-06 ***
## sj.ts...12..l2 1.05060 0.97004 1.083 0.27916
## sj.ts...13..l2 -0.71676 1.03189 -0.695 0.48753
## sj.ts...17..l2 0.25083 0.21546 1.164 0.24475
## sj.ts...12..l3 -0.57766 0.97476 -0.593 0.55363
## sj.ts...13..l3 0.81071 1.03881 0.780 0.43541
## sj.ts...17..l3 -0.18146 0.21657 -0.838 0.40238
## sj.ts...12..l4 0.06258 0.99030 0.063 0.94963
## sj.ts...13..l4 -0.36834 1.05524 -0.349 0.72715
## sj.ts...17..l4 0.18080 0.22023 0.821 0.41194
## sj.ts...12..l5 -2.37393 0.99569 -2.384 0.01738 *
## sj.ts...13..l5 2.74776 1.05972 2.593 0.00972 **
## sj.ts...17..l5 -0.55347 0.22149 -2.499 0.01269 *
## sj.ts...12..l6 0.85025 1.00046 0.850 0.39570
## sj.ts...13..l6 -1.18724 1.06569 -1.114 0.26564
## sj.ts...17..l6 0.22176 0.22269 0.996 0.31969
## sj.ts...12..l7 -0.24353 1.05476 -0.231 0.81747
## sj.ts...13..l7 0.25205 1.12155 0.225 0.82225
## sj.ts...17..l7 -0.03443 0.23542 -0.146 0.88375
## sj.ts...12..l8 2.01637 1.29996 1.551 0.12133
## sj.ts...13..l8 -2.15170 1.38068 -1.558 0.11959
## sj.ts...17..l8 0.50692 0.29377 1.726 0.08487 .
## sj.ts...12..l9 -3.75014 1.30392 -2.876 0.00415 **
## sj.ts...13..l9 3.92979 1.38776 2.832 0.00476 **
## sj.ts...17..l9 -0.84066 0.29520 -2.848 0.00453 **
## sj.ts...12..l10 -0.28368 1.28384 -0.221 0.82519
## sj.ts...13..l10 -0.19986 1.33510 -0.150 0.88105
## sj.ts...17..l10 0.08310 0.28676 0.290 0.77206
## const 33.61516 56.09758 0.599 0.54922
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
##
## Residual standard error: 2.86 on 694 degrees of freedom
## Multiple R-Squared: 0.3671, Adjusted R-squared: 0.3398
## F-statistic: 13.42 on 30 and 694 DF, p-value: < 2.2e-16
##
##
##
## Covariance matrix of residuals:
## sj.ts...12. sj.ts...13. sj.ts...17.
## sj.ts...12. 0.2875 0.2370 -0.1391
## sj.ts...13. 0.2370 0.5452 1.5330
## sj.ts...17. -0.1391 1.5330 8.1803
##
## Correlation matrix of residuals:
## sj.ts...12. sj.ts...13. sj.ts...17.
## sj.ts...12. 1.00000 0.5987 -0.09073
## sj.ts...13. 0.59867 1.0000 0.72588
## sj.ts...17. -0.09073 0.7259 1.00000autoplot(forecast(sj.var10))
sj.var4 = VAR(sj.tsint, p = 4)
summary(sj.var4)##
## VAR Estimation Results:
## =========================
## Endogenous variables: sj.ts...12., sj.ts...13., sj.ts...17.
## Deterministic variables: const
## Sample size: 731
## Log Likelihood: -1917.909
## Roots of the characteristic polynomial:
## 0.9554 0.9554 0.7746 0.547 0.547 0.5168 0.5168 0.4555 0.4555 0.2992 0.2992 0.2783
## Call:
## VAR(y = sj.tsint, p = 4)
##
##
## Estimation results for equation sj.ts...12.:
## ============================================
## sj.ts...12. = sj.ts...12..l1 + sj.ts...13..l1 + sj.ts...17..l1 + sj.ts...12..l2 + sj.ts...13..l2 + sj.ts...17..l2 + sj.ts...12..l3 + sj.ts...13..l3 + sj.ts...17..l3 + sj.ts...12..l4 + sj.ts...13..l4 + sj.ts...17..l4 + const
##
## Estimate Std. Error t value Pr(>|t|)
## sj.ts...12..l1 0.48874 0.16167 3.023 0.002591 **
## sj.ts...13..l1 0.24755 0.16973 1.459 0.145126
## sj.ts...17..l1 -0.04208 0.03538 -1.189 0.234676
## sj.ts...12..l2 -0.29989 0.18770 -1.598 0.110547
## sj.ts...13..l2 0.46985 0.19943 2.356 0.018744 *
## sj.ts...17..l2 -0.09632 0.04167 -2.311 0.021104 *
## sj.ts...12..l3 0.20800 0.18853 1.103 0.270279
## sj.ts...13..l3 -0.15199 0.20069 -0.757 0.449082
## sj.ts...17..l3 0.02553 0.04190 0.609 0.542550
## sj.ts...12..l4 0.68232 0.17830 3.827 0.000141 ***
## sj.ts...13..l4 -0.75952 0.18561 -4.092 4.76e-05 ***
## sj.ts...17..l4 0.16731 0.03888 4.303 1.92e-05 ***
## const 29.31439 6.90339 4.246 2.46e-05 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
##
## Residual standard error: 0.563 on 718 degrees of freedom
## Multiple R-Squared: 0.7923, Adjusted R-squared: 0.7888
## F-statistic: 228.2 on 12 and 718 DF, p-value: < 2.2e-16
##
##
## Estimation results for equation sj.ts...13.:
## ============================================
## sj.ts...13. = sj.ts...12..l1 + sj.ts...13..l1 + sj.ts...17..l1 + sj.ts...12..l2 + sj.ts...13..l2 + sj.ts...17..l2 + sj.ts...12..l3 + sj.ts...13..l3 + sj.ts...17..l3 + sj.ts...12..l4 + sj.ts...13..l4 + sj.ts...17..l4 + const
##
## Estimate Std. Error t value Pr(>|t|)
## sj.ts...12..l1 0.30784 0.22150 1.390 0.16503
## sj.ts...13..l1 0.51593 0.23255 2.219 0.02682 *
## sj.ts...17..l1 -0.01591 0.04847 -0.328 0.74289
## sj.ts...12..l2 -0.21502 0.25717 -0.836 0.40338
## sj.ts...13..l2 0.43967 0.27325 1.609 0.10805
## sj.ts...17..l2 -0.06428 0.05710 -1.126 0.26063
## sj.ts...12..l3 -0.09691 0.25831 -0.375 0.70765
## sj.ts...13..l3 0.21061 0.27497 0.766 0.44395
## sj.ts...17..l3 -0.05154 0.05741 -0.898 0.36963
## sj.ts...12..l4 0.42154 0.24429 1.726 0.08486 .
## sj.ts...13..l4 -0.62451 0.25431 -2.456 0.01429 *
## sj.ts...17..l4 0.15938 0.05327 2.992 0.00287 **
## const 8.14222 9.45847 0.861 0.38961
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
##
## Residual standard error: 0.7714 on 718 degrees of freedom
## Multiple R-Squared: 0.7602, Adjusted R-squared: 0.7562
## F-statistic: 189.7 on 12 and 718 DF, p-value: < 2.2e-16
##
##
## Estimation results for equation sj.ts...17.:
## ============================================
## sj.ts...17. = sj.ts...12..l1 + sj.ts...13..l1 + sj.ts...17..l1 + sj.ts...12..l2 + sj.ts...13..l2 + sj.ts...17..l2 + sj.ts...12..l3 + sj.ts...13..l3 + sj.ts...17..l3 + sj.ts...12..l4 + sj.ts...13..l4 + sj.ts...17..l4 + const
##
## Estimate Std. Error t value Pr(>|t|)
## sj.ts...12..l1 1.96372 0.82840 2.370 0.0180 *
## sj.ts...13..l1 -1.63197 0.86969 -1.876 0.0610 .
## sj.ts...17..l1 0.73402 0.18129 4.049 5.71e-05 ***
## sj.ts...12..l2 0.69695 0.96179 0.725 0.4689
## sj.ts...13..l2 -0.34996 1.02192 -0.342 0.7321
## sj.ts...17..l2 0.19524 0.21354 0.914 0.3609
## sj.ts...12..l3 -0.51119 0.96604 -0.529 0.5969
## sj.ts...13..l3 0.74261 1.02835 0.722 0.4704
## sj.ts...17..l3 -0.15594 0.21469 -0.726 0.4679
## sj.ts...12..l4 -0.69098 0.91363 -0.756 0.4497
## sj.ts...13..l4 0.06979 0.95107 0.073 0.9415
## sj.ts...17..l4 0.08562 0.19924 0.430 0.6675
## const -80.25238 35.37362 -2.269 0.0236 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
##
## Residual standard error: 2.885 on 718 degrees of freedom
## Multiple R-Squared: 0.3392, Adjusted R-squared: 0.3282
## F-statistic: 30.72 on 12 and 718 DF, p-value: < 2.2e-16
##
##
##
## Covariance matrix of residuals:
## sj.ts...12. sj.ts...13. sj.ts...17.
## sj.ts...12. 0.3170 0.273 -0.1045
## sj.ts...13. 0.2730 0.595 1.6001
## sj.ts...17. -0.1045 1.600 8.3226
##
## Correlation matrix of residuals:
## sj.ts...12. sj.ts...13. sj.ts...17.
## sj.ts...12. 1.00000 0.6286 -0.06432
## sj.ts...13. 0.62855 1.0000 0.71903
## sj.ts...17. -0.06432 0.7190 1.00000autoplot(forecast(sj.var4))
sj.var1 = VAR(sj.tsint, p = 1)
summary(sj.var1)##
## VAR Estimation Results:
## =========================
## Endogenous variables: sj.ts...12., sj.ts...13., sj.ts...17.
## Deterministic variables: const
## Sample size: 734
## Log Likelihood: -1996.246
## Roots of the characteristic polynomial:
## 0.9075 0.7662 0.4411
## Call:
## VAR(y = sj.tsint, p = 1)
##
##
## Estimation results for equation sj.ts...12.:
## ============================================
## sj.ts...12. = sj.ts...12..l1 + sj.ts...13..l1 + sj.ts...17..l1 + const
##
## Estimate Std. Error t value Pr(>|t|)
## sj.ts...12..l1 0.60110 0.11881 5.059 5.32e-07 ***
## sj.ts...13..l1 0.26898 0.11691 2.301 0.0217 *
## sj.ts...17..l1 -0.04096 0.02418 -1.694 0.0906 .
## const 43.22769 5.81719 7.431 3.02e-13 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
##
## Residual standard error: 0.5766 on 730 degrees of freedom
## Multiple R-Squared: 0.7787, Adjusted R-squared: 0.7778
## F-statistic: 856.1 on 3 and 730 DF, p-value: < 2.2e-16
##
##
## Estimation results for equation sj.ts...13.:
## ============================================
## sj.ts...13. = sj.ts...12..l1 + sj.ts...13..l1 + sj.ts...17..l1 + const
##
## Estimate Std. Error t value Pr(>|t|)
## sj.ts...12..l1 0.32133 0.16327 1.968 0.0494 *
## sj.ts...13..l1 0.65760 0.16067 4.093 4.74e-05 ***
## sj.ts...17..l1 -0.02529 0.03323 -0.761 0.4468
## const 6.87335 7.99440 0.860 0.3902
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
##
## Residual standard error: 0.7924 on 730 degrees of freedom
## Multiple R-Squared: 0.7429, Adjusted R-squared: 0.7419
## F-statistic: 703.2 on 3 and 730 DF, p-value: < 2.2e-16
##
##
## Estimation results for equation sj.ts...17.:
## ============================================
## sj.ts...17. = sj.ts...12..l1 + sj.ts...13..l1 + sj.ts...17..l1 + const
##
## Estimate Std. Error t value Pr(>|t|)
## sj.ts...12..l1 2.2882 0.6047 3.784 0.000167 ***
## sj.ts...13..l1 -1.8403 0.5951 -3.093 0.002059 **
## sj.ts...17..l1 0.8562 0.1231 6.958 7.70e-12 ***
## const -130.3805 29.6075 -4.404 1.22e-05 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
##
## Residual standard error: 2.935 on 730 degrees of freedom
## Multiple R-Squared: 0.306, Adjusted R-squared: 0.3032
## F-statistic: 107.3 on 3 and 730 DF, p-value: < 2.2e-16
##
##
##
## Covariance matrix of residuals:
## sj.ts...12. sj.ts...13. sj.ts...17.
## sj.ts...12. 0.33246 0.2906 -0.07931
## sj.ts...13. 0.29060 0.6279 1.68328
## sj.ts...17. -0.07931 1.6833 8.61235
##
## Correlation matrix of residuals:
## sj.ts...12. sj.ts...13. sj.ts...17.
## sj.ts...12. 1.00000 0.6360 -0.04687
## sj.ts...13. 0.63604 1.0000 0.72385
## sj.ts...17. -0.04687 0.7239 1.00000autoplot(forecast(sj.var1))
Neural Network
sj.n1 = nnetar(sj.ts[,4], size = 1)
sj.n2 = nnetar(sj.ts[,4], size = 2)
sj.n3 = nnetar(sj.ts[,4], size = 3)
sj.n4 = nnetar(sj.ts[,4], size = 4)sj.n1.fcast = forecast(sj.n1)
sj.n2.fcast = forecast(sj.n2)
sj.n3.fcast = forecast(sj.n3)
sj.n4.fcast = forecast(sj.n4)autoplot(sj.ts[,4]) +
autolayer(sj.n1.fcast,series="Size 1") +
autolayer(sj.n2.fcast,series="Size 2") +
autolayer(sj.n3.fcast,series="Size 3") +
autolayer(sj.n4.fcast,series="Size 4") +
ylab("Cases") +
xlab("Time") +
ggtitle("San Juan Neural Net")
The forecasts all share a similar pattern. Neural nets with Sizes 2-4 all capture a downward trend after a significant spike; all Sizes capture a bump in the beginning of the forecast followed by a spike.
checkresiduals(sj.n1.fcast)## Warning in modeldf.default(object): Could not find appropriate degrees of
## freedom for this model.
accuracy(sj.n1.fcast)## ME RMSE MAE MPE MAPE MASE ACF1
## Training set -0.02968979 11.80618 7.66867 -Inf Inf 0.2234228 0.003317925sj.n1.fcast## Point Forecast
## 2004.192 2.863876
## 2004.212 6.016093
## 2004.231 9.296859
## 2004.250 12.027797
## 2004.269 15.063115
## 2004.288 17.882773
## 2004.308 20.282907
## 2004.327 22.328893
## 2004.346 24.040261
## 2004.365 25.549038
## 2004.385 26.625867
## 2004.404 27.409386
## 2004.423 27.701037
## 2004.442 28.027705
## 2004.462 28.009507
## 2004.481 27.776507
## 2004.500 27.362731
## 2004.519 27.068701
## 2004.538 25.965649
## 2004.558 24.735829
## 2004.577 23.118423
## 2004.596 21.305390
## 2004.615 19.557448
## 2004.635 16.486784
## 2004.654 14.223476
## 2004.673 12.719799
## 2004.692 12.820097
## 2004.712 13.002250
## 2004.731 13.956421
## 2004.750 14.970629
## 2004.769 16.605367
## 2004.788 18.074083
## 2004.808 19.626185
## 2004.827 21.135140
## 2004.846 22.788810
## 2004.865 24.200654
## 2004.885 25.448726
## 2004.904 26.681186
## 2004.923 27.796988
## 2004.942 28.746251
## 2004.962 29.534346
## 2004.981 30.238212
## 2005.000 30.882098
## 2005.019 31.346146
## 2005.038 31.824597
## 2005.058 32.233635
## 2005.077 32.596909
## 2005.096 32.888011
## 2005.115 33.145132
## 2005.135 33.358591
## 2005.154 33.539376
## 2005.173 33.685286
## 2005.192 33.744268
## 2005.212 33.706720
## 2005.231 33.570605
## 2005.250 33.353770
## 2005.269 33.049238
## 2005.288 32.667896
## 2005.308 32.224789
## 2005.327 31.739079
## 2005.346 31.224906
## 2005.365 30.695285
## 2005.385 30.167072
## 2005.404 29.653995
## 2005.423 29.172178
## 2005.442 28.724769
## 2005.462 28.321908
## 2005.481 27.968846
## 2005.500 27.670356
## 2005.519 27.421354
## 2005.538 27.236827
## 2005.558 27.116646
## 2005.577 27.066685
## 2005.596 27.085251
## 2005.615 27.168395
## 2005.635 27.338486
## 2005.654 27.574925
## 2005.673 27.855823
## 2005.692 28.136562
## 2005.712 28.410355
## 2005.731 28.651471
## 2005.750 28.853666
## 2005.769 28.995169
## 2005.788 29.079621
## 2005.808 29.103235
## 2005.827 29.070849
## 2005.846 28.980257
## 2005.865 28.842192
## 2005.885 28.664503
## 2005.904 28.454016
## 2005.923 28.217374
## 2005.942 27.963826
## 2005.962 27.701670
## 2005.981 27.436922
## 2006.000 27.174084
## 2006.019 26.919806
## 2006.038 26.676001
## 2006.058 26.445451
## 2006.077 26.229433
## 2006.096 26.029983
## 2006.115 25.847257
## 2006.135 25.681580
## 2006.154 25.532529
## 2006.173 25.399818checkresiduals(sj.n2.fcast)## Warning in modeldf.default(object): Could not find appropriate degrees of
## freedom for this model.
accuracy(sj.n2.fcast)## ME RMSE MAE MPE MAPE MASE ACF1
## Training set -0.06673523 10.76818 6.983735 -Inf Inf 0.2034676 0.003897812sj.n2.fcast## Point Forecast
## 2004.192 2.153358
## 2004.212 4.180864
## 2004.231 6.383274
## 2004.250 7.793618
## 2004.269 9.979473
## 2004.288 11.977475
## 2004.308 13.272749
## 2004.327 14.559217
## 2004.346 15.546916
## 2004.365 16.591889
## 2004.385 17.067913
## 2004.404 17.519546
## 2004.423 17.574159
## 2004.442 17.887912
## 2004.462 17.821878
## 2004.481 17.772338
## 2004.500 17.638068
## 2004.519 17.620342
## 2004.538 17.257477
## 2004.558 17.012082
## 2004.577 16.712228
## 2004.596 16.460517
## 2004.615 16.216269
## 2004.635 15.710000
## 2004.654 15.631709
## 2004.673 15.520677
## 2004.692 15.410622
## 2004.712 15.299120
## 2004.731 15.508691
## 2004.750 15.552825
## 2004.769 16.174794
## 2004.788 16.452044
## 2004.808 16.938425
## 2004.827 17.550969
## 2004.846 18.641195
## 2004.865 19.319786
## 2004.885 20.007910
## 2004.904 21.107823
## 2004.923 22.090781
## 2004.942 22.961569
## 2004.962 23.790662
## 2004.981 24.738030
## 2005.000 25.770888
## 2005.019 26.500019
## 2005.038 27.677659
## 2005.058 28.790379
## 2005.077 29.982846
## 2005.096 31.184303
## 2005.115 32.530023
## 2005.135 33.923765
## 2005.154 35.441749
## 2005.173 37.089371
## 2005.192 38.546059
## 2005.212 39.879287
## 2005.231 41.034702
## 2005.250 42.088584
## 2005.269 42.900599
## 2005.288 43.532393
## 2005.308 44.017909
## 2005.327 44.367835
## 2005.346 44.585893
## 2005.365 44.681933
## 2005.385 44.707821
## 2005.404 44.663986
## 2005.423 44.588161
## 2005.442 44.461009
## 2005.462 44.327506
## 2005.481 44.181663
## 2005.500 44.040828
## 2005.519 43.892096
## 2005.538 43.772482
## 2005.558 43.667061
## 2005.577 43.587263
## 2005.596 43.525760
## 2005.615 43.489616
## 2005.635 43.494821
## 2005.654 43.509113
## 2005.673 43.538980
## 2005.692 43.582541
## 2005.712 43.642084
## 2005.731 43.686898
## 2005.750 43.734479
## 2005.769 43.735534
## 2005.788 43.723039
## 2005.808 43.668407
## 2005.827 43.570038
## 2005.846 43.383568
## 2005.865 43.151496
## 2005.885 42.861048
## 2005.904 42.488141
## 2005.923 42.040780
## 2005.942 41.534928
## 2005.962 40.973025
## 2005.981 40.348743
## 2006.000 39.664821
## 2006.019 38.947964
## 2006.038 38.172085
## 2006.058 37.352654
## 2006.077 36.489182
## 2006.096 35.593125
## 2006.115 34.658815
## 2006.135 33.696495
## 2006.154 32.707153
## 2006.173 31.696422checkresiduals(sj.n3.fcast)## Warning in modeldf.default(object): Could not find appropriate degrees of
## freedom for this model.
accuracy(sj.n3.fcast)## ME RMSE MAE MPE MAPE MASE ACF1
## Training set 0.02416489 9.892929 6.631209 -Inf Inf 0.1931969 -0.003795321sj.n3.fcast## Point Forecast
## 2004.192 1.623465
## 2004.212 2.803540
## 2004.231 4.365614
## 2004.250 5.485757
## 2004.269 7.187762
## 2004.288 8.673815
## 2004.308 9.799751
## 2004.327 11.034191
## 2004.346 12.002567
## 2004.365 13.052858
## 2004.385 13.677144
## 2004.404 14.304278
## 2004.423 14.524470
## 2004.442 15.018683
## 2004.462 15.105125
## 2004.481 15.185897
## 2004.500 15.172882
## 2004.519 15.279234
## 2004.538 14.942068
## 2004.558 14.759483
## 2004.577 14.513230
## 2004.596 14.333202
## 2004.615 14.125699
## 2004.635 15.140561
## 2004.654 15.228469
## 2004.673 14.821148
## 2004.692 14.902273
## 2004.712 14.868542
## 2004.731 15.090183
## 2004.750 15.126058
## 2004.769 15.823993
## 2004.788 16.108605
## 2004.808 16.659374
## 2004.827 17.347842
## 2004.846 18.509569
## 2004.865 19.278525
## 2004.885 20.151803
## 2004.904 21.487248
## 2004.923 22.704645
## 2004.942 23.909871
## 2004.962 25.154054
## 2004.981 26.585024
## 2005.000 28.187926
## 2005.019 29.542641
## 2005.038 31.504023
## 2005.058 33.500871
## 2005.077 35.702509
## 2005.096 38.052973
## 2005.115 40.704455
## 2005.135 43.559569
## 2005.154 46.692424
## 2005.173 50.105232
## 2005.192 53.608400
## 2005.212 57.262511
## 2005.231 60.999981
## 2005.250 64.897592
## 2005.269 68.812610
## 2005.288 72.785999
## 2005.308 76.775796
## 2005.327 80.565833
## 2005.346 84.250104
## 2005.365 87.792640
## 2005.385 91.231910
## 2005.404 94.468546
## 2005.423 97.566826
## 2005.442 100.394143
## 2005.462 103.081510
## 2005.481 107.857213
## 2005.500 112.787922
## 2005.519 117.488335
## 2005.538 121.393289
## 2005.558 124.645113
## 2005.577 127.046898
## 2005.596 128.680907
## 2005.615 129.565079
## 2005.635 129.446170
## 2005.654 128.674221
## 2005.673 127.637648
## 2005.692 126.418829
## 2005.712 125.142145
## 2005.731 123.870395
## 2005.750 122.815920
## 2005.769 121.770842
## 2005.788 120.890939
## 2005.808 120.051789
## 2005.827 119.224442
## 2005.846 118.136073
## 2005.865 116.942194
## 2005.885 115.610754
## 2005.904 114.033010
## 2005.923 112.234945
## 2005.942 110.319261
## 2005.962 106.907323
## 2005.981 102.397580
## 2006.000 97.889699
## 2006.019 93.735276
## 2006.038 89.584637
## 2006.058 85.473511
## 2006.077 81.522031
## 2006.096 77.743423
## 2006.115 74.004491
## 2006.135 70.264644
## 2006.154 66.505317
## 2006.173 62.785162checkresiduals(sj.n4.fcast)## Warning in modeldf.default(object): Could not find appropriate degrees of
## freedom for this model.
accuracy(sj.n4.fcast)## ME RMSE MAE MPE MAPE MASE ACF1
## Training set -0.01262545 9.361571 6.421832 -Inf Inf 0.1870968 -0.01516512sj.n4.fcast## Point Forecast
## 2004.192 1.713198
## 2004.212 3.037005
## 2004.231 4.629786
## 2004.250 5.597591
## 2004.269 7.330493
## 2004.288 8.888608
## 2004.308 9.788867
## 2004.327 10.949356
## 2004.346 11.767011
## 2004.365 12.706200
## 2004.385 13.218102
## 2004.404 13.761281
## 2004.423 13.694947
## 2004.442 14.176138
## 2004.462 14.031144
## 2004.481 13.830394
## 2004.500 13.574832
## 2004.519 13.561185
## 2004.538 12.716193
## 2004.558 12.200309
## 2004.577 11.588408
## 2004.596 11.176546
## 2004.615 10.671486
## 2004.635 11.623065
## 2004.654 11.189595
## 2004.673 10.772242
## 2004.692 10.694182
## 2004.712 10.562266
## 2004.731 10.963337
## 2004.750 11.094119
## 2004.769 11.808791
## 2004.788 12.211616
## 2004.808 12.790973
## 2004.827 13.513925
## 2004.846 14.755184
## 2004.865 15.512001
## 2004.885 16.310806
## 2004.904 17.665346
## 2004.923 18.806399
## 2004.942 19.887469
## 2004.962 20.973768
## 2004.981 22.218895
## 2005.000 23.600133
## 2005.019 24.640751
## 2005.038 26.326756
## 2005.058 27.985626
## 2005.077 29.813577
## 2005.096 31.785748
## 2005.115 34.050230
## 2005.135 36.525331
## 2005.154 39.308346
## 2005.173 42.427615
## 2005.192 45.722235
## 2005.212 49.276962
## 2005.231 53.055190
## 2005.250 57.162822
## 2005.269 61.485566
## 2005.288 66.143923
## 2005.308 71.258004
## 2005.327 76.912263
## 2005.346 83.408739
## 2005.365 91.129063
## 2005.385 100.548372
## 2005.404 111.423268
## 2005.423 123.052597
## 2005.442 133.884284
## 2005.462 141.935510
## 2005.481 145.627199
## 2005.500 145.911122
## 2005.519 143.848342
## 2005.538 140.898765
## 2005.558 137.225313
## 2005.577 133.208249
## 2005.596 129.141494
## 2005.615 125.645605
## 2005.635 122.519354
## 2005.654 120.211622
## 2005.673 118.697484
## 2005.692 118.060528
## 2005.712 118.131082
## 2005.731 118.833339
## 2005.750 120.046445
## 2005.769 121.519650
## 2005.788 123.109148
## 2005.808 124.598891
## 2005.827 125.809272
## 2005.846 126.461648
## 2005.865 126.607760
## 2005.885 126.215699
## 2005.904 125.211302
## 2005.923 123.679960
## 2005.942 121.782032
## 2005.962 119.647934
## 2005.981 117.335759
## 2006.000 114.918307
## 2006.019 112.555360
## 2006.038 110.160840
## 2006.058 107.753394
## 2006.077 105.267907
## 2006.096 102.664576
## 2006.115 99.784272
## 2006.135 96.514269
## 2006.154 92.711368
## 2006.173 88.321871Evaluating all of the Neural Net models as a whole, they paint a more comprehensive picture than the previous models that we have examined. Considering the patterns in the forecasts, we observe a small bump followed by a significant spike with is in-line with pervious patterns that we have seen in the data.
Iquitos Models & Evaluation
ARIMA
iq.arima = auto.arima(iq.ts[,4])
forecast(iq.arima)## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## 2010.115 4.254836 -5.735669 14.24534 -11.02432 19.53399
## 2010.135 4.293675 -6.744911 15.33226 -12.58839 21.17574
## 2010.154 5.431608 -6.705463 17.56868 -13.13044 23.99366
## 2010.173 6.066634 -6.763518 18.89679 -13.55539 25.68866
## 2010.192 6.639917 -6.640693 19.92053 -13.67102 26.95086
## 2010.212 6.488786 -7.089484 20.06706 -14.27739 27.25496
## 2010.231 6.761419 -7.015504 20.53834 -14.30857 27.83141
## 2010.250 7.157613 -6.752719 21.06795 -14.11640 28.43163
## 2010.269 7.599464 -6.400823 21.59975 -13.81213 29.01105
## 2010.288 7.853893 -6.207208 21.91500 -13.65071 29.35849
## 2010.308 7.735849 -6.366439 21.83814 -13.83174 29.30344
## 2010.327 7.751150 -6.379066 21.88137 -13.85915 29.36145
## 2010.346 8.012669 -6.136498 22.16184 -13.62661 29.65195
## 2010.365 7.913764 -6.248270 22.07580 -13.74520 29.57273
## 2010.385 8.057284 -6.113489 22.22806 -13.61504 29.72961
## 2010.404 8.288199 -5.888511 22.46491 -13.39321 29.96961
## 2010.423 8.513911 -5.666833 22.69466 -13.17367 30.20149
## 2010.442 8.796442 -5.387043 22.97993 -12.89533 30.48821
## 2010.462 8.739003 -5.446346 22.92435 -12.95562 30.43362
## 2010.481 8.460781 -5.725834 22.64740 -13.23577 30.15734
## 2010.500 8.554104 -5.633371 22.74158 -13.14377 30.25198
## 2010.519 8.550173 -5.637887 22.73823 -13.14859 30.24894
## 2010.538 8.383847 -5.804611 22.57230 -13.31553 30.08322
## 2010.558 8.691491 -5.497237 22.88022 -13.00830 30.39128
## 2010.577 8.884298 -5.304614 23.07321 -12.81577 30.58437
## 2010.596 8.902554 -5.286483 23.09159 -12.79771 30.60281
## 2010.615 9.260113 -4.929008 23.44923 -12.44028 30.96050
## 2010.635 9.498243 -4.690936 23.68742 -12.20223 31.19872
## 2010.654 10.461064 -3.728154 24.65028 -11.23947 32.16160
## 2010.673 10.353381 -3.835864 24.54263 -11.34720 32.05396
## 2010.692 10.266841 -3.922423 24.45610 -11.43376 31.96745
## 2010.712 9.754020 -4.435255 23.94330 -11.94660 31.45464
## 2010.731 9.148432 -5.040851 23.33772 -12.55220 30.84907
## 2010.750 9.163653 -5.025636 23.35294 -12.53699 30.86430
## 2010.769 8.821410 -5.367884 23.01070 -12.87924 30.52206
## 2010.788 8.573644 -5.615652 22.76294 -13.12701 30.27430
## 2010.808 9.548251 -4.641047 23.73755 -12.15241 31.24891
## 2010.827 9.910978 -4.278321 24.10028 -11.78968 31.61164
## 2010.846 9.651044 -4.538256 23.84034 -12.04962 31.35171
## 2010.865 9.531057 -4.658243 23.72036 -12.16960 31.23172
## 2010.885 9.716234 -4.473067 23.90554 -11.98443 31.41690
## 2010.904 10.839924 -3.349377 25.02922 -10.86074 32.54059
## 2010.923 10.617735 -3.571566 24.80704 -11.08293 32.31840
## 2010.942 10.221644 -3.967657 24.41095 -11.47902 31.92231
## 2010.962 9.982895 -4.206406 24.17220 -11.71777 31.68356
## 2010.981 10.478238 -3.711064 24.66754 -11.22243 32.17890
## 2011.000 9.742020 -4.447281 23.93132 -11.95864 31.44268
## 2011.019 9.212091 -4.977211 23.40139 -12.48857 30.91275
## 2011.038 9.202163 -4.987139 23.39146 -12.49850 30.90283
## 2011.058 9.882128 -4.307173 24.07143 -11.81854 31.58279
## 2011.077 9.050359 -5.138942 23.23966 -12.65030 30.75102
## 2011.096 9.238143 -4.951158 23.42744 -12.46252 30.93881
## 2011.115 9.174743 -5.037563 23.38705 -12.56110 30.91059
## 2011.135 9.049447 -5.167937 23.26683 -12.69417 30.79306
## 2011.154 9.229590 -4.993655 23.45284 -12.52299 30.98217
## 2011.173 9.240333 -4.986895 23.46756 -12.51833 30.99900
## 2011.192 9.271795 -4.958137 23.50173 -12.49101 31.03460
## 2011.212 9.014234 -5.217537 23.24601 -12.75138 30.77985
## 2011.231 8.982427 -5.250594 23.21545 -12.78510 30.74996
## 2011.250 9.034114 -5.199758 23.26798 -12.73471 30.80294
## 2011.269 9.129088 -5.105361 23.36354 -12.64062 30.89880
## 2011.288 9.157462 -5.077380 23.39230 -12.61285 30.92777
## 2011.308 9.031869 -5.203239 23.26698 -12.73885 30.80259
## 2011.327 8.979509 -5.255780 23.21480 -12.79149 30.75051
## 2011.346 9.049466 -5.185946 23.28488 -12.72172 30.82065
## 2011.365 8.964198 -5.271298 23.19969 -12.80711 30.73551
## 2011.385 8.996188 -5.239365 23.23174 -12.77521 30.76759
## 2011.404 9.073733 -5.161859 23.30933 -12.69773 30.84519
## 2011.423 9.153724 -5.081894 23.38934 -12.61778 30.92522
## 2011.442 9.263505 -4.972131 23.49914 -12.50802 31.03503
## 2011.462 9.222043 -5.013605 23.45769 -12.54950 30.99359
## 2011.481 9.082933 -5.152724 23.31859 -12.68863 30.85449
## 2011.500 9.114909 -5.120753 23.35057 -12.65666 30.88648
## 2011.519 9.104517 -5.131149 23.34018 -12.66706 30.87609
## 2011.538 9.021828 -5.213841 23.25750 -12.74975 30.79340
## 2011.558 9.155800 -5.079871 23.39147 -12.61578 30.92738
## 2011.577 9.238606 -4.997066 23.47428 -12.53298 31.01019
## 2011.596 9.242927 -4.992746 23.47860 -12.52866 31.01451
## 2011.615 9.402157 -4.833516 23.63783 -12.36943 31.17374
## 2011.635 9.507682 -4.727992 23.74336 -12.26390 31.27927
## 2011.654 9.943047 -4.292627 24.17872 -11.82854 31.71463
## 2011.673 9.892269 -4.343405 24.12794 -11.87932 31.66385
## 2011.692 9.851422 -4.384252 24.08710 -11.92016 31.62301
## 2011.712 9.617102 -4.618572 23.85278 -12.15448 31.38869
## 2011.731 9.340838 -4.894836 23.57651 -12.43075 31.11242
## 2011.750 9.346907 -4.888767 23.58258 -12.42468 31.11849
## 2011.769 9.190661 -5.045013 23.42634 -12.58092 30.96225
## 2011.788 9.077477 -5.158197 23.31315 -12.69411 30.84906
## 2011.808 9.519951 -4.715723 23.75563 -12.25163 31.29154
## 2011.827 9.684415 -4.551259 23.92009 -12.08717 31.45600
## 2011.846 9.565954 -4.669720 23.80163 -12.20563 31.33754
## 2011.865 9.511155 -4.724519 23.74683 -12.26043 31.28274
## 2011.885 9.595096 -4.640578 23.83077 -12.17649 31.36668
## 2011.904 10.105622 -4.130053 24.34130 -11.66596 31.87721
## 2011.923 10.004489 -4.231185 24.24016 -11.76710 31.77607
## 2011.942 9.824347 -4.411327 24.06002 -11.94724 31.59593
## 2011.962 9.715736 -4.519938 23.95141 -12.05585 31.48732
## 2011.981 9.940781 -4.294893 24.17646 -11.83080 31.71237
## 2012.000 9.606108 -4.629566 23.84178 -12.16548 31.37769
## 2012.019 9.365203 -4.870471 23.60088 -12.40638 31.13679
## 2012.038 9.360644 -4.875030 23.59632 -12.41094 31.13223
## 2012.058 9.669643 -4.566031 23.90532 -12.10194 31.44123
## 2012.077 9.291580 -4.944094 23.52725 -12.48000 31.06317
## 2012.096 9.376900 -4.858774 23.61257 -12.39468 31.14848autoplot(forecast(iq.arima))
Auto.arima chose Arima(1,0,2). The forecast is quite broad, and obviously, anything below 0 is unreasonable.
checkresiduals(forecast(iq.arima))
##
## Ljung-Box test
##
## data: Residuals from ARIMA(1,0,2)(1,0,1)[52] with non-zero mean
## Q* = 25.177, df = 45, p-value = 0.9926
##
## Model df: 6. Total lags used: 51summary(iq.arima)## Series: iq.ts[, 4]
## ARIMA(1,0,2)(1,0,1)[52] with non-zero mean
##
## Coefficients:
## ar1 ma1 ma2 sar1 sma1 mean
## 0.8244 -0.3545 0.1176 0.4545 -0.3736 9.4923
## s.e. 0.0509 0.0776 0.0740 0.5889 0.6057 2.2616
##
## sigma^2 estimated as 60.77: log likelihood=-886.57
## AIC=1787.14 AICc=1787.59 BIC=1811.96
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE ACF1
## Training set -0.1017739 7.703724 4.53396 -Inf Inf 0.4654896 0.01575443The residuals plot is mostly white noise with a couple of spikes; the ACF performs well; the graph is normally distrubuted but with a slight right skew.
STL
iq.stl = stl(iq.ts[,4], s.window = "periodic", robust = TRUE)
forecast(iq.stl)## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## 2010.115 3.08905850 -7.538237 13.71635 -13.163988 19.34210
## 2010.135 2.44055722 -9.765479 14.64659 -16.226964 21.10808
## 2010.154 1.84153902 -11.761226 15.44430 -18.962095 22.64517
## 2010.173 4.52704113 -10.341823 19.39590 -18.212924 27.26701
## 2010.192 14.59314538 -1.442160 30.62845 -9.930738 39.11703
## 2010.212 1.72720125 -15.395267 18.84967 -24.459354 27.91376
## 2010.231 3.85994505 -14.284663 22.00455 -23.889839 31.60973
## 2010.250 1.78117903 -17.330982 20.89334 -27.448349 31.01071
## 2010.269 0.71607639 -19.316962 20.74911 -29.921811 31.35396
## 2010.288 1.13643125 -19.776974 22.04984 -30.847862 33.12072
## 2010.308 0.89940369 -20.858777 22.65758 -32.376861 34.17567
## 2010.327 0.92676759 -21.644593 23.49813 -33.593149 35.44668
## 2010.346 1.12921405 -22.227031 24.48546 -34.591081 36.84951
## 2010.365 -1.10255924 -25.218158 23.01304 -37.984185 35.77907
## 2010.385 0.93314495 -23.918616 25.78491 -37.074343 38.94063
## 2010.404 0.48959287 -25.077141 26.05633 -38.611353 39.59054
## 2010.423 1.30687164 -24.955379 27.56912 -38.857775 41.47152
## 2010.442 2.05500086 -24.884815 28.99482 -39.145893 43.25589
## 2010.462 1.82607889 -25.774675 29.42683 -40.385631 44.03779
## 2010.481 0.62460765 -27.621622 28.87084 -42.574273 43.82349
## 2010.500 2.12801530 -26.749266 31.00530 -42.035976 46.29201
## 2010.519 0.46903019 -29.025805 29.96387 -44.639427 45.57749
## 2010.538 -0.07235503 -30.172076 30.02737 -46.105906 45.96120
## 2010.558 1.55706899 -29.135619 32.24976 -45.383347 48.49748
## 2010.577 1.35515328 -29.919261 32.62957 -46.474936 49.18524
## 2010.596 -1.87774114 -33.723257 29.96777 -50.581255 46.82577
## 2010.615 0.94472800 -31.461827 33.35128 -48.616821 50.50628
## 2010.635 -1.27953958 -34.237584 31.67850 -51.684519 49.12544
## 2010.654 1.12740680 -32.373050 34.62786 -50.107120 52.36193
## 2010.673 1.18021393 -32.854011 35.21444 -50.870642 53.23107
## 2010.692 0.79342852 -33.766323 35.35318 -52.061150 53.64801
## 2010.712 0.10016324 -34.977241 35.17757 -53.546097 53.74642
## 2010.731 0.46550210 -35.122026 36.05303 -53.960926 54.89193
## 2010.750 1.06709961 -35.023344 37.15754 -54.128470 56.26267
## 2010.769 1.18311443 -35.403331 37.76956 -54.771025 57.13725
## 2010.788 -0.40580671 -37.481619 36.67001 -57.108369 56.29676
## 2010.808 3.73673694 -33.822067 41.29554 -53.704497 61.17797
## 2010.827 2.71097595 -35.324687 40.74664 -55.459551 60.88150
## 2010.846 0.50266127 -38.003956 39.00928 -58.388128 59.39345
## 2010.865 3.23243526 -35.739445 42.20432 -56.369913 62.83478
## 2010.885 4.74016806 -34.691487 44.17182 -55.565344 65.04568
## 2010.904 6.65960820 -33.226521 46.54574 -54.340962 67.66018
## 2010.923 1.12980395 -39.205679 41.46529 -60.557994 62.81760
## 2010.942 1.88417994 -38.895706 42.66407 -60.483273 64.25163
## 2010.962 5.35324931 -35.866248 46.57275 -57.686532 68.39303
## 2010.981 16.38619391 -25.268276 58.04066 -47.318821 80.09121
## 2011.000 5.99715768 -36.087790 48.08210 -58.366215 70.36053
## 2011.019 2.70720288 -39.803862 45.21827 -62.307861 67.72227
## 2011.038 3.04444748 -39.888507 45.97740 -62.615840 68.70474
## 2011.058 4.51669832 -38.834040 47.86744 -61.782534 70.81593
## 2011.077 -0.75408721 -44.518621 43.01045 -67.686165 66.17799
## 2011.096 5.10620321 -39.068250 49.28066 -62.452792 72.66520
## 2011.115 3.08905850 -41.491545 47.66966 -65.091090 71.26921
## 2011.135 2.44055722 -42.542529 47.42364 -66.355137 71.23625
## 2011.154 1.84153902 -43.540462 47.22354 -67.564241 71.24732
## 2011.173 4.52704113 -41.250397 50.30448 -65.483509 74.53759
## 2011.192 14.59314538 -31.576344 60.76263 -56.016996 85.20329
## 2011.212 1.72720125 -44.831038 48.28544 -69.477482 72.93188
## 2011.231 3.85994505 -43.083825 50.80372 -67.934356 75.65425
## 2011.250 1.78117903 -45.544981 49.10734 -70.597938 74.16030
## 2011.269 0.71607639 -46.989409 48.42156 -72.243168 73.67532
## 2011.288 1.13643125 -46.945387 49.21825 -72.398364 74.67123
## 2011.308 0.89940369 -47.555825 49.35463 -73.206473 75.00528
## 2011.327 0.92676759 -47.899015 49.75255 -73.745823 75.59936
## 2011.346 1.12921405 -48.064332 50.32276 -74.105822 76.36425
## 2011.365 -1.10255924 -50.661139 48.45602 -76.895867 74.69075
## 2011.385 0.93314495 -48.987800 50.85409 -75.414352 77.28064
## 2011.404 0.48959287 -49.791106 50.77029 -76.408100 77.38729
## 2011.423 1.30687164 -49.331025 51.94477 -76.137108 78.75085
## 2011.442 2.05500086 -48.937591 53.04759 -75.931439 80.04144
## 2011.462 1.82607889 -49.518759 53.17092 -76.699074 80.35123
## 2011.481 0.62460765 -51.070075 52.31929 -78.435588 79.68480
## 2011.500 2.12801530 -49.914161 54.17019 -77.463626 81.71966
## 2011.519 0.46903019 -51.918335 52.85640 -79.650531 80.58859
## 2011.538 -0.07235503 -52.802650 52.65794 -80.716381 80.57167
## 2011.558 1.55706899 -51.513939 54.62808 -79.608033 82.72217
## 2011.577 1.35515328 -52.054394 54.76470 -80.327701 83.03801
## 2011.596 -1.87774114 -55.623696 51.86821 -84.075086 80.31960
## 2011.615 0.94472800 -53.135542 55.02500 -81.763907 83.65336
## 2011.635 -1.27953958 -55.692070 53.13299 -84.496323 81.93724
## 2011.654 1.12740680 -53.615368 55.87018 -82.594442 84.84926
## 2011.673 1.18021393 -53.890824 56.25125 -83.043671 85.40410
## 2011.692 0.79342852 -54.603928 56.19079 -83.929518 85.51637
## 2011.712 0.10016324 -55.621601 55.82193 -85.118922 85.31925
## 2011.731 0.46550210 -55.578792 56.50980 -85.246850 86.17785
## 2011.750 1.06709961 -55.297879 57.43208 -85.135697 87.26990
## 2011.769 1.18311443 -55.500735 57.86696 -85.507352 87.87358
## 2011.788 -0.40580671 -57.406742 56.59513 -87.581215 86.76960
## 2011.808 3.73673694 -53.579531 61.05300 -83.920930 91.39440
## 2011.827 2.71097595 -54.918899 60.34085 -85.426311 90.84826
## 2011.846 0.50266127 -57.439123 58.44445 -88.111651 89.11697
## 2011.865 3.23243526 -55.019588 61.48446 -85.856347 92.32122
## 2011.885 4.74016806 -53.820451 63.30079 -84.820571 94.30091
## 2011.904 6.65960820 -52.207989 65.52721 -83.370613 96.68983
## 2011.923 1.12980395 -58.043179 60.30279 -89.367464 91.62707
## 2011.942 1.88417994 -57.592621 61.36098 -89.077737 92.84610
## 2011.962 5.35324931 -54.425825 65.13232 -86.070956 96.77745
## 2011.981 16.38619391 -43.693633 76.46602 -75.497973 108.27036
## 2012.000 5.99715768 -54.381924 66.37624 -86.344680 98.33899
## 2012.019 2.70720288 -57.969658 63.38406 -90.090048 95.50445
## 2012.038 3.04444748 -57.928738 64.01763 -90.205993 96.29489
## 2012.058 4.51669832 -56.751378 65.78478 -89.184739 98.21814
## 2012.077 -0.75408721 -62.315643 60.80747 -94.904362 93.39619
## 2012.096 5.10620321 -56.747439 66.95985 -89.490779 99.70319autoplot(forecast(iq.stl)) +
ylab("Dengue Cases") +
ggtitle("Iquitos STL + ETS(A,N,N") +
xlab("Date")
autoplot(iq.stl)
The STL forecast seems to pick up on the off-season and peak-season well. Visiaully, this model performs significantly better for Iquitos than San Juan. However, as with the San Juan model, this one also predicts negative cases in some years.
checkresiduals(forecast(iq.stl))## Warning in checkresiduals(forecast(iq.stl)): The fitted degrees of freedom
## is based on the model used for the seasonally adjusted data.
##
## Ljung-Box test
##
## data: Residuals from STL + ETS(A,N,N)
## Q* = 43.087, df = 49, p-value = 0.7106
##
## Model df: 2. Total lags used: 51summary(forecast(iq.stl))##
## Forecast method: STL + ETS(A,N,N)
##
## Model Information:
## ETS(A,N,N)
##
## Call:
## ets(y = na.interp(x), model = etsmodel, allow.multiplicative.trend = allow.multiplicative.trend)
##
## Smoothing parameters:
## alpha = 0.565
##
## Initial states:
## l = 21.0606
##
## sigma: 8.2925
##
## AIC AICc BIC
## 2506.619 2506.714 2517.254
##
## Error measures:
## ME RMSE MAE MPE MAPE MASE ACF1
## Training set -0.1295968 8.260067 4.757433 NaN Inf 0.488433 0.002786997
##
## Forecasts:
## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## 2010.115 3.08905850 -7.538237 13.71635 -13.163988 19.34210
## 2010.135 2.44055722 -9.765479 14.64659 -16.226964 21.10808
## 2010.154 1.84153902 -11.761226 15.44430 -18.962095 22.64517
## 2010.173 4.52704113 -10.341823 19.39590 -18.212924 27.26701
## 2010.192 14.59314538 -1.442160 30.62845 -9.930738 39.11703
## 2010.212 1.72720125 -15.395267 18.84967 -24.459354 27.91376
## 2010.231 3.85994505 -14.284663 22.00455 -23.889839 31.60973
## 2010.250 1.78117903 -17.330982 20.89334 -27.448349 31.01071
## 2010.269 0.71607639 -19.316962 20.74911 -29.921811 31.35396
## 2010.288 1.13643125 -19.776974 22.04984 -30.847862 33.12072
## 2010.308 0.89940369 -20.858777 22.65758 -32.376861 34.17567
## 2010.327 0.92676759 -21.644593 23.49813 -33.593149 35.44668
## 2010.346 1.12921405 -22.227031 24.48546 -34.591081 36.84951
## 2010.365 -1.10255924 -25.218158 23.01304 -37.984185 35.77907
## 2010.385 0.93314495 -23.918616 25.78491 -37.074343 38.94063
## 2010.404 0.48959287 -25.077141 26.05633 -38.611353 39.59054
## 2010.423 1.30687164 -24.955379 27.56912 -38.857775 41.47152
## 2010.442 2.05500086 -24.884815 28.99482 -39.145893 43.25589
## 2010.462 1.82607889 -25.774675 29.42683 -40.385631 44.03779
## 2010.481 0.62460765 -27.621622 28.87084 -42.574273 43.82349
## 2010.500 2.12801530 -26.749266 31.00530 -42.035976 46.29201
## 2010.519 0.46903019 -29.025805 29.96387 -44.639427 45.57749
## 2010.538 -0.07235503 -30.172076 30.02737 -46.105906 45.96120
## 2010.558 1.55706899 -29.135619 32.24976 -45.383347 48.49748
## 2010.577 1.35515328 -29.919261 32.62957 -46.474936 49.18524
## 2010.596 -1.87774114 -33.723257 29.96777 -50.581255 46.82577
## 2010.615 0.94472800 -31.461827 33.35128 -48.616821 50.50628
## 2010.635 -1.27953958 -34.237584 31.67850 -51.684519 49.12544
## 2010.654 1.12740680 -32.373050 34.62786 -50.107120 52.36193
## 2010.673 1.18021393 -32.854011 35.21444 -50.870642 53.23107
## 2010.692 0.79342852 -33.766323 35.35318 -52.061150 53.64801
## 2010.712 0.10016324 -34.977241 35.17757 -53.546097 53.74642
## 2010.731 0.46550210 -35.122026 36.05303 -53.960926 54.89193
## 2010.750 1.06709961 -35.023344 37.15754 -54.128470 56.26267
## 2010.769 1.18311443 -35.403331 37.76956 -54.771025 57.13725
## 2010.788 -0.40580671 -37.481619 36.67001 -57.108369 56.29676
## 2010.808 3.73673694 -33.822067 41.29554 -53.704497 61.17797
## 2010.827 2.71097595 -35.324687 40.74664 -55.459551 60.88150
## 2010.846 0.50266127 -38.003956 39.00928 -58.388128 59.39345
## 2010.865 3.23243526 -35.739445 42.20432 -56.369913 62.83478
## 2010.885 4.74016806 -34.691487 44.17182 -55.565344 65.04568
## 2010.904 6.65960820 -33.226521 46.54574 -54.340962 67.66018
## 2010.923 1.12980395 -39.205679 41.46529 -60.557994 62.81760
## 2010.942 1.88417994 -38.895706 42.66407 -60.483273 64.25163
## 2010.962 5.35324931 -35.866248 46.57275 -57.686532 68.39303
## 2010.981 16.38619391 -25.268276 58.04066 -47.318821 80.09121
## 2011.000 5.99715768 -36.087790 48.08210 -58.366215 70.36053
## 2011.019 2.70720288 -39.803862 45.21827 -62.307861 67.72227
## 2011.038 3.04444748 -39.888507 45.97740 -62.615840 68.70474
## 2011.058 4.51669832 -38.834040 47.86744 -61.782534 70.81593
## 2011.077 -0.75408721 -44.518621 43.01045 -67.686165 66.17799
## 2011.096 5.10620321 -39.068250 49.28066 -62.452792 72.66520
## 2011.115 3.08905850 -41.491545 47.66966 -65.091090 71.26921
## 2011.135 2.44055722 -42.542529 47.42364 -66.355137 71.23625
## 2011.154 1.84153902 -43.540462 47.22354 -67.564241 71.24732
## 2011.173 4.52704113 -41.250397 50.30448 -65.483509 74.53759
## 2011.192 14.59314538 -31.576344 60.76263 -56.016996 85.20329
## 2011.212 1.72720125 -44.831038 48.28544 -69.477482 72.93188
## 2011.231 3.85994505 -43.083825 50.80372 -67.934356 75.65425
## 2011.250 1.78117903 -45.544981 49.10734 -70.597938 74.16030
## 2011.269 0.71607639 -46.989409 48.42156 -72.243168 73.67532
## 2011.288 1.13643125 -46.945387 49.21825 -72.398364 74.67123
## 2011.308 0.89940369 -47.555825 49.35463 -73.206473 75.00528
## 2011.327 0.92676759 -47.899015 49.75255 -73.745823 75.59936
## 2011.346 1.12921405 -48.064332 50.32276 -74.105822 76.36425
## 2011.365 -1.10255924 -50.661139 48.45602 -76.895867 74.69075
## 2011.385 0.93314495 -48.987800 50.85409 -75.414352 77.28064
## 2011.404 0.48959287 -49.791106 50.77029 -76.408100 77.38729
## 2011.423 1.30687164 -49.331025 51.94477 -76.137108 78.75085
## 2011.442 2.05500086 -48.937591 53.04759 -75.931439 80.04144
## 2011.462 1.82607889 -49.518759 53.17092 -76.699074 80.35123
## 2011.481 0.62460765 -51.070075 52.31929 -78.435588 79.68480
## 2011.500 2.12801530 -49.914161 54.17019 -77.463626 81.71966
## 2011.519 0.46903019 -51.918335 52.85640 -79.650531 80.58859
## 2011.538 -0.07235503 -52.802650 52.65794 -80.716381 80.57167
## 2011.558 1.55706899 -51.513939 54.62808 -79.608033 82.72217
## 2011.577 1.35515328 -52.054394 54.76470 -80.327701 83.03801
## 2011.596 -1.87774114 -55.623696 51.86821 -84.075086 80.31960
## 2011.615 0.94472800 -53.135542 55.02500 -81.763907 83.65336
## 2011.635 -1.27953958 -55.692070 53.13299 -84.496323 81.93724
## 2011.654 1.12740680 -53.615368 55.87018 -82.594442 84.84926
## 2011.673 1.18021393 -53.890824 56.25125 -83.043671 85.40410
## 2011.692 0.79342852 -54.603928 56.19079 -83.929518 85.51637
## 2011.712 0.10016324 -55.621601 55.82193 -85.118922 85.31925
## 2011.731 0.46550210 -55.578792 56.50980 -85.246850 86.17785
## 2011.750 1.06709961 -55.297879 57.43208 -85.135697 87.26990
## 2011.769 1.18311443 -55.500735 57.86696 -85.507352 87.87358
## 2011.788 -0.40580671 -57.406742 56.59513 -87.581215 86.76960
## 2011.808 3.73673694 -53.579531 61.05300 -83.920930 91.39440
## 2011.827 2.71097595 -54.918899 60.34085 -85.426311 90.84826
## 2011.846 0.50266127 -57.439123 58.44445 -88.111651 89.11697
## 2011.865 3.23243526 -55.019588 61.48446 -85.856347 92.32122
## 2011.885 4.74016806 -53.820451 63.30079 -84.820571 94.30091
## 2011.904 6.65960820 -52.207989 65.52721 -83.370613 96.68983
## 2011.923 1.12980395 -58.043179 60.30279 -89.367464 91.62707
## 2011.942 1.88417994 -57.592621 61.36098 -89.077737 92.84610
## 2011.962 5.35324931 -54.425825 65.13232 -86.070956 96.77745
## 2011.981 16.38619391 -43.693633 76.46602 -75.497973 108.27036
## 2012.000 5.99715768 -54.381924 66.37624 -86.344680 98.33899
## 2012.019 2.70720288 -57.969658 63.38406 -90.090048 95.50445
## 2012.038 3.04444748 -57.928738 64.01763 -90.205993 96.29489
## 2012.058 4.51669832 -56.751378 65.78478 -89.184739 98.21814
## 2012.077 -0.75408721 -62.315643 60.80747 -94.904362 93.39619
## 2012.096 5.10620321 -56.747439 66.95985 -89.490779 99.70319There is a light pattern in the residual graph, and the ACF breaks at Lags = 3 & 4. However the residuals are still normally distrubuted with a slight right skew.
VAR
Some factors that may affect mosquito population include precipitation, temperature, dew point, and humidity.
Thus the linear model:
Total cases = Average temperature(k) + Dew point(k) + Precipitation + Humidity(%)
iq.tslm = tslm(iq.ts[,4]~iq.ts[,10]+iq.ts[,12]+iq.ts[,13]+iq.ts[,17])
summary(iq.tslm)##
## Call:
## tslm(formula = iq.ts[, 4] ~ iq.ts[, 10] + iq.ts[, 12] + iq.ts[,
## 13] + iq.ts[, 17])
##
## Residuals:
## Min 1Q Median 3Q Max
## -11.330 -5.943 -3.275 1.383 72.257
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -588.27881 223.52810 -2.632 0.00902 **
## iq.ts[, 10] 0.02178 0.02086 1.044 0.29727
## iq.ts[, 12] -1.57977 2.39838 -0.659 0.51070
## iq.ts[, 13] 3.76451 2.86641 1.313 0.19028
## iq.ts[, 17] -0.49990 0.62757 -0.797 0.42646
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 10.73 on 251 degrees of freedom
## Multiple R-squared: 0.0504, Adjusted R-squared: 0.03527
## F-statistic: 3.33 on 4 and 251 DF, p-value: 0.01112The linear model shows us that none of the variables chosen are necessarily significant, unlike the linear model for San Juan. However I have decided to proceed in a similar manner, since the purpose of this project is to use these environmental variables to try to predict the spread of Dengue.
iq.tsint = ts.intersect(iq.ts[,12], iq.ts[,13], iq.ts[,17])
VARselect(iq.tsint)## $selection
## AIC(n) HQ(n) SC(n) FPE(n)
## 4 1 1 4
##
## $criteria
## 1 2 3 4 5
## AIC(n) -0.08909472 -0.128363752 -0.13421663 -0.17714232 -0.1673213
## HQ(n) -0.02024428 -0.007875485 0.03790947 0.04662161 0.1080805
## SC(n) 0.08189706 0.170871867 0.29326282 0.37858097 0.5166459
## FPE(n) 0.91476679 0.879573939 0.87451822 0.83790826 0.8463946
## 6 7 8 9 10
## AIC(n) -0.1436316 -0.1120790 -0.07229027 -0.04660093 -0.01304136
## HQ(n) 0.1834080 0.2665984 0.35802496 0.43535214 0.52054953
## SC(n) 0.6685794 0.8283758 0.99640836 1.15034155 1.31214495
## FPE(n) 0.8670082 0.8952596 0.93222786 0.95730941 0.99103989The following models will be named according to the number of lags used.
VARselect suggests lags = 4 & 1.
iq.var4 = VAR(iq.tsint, p = 4)
summary(iq.var4)##
## VAR Estimation Results:
## =========================
## Endogenous variables: iq.ts...12., iq.ts...13., iq.ts...17.
## Deterministic variables: const
## Sample size: 252
## Log Likelihood: -1011.659
## Roots of the characteristic polynomial:
## 0.898 0.898 0.712 0.6682 0.6682 0.6173 0.6173 0.602 0.5705 0.4916 0.4916 0.0489
## Call:
## VAR(y = iq.tsint, p = 4)
##
##
## Estimation results for equation iq.ts...12.:
## ============================================
## iq.ts...12. = iq.ts...12..l1 + iq.ts...13..l1 + iq.ts...17..l1 + iq.ts...12..l2 + iq.ts...13..l2 + iq.ts...17..l2 + iq.ts...12..l3 + iq.ts...13..l3 + iq.ts...17..l3 + iq.ts...12..l4 + iq.ts...13..l4 + iq.ts...17..l4 + const
##
## Estimate Std. Error t value Pr(>|t|)
## iq.ts...12..l1 0.08523 0.22255 0.383 0.702
## iq.ts...13..l1 0.32082 0.28284 1.134 0.258
## iq.ts...17..l1 -0.09114 0.05864 -1.554 0.121
## iq.ts...12..l2 -0.17029 0.22150 -0.769 0.443
## iq.ts...13..l2 0.44348 0.28574 1.552 0.122
## iq.ts...17..l2 -0.07151 0.05904 -1.211 0.227
## iq.ts...12..l3 0.06110 0.21997 0.278 0.781
## iq.ts...13..l3 -0.07036 0.28557 -0.246 0.806
## iq.ts...17..l3 0.01173 0.05901 0.199 0.843
## iq.ts...12..l4 0.15451 0.21738 0.711 0.478
## iq.ts...13..l4 -0.10040 0.27684 -0.363 0.717
## iq.ts...17..l4 -0.01700 0.05828 -0.292 0.771
## const 99.65304 23.94749 4.161 4.42e-05 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
##
## Residual standard error: 0.9549 on 239 degrees of freedom
## Multiple R-Squared: 0.4068, Adjusted R-squared: 0.377
## F-statistic: 13.66 on 12 and 239 DF, p-value: < 2.2e-16
##
##
## Estimation results for equation iq.ts...13.:
## ============================================
## iq.ts...13. = iq.ts...12..l1 + iq.ts...13..l1 + iq.ts...17..l1 + iq.ts...12..l2 + iq.ts...13..l2 + iq.ts...17..l2 + iq.ts...12..l3 + iq.ts...13..l3 + iq.ts...17..l3 + iq.ts...12..l4 + iq.ts...13..l4 + iq.ts...17..l4 + const
##
## Estimate Std. Error t value Pr(>|t|)
## iq.ts...12..l1 -0.1354917 0.2027256 -0.668 0.50456
## iq.ts...13..l1 0.6867163 0.2576476 2.665 0.00822 **
## iq.ts...17..l1 -0.0635551 0.0534210 -1.190 0.23534
## iq.ts...12..l2 0.0410781 0.2017685 0.204 0.83885
## iq.ts...13..l2 0.1498824 0.2602886 0.576 0.56527
## iq.ts...17..l2 -0.0006325 0.0537848 -0.012 0.99063
## iq.ts...12..l3 -0.1829216 0.2003791 -0.913 0.36223
## iq.ts...13..l3 0.4450298 0.2601391 1.711 0.08843 .
## iq.ts...17..l3 -0.0701198 0.0537543 -1.304 0.19333
## iq.ts...12..l4 0.1629231 0.1980236 0.823 0.41147
## iq.ts...13..l4 -0.1358648 0.2521826 -0.539 0.59056
## iq.ts...17..l4 0.0420633 0.0530910 0.792 0.42898
## const -0.5985053 21.8146679 -0.027 0.97813
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
##
## Residual standard error: 0.8699 on 239 degrees of freedom
## Multiple R-Squared: 0.5373, Adjusted R-squared: 0.514
## F-statistic: 23.13 on 12 and 239 DF, p-value: < 2.2e-16
##
##
## Estimation results for equation iq.ts...17.:
## ============================================
## iq.ts...17. = iq.ts...12..l1 + iq.ts...13..l1 + iq.ts...17..l1 + iq.ts...12..l2 + iq.ts...13..l2 + iq.ts...17..l2 + iq.ts...12..l3 + iq.ts...13..l3 + iq.ts...17..l3 + iq.ts...12..l4 + iq.ts...13..l4 + iq.ts...17..l4 + const
##
## Estimate Std. Error t value Pr(>|t|)
## iq.ts...12..l1 -0.97830 1.17675 -0.831 0.406605
## iq.ts...13..l1 1.80845 1.49556 1.209 0.227776
## iq.ts...17..l1 0.04605 0.31009 0.149 0.882073
## iq.ts...12..l2 0.83842 1.17120 0.716 0.474775
## iq.ts...13..l2 -1.04386 1.51089 -0.691 0.490304
## iq.ts...17..l2 0.27574 0.31220 0.883 0.378008
## iq.ts...12..l3 -1.16228 1.16313 -0.999 0.318676
## iq.ts...13..l3 2.25587 1.51002 1.494 0.136513
## iq.ts...17..l3 -0.37865 0.31203 -1.214 0.226123
## iq.ts...12..l4 0.61069 1.14946 0.531 0.595715
## iq.ts...13..l4 -0.66484 1.46383 -0.454 0.650115
## iq.ts...17..l4 0.38325 0.30818 1.244 0.214859
## const -429.48801 126.62675 -3.392 0.000813 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
##
## Residual standard error: 5.049 on 239 degrees of freedom
## Multiple R-Squared: 0.3942, Adjusted R-squared: 0.3637
## F-statistic: 12.96 on 12 and 239 DF, p-value: < 2.2e-16
##
##
##
## Covariance matrix of residuals:
## iq.ts...12. iq.ts...13. iq.ts...17.
## iq.ts...12. 0.9118 0.1103 -2.769
## iq.ts...13. 0.1103 0.7567 3.116
## iq.ts...17. -2.7687 3.1162 25.495
##
## Correlation matrix of residuals:
## iq.ts...12. iq.ts...13. iq.ts...17.
## iq.ts...12. 1.0000 0.1328 -0.5742
## iq.ts...13. 0.1328 1.0000 0.7095
## iq.ts...17. -0.5742 0.7095 1.0000autoplot(forecast(iq.var4))
iq.var1 = VAR(iq.tsint, p = 1)
summary(iq.var4)##
## VAR Estimation Results:
## =========================
## Endogenous variables: iq.ts...12., iq.ts...13., iq.ts...17.
## Deterministic variables: const
## Sample size: 252
## Log Likelihood: -1011.659
## Roots of the characteristic polynomial:
## 0.898 0.898 0.712 0.6682 0.6682 0.6173 0.6173 0.602 0.5705 0.4916 0.4916 0.0489
## Call:
## VAR(y = iq.tsint, p = 4)
##
##
## Estimation results for equation iq.ts...12.:
## ============================================
## iq.ts...12. = iq.ts...12..l1 + iq.ts...13..l1 + iq.ts...17..l1 + iq.ts...12..l2 + iq.ts...13..l2 + iq.ts...17..l2 + iq.ts...12..l3 + iq.ts...13..l3 + iq.ts...17..l3 + iq.ts...12..l4 + iq.ts...13..l4 + iq.ts...17..l4 + const
##
## Estimate Std. Error t value Pr(>|t|)
## iq.ts...12..l1 0.08523 0.22255 0.383 0.702
## iq.ts...13..l1 0.32082 0.28284 1.134 0.258
## iq.ts...17..l1 -0.09114 0.05864 -1.554 0.121
## iq.ts...12..l2 -0.17029 0.22150 -0.769 0.443
## iq.ts...13..l2 0.44348 0.28574 1.552 0.122
## iq.ts...17..l2 -0.07151 0.05904 -1.211 0.227
## iq.ts...12..l3 0.06110 0.21997 0.278 0.781
## iq.ts...13..l3 -0.07036 0.28557 -0.246 0.806
## iq.ts...17..l3 0.01173 0.05901 0.199 0.843
## iq.ts...12..l4 0.15451 0.21738 0.711 0.478
## iq.ts...13..l4 -0.10040 0.27684 -0.363 0.717
## iq.ts...17..l4 -0.01700 0.05828 -0.292 0.771
## const 99.65304 23.94749 4.161 4.42e-05 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
##
## Residual standard error: 0.9549 on 239 degrees of freedom
## Multiple R-Squared: 0.4068, Adjusted R-squared: 0.377
## F-statistic: 13.66 on 12 and 239 DF, p-value: < 2.2e-16
##
##
## Estimation results for equation iq.ts...13.:
## ============================================
## iq.ts...13. = iq.ts...12..l1 + iq.ts...13..l1 + iq.ts...17..l1 + iq.ts...12..l2 + iq.ts...13..l2 + iq.ts...17..l2 + iq.ts...12..l3 + iq.ts...13..l3 + iq.ts...17..l3 + iq.ts...12..l4 + iq.ts...13..l4 + iq.ts...17..l4 + const
##
## Estimate Std. Error t value Pr(>|t|)
## iq.ts...12..l1 -0.1354917 0.2027256 -0.668 0.50456
## iq.ts...13..l1 0.6867163 0.2576476 2.665 0.00822 **
## iq.ts...17..l1 -0.0635551 0.0534210 -1.190 0.23534
## iq.ts...12..l2 0.0410781 0.2017685 0.204 0.83885
## iq.ts...13..l2 0.1498824 0.2602886 0.576 0.56527
## iq.ts...17..l2 -0.0006325 0.0537848 -0.012 0.99063
## iq.ts...12..l3 -0.1829216 0.2003791 -0.913 0.36223
## iq.ts...13..l3 0.4450298 0.2601391 1.711 0.08843 .
## iq.ts...17..l3 -0.0701198 0.0537543 -1.304 0.19333
## iq.ts...12..l4 0.1629231 0.1980236 0.823 0.41147
## iq.ts...13..l4 -0.1358648 0.2521826 -0.539 0.59056
## iq.ts...17..l4 0.0420633 0.0530910 0.792 0.42898
## const -0.5985053 21.8146679 -0.027 0.97813
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
##
## Residual standard error: 0.8699 on 239 degrees of freedom
## Multiple R-Squared: 0.5373, Adjusted R-squared: 0.514
## F-statistic: 23.13 on 12 and 239 DF, p-value: < 2.2e-16
##
##
## Estimation results for equation iq.ts...17.:
## ============================================
## iq.ts...17. = iq.ts...12..l1 + iq.ts...13..l1 + iq.ts...17..l1 + iq.ts...12..l2 + iq.ts...13..l2 + iq.ts...17..l2 + iq.ts...12..l3 + iq.ts...13..l3 + iq.ts...17..l3 + iq.ts...12..l4 + iq.ts...13..l4 + iq.ts...17..l4 + const
##
## Estimate Std. Error t value Pr(>|t|)
## iq.ts...12..l1 -0.97830 1.17675 -0.831 0.406605
## iq.ts...13..l1 1.80845 1.49556 1.209 0.227776
## iq.ts...17..l1 0.04605 0.31009 0.149 0.882073
## iq.ts...12..l2 0.83842 1.17120 0.716 0.474775
## iq.ts...13..l2 -1.04386 1.51089 -0.691 0.490304
## iq.ts...17..l2 0.27574 0.31220 0.883 0.378008
## iq.ts...12..l3 -1.16228 1.16313 -0.999 0.318676
## iq.ts...13..l3 2.25587 1.51002 1.494 0.136513
## iq.ts...17..l3 -0.37865 0.31203 -1.214 0.226123
## iq.ts...12..l4 0.61069 1.14946 0.531 0.595715
## iq.ts...13..l4 -0.66484 1.46383 -0.454 0.650115
## iq.ts...17..l4 0.38325 0.30818 1.244 0.214859
## const -429.48801 126.62675 -3.392 0.000813 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
##
## Residual standard error: 5.049 on 239 degrees of freedom
## Multiple R-Squared: 0.3942, Adjusted R-squared: 0.3637
## F-statistic: 12.96 on 12 and 239 DF, p-value: < 2.2e-16
##
##
##
## Covariance matrix of residuals:
## iq.ts...12. iq.ts...13. iq.ts...17.
## iq.ts...12. 0.9118 0.1103 -2.769
## iq.ts...13. 0.1103 0.7567 3.116
## iq.ts...17. -2.7687 3.1162 25.495
##
## Correlation matrix of residuals:
## iq.ts...12. iq.ts...13. iq.ts...17.
## iq.ts...12. 1.0000 0.1328 -0.5742
## iq.ts...13. 0.1328 1.0000 0.7095
## iq.ts...17. -0.5742 0.7095 1.0000autoplot(forecast(iq.var4))
Neural Network
iq.n1 = nnetar(iq.ts[,4], size = 1)
iq.n2 = nnetar(iq.ts[,4], size = 2)
iq.n3 = nnetar(iq.ts[,4], size = 3)
iq.n4 = nnetar(iq.ts[,4], size = 4)iq.n1.fcast = forecast(iq.n1)
iq.n2.fcast = forecast(iq.n2)
iq.n3.fcast = forecast(iq.n3)
iq.n4.fcast = forecast(iq.n4)autoplot(iq.ts[,4]) +
autolayer(iq.n1.fcast,series="Size 1") +
autolayer(iq.n2.fcast,series="Size 2") +
autolayer(iq.n3.fcast,series="Size 3") +
autolayer(iq.n4.fcast,series="Size 4") +
ylab("Cases") +
xlab("Time") +
ggtitle("Iquitos Neural Net")
I am not sure why the Neural net models perform so differently for San Juan and Iquitos. For San Juan, the model picked up nuance in the data, whereas with Iquitos, the models produce white noise or flat predictions at every Size. Perhaps the number of observations or time frame has a significant impact. Needless to say the performance here is sub-par.
checkresiduals(iq.n1.fcast)## Warning in modeldf.default(object): Could not find appropriate degrees of
## freedom for this model.
accuracy(iq.n1.fcast)## ME RMSE MAE MPE MAPE MASE
## Training set -0.002789042 6.46119 4.071806 -Inf Inf 0.4180415
## ACF1
## Training set -0.006693787iq.n1.fcast## Point Forecast
## 2010.115 3.791414
## 2010.135 3.812863
## 2010.154 4.729203
## 2010.173 5.623150
## 2010.192 6.168889
## 2010.212 6.778439
## 2010.231 7.215695
## 2010.250 7.561193
## 2010.269 7.851504
## 2010.288 8.068321
## 2010.308 8.232043
## 2010.327 8.357674
## 2010.346 8.453944
## 2010.365 8.521798
## 2010.385 8.574285
## 2010.404 8.620262
## 2010.423 8.658044
## 2010.442 8.692066
## 2010.462 8.713620
## 2010.481 8.724756
## 2010.500 8.734280
## 2010.519 8.739733
## 2010.538 8.738673
## 2010.558 8.743355
## 2010.577 8.742492
## 2010.596 8.742519
## 2010.615 8.745065
## 2010.635 8.745916
## 2010.654 8.749645
## 2010.673 8.766645
## 2010.692 8.767866
## 2010.712 8.773467
## 2010.731 8.775160
## 2010.750 8.775560
## 2010.769 8.769251
## 2010.788 8.762627
## 2010.808 8.778252
## 2010.827 8.804496
## 2010.846 8.822525
## 2010.865 8.830303
## 2010.885 8.841757
## 2010.904 8.865086
## 2010.923 8.870562
## 2010.942 8.872756
## 2010.962 8.875352
## 2010.981 8.884345
## 2011.000 8.872964
## 2011.019 8.856579
## 2011.038 8.848609
## 2011.058 8.842271
## 2011.077 8.827269
## 2011.096 8.813536
## 2011.115 8.806165
## 2011.135 8.798003
## 2011.154 8.794525
## 2011.173 8.793728
## 2011.192 8.794120
## 2011.212 8.796483
## 2011.231 8.799463
## 2011.250 8.802852
## 2011.269 8.806475
## 2011.288 8.809949
## 2011.308 8.813212
## 2011.327 8.816176
## 2011.346 8.818793
## 2011.365 8.821060
## 2011.385 8.822997
## 2011.404 8.824645
## 2011.423 8.826036
## 2011.442 8.827211
## 2011.462 8.828185
## 2011.481 8.828978
## 2011.500 8.829626
## 2011.519 8.830143
## 2011.538 8.830544
## 2011.558 8.830869
## 2011.577 8.831116
## 2011.596 8.831306
## 2011.615 8.831460
## 2011.635 8.831577
## 2011.654 8.831676
## 2011.673 8.831792
## 2011.692 8.831878
## 2011.712 8.831964
## 2011.731 8.832037
## 2011.750 8.832095
## 2011.769 8.832128
## 2011.788 8.832141
## 2011.808 8.832189
## 2011.827 8.832278
## 2011.846 8.832384
## 2011.865 8.832491
## 2011.885 8.832611
## 2011.904 8.832762
## 2011.923 8.832891
## 2011.942 8.833007
## 2011.962 8.833111
## 2011.981 8.833214
## 2012.000 8.833266
## 2012.019 8.833277
## 2012.038 8.833271
## 2012.058 8.833244
## 2012.077 8.833184
## 2012.096 8.833106checkresiduals(iq.n2.fcast)## Warning in modeldf.default(object): Could not find appropriate degrees of
## freedom for this model.
accuracy(iq.n2.fcast)## ME RMSE MAE MPE MAPE MASE ACF1
## Training set 0.0008275682 5.676093 3.624278 -Inf Inf 0.372095 -0.03051129iq.n2.fcast## Point Forecast
## 2010.115 4.256890
## 2010.135 4.369391
## 2010.154 4.215238
## 2010.173 4.372372
## 2010.192 4.276714
## 2010.212 4.249860
## 2010.231 4.248447
## 2010.250 4.231633
## 2010.269 4.225171
## 2010.288 4.226491
## 2010.308 4.219201
## 2010.327 4.217488
## 2010.346 4.215579
## 2010.365 4.217427
## 2010.385 4.217617
## 2010.404 4.215353
## 2010.423 4.220189
## 2010.442 4.230865
## 2010.462 4.224984
## 2010.481 4.218757
## 2010.500 4.217064
## 2010.519 4.215202
## 2010.538 4.217220
## 2010.558 4.214293
## 2010.577 4.214145
## 2010.596 4.213968
## 2010.615 4.212955
## 2010.635 4.212647
## 2010.654 4.213238
## 2010.673 4.241571
## 2010.692 4.227296
## 2010.712 4.226530
## 2010.731 4.223650
## 2010.750 4.220768
## 2010.769 4.217800
## 2010.788 4.218862
## 2010.808 4.244100
## 2010.827 4.339208
## 2010.846 4.359020
## 2010.865 4.323269
## 2010.885 4.330972
## 2010.904 4.411550
## 2010.923 4.378236
## 2010.942 4.349504
## 2010.962 4.341765
## 2010.981 4.372099
## 2011.000 4.310236
## 2011.019 4.272010
## 2011.038 4.262110
## 2011.058 4.257956
## 2011.077 4.237738
## 2011.096 4.228775
## 2011.115 4.226473
## 2011.135 4.223374
## 2011.154 4.221471
## 2011.173 4.221010
## 2011.192 4.220152
## 2011.212 4.219676
## 2011.231 4.219434
## 2011.250 4.219192
## 2011.269 4.219053
## 2011.288 4.218977
## 2011.308 4.218901
## 2011.327 4.218859
## 2011.346 4.218828
## 2011.365 4.218815
## 2011.385 4.218806
## 2011.404 4.218793
## 2011.423 4.218802
## 2011.442 4.218834
## 2011.462 4.218827
## 2011.481 4.218811
## 2011.500 4.218803
## 2011.519 4.218792
## 2011.538 4.218793
## 2011.558 4.218783
## 2011.577 4.218778
## 2011.596 4.218776
## 2011.615 4.218771
## 2011.635 4.218768
## 2011.654 4.218768
## 2011.673 4.218850
## 2011.692 4.218833
## 2011.712 4.218835
## 2011.731 4.218833
## 2011.750 4.218820
## 2011.769 4.218808
## 2011.788 4.218806
## 2011.808 4.218876
## 2011.827 4.219180
## 2011.846 4.219341
## 2011.865 4.219325
## 2011.885 4.219387
## 2011.904 4.219656
## 2011.923 4.219637
## 2011.942 4.219582
## 2011.962 4.219560
## 2011.981 4.219631
## 2012.000 4.219458
## 2012.019 4.219299
## 2012.038 4.219208
## 2012.058 4.219132
## 2012.077 4.219030
## 2012.096 4.218960checkresiduals(iq.n3.fcast)## Warning in modeldf.default(object): Could not find appropriate degrees of
## freedom for this model.
accuracy(iq.n3.fcast)## ME RMSE MAE MPE MAPE MASE ACF1
## Training set -0.02248599 4.956641 3.334689 -Inf Inf 0.3423636 -0.01034411iq.n3.fcast## Point Forecast
## 2010.115 4.459031
## 2010.135 4.411400
## 2010.154 4.166678
## 2010.173 4.349768
## 2010.192 4.234116
## 2010.212 4.216010
## 2010.231 4.224030
## 2010.250 4.212775
## 2010.269 4.209912
## 2010.288 4.191694
## 2010.308 4.207935
## 2010.327 4.206508
## 2010.346 4.208825
## 2010.365 4.202230
## 2010.385 4.200741
## 2010.404 4.205362
## 2010.423 4.187510
## 2010.442 4.165007
## 2010.462 4.178335
## 2010.481 4.198567
## 2010.500 4.202122
## 2010.519 4.207130
## 2010.538 4.200821
## 2010.558 4.209137
## 2010.577 4.207650
## 2010.596 4.207340
## 2010.615 4.211108
## 2010.635 4.211355
## 2010.654 4.208272
## 2010.673 4.116350
## 2010.692 4.171122
## 2010.712 4.171081
## 2010.731 4.178596
## 2010.750 4.189065
## 2010.769 4.198788
## 2010.788 4.196077
## 2010.808 4.111580
## 2010.827 4.066749
## 2010.846 4.047235
## 2010.865 4.042918
## 2010.885 4.035541
## 2010.904 4.028065
## 2010.923 4.025248
## 2010.942 4.029959
## 2010.962 4.030324
## 2010.981 4.023854
## 2011.000 4.048344
## 2011.019 4.135520
## 2011.038 4.135498
## 2011.058 4.377710
## 2011.077 4.225397
## 2011.096 4.227245
## 2011.115 4.220944
## 2011.135 4.195384
## 2011.154 4.197969
## 2011.173 4.193018
## 2011.192 4.191173
## 2011.212 4.191657
## 2011.231 4.190871
## 2011.250 4.190890
## 2011.269 4.190951
## 2011.288 4.191044
## 2011.308 4.190946
## 2011.327 4.190956
## 2011.346 4.190931
## 2011.365 4.190978
## 2011.385 4.191003
## 2011.404 4.190967
## 2011.423 4.191139
## 2011.442 4.191386
## 2011.462 4.191324
## 2011.481 4.191165
## 2011.500 4.191109
## 2011.519 4.191015
## 2011.538 4.191036
## 2011.558 4.190951
## 2011.577 4.190943
## 2011.596 4.190940
## 2011.615 4.190893
## 2011.635 4.190884
## 2011.654 4.190906
## 2011.673 4.191783
## 2011.692 4.191442
## 2011.712 4.191479
## 2011.731 4.191460
## 2011.750 4.191292
## 2011.769 4.191179
## 2011.788 4.191160
## 2011.808 4.191933
## 2011.827 4.192506
## 2011.846 4.192898
## 2011.865 4.193161
## 2011.885 4.193355
## 2011.904 4.193511
## 2011.923 4.193603
## 2011.942 4.193604
## 2011.962 4.193620
## 2011.981 4.193686
## 2012.000 4.193469
## 2012.019 4.192611
## 2012.038 4.192418
## 2012.058 4.189910
## 2012.077 4.190787
## 2012.096 4.190669checkresiduals(iq.n4.fcast)## Warning in modeldf.default(object): Could not find appropriate degrees of
## freedom for this model.
accuracy(iq.n4.fcast)## ME RMSE MAE MPE MAPE MASE ACF1
## Training set 0.01159781 4.306568 3.058338 -Inf Inf 0.3139914 0.03443965iq.n4.fcast## Point Forecast
## 2010.115 4.437469
## 2010.135 5.330118
## 2010.154 4.217758
## 2010.173 4.260243
## 2010.192 4.274966
## 2010.212 4.158213
## 2010.231 4.193563
## 2010.250 4.153089
## 2010.269 4.121580
## 2010.288 4.097215
## 2010.308 4.121448
## 2010.327 4.136242
## 2010.346 4.122950
## 2010.365 4.160987
## 2010.385 4.167289
## 2010.404 4.120351
## 2010.423 4.095742
## 2010.442 4.069891
## 2010.462 4.075322
## 2010.481 4.108340
## 2010.500 4.111704
## 2010.519 4.116616
## 2010.538 4.156649
## 2010.558 4.127774
## 2010.577 4.143113
## 2010.596 4.147614
## 2010.615 4.129454
## 2010.635 4.129507
## 2010.654 4.114142
## 2010.673 4.054087
## 2010.692 4.077098
## 2010.712 4.072669
## 2010.731 4.078471
## 2010.750 4.083302
## 2010.769 4.124461
## 2010.788 4.151844
## 2010.808 4.056642
## 2010.827 3.715693
## 2010.846 3.678489
## 2010.865 3.889232
## 2010.885 3.838082
## 2010.904 3.613445
## 2010.923 3.723145
## 2010.942 3.845778
## 2010.962 3.855737
## 2010.981 3.690542
## 2011.000 3.955064
## 2011.019 4.009403
## 2011.038 4.007558
## 2011.058 4.032273
## 2011.077 4.065452
## 2011.096 4.096042
## 2011.115 4.076825
## 2011.135 4.071234
## 2011.154 4.082057
## 2011.173 4.079558
## 2011.192 4.080037
## 2011.212 4.082518
## 2011.231 4.081944
## 2011.250 4.082671
## 2011.269 4.083336
## 2011.288 4.083636
## 2011.308 4.083578
## 2011.327 4.083490
## 2011.346 4.083631
## 2011.365 4.083211
## 2011.385 4.083085
## 2011.404 4.083559
## 2011.423 4.083836
## 2011.442 4.084208
## 2011.462 4.084288
## 2011.481 4.083989
## 2011.500 4.083949
## 2011.519 4.083858
## 2011.538 4.083364
## 2011.558 4.083603
## 2011.577 4.083413
## 2011.596 4.083305
## 2011.615 4.083512
## 2011.635 4.083503
## 2011.654 4.083684
## 2011.673 4.084401
## 2011.692 4.084263
## 2011.712 4.084382
## 2011.731 4.084398
## 2011.750 4.084322
## 2011.769 4.083876
## 2011.788 4.083500
## 2011.808 4.084441
## 2011.827 4.088369
## 2011.846 4.089455
## 2011.865 4.087689
## 2011.885 4.088531
## 2011.904 4.091106
## 2011.923 4.090059
## 2011.942 4.088883
## 2011.962 4.088760
## 2011.981 4.090342
## 2012.000 4.087428
## 2012.019 4.086555
## 2012.038 4.086345
## 2012.058 4.085575
## 2012.077 4.085058
## 2012.096 4.084569